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Rotating stars in relativity

The Original Version of this article was published on 16 June 2003

Abstract

Rotating relativistic stars have been studied extensively in recent years, both theoretically and observationally, because of the information they might yield about the equation of state of matter at extremely high densities and because they are considered to be promising sources of gravitational waves. The latest theoretical understanding of rotating stars in relativity is reviewed in this updated article. The sections on equilibrium properties and on nonaxisymmetric oscillations and instabilities in f-modes and r-modes have been updated. Several new sections have been added on equilibria in modified theories of gravity, approximate universal relationships, the one-arm spiral instability, on analytic solutions for the exterior spacetime, rotating stars in LMXBs, rotating strange stars, and on rotating stars in numerical relativity including both hydrodynamic and magnetohydrodynamic studies of these objects.

Introduction

Rotating relativistic stars are of fundamental interest in physics. Their bulk properties constrain the proposed equations of state for densities greater than the nuclear saturation density. Accreted matter in their gravitational fields undergoes high-frequency oscillations that could become a sensitive probe for general relativistic effects. Temporal changes in the rotational period of millisecond pulsars can also reveal a wealth of information about important physical processes inside the stars or of cosmological relevance. In addition, rotational instabilities can result in the generation of copious amounts of gravitational waves the detection of which would initiate a new field of observational asteroseismology of relativistic stars. The latter is of particular importance because with the first direct detections of gravitational waves by the LIGO and VIRGO collaborations (Abbott et al. 2016a, b) the era of gravitational wave astronomy has arrived.

There exist several independent numerical codes for obtaining accurate models of rotating neutron stars in full general relativity, including two that are publicly available. The uncertainty in the high-density equation of state still allows numerically constructed maximum mass models to differ by more than 50% in mass, radius and angular velocity, and by a larger factor in the moment of inertia. Given these uncertainties, an absolute upper limit on the rotation of relativistic stars can be obtained by imposing causality as the only requirement on the equation of state. It then follows that gravitationally bound stars cannot rotate faster than 0.41 ms.

In rotating stars, nonaxisymmetric oscillations have been studied in various approximations (the Newtonian limit and the post-Newtonian approximation, the slow rotation limit, the Cowling approximation, the spatial conformal flatness approximation) as an eigenvalue problem. Normal modes in full general relativity have been obtained through numerical simulations only. Time evolutions of the linearized equations have improved our understanding of the spectrum of axial and hybrid modes in relativistic stars.

Nonaxisymmetric instabilities in rotating stars can be driven by the emission of gravitational waves [Chandrasekhar–Friedman–Schutz (CFS) instability] or by viscosity. Relativity strengthens the former, but weakens the latter. Nascent neutron stars can be subject to the \(l=2\) bar mode CFS instability, which would turn them into a strong gravitational wave source. Axial fluid modes in rotating stars (r-modes) have received considerable attention since the discovery that they are generically unstable to the emission of gravitational waves. The r-mode instability could slow down newly-born relativistic stars and limit their spin during accretion-induced spin-up, which would explain the absence of millisecond pulsars with rotational periods \(<\,\sim \) 1.5 ms. Gravitational waves from the r-mode instability could become detectable if the amplitude of r-modes is sufficiently large, however, nonlinear effects seem to set a small saturation amplitude on long timescales. Nevertheless, if the signal persists for a long time, even a small amplitude could become detectable. Highly differentially rotating neutron stars are also subject to the development of a one-arm (\(m=1\)) instability, as well as to the development of a dynamical bar-mode (\(m=2\)) instability which both act as emitters of potentially detectable gravitational waves.

Recent advances in numerical relativity have enabled the long-term dynamical evolution of rotating stars. Several interesting phenomena, such as dynamical instabilities, pulsation modes, and neutron star and black hole formation in rotating collapse have now been studied in full general relativity. The latest studies include realistic equations of state and also magnetic fields.

The aim of this article is to present a summary of theoretical and numerical methods that are used to describe the equilibrium properties of rotating relativistic stars, their oscillations and dynamical evolution. The focus is on the most recently available publications in the field, in order to rapidly communicate new methods and results. At the end of some sections, the reader is directed to papers that could not be presented in detail here, or to other review articles. As new developments in the field occur, updated versions of this article will appear. Another review on rotating relativistic stars has appeared by Gourgoulhon (2010), while monographs appeared by Meinel et al. (2008) and Friedman and Stergioulas (2013). In several sections, our Living Review article updates and extends previous versions (Stergioulas 1998, 2003) using also abridged discussions of topics from Friedman and Stergioulas (2013).

Notation and conventions Throughout the article, gravitational units, where \(G = c = 1\) (these units are also referred to as geometrized), will be adopted in writing the equations governing stellar structure and dynamics, while numerical properties of stellar models will be listed in cgs units, unless otherwise noted. We use the conventions of Misner et al. (1973) for the signature of the spacetime metric \((-+++)\) and for signs of the curvature tensor and its contractions. Spacetime indices will be denoted by Greek letters, \(\alpha \), \(\beta , \ldots \), while Latin \(a, b, \ldots \) characters will be reserved to denote spatial indices. (Readers familiar with abstract indices can regard indices early in the alphabet as abstract, while indices \(\mu , \nu , \lambda \) and ijk will be concrete, taking values \(\mu = 0,1,2,3\), \(i=1,2,3\).) Components of a vector \(u^\alpha \) in an orthonormal frame, \(\{ \mathbf{e}_{\hat{0}}, \ldots , \mathbf{e}_{\hat{3}}\} \), will be written as \(\{u^{\hat{0}}, \ldots , u^{\hat{3}}\}\). Parentheses enclosing a set of indices indicate symmetrization, while square brackets indicate anti-symmetrization.

Numbers that rely on physical constants are based on the values \(c = 2.9979 \times 10^{10}\,\mathrm {cm\ s^{-1}}\), \(G = 6.670\times 10^{-8}\,\mathrm {g^{-1}\ cm^{3}\ s^{-2}}\), \(\hbar = 1.0545\times 10^{-27}\,\mathrm {g\ cm^{2}\ s^{-1}}\), baryon mass \(m_B = 1.659\times 10^{-24}\,\mathrm {g}\), and \(M_\odot = 1.989\times 10^{33}\,\mathrm {g} = 1.477\,\mathrm {km}\).

The equilibrium structure of rotating relativistic stars

Assumptions

A relativistic star can have a complicated structure (such as a solid crust, magnetic field, possible superfluid interior, possible quark core, etc.). Depending on which phase in the lifetime of the star one wants to study, a number of physical effects can be ignored, so that the description becomes significantly simplified. In the following, we will take the case of a cold, uniformly rotating relativistic star as a reference case and mention additional assumptions for other cases, where necessary.

The matter can be modeled as a perfect fluid because observations of pulsar glitches are consistent with departures from a perfect fluid equilibrium (due to the presence of a solid crust) of order \(10^{-5}\) (see Friedman and Ipser 1992). The temperature of a cold neutron star has a negligible affect on its bulk properties and can be assumed to be 0 K, because its thermal energy (\( \ll 1 \,\mathrm {MeV} \sim 10^{10}\,\mathrm {K} \)) is much smaller than Fermi energies of the interior (\(>\,60 \,\mathrm {MeV}\)). One can then use a one-parameter, barotropic equation of state (EOS) to describe the matter:

$$\begin{aligned} \varepsilon = \varepsilon (P), \end{aligned}$$
(1)

where \(\varepsilon \) is the energy density and P is the pressure. At birth, a neutron star is expected to be rotating differentially, but as the neutron star cools, several mechanisms can act to enforce uniform rotation. Kinematical shear viscosity is acting against differential rotation on a timescale that has been estimated to be (Flowers and Itoh 1976, 1979; Cutler and Lindblom 1987)

$$\begin{aligned} \tau \sim 18 \left( \frac{\rho }{10^{15}\,\mathrm {g\ cm}^{-3}}\right) ^{-5/4} \left( \frac{T}{10^9\,\mathrm {K}}\right) ^2 \left( \frac{R}{10^6\,\mathrm {cm}}\right) \,\mathrm {yr}, \end{aligned}$$
(2)

where \(\rho \), T and R are the central density, temperature, and radius of the star. It has also been suggested that convective and turbulent motions may enforce uniform rotation on a timescale of the order of days (Hegyi 1977). Shapiro (2000) suggested that magnetic braking of differential rotation by Alfvén waves could be the most effective damping mechanism, acting on short timescales, possibly of the order of minutes.

Within a short time after its formation, the temperature of a neutron star becomes less than \(10^{10}\,\mathrm {K}\) (due to neutrino emission). When the temperature drops further, below roughly \(10^{9} \,\mathrm {K}\), its outer core is expected to become superfluid (see Mendell 1998 and references therein). Rotation causes superfluid neutrons to form an array of quantized vortices, with an intervortex spacing of

$$\begin{aligned} d_{\mathrm {n}} \sim 3.4 \times 10^{-3} \varOmega _2^{-1/2} \,\mathrm {cm}, \end{aligned}$$
(3)

where \(\varOmega _2\) is the angular velocity of the star in \(10^2 \,\mathrm {s}^{-1}\). On scales much larger than the intervortex spacing, e.g., of the order of centimeters or meters, the fluid motions can be averaged and the rotation can be considered to be uniform (Sonin 1987). With such an assumption, the error in computing the metric is of order

$$\begin{aligned} \left( \frac{1 \,\mathrm {cm}}{R} \right) ^2 \sim 10^{-12}, \end{aligned}$$
(4)

assuming \(R\sim 10 \,\mathrm {km}\) to be a typical neutron star radius.

The above arguments show that the bulk properties of a cold, isolated rotating relativistic star can be modeled accurately by a uniformly rotating, one-parameter perfect fluid. Effects of differential rotation and of finite temperature need only be considered during the first year (or less) after the formation of a relativistic star. Furthermore, magnetic fields, while important for high-energy phenomena in the magnetosphere and for the damping of differential rotation and oscillations, do not alter the structure of the star, unless one assumes magnetic field strengths significantly higher than typical observed values.

Geometry of spacetime

In general relativity, the spacetime geometry of a rotating star in equilibrium can be described by a stationary and axisymmetric metric \(g_{\alpha \beta }\) of the form

$$\begin{aligned} ds^2 = -e^{2 \nu } \, dt^2 + e^{2 \psi } (d \phi - \omega \, dt)^2 + e^{2 \mu } (dr^2+r^2 d \theta ^2), \end{aligned}$$
(5)

where \(\nu \), \(\psi \), \(\omega \) and \(\mu \) are four metric functions that depend on the coordinates r and \(\theta \) only (see, e.g., Bardeen and Wagoner 1971). For a discussion and historical overview of other coordinate choices for axisymmetric rotating spacetimes see Gourgoulhon (2010), Friedman and Stergioulas (2013). In the exterior vacuum, it is possible to reduce the number of metric functions to three, but as long as one is interested in describing the whole spacetime (including the source-region of nonzero pressure), four different metric functions are required. It is convenient to write \(e^\psi \) in the form

$$\begin{aligned} e^\psi =r \sin \theta B e^{-\nu }, \end{aligned}$$
(6)

where B is again a function of r and \(\theta \) only (Bardeen 1973).

One arrives at the above form of the metric assuming that

  1. 1.

    The spacetime is stationary and axisymmetric: There exist an asymptotically timelike symmetry vector \(t^\alpha \) and a rotational symmetry vector \(\phi ^\alpha \).

    The spacetime is said to be strictly stationary if \(t^\alpha \) is everywhere timelike. (Some rapidly rotating stellar models, as well as black-hole spacetimes, have ergospheres, regions in which \(t^\alpha \) is spacelike.)

  2. 2.

    The Killing vectors commute,

    $$\begin{aligned}{}[t, \phi ]=0, \end{aligned}$$
    (7)

    and there is an isometry of the spacetime that simultaneously reverses the direction of \(t^\alpha \) and \(\phi ^\alpha \),

    $$\begin{aligned} t^\alpha \rightarrow -t^\alpha ,\quad \phi ^\alpha \rightarrow - \phi ^\alpha . \end{aligned}$$
    (8)
  3. 3.

    The spacetime is asymptotically flat, i.e., \(t_at^a=-1\), \(\phi _a\phi ^a=+\infty \) and \(t_a\phi ^a=0\) at spatial infinity

  4. 4.

    The spacetime is circular (there are no meridional currents in the fluid).

If the spacetime is strictly stationary, one does not need (7) as a separate assumption: A theorem by Carter (1970) shows that \([t, \phi ]=0\). The Frobenius theorem now implies the existence of scalars t and \(\phi \) (Kundt and Trümper 1966; Carter 1969) for which there exists a family of 2-surfaces orthogonal to \(t^\alpha \) and \(\phi ^\alpha \), the surfaces of constant t and \(\phi \); and it is natural to choose as coordinates \(x^0=t\) and \(x^3=\phi \). In the absence of meridional currents, the 2-surfaces orthogonal to \(t^\alpha \) and \(\phi ^\alpha \) can be described by the remaining two coordinates \(x^1\) and \(x^2\) (Carter 1970). Requiring that these are Lie derived by \(t^\alpha \) and \(\phi ^\alpha \), we have

$$\begin{aligned} t^\alpha= & {} {\varvec{\partial }}_t, \end{aligned}$$
(9)
$$\begin{aligned} \phi ^\alpha= & {} {\varvec{\partial }}_\phi . \end{aligned}$$
(10)

With coordinates chosen in this way, the metric components are independent of t and \(\phi \).

Because time reversal inverts the direction of rotation, the fluid is not invariant under \(t\rightarrow -t\) alone, implying that \(t^\alpha \) and \(\phi ^\alpha \) are not orthogonal to each other. The lack of orthogonality is measured by the metric function \(\omega \) that describes the dragging of inertial frames.

In a fluid with meridional convective currents one loses both time-reversal invariance and invariance under the simultaneous inversion \(t\rightarrow -t, \phi \rightarrow -\phi \), because the inversion changes the direction of the circulation. In this case, the spacetime metric will have additional off-diagonal components (Gourgoulhon and Bonazzola 1993; Birkl et al. 2011).

A common choice for \(x^1\) and \(x^2\) are quasi-isotropic coordinates, for which \(g_{r\theta }=0\) and \(g_{\theta \theta }=r^2 g_{rr}\) (in spherical polar coordinates), or \(g_{\varpi z}=0\) and \(g_{zz}=r^2 g_{\varpi \varpi }\) (in cylindrical coordinates). In the nonrotating limit, the metric (5) reduces to the metric of a nonrotating relativistic star in isotropic coordinates (see Weinberg 1972 for the definition of these coordinates). In the slow rotation formalism by Hartle (1967), a different form of the metric is used, requiring \(g_{\theta \theta }=g_{\phi \phi }/ \sin ^2 \theta \), which corresponds to the choice of Schwarzschild coordinates in the vacuum region.

The three metric functions \(\nu \), \(\psi \) and \(\omega \) can be written as invariant combinations of the two Killing vectors \(t^\alpha \) and \(\phi ^\alpha \), through the relations

$$\begin{aligned} t_\alpha t^\alpha= & {} g_{tt}= -e^{2\nu } + \omega ^2 e^{2\psi }, \end{aligned}$$
(11)
$$\begin{aligned} \phi _\alpha \phi ^\alpha= & {} g_{\phi \phi }= e^{2\psi }, \end{aligned}$$
(12)
$$\begin{aligned} t_\alpha \phi ^\alpha= & {} g_{t\phi }= -\omega e^{2\psi }, \end{aligned}$$
(13)

The corresponding components of the contravariant metric are

$$\begin{aligned} g^{tt}= & {} \nabla _\alpha t \nabla ^\alpha t = -e^{-2\nu }, \end{aligned}$$
(14)
$$\begin{aligned} g^{\phi \phi }= & {} \nabla _\alpha \phi \nabla ^\alpha \phi = e^{-2\psi }-\omega ^2 e^{-2\nu }, \end{aligned}$$
(15)
$$\begin{aligned} g^{t\phi }= & {} \nabla _\alpha t \nabla ^\alpha \phi = -\omega e^{-2\nu }. \end{aligned}$$
(16)

The fourth metric function \(\mu \) determines the conformal factor \(e^{2\mu }\) that characterizes the geometry of the orthogonal 2-surfaces.

There are two main effects that distinguish a rotating relativistic star from its nonrotating counterpart: The shape of the star is flattened by centrifugal forces (an effect that first appears at second order in the rotation rate), and the local inertial frames are dragged by the rotation of the source of the gravitational field. While the former effect is also present in the Newtonian limit, the latter is a purely relativistic effect.

The study of the dragging of inertial frames in the spacetime of a rotating star is assisted by the introduction of the local Zero-Angular-Momentum-Observers (ZAMO) (Bardeen 1970, 1973). These are observers whose worldlines are normal to the \(t=\,\mathrm {const.}\) hypersurfaces (also called Eulerian or normal observers in the 3+1 formalism Arnowitt et al. 2008). Then, the metric function \(\omega \) is the angular velocity \(d\phi /dt\) of the local ZAMO with respect to an observer at rest at infinity. Also, \(e^{-\nu }\) is the time dilation factor between the proper time of the local ZAMO and coordinate time t (proper time at infinity) along a radial coordinate line. The metric function \(\psi \) has a geometrical meaning: \(e^\psi \) is the proper circumferential radius of a circle around the axis of symmetry.

In rapidly rotating models, an ergosphere can appear, where \(g_{tt}>0\) (as long as we are using the Killing coordinates described above). In this region, the rotational frame-dragging is strong enough to prohibit counter-rotating time-like or null geodesics to exist, and particles can have negative energy with respect to a stationary observer at infinity. Radiation fields (scalar, electromagnetic, or gravitational) can become unstable in the ergosphere (Friedman 1978), but the associated growth time is comparable to the age of the universe (Comins and Schutz 1978).

The lowest-order asymptotic behaviour of the metric functions \(\nu \) and \(\omega \) is

$$\begin{aligned} \nu\sim & {} -{M \over r}, \end{aligned}$$
(17)
$$\begin{aligned} \omega\sim & {} {2J \over r^3}, \end{aligned}$$
(18)

where M and J are the total gravitational mass and angular momentum (see Sect. 2.5 for definitions). The asymptotic expansion of the dragging potential \(\omega \) shows that it decays rapidly far from the star, so that its effect will be significant mainly in the vicinity of the star.

The rotating fluid

When sources of non-isotropic stresses (such as a magnetic field or a solid state of parts of the star), viscous stresses, and heat transport are neglected in constructing an equilibrium model of a relativistic star, then the matter can be modeled as a perfect fluid, described by the stress-energy tensor

$$\begin{aligned} T^{\alpha \beta } = (\varepsilon +P)u^\alpha u^\beta + P g^{\alpha \beta }, \end{aligned}$$
(19)

where \(u^\alpha \) is the fluid’s 4-velocity. In terms of the two Killing vectors \(t^\alpha \) and \(\phi ^\alpha \), the 4-velocity can be written as

$$\begin{aligned} u^\alpha = \frac{e^{-\nu }}{\sqrt{1-v^2}} (t^\alpha + \varOmega \phi ^\alpha ), \end{aligned}$$
(20)

where v is the 3-velocity of the fluid with respect to a local ZAMO, given by

$$\begin{aligned} v = (\varOmega -\omega )e^{\psi -\nu }, \end{aligned}$$
(21)

and \(\varOmega \equiv u^\phi / u^t=d\phi / dt\) is the angular velocity of the fluid in the coordinate frame, which is equivalent to the angular velocity of the fluid as seen by an observer at rest at infinity. Stationary configurations can be differentially rotating, while uniform rotation (\(\varOmega = \,\mathrm {const.}\)) is a special case (see Sect. 2.5).

The covariant components of the 4-velocity take the form

$$\begin{aligned} u_t= & {} -\frac{e^\nu }{\sqrt{1-v^2}}(1+e^{\psi -\nu }\omega v ), \qquad u_{\phi } = \frac{e^\psi v}{\sqrt{1-v^2}}. \end{aligned}$$
(22)

Notice that the components of the 4-velocity are proportional to the Lorentz factor \(W:=(1-v^2)^{-1/2}\).

Equations of structure

Having specified an equation of state of the form \(\varepsilon = \varepsilon (P)\), the structure of the star is determined by solving four components of Einstein’s gravitational field equation

$$\begin{aligned} R_{\alpha \beta } = 8 \pi \left( T_{\alpha \beta }- \frac{1}{2} g_{\alpha \beta }T \right) , \end{aligned}$$
(23)

(where \(R_{\alpha \beta }\) is the Ricci tensor and \(T=T_\alpha {}^\alpha \)) and the equation of hydrostationary equilibrium. Setting \(\zeta = \mu + \nu \), one common choice (Butterworth and Ipser 1976) for the components of the gravitational field equation are the three equations of elliptic type

$$\begin{aligned} \varvec{\nabla }\cdot (B \varvec{\nabla }\nu )= & {} \frac{1}{2} r^2\sin ^2\theta B^3e^{-4\nu } \varvec{\nabla }\omega \cdot \varvec{\nabla }\omega \nonumber \\&+\,4 \pi B e^{2\zeta -2\nu } \left[ \frac{(\varepsilon +P)(1+v^2)}{1-v^2} +2P \right] , \end{aligned}$$
(24)
$$\begin{aligned} \varvec{\nabla }\cdot (r^2\sin ^2\theta B^3e^{-4\nu }\varvec{\nabla }\omega )= & {} -16 \pi r \sin \theta B^2e^{2\zeta -4 \nu } \frac{(\varepsilon +P)v}{1-v^2}, \end{aligned}$$
(25)
$$\begin{aligned} \varvec{\nabla }\cdot (r \sin \theta \varvec{\nabla }B)= & {} 16 \pi r \sin \theta Be^{2\zeta -2\nu }P, \end{aligned}$$
(26)

supplemented by a first order differential equation for \(\zeta \)

$$\begin{aligned} \frac{1}{\varpi }\zeta ,_\varpi + \frac{1}{B}(B,_\varpi \zeta ,_\varpi -B,_z\zeta ,_z)= & {} \frac{1}{2\varpi ^2B}(\varpi ^2B,_\varpi ),_\varpi -\frac{1}{2B}B,_{zz}+(\nu ,_\varpi )^2 \nonumber \\&-(\nu ,_z)^2 -\frac{1}{4}\varpi ^2B^2e^{-4\nu }\left[ (\omega ,_\varpi )^2-(\omega ,_z)^2\right] . \nonumber \\&\end{aligned}$$
(27)

Above, \(\varvec{\nabla }\) is the 3-dimensional derivative operator in a flat 3-space with spherical polar coordinates r, \(\theta \), \(\phi \). The remaining nonzero components of the gravitational field equation yield two more elliptic equations and one first order partial differential equation, which are consistent with the above set of four equations.

The equation of motion (Euler equation) follows from the projection of the conservation of the stress-energy tensor orthogonal to the 4-velocity \((\delta ^\gamma {}_\beta +u^\gamma u_\beta )\nabla _\alpha T^{\alpha \beta }=0\)

$$\begin{aligned} \frac{\nabla _\alpha p}{(\epsilon +p)}= & {} -u^\beta \nabla _\beta u_\alpha \nonumber \\= & {} \nabla _\alpha \ln u^t -u^tu_\phi \nabla _\alpha \varOmega . \end{aligned}$$
(28)

In the \(r-\theta \) subspace, one can find the following equivalent forms

$$\begin{aligned} \frac{\nabla p}{(\epsilon +p)}= & {} -\frac{1}{1-v^2} \left( \nabla \nu -v^2\nabla \psi +e^{\psi -\nu } v \nabla \omega \right) , \end{aligned}$$
(29)
$$\begin{aligned}= & {} \nabla \ln u^t - u^t u_\phi \nabla \varOmega , \end{aligned}$$
(30)
$$\begin{aligned}= & {} \nabla \ln u^t - \frac{l}{1-\varOmega l} \nabla \varOmega , \end{aligned}$$
(31)
$$\begin{aligned}= & {} -\nabla \ln (-u_t) + \frac{\varOmega }{1-\varOmega l} \nabla l, \end{aligned}$$
(32)
$$\begin{aligned}= & {} -\nabla \nu + \frac{1}{1-v^2} \left( v \nabla v - \frac{v^2\nabla \varOmega }{\varOmega -\omega } \right) , \end{aligned}$$
(33)

where \(l:=-u_\phi /u_t\) is conserved along fluid trajectories (since \(h u_t\) and \(h u_\phi \) are conserved, so is their ratio and l is the angular momentum per unit energy).

For barotropes, one can arrive at a first integral of the equations of motion in the following way. Since \(\epsilon =\epsilon (p)\), one can define a function

$$\begin{aligned} H(p) := \int _0^p \frac{dp'}{\epsilon (p')+p'}, \end{aligned}$$
(34)

so that (28) becomes

$$\begin{aligned} \nabla (H-\ln u^t) = -F \nabla \varOmega , \end{aligned}$$
(35)

where we have set \(F:=u^t u_\phi =l/(1-l\varOmega )\). For homentropic stars (stars with a homogeneous entropy distribution) one obtains \(H=\ln h\) (where h is the specific enthalpy) and the equation of hydrostationary equilibrium takes the form

$$\begin{aligned} \nabla \left( \ln \frac{h}{u^t} \right) = -F \nabla \varOmega . \end{aligned}$$
(36)

Taking the curl of (35) one finds that either

$$\begin{aligned} \varOmega =\mathrm{constant}, \end{aligned}$$
(37)

(uniform rotation), or

$$\begin{aligned} F=F(\varOmega ), \end{aligned}$$
(38)

in the case of differential rotation. In the latter case, (35) becomes

$$\begin{aligned} H-\ln u^t+\int ^\varOmega _{\varOmega _\mathrm{pole}} F(\varOmega ')d\varOmega ' =\nu |_\mathrm{pole}, \end{aligned}$$
(39)

where the lower limit, \(\varOmega _0\) is chosen as the value of \(\varOmega \) at the pole, where H and v vanish. The above global first integral of the hydrostationary equilibrium equations is useful in constructing numerical models of rotating stars.

For a uniformly rotating star, (39) can be written as

$$\begin{aligned} H-\ln u^t = \nu |_\mathrm{pole}, \end{aligned}$$
(40)

which, in the case of a homentropic star, becomes

$$\begin{aligned} \frac{h}{u^t} = \mathcal{E}, \end{aligned}$$
(41)

with \(\mathcal{E}=\left. e^\nu \right. |_\mathrm{pole}\) constant over the star (\(\mathcal E\) has the meaning of the injection energy (Friedman and Stergioulas 2013), the increase in a star’s mass when a unit mass of baryons is injected at a point in the star).

In the Newtonian limit \(\displaystyle e^\psi = \varpi + O(\lambda ^2), \quad e^\nu = 1+O(\lambda ^2)\), so to Newtonian order we have

$$\begin{aligned} u^t u_\phi = v\varpi = \varpi ^2\varOmega , \end{aligned}$$
(42)

and the functional dependence of \(\varOmega \) implied by Eq. (38) becomes the familiar requirement that, for a barotropic equation of state, \(\varOmega \) be stratified on cylinders,

$$\begin{aligned} \varOmega = \varOmega (\varpi ), \end{aligned}$$
(43)

where \(\varpi =r\sin (\theta )\). The Newtonian limit of the integral of motion (39) is

$$\begin{aligned} h_\mathrm{Newtonian} -\frac{1}{2} v^2 + \varPhi = \text{ constant }, \end{aligned}$$
(44)

where, in the Newtonian limit, \(h_\mathrm{Newtonian}=h-1\) differs from the relativistic definition by the rest mass per unit rest mass.

Rotation law and equilibrium quantities

A special case of rotation law is uniform rotation (uniform angular velocity in the coordinate frame), which minimizes the total mass–energy of a configuration for a given baryon number and total angular momentum (Boyer and Lindquist 1966; Hartle and Sharp 1967). In this case, the term involving \(F(\varOmega )\) in (39) vanishes.

More generally, a simple, one-parameter choice of a differential-rotation law is

$$\begin{aligned} F(\varOmega )= A^2(\varOmega _{\mathrm {c}}-\varOmega ) = \frac{(\varOmega -\omega )r^2\sin ^2\theta ~e^{2(\beta -\nu )}}{1-(\varOmega -\omega )^2r^2\sin ^2\theta ~e^{2(\beta -\nu )}}, \end{aligned}$$
(45)

where A is a constant (Komatsu et al. 1989a, b). When \(A \rightarrow \infty \), the above rotation law reduces to the uniform rotation case. In the Newtonian limit and when \(A \rightarrow 0\), the rotation law becomes a so-called j-constant rotation law (with specific angular momentum j, angular momentum per unit mass, being constant in space), which satisfies the Rayleigh criterion for local dynamical stability against axisymmetric disturbances (j should not decrease outwards, \(dj/d\varOmega <0\)). The same criterion is also satisfied in the relativistic case, but with \(j\rightarrow \tilde{j}=u_\phi (\varepsilon +P)/\rho \) (Komatsu et al. 1989b), where \(\rho \) is the fluid rest-mass density. It should be noted that differentially rotating stars may also be subject to a shear instability that tends to suppress differential rotation (Zahn 1993).

The above rotation law is a simple choice that has proven to be computationally convenient. A new, 3-parameter generalization of the above rotation law was recently proposed in Galeazzi et al. (2012) and is defined by

$$\begin{aligned} F(\varOmega )= \frac{\frac{R_0^2}{\varOmega _c^\alpha }\varOmega (\varOmega ^\alpha -\varOmega _c^\alpha )}{1-\frac{R_0^2}{\varOmega _c^\alpha }\varOmega ^2(\varOmega ^\alpha -\varOmega _c^\alpha )} \end{aligned}$$
(46)

where \(\alpha ,\ R_0\) and \(\varOmega _c\) are constants. The specific angular momentum corresponding to this law is

$$\begin{aligned} l= \frac{R_0^2}{\varOmega _c^\alpha }\varOmega (\varOmega ^\alpha -\varOmega _c^\alpha ) . \end{aligned}$$
(47)

The Newtonian limit for this law yields an angular frequency of

$$\begin{aligned} \varOmega = \varOmega _c\left[ 1+\left( \frac{\varpi }{R_0}\right) ^2\right] ^{\frac{1}{\alpha }}, \end{aligned}$$
(48)

thus, for \(\varpi \ll R_0\), \(\varOmega \sim \varOmega _c\), whereas for \(\varpi \gg R_0\), \(\varOmega _c\sim \varOmega (\varpi /R_0)^{2/\alpha }\). A more recent 4-parameter family of rotation laws was proposed in (Mach and Malec 2015) mainly for accretion tori (it has not yet been applied to models of rotating neutron stars). It remains to be seen how well the above laws can match the angular velocity profiles of proto-neutron stars and remnants of binary neutron star mergers formed in numerical simulations.

Table 1 Equilibrium properties

Equilibrium quantities for rotating stars, such as gravitational mass, baryon mass, or angular momentum, for example, can be obtained as integrals over the source of the gravitational field. A list of the most important equilibrium quantities that can be computed for axisymmetric models, along with the equations that define them, is displayed in Table 1. There, \(\rho \) is the rest-mass density, \(u=\varepsilon -\rho c^2\) is the internal energy density, \(\hat{n}^a= \nabla _at/|\nabla _bt\nabla ^bt|^{1/2}\) is the unit normal vector to the \(t= \,\mathrm {const.}\) spacelike hypersurfaces, and \(dV=\sqrt{|{}^3g|} \, d^3x\) is the proper 3-volume element (with \({}^3g\) being the determinant of the 3-metric of spacelike hypersurfaces). It should be noted that the moment of inertia cannot be computed directly as an integral quantity over the source of the gravitational field. In addition, there exists no unique generalization of the Newtonian definition of the moment of inertia in general relativity and \(I=J/\varOmega \) is a common choice.

Equations of state

Relativistic polytropes

Because old neutron-stars have temperatures much smaller than the Fermi energy of their constituent particles, one can ignore entropy gradients and assume a uniform specific entropy s. The increase in pressure and density toward the star’s center are therefore adiabatic, if one neglects the slow change in composition. That is, they are related by the first law of thermodynamics, with \(ds=0\),

$$\begin{aligned} d\epsilon = \frac{\epsilon +p}{\rho } d\rho , \end{aligned}$$
(49)

with p given in terms of \(\rho \) by

$$\begin{aligned} \frac{\rho }{p} \frac{dp}{d\rho } = \frac{\epsilon +p}{p}\frac{dp}{d \epsilon } = \varGamma _1. \end{aligned}$$
(50)

Here \(\varGamma _1\) is the adiabatic index, the fractional change in pressure per fractional change in comoving volume, at constant entropy and composition. In an ideal degenerate Fermi gas, in the nonrelativistic and ultrarelativistic regimes, \(\varGamma _1\) has the constant values 5 / 3 and 4 / 3, respectively. Except in the outer crust, neutron-star matter is far from an ideal Fermi gas, but models often assume a constant effective adiabatic index, chosen to match an average stellar compressibility. An equation of state of the form

$$\begin{aligned} p = K \rho ^\varGamma , \end{aligned}$$
(51)

with K and \(\varGamma \) constants, is called polytropic; K and \(\varGamma \) are the polytropic constant and polytropic exponent, respectively. The corresponding relation between \(\epsilon \) and p follows from (49)

$$\begin{aligned} \epsilon = \rho + \frac{p}{\varGamma -1}. \end{aligned}$$
(52)

The polytropic exponent \(\varGamma \) is commonly replaced by a polytropic index N, given by

$$\begin{aligned} \varGamma =1+\frac{1}{N}. \end{aligned}$$
(53)

For the above polytropic EOS, the quantity \(c^{(\varGamma -2)/(\varGamma -1)} \sqrt{K^{1/(\varGamma -1)}/G}\) has units of length. In gravitational units one can thus use \(K^{N/2}\) as a fundamental length scale to define dimensionless quantities. Equilibrium models are then characterized by the polytropic index N and their dimensionless central energy density. Equilibrium properties can be scaled to different dimensional values, using appropriate values for K. For \(N<1.0\) (\(N>1.0\)) one obtains stiff (soft) models, while for \(N\sim 0.5\)–1.0, one obtains models whose masses and radii are roughly consistent with observed neutron-star masses and with the weak constraints on radius imposed by present observations and by candidate equations of state.

The definition (51), (52) of the relativistic polytropic EOS was introduced by Tooper (1965), to allow a polytropic exponent \(\varGamma \) that coincides with the adiabatic index of a relativistic fluid with constant entropy per baryon (a homentropic fluid). A different form, \(p=K\epsilon ^\varGamma \), previously also introduced by Tooper (1964), does not satisfy Eq. (49) and therefore it is not consistent with the first law of thermodynamics for a fluid with uniform entropy.

Hadronic equations of state

Cold matter below the nuclear saturation density, \(\rho _0 = 2.7\times 10^{14}\,\mathrm {g/cm}^{3}\) (or \(n_0 = 0.16\,\mathrm {fm}^{-3}\)), is thought to be well understood. A derivation of a sequence of equations of state at increasing densities, beginning with the semi-empirical mass formula for nuclei, can be found in Shapiro and Teukolsky (1983) (see also Haensel et al. 2007). Another treatment, using experimental data on neutron-rich nuclei was given in Haensel and Pichon (1994). In a neutron star, matter below nuclear density forms a crust, whose outer part is a lattice of nuclei in a relativistic electron gas. At \(4\times 10^{11}\,\mathrm {g/cm}^{3}\), the electron Fermi energy is high enough to induce neutron drip: Above this density nucleons begin leaving their nuclei to become free neutrons. The inner crust is then a two-phase equilibrium of the lattice nuclei and electrons and a gas of free neutrons. The emergence of a free-neutron phase means that the equation of state softens immediately above neutron drip: Increasing the density leads to an increase in free neutrons and to a correspondingly smaller increase in pressure. Melting of the Coulomb lattice, marking the transition from crust to a liquid core of neutrons, protons and electrons occurs between \(10^{14}\,\mathrm {g/cm}^{3}\) and \(\rho _0\).

A review by Heiselberg and Pandharipande (2000) describes the partly phenomenological construction of a primarily nonrelativistic many-body theory that gives the equation of state at and slightly below nuclear density. Two-nucleon interactions are matched to neutron–neutron scattering data and the experimentally determined structure of the deuteron. Parameters of the three-nucleon interaction are fixed by matching the observed energy levels of light nuclei.

Above nuclear density, however, the equation of state is still beset by substantial uncertainties. For a typical range of current candidate equations of state, values of the pressure differ by more than a factor of 5 at \(2\rho _0 \sim 5\times 10^{14}\) g/cm\(^3\), and by at least that much at higher densities (Haensel 2003). Although scattering experiments probe the interactions of nucleons (and quarks) at distances small compared to the radius of a nucleon, the many-body theory required to deduce the equation of state from fundamental interactions is poorly understood. Heavy ion collisions do produce collections of nucleons at supranuclear densities, but here the unknown extrapolation is from the high temperature of the experiment to the low temperature of neutron-star matter.

Observations of neutron stars provide a few additional constraints, of which, two are unambiguous and precise: The equation of state must allow a mass at least as large as \(1.97\,M_{\odot }\), the largest accurately determined mass of a neutron star. (The observation by Antoniadis et al. (2013) is of a \(2.01 \pm 0.04\) neutron star. There is also an observation by Demorest et al. (2010) of a \(1.97 \pm 0.04\,M_{\odot }\) neutron star). The equation of state must also allow a rotational period at least as small as 1.4 ms, the period of the fastest confirmed millisecond pulsar (Hessels et al. 2006). Observations of neutron star radii are much less precise, but a large number of observations of type I X-ray bursts or transient X-ray binaries may allow for the reconstruction of the neutron star equation of state (Özel and Psaltis 2009; Özel et al. 2010; Steiner et al. 2010).

The uncertainty in the equation of state above nuclear density is dramatically seen in the array of competing alternatives for the nature of matter in neutron star cores: Cores that are dominantly neutron matter may have sharply different equations of state, depending on the presence or absence of pion or kaon condensates, of hyperons, and of droplets of strange quark matter (described below). The inner core of the most massive neutron stars may be entirely strange quark matter. Other differences in candidate equations of state arise from constructions based on relativistic and on nonrelativistic many-body theory. A classic collection of early proposed EOSs was compiled by Arnett and Bowers (1977), while reviews of many modern EOSs have been compiled by Haensel (2003) and Lattimer and Prakash (2007). Detailed descriptions and tables of several modern EOSs, especially EOSs with phase transitions, can be found in Glendenning (1997); his treatment is particularly helpful in showing how one constructs an equation of state from a relativistic field theory. The review by Heiselberg and Pandharipande (2000), in contrast, presents a more phenomenological construction of equations of state that match experimental data. Detailed theoretical derivations of equations of state are presented in the book by Haensel et al. (2007). For recent reviews on nuclear EOSs see Sagert et al. (2010), Lattimer (2012), Fischer et al. (2014), Lattimer and Prakash (2016), Oertel et al. (2017).

Candidate EOSs are supplied in the form of an energy density versus pressure table and intermediate values are interpolated. This results in some loss of accuracy because the usual interpolation methods do not preserve thermodynamic consistency. Swesty (1996) devised a cubic Hermite interpolation scheme that does preserve thermodynamical consistency and the scheme has been shown to indeed produce higher-accuracy neutron star models (Nozawa et al. 1998).

High density equations of state with pion condensation were proposed in Migdal (1971) and Sawyer and Scalapino (1972) (see also Kunihiro et al. 1993). Beyond nuclear density, the electron chemical potential could exceed the rest mass of \(\pi ^-\) (139 MeV) by a margin large enough to overcome a pion-neutron repulsion and thus allow a condensate of zero-momentum pions. The critical density is thought to be \(2\rho _0\) or higher, but the uncertainty is greater than a factor of 2; and a condensate with both \(\pi ^0\) and \(\pi ^-\) has also been suggested. Because the s-wave kaon-neutron interaction is attractive, kaon condensation may also occur, despite the higher kaon mass, a possibility suggested in Kaplan and Nelson (1986) (for discussions with differing viewpoints see Brown and Bethe 1994; Pandharipande et al. 1995; Heiselberg and Pandharipande 2000). Pion and kaon condensates lead to significant softening of the equation of state.

As initially suggested in Ambartsumyan and Saakyan (1960), when the Fermi energy of the degenerate neutrons exceeds the mass of a \(\varLambda \) or \(\varSigma \), weak interactions convert neutrons to these hyperons: Examples are \(2n\leftrightarrow p+\varSigma ^-\), \(n+p^+\rightarrow p^++\varLambda \). Reviews and further references can be found in Glendenning (1997), Balberg and Gal (1997), Prakash et al. (1997), and more recent work, spurred by the r-mode instability (see Sect. 4.5.3), is reported in Lindblom and Owen 2002; Haensel et al. 2002; Lackey et al. 2006. The critical density above which hyperons appear is estimated at 2 or 3 times nuclear density. Above that density, the presence of copious hyperons can significantly soften the equation of state. Because a softer core equation of state can support less mass against collapse, the larger the observed maximum mass, the less likely that neutron stars have cores with hyperons (or with pion or kaon condensates). In particular, a measured mass of \(1.97\pm 0.04\,M_{\odot }\) for the pulsar PSR J1614-2230 with a white dwarf companion (Demorest et al. 2010) limits the equation of state parameter space (Read et al. 2009), ruling out several candidate equations of states with hyperons (Özel et al. 2010). Whether a hyperon core is consistent with a mass this large remains an open question (Stone et al. 2010).

A new hadron-quark hybrid equation of state was recently introduced by Benić et al. (2015) (see also Bejger et al. 2016 for potential observational signatures of these objects). The quark matter description is based on a quantum chromodynamics approach, while the hadronic matter is modeled by means of a relativistic mean-field method with an excluded volume correction at supranuclear densities to treat the finite size of the nucleons. The excluded volume correction in conjunction with the quark repulsive interactions, result in a first-order phase transition, which leads to a new family of compact stars in a mass-radius relationship plot whose masses can exceed the \(2\,M_{\odot }\) limit that is set by observations. These new stars are termed “twin” stars. The twin star phenomenon was predicted a long time ago by Gerlach (1968) (see also Kampfer 1981; Schertler et al. 2000; Glendenning and Kettner 2000). Twin stars consist of a quark core with a shell made of hadrons and a first-order phase transition at their interface. Recently, rotating twin star solutions were constructed by Haensel et al. (2016).

Strange quark equations of state

Before a density of about \(6\rho _0\) is reached, lattice QCD calculations indicate a phase transition from quarks confined to nucleons (or hyperons) to a collection of free quarks (and gluons). Heavy ion collisions at CERN and RHIC show evidence of the formation of such a quark-gluon plasma. A density for the phase transition higher than that needed for strange quarks in hyperons is similarly high enough to give a mixture of up, down and strange quarks in quark matter, and the expected strangeness per unit baryon number is \(\simeq -1\). If densities become high enough for a phase transition to quark matter to occur, neutron-star cores may contain a transition region with a mixed phase of quark droplets in neutron matter (Glendenning 1997).

Bodmer (1971) and, later, Witten (1984) pointed out that experimental data do not rule out the possibility that the ground state of matter at zero pressure and large baryon number is not iron but strange quark matter. If this is the case, all “neutron stars” may be strange quark stars, a lower density version of the quark-gluon plasma, again with roughly equal numbers of up, down and strange quarks, together with electrons to give overall charge neutrality (Bodmer 1971; Farhi and Jaffe 1984). The first extensive study of strange quark star properties is due to Witten (1984) (but, see also Ipser et al. 1975; Brecher and Caporaso 1976), while hybrid stars that have a mixed-phase region of quark and hadronic matter, have also been studied extensively (see, e.g., the review by Glendenning 1997).

The strange quark matter equation of state can be represented by the following linear relation between pressure and energy density

$$\begin{aligned} p=a(\epsilon -\epsilon _0), \end{aligned}$$
(54)

where \(\epsilon _0\) is the energy density at the surface of a bare strange star (neglecting a possible thin crust of normal matter). The MIT bag model of strange quark matter involves three parameters, the bag constant, \(\mathcal{B}=\epsilon _0/4\), the mass of the strange quark, \(m_s\), and the QCD coupling constant, \(\alpha _c\). The constant a in (54) is equal to 1 / 3 if one neglects the mass of the strange quark, while it takes the value of \(a=0.289\) for \(m_s=250\,\mathrm {MeV}\). When measured in units of \(\mathcal{B}_{60}=\mathcal{B}/(60\,\mathrm {MeV\ fm}^{-3})\), the constant \(\mathcal{B}\) is restricted to be in the range

$$\begin{aligned} 0.9821<\mathcal{B}_{60}<1.525, \end{aligned}$$
(55)

assuming \(m_s=0\). The lower limit is set by the requirement of stability of neutrons with respect to a spontaneous fusion into strangelets, while the upper limit is determined by the energy per baryon of \({}^{56}\)Fe at zero pressure (930.4 MeV). For other values of \(m_s\) the above limits are modified somewhat (see also Dey et al. 1998; Gondek-Rosińska et al. 2000 for other attempts to describe deconfined strange quark matter).

Numerical schemes

All available methods for solving the system of equations describing the equilibrium of rotating relativistic stars are numerical, as no self-consistent solution for both the interior and exterior spacetime in an algebraic closed form has been found. The first numerical solutions were obtained by Wilson (1972) and by Bonazzola and Schneider (1974). In the following, we give a description of several available numerical methods and their various implementations (codes) and extensions.

Hartle

To order \(\mathcal{O}(\varOmega ^2)\), the structure of a star changes only by quadrupole terms and the equilibrium equations become a set of ordinary differential equations. Hartle’s (1967; 1968) method computes rotating stars in this slow rotation approximation, and a review of slowly rotating models has been compiled by Datta (1988). Weber and Glendenning (1991) and Weber et al. (1991) also implement Hartle’s formalism to explore the rotational properties of four new EOSs.

Weber and Glendenning (1992) improve on Hartle’s formalism in order to obtain a more accurate estimate of the angular velocity at the mass-shedding limit, but their models still show large discrepancies compared to corresponding models computed without the slow rotation approximation (Salgado et al. 1994a). Thus, Hartle’s formalism is appropriate for typical pulsar (and most millisecond pulsar) rotational periods, but it is not the method of choice for computing models of rapidly rotating relativistic stars near the mass-shedding limit. An extension of Hartle’s scheme to 3rd order was presented by Benhar et al. (2005).

Butterworth and Ipser (BI)

The BI scheme (Butterworth and Ipser 1976) solves the four field equations following a Newton–Raphson-like linearization and iteration procedure. One starts with a nonrotating model and increases the angular velocity in small steps, treating a new rotating model as a linear perturbation of the previously computed rotating model. Each linearized field equation is discretized and the resulting linear system is solved. The four field equations and the hydrostationary equilibrium equation are solved separately and iteratively until convergence is achieved.

Space is truncated at a finite distance from the star and the boundary conditions there are imposed by expanding the metric potentials in powers of 1 / r. Angular derivatives are approximated by high-accuracy formulae and models with density discontinuities are treated specially at the surface. An equilibrium model is specified by fixing its rest mass and angular velocity.

The original BI code was used to construct uniform density models and polytropic models (Butterworth and Ipser 1976; Butterworth 1976). Friedman et al. (1986, 1989) (FIP) extend the BI code to obtain a large number of rapidly rotating models based on a variety of realistic EOSs. Lattimer et al. (1990) used a code that was also based on the BI scheme to construct rotating stars using “exotic” and schematic EOSs, including pion or kaon condensation and strange quark matter.

Komatsu, Eriguchi, and Hachisu (KEH)

In the KEH scheme (Komatsu et al. 1989a, b), the same set of field equations as in BI is used, but the three elliptic-type field equations are converted into integral equations using appropriate Green’s functions. The boundary conditions at large distance from the star are thus incorporated into the integral equations, but the region of integration is truncated at a finite distance from the star. The fourth field equation is an ordinary first order differential equation. The field equations and the equation of hydrostationary equilibrium are solved iteratively, fixing the maximum energy density and the ratio of the polar radius to the equatorial radius, until convergence is achieved. In Komatsu et al. (1989a, b) and Eriguchi et al. (1994), the original KEH code is used to construct uniformly and differentially rotating stars for both polytropic and realistic EOSs.

Cook, Shapiro, and Teukolsky (CST) improve on the KEH scheme by introducing a new radial variable that maps the semi-infinite region \([0,\infty )\) to the closed region [0, 1]. In this way, the region of integration is not truncated and the model converges to a higher accuracy. Details of the code are presented in Cook et al. (1992) and polytropic and realistic models are computed in Cook et al. (1994b) and Cook et al. (1994a).

Stergioulas and Friedman (SF) implement their own KEH code following the CST scheme. They improve on the accuracy of the code by a special treatment of the second order radial derivative that appears in the source term of the first order differential equation for one of the metric functions. This derivative was introducing a numerical error of 1–2% in the bulk properties of the most rapidly rotating stars computed in the original implementation of the KEH scheme. The SF code is presented in Stergioulas and Friedman (1995) and in Stergioulas (1996). It is available as a public domain code, named RNS, and can be downloaded from Stergioulas (1999).

A generalized KEH-type numerical code, suitable also for binary compact objects, was presented by Uryū and Tsokaros (2012); Uryū et al. (2012). The COCAL code has been applied to black hole models, and was recently extended to neutron star models, either in isolation (Uryū et al. 2014, 2016b) or in binaries (Tsokaros et al. 2015). The extended COCAL code allows for the generation of (quasi)equilribrium, magnetized, and rotating axisymmetric neutron star models, as well as quasiequilibrium corotational, irrotational, and spinning neutron star binaries. The code can also build models of isolated, quasiequilibrium, triaxial neutron stars (Uryū et al. 2016b, a)—a generalization of Jacobi ellipsoids in general relativity. Such configurations were recently studied dynamically in Tsokaros et al. (2017) and were found to be dynamically stable, though their secular stability still remains an open question.

Bonazzola et al. (BGSM)

In the BGSM scheme (Bonazzola et al. 1993), the field equations are derived in the 3\(+\)1 formulation. All four chosen equations that describe the gravitational field are of elliptic type. This avoids the problem with the second order radial derivative in the source term of the ODE used in BI and KEH. The equations are solved using a spectral method, i.e., all functions are expanded in terms of trigonometric functions in both the angular and radial directions and a Fast Fourier Transform (FFT) is used to obtain coefficients. Outside the star a redefined radial variable is used, which maps infinity to a finite distance.

In Salgado et al. (1994a, b), the code is used to construct a large number of models based on recent EOSs. The accuracy of the computed models is estimated using two general relativistic virial identities, valid for general asymptotically flat spacetimes (Gourgoulhon and Bonazzola 1994; Bonazzola and Gourgoulhon 1994) (see Sect. 2.7.8).

While the field equations used in the BI and KEH schemes assume a perfect fluid, isotropic stress-energy tensor, the BGSM formulation makes no assumption about the isotropy of \(T_{ab}\). Thus, the BGSM code can compute stars with a magnetic field, a solid crust, or a solid interior, and it can also be used to construct rotating boson stars.

LORENE/rotstar

Bonazzola et al. (1998) have improved the BGSM spectral method by allowing for several domains of integration. One of the domain boundaries is chosen to coincide with the surface of the star and a regularization procedure is introduced for the divergent derivatives at the surface (that appear in the density field when stiff equations of state are used). This allows models to be computed that are nearly free of Gibbs phenomena at the surface. The same method is also suitable for constructing quasi-stationary models of binary neutron stars. The new method has been used in Gourgoulhon et al. (1999) for computing models of rapidly rotating strange stars and it has also been used in 3D computations of the onset of the viscosity-driven instability to bar-mode formation (Gondek-Rosińska and Gourgoulhon 2002).

The LORENE library is available as public domain software (Gourgoulhon et al. 2008). It has also been used to construct equilibrium models of rotating stars as initial data for a fully constraint evolution scheme in the Dirac gauge and with maximal slicing (Lin and Novak 2006).

Ansorg et al. (AKM)

Another multi-domain spectral scheme was introduced in Ansorg et al. (2002, 2003). The scheme can use several domains inside the star, one for each possible phase transition in the equation of state. Surface-adapted coordinates are used and approximated by a two-dimensional Chebyshev-expansion. Transition conditions are satisfied at the boundary of each domain, and the field and fluid equations are solved as a free boundary value problem by a full Newton–Raphson method, starting from an initial guess. The field-equation components are simplified by using a corotating reference frame. Applying this new method to the computation of rapidly rotating homogeneous relativistic stars, Ansorg et al. achieve near machine accuracy, when about 24 expansion coefficients are used (see Sect. 2.7.9). For configurations near the mass-shedding limit the relative error increases to about \(10^{-5}\), even with 24 expansion coefficients, due to the low differentiability of the solution at the surface. The AKM code has been used in systematic studies of uniformly rotating homogeneous stars (Schöbel and Ansorg 2003) and differentially rotating polytropes (Ansorg et al. 2009). A detailed description of the numerical method and a review of the results is given in Meinel et al. (2008).

A public domain library which implements spectral methods for solving nonlinear systems of partial differential equations with a Newton–Rapshon method was presented by Grandclément (2010, 2009). The KADATH library could be used to construct equilibrium models of rotating relativistic stars in a similar manner as in Ansorg et al. (2002, 2003).

IWM-CFC approximation

The spatial conformal flatness condition (IWM-CFC) (Isenberg 2008; Wilson et al. 1996) is an approximation, in which the spatial part of the metric is assumed to be conformally flat. Computationally, one has to solve one equation less than in full GR, for isolated stars. The accuracy of this approximation has been tested for uniformly rotating stars by Cook et al. (1996) and it is satisfactory for many applications. Nonaxisymmetric configurations in the IWM-CFC approximation were obtained in Huang et al. (2008). The accuracy of the IWM-CFC approximation was also tested for initial data of strongly differentially rotating neutron star models (Iosif and Stergioulas 2013).

The conformal flatness approach has been extended to avoid non-uniqueness issues arising in the solution of the standard CFC equations by Cordero-Carrión et al. (2009). This method has also been termed the “extended CFC” approach (Bucciantini and Del Zanna 2011) and has been applied to the construction of general relativistic magnetodydrodynamic equilibria (Pili et al. 2014, 2017).

The virial identities

Equilibrium configurations in Newtonian gravity satisfy the well-known virial relation (assuming a polytropic equation of state)

$$\begin{aligned} 2T+3(\varGamma -1)U+W=0. \end{aligned}$$
(56)

This can be used to check the accuracy of computed numerical solutions. In general relativity, a different identity, valid for a stationary and axisymmetric spacetime, was found in Bonazzola (1973). More recently, two relativistic virial identities, valid for general asymptotically flat spacetimes, have been derived by Gourgoulhon and Bonazzola (1994); Bonazzola and Gourgoulhon (1994). The 3-dimensional virial identity (GRV3) (Gourgoulhon and Bonazzola 1994) is the extension of the Newtonian virial identity (56) to general relativity. The 2-dimensional (GRV2) (Bonazzola and Gourgoulhon 1994) virial identity is the generalization of the identity found in Bonazzola (1973) (for axisymmetric spacetimes) to general asymptotically flat spacetimes. In Bonazzola and Gourgoulhon (1994), the Newtonian limit of GRV2, in axisymmetry, is also derived. Previously, such a Newtonian identity had only been known for spherical configurations (Chandrasekhar 1939).

The two virial identities are an important tool for checking the accuracy of numerical models and have been repeatedly used by several authors (see, e.g., Bonazzola et al. 1993; Salgado et al. 1994a, b; Nozawa et al. 1998; Ansorg et al. 2002).

Direct comparison of numerical codes

The accuracy of the above numerical codes can be estimated, if one constructs exactly the same models with different codes and compares them directly. The first such comparison of rapidly rotating models constructed with the FIP and SF codes is presented in Stergioulas and Friedman (1995). Rapidly rotating models constructed with several EOSs agree to 0.1–1.2% in the masses and radii and to better than 2% in any other quantity that was compared (angular velocity and momentum, central values of metric functions, etc.). This is a very satisfactory agreement, considering that the BI code was using relatively few grid points, due to limitations of computing power at the time of its implementation.

Table 2 Detailed comparison of the accuracy of different numerical codes in computing a rapidly rotating, uniform density model

In Stergioulas and Friedman (1995), it is also shown that a large discrepancy between certain rapidly rotating models (constructed with the FIP and KEH codes) that was reported by Eriguchi et al. (1994), resulted from the fact that Eriguchi et al. and FIP used different versions of a tabulated EOS.

Nozawa et al. (1998) have completed an extensive direct comparison of the BGSM, SF, and the original KEH codes, using a large number of models and equations of state. More than twenty different quantities for each model are compared and the relative differences range from \(10^{-3}\) to \(10^{-4}\) or better, for smooth equations of state. The agreement is also excellent for soft polytropes. These checks show that all three codes are correct and successfully compute the desired models to an accuracy that depends on the number of grid points used to represent the spacetime.

If one makes the extreme assumption of uniform density, the agreement is at the level of \(10^{-2}\). In the BGSM code this is due to the fact that the spectral expansion in terms of trigonometric functions cannot accurately represent functions with discontinuous first order derivatives at the surface of the star. In the KEH and SF codes, the three-point finite-difference formulae cannot accurately represent derivatives across the discontinuous surface of the star.

The accuracy of the three codes is also estimated by the use of the two virial identities. Overall, the BGSM and SF codes show a better and more consistent agreement than the KEH code with BGSM or SF. This is largely due to the fact that the KEH code does not integrate over the whole spacetime but within a finite region around the star, which introduces some error in the computed models.

A direct comparison of different codes is also presented by Ansorg et al. (2002). Their multi-domain spectral code is compared to the BGSM, KEH, and SF codes for a particular uniform density model of a rapidly rotating relativistic star. An extension of the detailed comparison in Ansorg et al. (2002), which includes results obtained by the LORENE/rotstar code in Gondek-Rosińska and Gourgoulhon (2002) and by the SF code with higher resolution than the resolution used in Nozawa et al. (1998), is shown in Table 2. The comparison confirms that the virial identity GRV3 is a good indicator of the accuracy of each code. For the particular model in Table 2, the AKM code achieves nearly double-precision accuracy, while the Lorene/rotstar code has a typical relative accuracy of \(2 \times 10^{-4}\)\(7\times 10^{-6}\) in various quantities. The SF code at high resolution comes close to the accuracy of the Lorene/rotstar code for this model. Lower accuracy is obtained with the SF, BGSM, and KEH codes at the resolutions used in Nozawa et al. (1998).

The AKM code converges to machine accuracy when a large number of about 24 expansion coefficients are used at a high computational cost. With significantly fewer expansion coefficients (and comparable computational cost to the SF code at high resolution) the achieved accuracy is comparable to the accuracy of the LORENE/rotstar and SF codes. Moreover, at the mass-shedding limit, the accuracy of the AKM code reduces to about 5 digits (which is still highly accurate, of course), even with 24 expansion coefficients, due to the nonanalytic behaviour of the solution at the surface. Nevertheless, the AKM method represents a great achievement, as it is the first method to converge to machine accuracy when computing rapidly rotating stars in general relativity.

Going further   A review of spectral methods in numerical relativity can be found in Grandclément and Novak (2009). Pseudo-Newtonian models of axisymmetric, rotating relativistic stars are treated in Kim et al. (2009), while a formulation for nonaxisymmetric, uniformly rotating equilibrium configurations in the second post-Newtonian approximation is presented in Asada and Shibata (1996). Slowly-rotating models of white dwarfs in general relativity are presented in Boshkayev et al. (2013). The validity of the slow-rotation approximation is examined in Berti et al. (2005). A minimal-surface scheme was presented in Neugebauer and Herold (1992). The convergence properties iterative self-consistent-field methods when applied to stellar equilibria are investigated in Price et al. (2009).

Analytic approximations to the exterior spacetime

The exterior metric of a rapidly rotating neutron star differs considerably from the Kerr metric. The two metrics agree only to lowest order in the rotational velocity (Hartle and Thorne 1969). At higher order, the multipole moments of the gravitational field created by a rapidly rotating compact star are different from the multipole moments of the Kerr field. There have been many attempts in the past to find analytic solutions to the Einstein equations in the stationary, axisymmetric case, that could describe a rapidly rotating neutron star.

In the vacuum region surrounding a stationary and axisymmetric star, the spacetime only depends on three metric functions (while four metric functions are needed for the interior). The most general form of the metric was given by Papapetrou (1953)

$$\begin{aligned} ds^2=-f(dt-\omega d\phi )^2+f^{-1}\left\{ e^{2\gamma } (d\tilde{\varpi }^2+d\tilde{z}^2)+\tilde{\varpi }^2d\phi ^2 \right\} . \end{aligned}$$
(57)

Here f, \(\omega \) and \(\gamma \) are functions of the quasi-cylindrical Weyl–Lewis–Papapetrou coordinates \((\tilde{\varpi },~\tilde{z})\). Starting from this metric, one can write the vacuum Einstein–Maxwell equations as two equations for two complex potentials \(\mathcal{E}\) and \(\varPhi \), following a procedure due to Ernst Ernst (1968a, b). Once the potentials are known, the metric can be reconstructed. Sibgatullin and Queen (1991) devised a powerful procedure for reducing the solution of the Ernst equations to simple integral equations. The exact solutions are generated as a series expansion, in terms of the physical multipole moments of the spacetime, by choosing the values of the Ernst potentials on the symmetry axis.

An interesting exact vacuum solution, given in a closed, algebraic form, was found by Manko et al. (2000a, b). For non-magnetized sources of zero net charge, it reduces to a 3-parameter solution, involving the gravitational mass, M, the specific angular momentum, \(a=J/M\), and a third parameter, b, related to the quadrupole moment of the source. The Ernst potential \(\mathcal E\) on the symmetry axis is

$$\begin{aligned} e(z)={(z-M-ia)(z+ib)+d-\delta -ab \over (z+M-ia)(z+ib)+d-\delta -ab}, \end{aligned}$$
(58)

where

$$\begin{aligned} \delta= & {} {-M^2b^2\over M^2-(a-b)^2}, \end{aligned}$$
(59)
$$\begin{aligned} d= & {} {1\over 4}\left[ M^2-(a-b)^2\right] . \end{aligned}$$
(60)

Since a and b are independent parameters, setting a equal to zero does not automatically imply a vanishing quadrupole moment. Instead, the nonrotating solution (\(a=0\)) has a quadrupole moment equal to

$$\begin{aligned} Q(a=0)=-{M \over 4} {\left( M^2+b^2 \right) ^2 \over \left( M^2-b^2 \right) }, \end{aligned}$$
(61)

and there exists no real value of the parameter b for which the quadrupole moment vanishes for a nonrotating star. Hence, the 3-parameter solution by Manko et al. does not reduce continuously to the Schwarzschild solution as the rotation vanishes and is not suitable for describing slowly rotating stars.

For rapidly rotating models, when the quadrupole deformation induced by rotation roughly exceeds the minimum nonvanishing oblate quadrupole deformation of the solution in the absence of rotation, the 3-parameter solution by Manko et al. is still relevant. A matching of the vacuum exterior solution to numerically-constructed interior solutions of rapidly rotating stars (by identifying three multipole moments) was presented by Berti and Stergioulas (2004). For a wide range of candidate EOSs, the critical rotation rate \(\varOmega _\mathrm{crit}/\varOmega _\mathrm{K}\) above which the Manko et al. 3-parameter solution is relevant, ranges from \(\sim 0.4\) to \(\sim 0.7\) for sequences of models with \(M=1.4\,M_{\odot }\), with the lower ratio corresponding to the stiffest EOS. For the maximum-mass sequence the ratio is \(\sim 0.9\), nearly independent of the EOS. In Manko et al. (2000a), the quadrupole moment was also used for matching the exact vacuum solution to numerical interior solutions, but only along a different solution branch which is not a good approximation to rotating stars.

A more versatile exact exterior vacuum solution found by Manko et al. (1995) involves (in the case of vanishing charge and magnetic field) four parameters, which can be directly related to the four lowest-order multipole moments of a source (mass, angular momentum, quadrupole moment and current octupole moment). The advantage of the above solution is that its four parameters are introduced linearly in the first moment it appears. For this reason, one can always match the exact solution to a numerical solution by identifying the four lowest-order multipole moments. Therefore, the 4-parameter Manko et al. (1995) solution is relevant for studying rotating relativistic stars at any rotation rate. Pappas (2009) compared the two Manko et al. solutions to numerical solutions of rapidly rotating relativistic stars, finding good agreement. In Pappas et al. (2013), a more detailed comparison is shown, using a corrected expression for the numerical computation of the quadrupole moment. Manko and Ruiz (2016a) express the Manko et al. 4-parameter solution explicitly in terms of only three potentials, and compare the multipole structure of the solution with physically realistic numerical models of Berti and Stergioulas (2004).

Another exact exterior solution (that is related to the 4-parameter Manko et al. solution) was presented by Pachón et al. (2006) and was applied to relativistic precession and oscillation frequencies of test particles around rotating compact stars. Furthermore, an exact vacuum solution (constructed via Bäcklund transformations), that can be matched to numerically constructed solutions with an arbitrary number of constants, was presented by Teichmüller et al. (2011), who found very good agreement with numerical solutions even for a small number of parameters.

A very recent analytic solution for the exterior spacetime is provided by Pappas (2017). The metric is constructed by adopting the Ernst formulation, it is written as an expansion in Weyl–Papapetrou coordinates and has 3 free parameters—multipole moments of the NS. The metric compares favourably with numerically computed general relativistic neutron star spacetimes. An extension of the approximate metric to scalar–tensor theories with massless fields is also provided.

Properties of equilibrium models

Bulk properties and sequences of equilibrium models

Neutron star models constructed with various realistic EOSs have considerably different bulk properties, due to the large uncertainties in the equation of state at high densities. Very compressible (soft) EOSs produce models with small maximum mass, small radius, and large rotation rate. On the other hand, less compressible (stiff) EOSs produce models with a large maximum mass, large radius, and low rotation rate. The sensitivity of the maximum mass to the compressibility of the neutron-star core is responsible for the strongest astrophysical constraint on the equation of state of cold matter above nuclear density. With the mass measurement of \(1.97 \pm 0.04\,M_{\odot }\) for PSR J1614-2230 (Demorest et al. 2010) and of \(2.01 \pm 0.04\) for PSR J0348+0432 (Antoniadis et al. 2013), several candidate EOSs that yielded models with maximum masses of nonrotating stars below this limit are ruled out, but the remaining range of candidate EOSs is still large, yielding compact objects with substantially different properties.

Not all properties of the maximum mass models between proposed EOSs differ considerably, at least not within groups of similar EOSs. For example, most realistic hadronic EOSs predict a maximum mass model with a ratio of rotational to gravitational energy T / |W| of \(0.11 \pm 0.02\), a dimensionless angular momentum \(cJ/GM^2\) of \(0.64 \pm 0.06\), and an eccentricity of \(0.66 \pm 0.04\) (Friedman and Ipser 1992). Hence, within the set of realistic hadronic EOSs, some properties are directly related to the stiffness of the EOS while other properties are rather insensitive to stiffness. On the other hand, if one considers strange quark EOSs, then for the maximum mass model, T / |W| can become more than 60% larger than for hadronic EOSs.

Fig. 1
figure1

The radius R of a uniformly rotating star increases sharply as the Kepler (mass-shedding) limit (\(\varOmega =\varOmega _K\)) is approached. The particular sequence of models shown here has a constant central energy density of \(\epsilon _c = 1.21 \times 10^{15}\,\mathrm {g\ cm}^{-3}\) and was constructed with EOS L. (Image reproduced with permission from Stergioulas and Friedman 1995, copyright by AAS)

Fig. 2
figure2

Representative sequences of rotating stars with fixed baryon mass, for EOS WFF3 (Wiringa et al. 1988). Above a rest mass of \(M_0=2.17\,M_{\odot }\) only supramassive stars exist, which reach the axisymmetric instability limit when spun down. The onset of axisymmetric instability approximately coincides with the minima of the constant rest mass sequences. (Image reproduced with permission from Friedman and Stergioulas 2013, copyright by the authors)

Compared to nonrotating stars, the effect of rotation is to increase the equatorial radius of the star and also to increase the mass that can be sustained at a given central energy density. As a result, the mass of the maximum-mass rigidly rotating model is roughly 15–20% higher than the mass of the maximum mass nonrotating model (Morrison et al. 2004), for typical realistic hadronic EOSs. The corresponding increase in radius is 30–40%. Figure 1 shows an example of a sequence of uniformly rotating equilibrium models with fixed central energy density,Footnote 1 constructed with EOS L (Pandharipande and Smith 1975; Pandharipande et al. 1976). Near the Kepler (mass-shedding) limit (\(\varOmega =\varOmega _K\)), the radius increases sharply. This leads to the appearance of a cusp in the equatorial plane. The effect of rotation in increasing the mass and radius becomes more pronounced in the case of strange quark EOSs (see Sect. 2.9.8).

For a given zero-temperature EOS, the uniformly rotating equilibrium models form a two-dimensional surface in the three-dimensional space of central energy density, gravitational mass, and angular momentum (Stergioulas and Friedman 1995). The surface is limited by the nonrotating models and by the models rotating at the mass-shedding (Kepler) limit. Cook et al. (1992, 1994b, 1994a) have shown that the model with maximum angular velocity does not coincide with the maximum mass model, but is generally very close to it in central density and mass. Stergioulas and Friedman (1995) showed that the maximum angular velocity and maximum baryon mass equilibrium models are also distinct. The distinction becomes significant in the case where the EOS has a large phase transition near the central density of the maximum mass model; otherwise the models of maximum mass, baryon mass, angular velocity, and angular momentum can be considered to coincide for most purposes.

In the two-dimensional parameter space of uniformly rotating models one can construct different one-dimensional sequences, depending on which quantity is held fixed. Examples are sequences of constant central energy density, constant angular momentum or constant rest mass. Figure 2 displays a representative sample of fixed rest mass sequences for EOS WFF3 (Wiringa et al. 1988) in a mass versus central energy density graph, where the sequence of nonrotating models and the sequence of models at the mass-shedding limit are also shown.Footnote 2 The rest mass of the maximum-mass nonrotating model is \(2.17\, M_{\odot }\). Below this value, all fixed rest mass sequences have a nonrotating member. Along such a sequence, the gravitational mass increases somewhat, since it also includes the rotational kinetic energy. Above \(M_0=2.17\,M_{\odot }\) none of the fixed rest mass sequences have a nonrotating member. Instead, the sequences terminate at the axisymmetric instability limit (see Sect. 4.3.1). The onset of the instability occurs just prior to the minimum of each fixed rest mass sequence, and models to the right of the instability line are unstable.

Models with \(M_0>2.17\,M_{\odot }\) have masses larger than the maximum-mass nonrotating model and are called supramassive (Cook et al. 1992). A millisecond pulsar spun up by accretion can become supramassive, in which case it would subsequently spin down along a sequence with approximately fixed rest mass, finally reaching the axisymmetric instability limit and collapsing to a black hole. Some relativistic stars could also be born supramassive or become so as the result of a binary merger; here, however, the star would be initially differentially rotating, and collapse would be triggered by a combination of spin-down and by viscosity (or magnetic-field braking) driving the star to uniform rotation. The maximum mass of differentially rotating supramassive neutron stars can be significantly larger than in the case of uniform rotation (Lyford et al. 2003) and typically 50% or more than the TOV limit (Morrison et al. 2004).

A supramassive relativistic star approaching the axisymmetric instability will actually spin up before collapse, even though it loses angular momentum (Cook et al. 1992, 1994b, a). This potentially observable effect is independent of the equation of state and it is more pronounced for rapidly rotating massive stars. Similarly, stars can be spun up by loss of angular momentum near the mass-shedding limit, if the equation of state is extremely stiff or extremely soft.

Multipole moments

The deformed shape of a rapidly rotating star creates a non-spherical distortion in the spacetime metric, and in the exterior vacuum region the metric is determined by a set of multipole moments, which arise at successively higher powers of \(r^{-1}\). As in electromagnetism, the asymptotic spacetime is characterized by two sets of multipoles, mass multipoles and current multipoles, analogs of the electromagnetic charge multipoles and current multipoles.

The dependence of metric components on the choice of coordinates leads to the complication that in coordinate choices natural for a rotating star (including the quasi-isotropic coordinates) the asymptotic form of the metric includes information about the coordinates as well as about the multipole structure of the geometry. Because the metric potentials \(\nu \), \(\omega \) and \(\psi \) are scalars constructed locally from the metric and the symmetry vectors \(t^\alpha \) and \(\phi ^\alpha \), as in Eqs. (1113), their definition is in this sense coordinate-independent. But, the functional forms, \(\nu (r,\theta )\), \(\omega (r,\theta )\), \(\psi (r,\theta )\), depend on r and \(\theta \) and one must disentangle the physical mass and current moments from the coordinate contributions.

Up to \(O(r^{-3})\), the only contributing multipoles are the monopole and quadrupole mass moments and the \(l=1\) current moment. Two approaches to asymptotic multipoles of stationary systems, developed by Thorne (1980) and by Geroch (1970b) and Hansen (1974) yield identical definitions for \(l\le 2\), while higher multipoles differ only in the normalization chosen. Ryan (1995) and Laarakkers and Poisson (1999) provide coordinate invariant definitions of multiple moments.

In the nonrotating limit, the quasi-isotropic metric (5) takes the isotropic form

$$\begin{aligned} ds^2 = -\left( \frac{1-M/2r}{1+M/2r}\right) ^2 \, dt^2 + \left( 1+\frac{M}{2r}\right) ^4(dr^2+ r^2\sin ^2\theta d\phi ^2 + r^2d\theta ^2), \qquad \end{aligned}$$
(62)

with asymptotic form

$$\begin{aligned} ds^2= & {} -\left[ 1-\frac{2M}{r}+2\frac{M^2}{r^2}-\frac{1}{4}\frac{M^3}{r^3}+O(r^{-5})\right] dt^2 \nonumber \\&+ \left[ 1+\frac{2M}{r}+\frac{3}{2}\frac{M^2}{r^2}+O(r^{-3})\right] (dr^2+r^2\sin ^2\theta d\phi ^2 + r^2d\theta ^2), \phantom {xxx} \end{aligned}$$
(63)

Thus, the metric potentials \(\nu \), \(\mu \) and \(\psi \) have asymptotic behavior

$$\begin{aligned} \nu= & {} -\frac{M}{r} -\frac{1}{12}\frac{M^3}{r^3} +O(r^{-5}), \end{aligned}$$
(64)
$$\begin{aligned} \mu= & {} \frac{M}{r} - \frac{1}{4}\frac{M^2}{r^2}+\frac{1}{12}\frac{M^3}{r^3} +O(r^{-4}),\end{aligned}$$
(65)
$$\begin{aligned} \psi= & {} \log (r\sin \theta ) +\mu . \end{aligned}$$
(66)

For a rotating star, the asymptotic metric differs from the nonrotating form already at \(O(r^{-2})\). Through \(O(r^{-3})\) there are three corrections due to rotation: (i) the frame dragging potential \(\displaystyle \omega \sim \frac{2J}{r^3}\); (ii) a quadrupole correction to the diagonal metric coefficients at \(O(r^{-3})\) associated with the mass quadrupole moment Q of the rotating star; and (iii) coordinate-dependent monopole and quadrupole corrections to the diagonal metric coefficients (reflecting the asymptotic shape of the r- and \(\theta \)- surfaces) which can be described by a dimensionless parameter a.

For convenience, one can define a dimensionless qudrupole moment parameter \(q := Q/M^3\). Then, Friedman and Stergioulas (2013) show that the asymptotic form of the metric is given in terms of the parameters M, J, q and a by:

$$\begin{aligned} \nu= & {} -\frac{M}{r} -\frac{1}{12}\frac{M^3}{r^3} +\left( a - 4aP_2 - qP_2\right) \frac{M^3}{r^3} +O(r^{-4}), \end{aligned}$$
(67)
$$\begin{aligned} \mu= & {} \frac{M}{r} - \frac{1}{4}\frac{M^2}{r^2}+\frac{1}{12}\frac{M^3}{r^3} -(a-4a P_2)\frac{M^2}{r^2} -(a-4a P_2 - qP_2)\frac{M^3}{r^3} \nonumber \\&+ O(r^{-4}),\end{aligned}$$
(68)
$$\begin{aligned} \psi= & {} \log (r\sin \theta ) +\mu + O(r^{-4}), \end{aligned}$$
(69)
$$\begin{aligned} \omega= & {} \frac{2J}{r^3} + O(r^{-4}), \end{aligned}$$
(70)

where \(P_2\) is the Legendre polynomial \(P_2(\cos \theta )\). The coefficient of \(-P_2/r^3\) in the expansion of the metric potential \(\nu \) is thus \(Q+4aM^3\), from which the quadrupole moment Q can be extracted, if the parameter a has been determined from the coefficient of \(P_2/r^2\) in the expansion of the metric potential \(\mu \). Notice that sometimes the coeffient of \(-P_2/r^3\) in the expansion of \(\nu \) is identified with Q (instead of \(Q+4aM^3\)), which can lead to a deviation of up to about \(20\%\) in the numerical values of the quadrupole moment. Pappas and Apostolatos (2012) have independently verified the correctness of the identification in Friedman and Stergioulas (2013) and also provide the correct identification of the current-octupole moment.

Laarakkers and Poisson (1999) found that along a sequence of fixed gravitational mass M, the quadrupole moment Q scales quadratically with the angular momentum, as

$$\begin{aligned} Q = -a_2 \frac{J^2}{Mc^2} = -a_2 \chi ^2 M^3, \end{aligned}$$
(71)

where \(a_2\) is a dimensionless coefficient that depends on the equation of state, and \(\chi :=J/M^2\). In Laarakkers and Poisson (1999), the coefficient \(a_2\) varied between \(a \sim 2\) for very soft EOSs and \(a \sim 8\) for very stiff EOSs, for sequences of \(M=1.4\,M_{\odot }\), but these values were computed with the erroneous identification of Q discussed above. Pappas and Apostolatos (2012) verify the simple form of (71) and provide corrected values for the parameter \(a_2\) as well as similar relations for other multipole moments. Pappas and Apostolatos (2014) and Yagi et al. (2014) have further found that in addition to Q, the spin octupole \(S_3\) and mass hexadecapole \(M_4\) also have scaling relationships for realistic equations of state as follows

$$\begin{aligned} S_3= & {} -\beta _3 \chi ^3 M^4, \end{aligned}$$
(72)
$$\begin{aligned} M_4= & {} \gamma _4 \chi ^4 M^5, \end{aligned}$$
(73)

where \(\beta _3\) and \(\gamma _4\) are dimensionless constants.

Mass-shedding limit and the empirical formula

Mass-shedding occurs when the angular velocity of the star reaches the angular velocity of a particle in a circular Keplerian orbit at the equator, i.e., when

$$\begin{aligned} \varOmega = \varOmega _{\mathrm {K}}, \end{aligned}$$
(74)

where

$$\begin{aligned} \varOmega _{\mathrm {K}}=\frac{\omega '}{2\psi '} + e^{\nu -\psi }\left[ c^2\frac{\nu '}{\psi '}+ \left( \frac{\omega '}{2\psi '}e^{\psi -\nu }\right) ^2\right] ^{1/2} \!\!\!+\omega , \end{aligned}$$
(75)

(a prime indicates radial differentiation). In differentially rotating stars, even a small amount of differential rotation can significantly increase the angular velocity required for mass-shedding. Thus, a newly-born, hot, differentially rotating neutron star or a massive, compact object formed in a binary neutron star merger could be sustained (temporarily) in equilibrium by differential rotation, even if a uniformly rotating configuration with the same rest mass does not exist.

In the Newtonian limit, one can use the Roche model to derive the maximum angular velocity for uniformly rotating polytropic stars, finding \(\varOmega _K \simeq (2/3)^{3/2} (GM/R^3)^{1/2}\) (see Shapiro and Teukolsky 1983). An identical result is obtained in the relativistic Roche model of Shapiro et al. (1983). For relativistic stars, the empirical formula (Haensel and Zdunik 1989; Friedman et al. 1989; Friedman 1990; Haensel et al. 1995)

$$\begin{aligned} \varOmega _K = 0.67 \sqrt{\frac{G M^{\max }_\mathrm{sph}}{(R^{\max }_\mathrm{sph})^3}}, \end{aligned}$$
(76)

gives the maximum angular velocity in terms of the mass and radius of the maximum mass nonrotating (spherical) model with an accuracy of 5–7%, without actually having to construct rotating models. Expressed in terms of the minimum period \(P_\mathrm{min}=2\pi /\varOmega _K\), the empirical formula reads

$$\begin{aligned} P_{\min } \simeq 0.82 \left( \frac{M_\odot }{M_\mathrm{sph}^{\max }} \right) ^{1/2} \left( \frac{R_\mathrm{sph}^{\max }}{10\,\mathrm {km}} \right) ^{3/2}\,\mathrm {ms}. \end{aligned}$$
(77)

The empirical formula results from universal proportionality relations that exist between the mass and radius of the maximum mass rotating model and those of the maximum mass nonrotating model for the same EOS. Lasota et al. (1996) found that, for most EOSs, the numerical coefficient in the empirical formula is an almost linear function of the parameter

$$\begin{aligned} \chi _s = \frac{2GM^{\max }_\mathrm{sph}}{R^{\max }_\mathrm{sph} c^2}. \end{aligned}$$
(78)

The Lasota et al. empirical formula

$$\begin{aligned} \varOmega _K = (0.468+0.378 \chi _s) \sqrt{\frac{G M^{\max }_\mathrm{sph}}{(R^{\max }_\mathrm{sph})^3}}, \end{aligned}$$
(79)

reproduces the exact values with a relative error of only \(1.5\%\). The corresponding formula for \(P_{\min }\) is

$$\begin{aligned} P_{\min } \simeq \frac{0.187}{(\chi _s)^{3/2}(1+0.808\chi _s)} \left( \frac{M_\odot }{M_\mathrm{sph}^{\max }} \right) \mathrm{ms}. \end{aligned}$$
(80)

The above empirical relations are specifically constructed for the most rapidly rotating model for a given EOS.

Lattimer and Prakash (2004) suggest the following empirical relation

$$\begin{aligned} P_{\min } \simeq 0.96 \left( \frac{M_\odot }{M} \right) ^{1/2} \left( \frac{R_\mathrm{sph}}{10\,\mathrm {km}} \right) ^{3/2}\,\mathrm {ms}, \end{aligned}$$
(81)

for any neutron star model with mass M and radius \(R_\mathrm{sph}\) of the nonrotating model with same mass, as long as its mass is not close to the maximum mass allowed by the EOS. Haensel et al. (2009) refine the above formula, giving a factor of 0.93 for hadronic EOSs and 0.87 for strange stars. A corresponding empirical relation between the radius at maximal rotation and the radius of a nonrotating configuration of same mass also exists.

Using the above relation, one can set an approximate constraint on the radius of a nonrotating star with mass M, given the minimum observed rotational period of pulsars.

Upper limits on mass and rotation: theory versus observation

Maximum mass: Candidate EOSs for high density matter predict vastly different maximum masses for nonrotating models. One of the stiffest proposed EOSs (EOS L) has a nonrotating maximum mass of \(3.3\,M_{\odot }\). Some core-collapse simulations suggest a bi-modal mass distribution of the remnant, with peaks at about \(1.3\,M_{\odot }\) and \(1.7\,M_{\odot }\) (Timmes et al. 1996).

Observationally, the masses of a large number of compact objects have been determined, but, in most cases, the observational error bars are still large. A recent review of masses and spins of neutron stars as determined by observations was presented by Miller and Miller (2015). The heaviest neutron stars with the most accurately determined masses ever observed are PSR J1614-2230, with \(M=1.97\pm 0.04\,M_{\odot }\) (Demorest et al. 2010) and PSR J0348+0432, with \(2.01\pm 0.04\) (Antoniadis et al. 2013), and there are indications for even higher masses (see Haensel et al. 2007 for a detailed account). Masses of compact objects have been measured in different types of binary systems: double neutron star binaries, neutron star-white dwarf binaries, X-ray binaries and binaries composed of a compact object around a main sequence star. For most double neutron star binaries, masses have already been determined with good precision and are restricted to a narrow range of about \(1.2-1.4\,M_{\odot }\) (Thorsett and Chakrabarty 1999). This narrow range of relatively small masses is probably associated with an upper mass limit on iron cores, which in turn is related to the stability of the core of each progenitor star. Masses determined for compact stars in X-ray binaries still have large error bars, but are consistently higher than \(1.4\,M_{\odot }\), which is probably the result of mass-accretion. A similar finding seems to apply to white dwarf–neutron star binaries (see Paschalidis et al. 2009 and references therein).

Minimum period: When magnetic-field effects are ignored, conservation of angular momentum can yield very rapidly rotating neutron stars at birth. Simulations of the rotational core collapse of evolved rotating progenitors (Heger et al. 2000; Fryer and Heger 2000) have demonstrated that rotational core collapse could result in the creation of neutron stars with rotational periods of the order of 1 ms (and similar initial rotation periods have been estimated for neutron stars created in the accretion-induced collapse of a white dwarf, Liu and Lindblom 2001). However, magnetic fields may complicate this picture. Spruit and Phinney (1998) have presented a model in which a strong internal magnetic field couples the angular velocity between core and surface during most evolutionary phases. The core rotation decouples from the rotation of the surface only after central carbon depletion takes place. Neutron stars born in this way would have very small initial rotation rates, even smaller than the ones that have been observed in pulsars associated with supernova remnants. In this model, an additional mechanism is required to spin up the neutron star to observed periods. On the other hand, Livio and Pringle (1998) argue for a much weaker rotational coupling between core and surface by a magnetic field, allowing for the production of more rapidly rotating neutron stars than in Spruit and Phinney (1998). In Heger et al. (2004), intermediate initial rotation rates were obtained. Clearly, more detailed studies of the role of magnetic fields are needed to resolve this important question.

Independently of their initial rotation rate, compact stars in binary systems are spun up by accretion, reaching high rotation rates. In principle, accretion could drive a compact star to its mass-shedding limit. For a wide range of candidates for the neutron-star EOS, the mass-shedding limit sets a minimum period of about 0.5–0.9 ms (Friedman 1995). However, there are a number of different processes that could limit the maximum spin to lower values. In one model, the minimum rotational period of pulsars could be set by the occurrence of the r-mode instability in accreting neutron stars in LMXBs (Bildsten 1998; Andersson et al. 2000), during which gravitational waves carry away angular momentum. Other models are based on the standard magnetospheric model for accretion-induced spin-up (White and Zhang 1997), or on the idea that the spin-up torque is balanced by gravitational radiation produced by an accretion-induced quadrupole deformation of the deep crust (Bildsten 1998; Ushomirsky et al. 2000), by deformations induced by a very strong toroidal field Cutler (2002) or by magnetically confined “mountains” (Melatos and Payne 2005; Vigelius and Melatos 2008). With the maximum observed pulsar spin frequency at 716 Hz (Hessels et al. 2006) and a few more pulsars at somewhat lower rotation rates (Chakrabarty 2008), it is likely that one of the above mechanisms ultimately dominates over the accretion-induced spin-up, setting an upper limit that may be somewhat dependent on the final mass, the magnetic field or the spin-up history of the star. This is consistent with the absence of sub-millisecond pulsars in pulsar surveys that were in principle sensitive down to a few tenths of a millisecond (Burderi and D’Amico 1997; D’Amico 2000; Crawford et al. 2000; Edwards et al. 2001).

EOS constraints: One can systematize the observational constraints on the neutron-star EOS by introducing a parameterized EOS above nuclear density with a set of parameters large enough to encompass the wide range of candidate EOSs and small enough that the number of parameters is smaller than the number of relevant observations. Read et al. (2009) found that one can match a representative set of EOSs to within about 3% rms accuracy with a 4-parameter EOS based on piecewise polytropes.

Using spectral modeling to simultaneously estimate the radius and mass of a set of neutron stars in transient low-mass X-ray binaries, Özel et al. (2010) and Steiner et al. (2010) find more stringent constraints. They also adopt piecewise-polytropic parametrizations to find the more restricted region of the EOS space. Future gravitational-wave observations of inspiraling neutron-star binaries (Flanagan and Hinderer 2008; Read et al. 2009; Markakis et al. 2009, 2012; Duez et al. 2010; Bernuzzi et al. 2012; Damour et al. 2012) and of oscillating, post-merger remnants (Shibata et al. 2005; Bauswein et al. 2012; Bauswein and Janka 2012; Bauswein et al. 2014, 2016; Clark et al. 2016) may yield comparable or more accurate constraints without the model-dependence of the current electromagnetic studies.

The existence of \(2.0\,M_{\odot }\) neutron stars in conjunction with nuclear physics place constraints on the neutron star EOS. For example, Hebeler et al. (2013) use microscopic calculations of neutron matter based on nuclear interactions derived from chiral effective field theory to constrain the equation of state of neutron-rich matter at sub- and supranuclear densities, arriving at a range of \(9.7{-}13.9\,\mathrm {km}\) for the radius of nonrotating neutron stars, which is somewhat smaller than the range that a large sample of various proposed EOSs allow (the authors use a piecewise polytropic approach to derive the constraints). The corresponding range of compactness is 0.149–0.213.

A review of efforts to observationally constrain the EOS is given by Lattimer (2001). For recent reviews and the most up-to-date constraints on the neutron star radii and masses from electromagnetic observations see Lattimer (2012) and Özel and Freire (2016) and references therein. Future observations with missions such as NICER (Gendreau et al. 2012) and the proposed LOFT (Feroci and Stella 2012) have the potential to determine the neutron star radius with \( \sim 5{-}10\%\) uncertainty, which will be useful in placing stringent (albeit model dependent) constraints on the EOS (see Psaltis et al. 2014).

Maximum mass set by causality

If one is interested in obtaining an upper limit on the mass, independent of the current uncertainty in the high-density part of the EOS for compact stars, one can construct a schematic EOS that satisfies only a minimal set of physical constraints and which yields a model of absolute maximum mass. The minimal set of constraints are

  1. (0)

    A relativistic star is described as a self-gravitating, uniformly rotating perfect fluid with a one-parameter EOS, an assumption that is satisfied to high accuracy by cold neutron stars.

  2. (1)

    Matter at high densities satisfies the causality constraint \(c_s\equiv \sqrt{dp/d\epsilon } < 1\), where \(c_\mathrm{s}\) is the sound speed. Relativistic fluids are governed by hyperbolic equations whose characteristics lie inside the light cone (consistent with the requirement of causality) only if \(c_\mathrm{s}< 1\)Geroch and Lindblom (1991).

  3. (2)

    The EOS is known at low densities One assumes that the EOS describing the crust of cold relativistic stars is accurately known up to a matching energy density \(\epsilon _m\).

For nonrotating stars, Rhoades and Ruffini (1974) showed that the EOS that satisfies the above constraints and yields the maximum mass consists of a high density region at the causal limit, \(dp/d \epsilon =1\) (as stiff as possible), that matches directly to the assumed low density EOS at \(\epsilon =\epsilon _m\)

$$\begin{aligned} p(\epsilon )= & {} {\left\{ \begin{array}{ll} p_\mathrm{crust}(\epsilon ) &{}\quad \epsilon < \epsilon _m, \\ &{} \\ p_m+ \epsilon -\epsilon _m &{}\quad \epsilon > \epsilon _m, \end{array} \right. } \end{aligned}$$
(82)

where \(p_m = p_\mathrm{crust}(\epsilon _m)\). For this maximum mass EOS and a specific value of the matching density, they computed a maximum mass of \(3.2\,M_{\odot }\). More generally, \(M_{\max }\) depends on \(\epsilon _m\) as (Hartle and Sabbadini 1977; Hartle 1978)

$$\begin{aligned} M_{\max } = 4.8 \ \left( \frac{2 \times 10^{14}\,\mathrm {g/cm}^3}{ \epsilon _m/c^2} \right) ^{1/2}\,M_{\odot }. \end{aligned}$$
(83)

In the case of uniformly rotating stars, one obtains the following limit on the mass, when matching to the FPS EOS at low densities

$$\begin{aligned} M^\mathrm{rot}_{\max } = 6.1 \ \left( \frac{2 \times 10^{14}\,\mathrm {g/cm}^3}{ \epsilon _m/c^2} \right) ^{1/2} M_{\odot }, \end{aligned}$$
(84)

(see Friedman and Ipser 1987; Koranda et al. 1997).

Minimum period set by causality

A rigorous limit on the minimum period of uniformly rotating, gravitationally bound stars, allowed by causality, has been obtained in Koranda et al. (1997) (hereafter KSF), extending previous results by Glendenning (1992). The same three minimal constraints (0), (1) and (2) of Sect. 2.9.5, as in the case of the maximum mass allowed by causality, yield the minimum period. However, the minimum period EOS is different from the maximum mass EOS (82). KSF found that just the two constraints (0), (1) (without matching to a known low-density part) suffice to yield a simpler, absolute minimum period EOS and an absolute lower bound on the minimum period.

Absolute minimum period, without matching to low-density EOS: Considering only assumptions (0) and (1), so that the EOS is constrained only by causality, the minimum period EOS is simply

$$\begin{aligned} p(\epsilon )= & {} {\left\{ \begin{array}{ll} 0 &{} \quad \epsilon \le \epsilon _C, \\ &{} \\ \epsilon -\epsilon _C &{}\quad \epsilon \ge \epsilon _C, \end{array} \right. } \end{aligned}$$
(85)

describing a star entirely at the causal limit \(dp/d\epsilon =1\), with surface energy density \(\epsilon _C\). This is not too surprising. A soft EOS yields stellar models with dense central cores and thus small rotational periods. Soft EOSs, however, cannot support massive stars. This suggests that the model with minimum period arises from an EOS which is maximally stiff (\(dp/d\epsilon =1\)) at high density, allowing stiff cores to support against collapse, but maximally soft at low density (\(dp/d\epsilon =0\)), allowing small radii and thus fast rotation, in agreement with (85). The minimum period EOS is depicted in Fig. 3 and yields an absolute lower bound on the period of uniformly rotating stars obeying the causality constraint, independent of any specific knowledge about the EOS for the matter composing the star. Choosing different values for \(\epsilon _C\), one constructs EOSs with different \(M_\mathrm{sph}^{\max }\). All properties of stars constructed with EOS (85) scale according to their dimensions in gravitational units and thus, the following relations hold between different maximally rotating stars computed from minimum-period EOSs with different \(\epsilon _C\):

$$\begin{aligned} P_\mathrm{min}\propto & {} M_\mathrm{sph}^{\max } \propto R_\mathrm{sph}^{\max }, \end{aligned}$$
(86)
$$\begin{aligned} \epsilon _\mathrm{sph}^{\max }\propto & {} \frac{1}{\bigl ( M_\mathrm{sph}^{\max } \bigr )^2}, \end{aligned}$$
(87)
$$\begin{aligned} M_\mathrm{rot}^{\max }\propto & {} M_\mathrm{sph}^{\max }, \end{aligned}$$
(88)
$$\begin{aligned} R_\mathrm{rot}^{\max }\propto & {} R_\mathrm{sph}^{\max }, \end{aligned}$$
(89)
$$\begin{aligned} \epsilon _\mathrm{rot}^{\max }\propto & {} \epsilon _\mathrm{sph}^{\max }. \end{aligned}$$
(90)
Fig. 3
figure3

Schematic representations of the minimum-period EOSs (85) and (92). For the minimum-period EOS (85) the pressure vanishes for \(\epsilon <\epsilon _C\). The minimum-period EOS (92) matches the FPS EOS to a constant pressure region at an energy density \(\epsilon _m\). For \(\epsilon > \epsilon _C\) both EOSs are at the causal limit with \(dp/d\epsilon =1\). (Image reproduced with permission from Koranda et al. 1997, copyright by AAS)

A fit to the numerical results, yields the following relation for the absolute minimum period

$$\begin{aligned} \frac{P_{\min }}{\mathrm{ms}} = 0.196 \left( \frac{M_{\mathrm{sph}}^{\max }}{{ M}_\odot } \right) . \end{aligned}$$
(91)

Thus, for \(M_\mathrm{sph}^{\max }=2\,M_\odot \) the absolute minimum period is 0.39 ms.

Minimum period when low-density EOS is known: Assuming all three constraints (0), (1) and (2) of Sect. 2.9.5 (so that the EOS matches to a known EOS at low density), the minimum-period EOS is

$$\begin{aligned} p(\epsilon )= & {} {\left\{ \begin{array}{ll} p_\mathrm{crust}(\epsilon ) &{} \quad \epsilon \le \epsilon _m, \\ &{} \\ p_m &{}\quad \epsilon _m\le \epsilon \le \epsilon _C, \\ &{} \\ p_m + \epsilon -\epsilon _C &{}\quad \epsilon \ge \epsilon _C. \end{array} \right. } \end{aligned}$$
(92)
Fig. 4
figure4

Minimum period \(P_{\min }\) allowed by causality for uniformly rotating, relativistic stars as a function of the mass \(M_\mathrm{sph}^{\max }\) of the maximum mass nonrotating model. Lower curve: constructed using the absolute minimum-period EOS (85), which does not match at low densities to a known EOS. Upper curve: constructed using the minimum-period EOS (92), which matches at low densities to the FPS EOS. Due to the causality constraint, the region below the curves is inaccessible to stars. (Image reproduced with permission from Koranda et al. 1997, copyright by AAS)

Between \(\epsilon _m\) and \(\epsilon _C\) the minimum period EOS has a constant pressure region (a first order phase transition) and is maximally soft, while above \(\epsilon _{C}\) the EOS is maximally stiff, see Fig. 3. For a matching number density of \(n_m=0.1\>\,\mathrm {fm}^{-3}\) to the FPS EOS, the minimum period allowed by causality is shown as a function of \(M^{\max }_\mathrm{sph}\) in Fig. 4. A quite accurate linear fit of the numerical results is

$$\begin{aligned} \frac{P_{\min }}{ ms} = 0.295 + 0.203 \ \left( \frac{M_\mathrm{sph}^{\max }}{{ M}_\odot }-1.442\right) . \end{aligned}$$
(93)

Thus, if \(M_\mathrm{sph}^{\max }=2\,M_{\odot }\), the minimum period is \(P_\mathrm{min}=0.41\,\mathrm {ms}\). This result is rather insensitive to \(n_m\), for \(n_m <0.2\,\mathrm {fm}^{-3}\), but starts to depend significantly on \(n_m\) for larger matching densities.

Comparing (93)–(91) it is evident that the currently trusted part of the nuclear EOS plays a negligible role in determining the minimum period due to causality. In addition, since matching to a known low-density EOS raises \(P_{\min }\), (91) represents an absolute minimum period.

Moment of inertia and ellipticity

The scalar moment of inertia of a neutron star, defined as the ratio \(I=J/\varOmega \), has been computed for polytropes and for a wide variety of candidate equations of state (see, e.g., Stergioulas et al. 1999; Cook et al. 1994a, b; Friedman et al. 1986). For a given equation of state the maximum value of the moment of inertia typically exceeds its maximum value for a spherical star by a factor of 1.5–1.6. For spherical models, Bejger et al. (2005) obtain an empirical formula for the maximum value of I for a given EOS in terms of the maximum mass for that EOS and the radius of that maximum-mass configuration,

$$\begin{aligned} I_{\max ,\varOmega =0}\approx & {} 0.97\times 10^{45} \left( \frac{M_{\max }}{M_\odot } \right) \left( \frac{R_{M_{\max }}}{10\,\mathrm {km}}\right) ^{2\,\mathrm {g\ cm}^2}. \end{aligned}$$
(94)

Neutron-star moments of inertia can in principle be measured by observing the periastron advance of a binary pulsar (Damour and Schäfer 1988). Because the mass of each star can be found to high accuracy, this would allow a simultaneous measurement of two properties of a single neutron star (Morrison et al. 2004; Lattimer and Schutz 2005; Bejger et al. 2005; Read et al. 2009).

The departure of the shape of a rotating neutron star from axisymmetry can be expressed in terms of its ellipticity \(\varepsilon \), defined in a Newtonian context by

$$\begin{aligned} \varepsilon := \frac{I_{xx}-I_{yy}}{I} = \sqrt{\frac{8\pi }{15}} \frac{Q_{22}}{I}, \end{aligned}$$
(95)

where \(I=I_{zz}\) is the moment of inertia about the star’s rotation axis and the \(m=2\) part of a neutron star’s quadrupole moment is given by

$$\begin{aligned} Q_{22} := \mathrm{Re}\int \rho Y_{22} r^2 \, dV, \end{aligned}$$
(96)

where \(Y_{22}\) is the \(l=2,m=2\) spherical harmonic.

Following Ushomirsky et al. (2000), Owen (2005) finds for the maximum value of a neutron star’s ellipticity the expression

$$\begin{aligned} \varepsilon _{\max }= & {} 3.3\times 10^{-7} \frac{\sigma _{\max }}{10^{-2}} \left( \frac{1.4\,M_{\odot }}{M}\right) ^{2.2} \left( \frac{R}{10\,\mathrm km}\right) ^{4.26} \nonumber \\&\times \left[ 1+0.7\left( \frac{M}{1.4\,M_{\odot }}\right) \left( \frac{10\,\mathrm km}{R}\right) \right] ^{-1}, \end{aligned}$$
(97)

where \(\sigma _{\max }\) is the breaking strain of the crust, with an estimated value of order \(10^{-2}\) for crusts below \(10^8K\) (Chugunov and Horowitz 2010).

Rotating strange quark stars

Most rotational properties of strange quark stars differ considerably from the properties of rotating stars constructed with hadronic EOSs. First models of rapidly rotating strange quark stars were computed by Friedman et al. (1989) and by Lattimer et al. (1990). Nonrotating strange stars obey relations that scale with the constant \(\mathcal{B}\) in the MIT bag-model of the strange quark matter EOS. In Gourgoulhon et al. (1999), scaling relations for the model with maximum rotation rate were also found. The maximum angular velocity scales as

$$\begin{aligned} \varOmega _{\max }=9.92 \times 10^3 \sqrt{\mathcal{B}_{60}} \ \mathrm{s}^{-1}, \end{aligned}$$
(98)

while the allowed range of \(\mathcal{B}\) implies an allowed range of \(0.513 \ \mathrm{ms}<P_\mathrm{min}<0.640 \ \mathrm{ms}\). The empirical formula (76) also holds for rotating strange stars with an accuracy of better than \(2\%\). Rotation increases the mass and radius of the maximum mass model by 44 and 54%, correspondingly, significantly more than for hadronic EOSs.

Accreting strange stars in LMXBs will follow different evolutionary paths in a mass versus central energy density diagram than accreting hadronic stars (Zdunik et al. 2002). When (and if) strange stars reach the mass-shedding limit, the ISCO still exists (Stergioulas et al. 1999) (while it disappears for most hadronic EOSs). In Stergioulas et al. (1999) it was shown that the radius and location of the ISCO for the sequence of mass-shedding models also scales as \(\mathcal{B}^{-1/2}\), while the angular velocity of particles in circular orbit at the ISCO scales as \(\mathcal{B}^{1/2}\). Additional scalings with the constant a in the strange quark EOS (54) (that were proposed in Lattimer et al. 1990) were found to hold within an accuracy of better than \(\sim 9\%\) for the mass-shedding sequence:

$$\begin{aligned} M \propto a^{1/2}, \qquad R \propto a^{1/4}, \qquad \varOmega \propto a^{-1/8}. \end{aligned}$$
(99)

In addition, it was found that models at the mass-shedding limit can have T / |W| as large as 0.28 for \(M=1.34\,M_{\odot }\).

If strange stars have a solid normal crust, then the density at the bottom of the crust is the neutron drip density \(\epsilon _\mathrm{ND}\simeq 4.1 \times 10^{11}\,\mathrm {g\ cm}^{-3}\), as neutrons are absorbed by strange quark matter. A strong electric field separates the nuclei of the crust from the quark plasma. In general, the mass of the crust that a strange star can support is very small, of the order of \(10^{-5}\,M_{\odot }\). Rapid rotation increases by a few times the mass of the crust and the thickness at the equator becomes much larger than the thickness at the poles (Zdunik et al. 2001). The mass \(M_\mathrm{crust}\) and thickness \(t_\mathrm{crust}\) of the crust can be expanded in powers of the spin frequency \(\nu _3=\nu /(10^3\,\mathrm {Hz})\) as

$$\begin{aligned} M_\mathrm{crust}= & {} M_\mathrm{crust,0}(1+0.24 \nu _3^2+0.16 \nu _3^8), \end{aligned}$$
(100)
$$\begin{aligned} t_\mathrm{crust}= & {} t_\mathrm{crust,0} (1+0.4 \nu _3^2+0.3\nu _3^6), \end{aligned}$$
(101)

where a subscript “0” denotes nonrotating values (Zdunik et al. 2001). For \(\nu \le 500\,\mathrm {Hz}\), the above expansion agrees well with a quadratic expansion derived previously in Glendenning and Weber (1992). The presence of the crust reduces the maximum angular momentum and ratio of T / |W| by about 20%, compared to corresponding bare strange star models.

Rotating magnetized neutron stars

The presence of a magnetic field has been ignored in the models of rapidly rotating relativistic stars that were considered in the previous sections. The reason is that the inferred surface dipole magnetic field strength of pulsars ranges between \(10^8\) and \(2 \times 10^{13}\,\mathrm {G}\). These values of the magnetic field strength imply a magnetic field energy density that is too small compared to the energy density of the fluid, to significantly affect the structure of a neutron star. However, there exists another class of compact objects with much stronger magnetic fields than normal pulsars—magnetars, that could have global fields up to the order of \(10^{15}\,\mathrm {G}\) (Duncan and Thompson 1992), possibly born initially with high spin (but quickly spinning down to rotational periods of a few seconds). In addition, even though moderate magnetic field strengths do not alter the bulk properties of neutron stars, they may have an effect on the damping or growth rate of various perturbations of an equilibrium star, affecting its stability. For these reasons, a fully relativistic description of magnetized neutron stars is necessary. However, for fields \(<\,10^{15}\,\mathrm {G}\) a passive description, where one ignores the influence of the magnetic field on the equilibrium properties of the fluid and the spacetime is sufficient for most practical purposes.

The equations of electromagnetism and magnetohydrodynamics (MHD) in general relativity have been discussed in a number of works; see, e.g., Lichnerowicz (1967), Misner et al. (1973), Bekenstein and Oron (1978), Anile (1989), Gourgoulhon et al. (2011) and references therein. The electromagnetic (E/M) field is described by a vector potential \(A_\alpha \), from which one constructs the antisymmetric Faraday tensor \( F_{\alpha \beta }= \nabla _\alpha A_\beta - \nabla _\beta A_\alpha , \) satisfying Maxwell’s equations

$$\begin{aligned} \nabla _\beta {}^*F^{\alpha \beta }= & {} 0, \end{aligned}$$
(102)
$$\begin{aligned} \nabla _\beta F^{\alpha \beta }= & {} 4\pi J^\alpha , \end{aligned}$$
(103)

where \({}^*F_{\alpha \beta }{}=\frac{1}{2}\epsilon _{\alpha \beta \gamma \delta }F^{\gamma \delta }\), with \(\epsilon _{\alpha \beta \gamma \delta }\) the totally antisymmetric Levi-Civita tensor. In (103), \(J^\alpha \) is the 4-current creating the E/M field and the Faraday tensor can be decomposed in terms of an electric 4-vector \(E_\alpha =F_{\alpha \beta }u^\beta \) and a magnetic 4-vector \(B_\alpha ={}^*F_{\beta \alpha }u^\beta \) which are measured by an observer comoving with the plasma and satisfy \(E_\alpha u^\alpha =B_\alpha u^\alpha =0\).

The stress–energy tensor of the E/M field is

$$\begin{aligned} T_{\alpha \beta }^\mathrm{(em)}=\frac{1}{4\pi } \left( F_{\alpha \gamma }F_\beta {}^\gamma -\frac{1}{4}F^{\gamma \delta }F_{\gamma \delta } g_{\alpha \beta }\right) , \end{aligned}$$
(104)

and the conservation of the total stress–energy tensor leads to the Euler equation in magnetohydrodynamics

$$\begin{aligned} (\epsilon +p)u^\beta \nabla _\beta u^\alpha = -q^{\alpha \beta }\nabla _\beta p + q^\alpha {}_\delta F^\delta {}_\gamma J^\gamma , \end{aligned}$$
(105)

where \(q_{\alpha \beta }:=g_{\alpha \beta }+u_\alpha u_\beta \). In the ideal MHD approximation, where the conductivity (\(\sigma \)) is assumed to be \(\sigma \rightarrow \infty \), the MHD Euler equation takes the form

$$\begin{aligned} \left( \epsilon +p+\frac{B_\gamma B^\gamma }{4\pi } \right) u^\beta \nabla _\beta u_\alpha = - q_\alpha {}^\beta \left[ \nabla _\beta \left( p+\frac{B_\gamma B^\gamma }{8\pi } \right) - \frac{1}{4\pi }\nabla _\gamma (B_\beta B^\gamma ) \right] . \nonumber \\ \end{aligned}$$
(106)

In general, a magnetized compact star will possess a magnetic field with both poloidal and toroidal components. Then its velocity field may include non-circular flows that give rise to the toroidal component. In such case, the spacetime metric will include additional non-vanishing components. The general formalism describing such a spacetime has been presented by Gourgoulhon and Bonazzola (1993), but no numerical solutions of equilibrium models have been constructed, so far. Instead, one can look for special cases, where the velocity field is circular or assume that it is approximately so.

If the current is purely toroidal, i.e., of the form \((J_t, 0,0, J_\phi )\), then a theorem by Carter (1973) allows for equilibrium solutions with circular velocity flows and a purely poloidal magnetic field, of the form \((0,B_r, B_\theta , 0)\). In ideal MHD, a purely toroidal magnetic field, \((B_t,0,0,B_\phi )\), is also allowed, generated by a current of the form \((0, J_r,J_\theta , 0)\) (Oron 2002).

For purely poloidal magnetic fields, rotating stars must be uniformly rotating in order to be in a stationary equilibrium and the Euler equation becomes

$$\begin{aligned} \nabla (H-\ln u^t) -\frac{1}{\epsilon +p}(j^\phi -\varOmega j^t) \nabla A_\phi =0, \end{aligned}$$
(107)

where \(j^\alpha \) is the conduction current (the component of \(J^\alpha \) normal to the fluid 4-velocity). The hydrostationary equilibrium equation has a first integral in three different cases. These are (a) \((j^\phi -\varOmega j^t)=0\), (b) \((\epsilon +p)^{-1}(j^\phi -\varOmega j^t)=\) const., and (c) \((\epsilon +p)^{-1}(j^\phi -\varOmega j^t)=f(A_\phi )\). The first case corresponds to a vanishing Lorentz force and has been considered in Bekenstein and Oron (1979), Oron (2002) (force-free field). The second case is difficult to realize, but has been considered as an approximation in, e.g., Colaiuda et al. (2008). The third case is more general and was first considered in Bonazzola et al. (1993); Bocquet et al. (1995). After making a choice for the current and for the total charge, the system consisting of the Einstein equations, the hydrostationary equilibrium equation and Maxwell’s equations can be solved for the spacetime metric, the hydrodynamical variables and the vector-potential components \(A_t\) and \(A_\phi \), from which the magnetic and electric fields in various observer frames are obtained.

For a purely toroidal magnetic field, the only non-vanishing component of the Faraday tensor is \(F_{r\theta }\). Then, the ideal MHD condition does not lead to a restriction on the angular velocity of the star. For uniformly rotating stars, the Euler equation becomes (Kiuchi and Yoshida 2008; Gourgoulhon et al. 2011)

$$\begin{aligned} \nabla ( H -\ln u^t) + \frac{1}{4\pi (\epsilon +p)g_2} \sqrt{\frac{g_2}{g_1}} F_{r\theta } \nabla \left( \sqrt{\frac{g_2}{g_1}} F_{r\theta } \right) =0, \end{aligned}$$
(108)

where \(g_1 = g_{rr} g_{\theta \theta }-(g_{r\theta })^2\), \(g_2 = -g_{tt} g_{\phi \phi }+(g_{t\phi })^2\), which implies the existence of solutions for which \(\sqrt{\frac{g_2}{g_1}} F_{r\theta }\) is a function of \((\epsilon +p)g_2\) (see Kiuchi and Yoshida 2008; Kiuchi et al. 2009 for representative numerical solutions). A detailed study of rapidly rotating equilibrium models with purely toroidal fields (in uniform rotation) was recently presented by Frieben and Rezzolla (2012) and Fig. 5 shows the isocontours of magnetic field strength in the meridional plane, for a representative case.

Fig. 5
figure5

Isocontours of magnetic field strength in the meridional plane, for a rapidly rotating model with a purely toroidal magnetic field. (Image from Frieben and Rezzolla 2012, copyright by the authors)

Gourgoulhon et al. (2011) find a general form of stationary axisymmetric magnetic fields, including non-circular equilibria.

Equilibria with purely toroidal or purely poloidal magnetic fields are unstable in nonrotating stars (and likely unstable in rotating stars), see, e.g., Wright (1973), Tayler (1973a), Markey and Tayler (1973), Lasky et al. (2011), Ciolfi and Rezzolla (2012) and Lasky et al. (2012). Some mixed poloidal/toroidal configurations seem more promising for stability (see, e.g., Duez et al. 2010, who infer stability of mixed equilibria in a Newtonian context from numerical evolutions).

A poloidal magnetic field in a differentially rotating star will be wound up, leading to the appearance of a toroidal component. This has several consequences, such as magnetic braking of the differential rotation, amplification of the magnetic field through dynamo action and the development of the magnetorotational instability see Sect. 2.12.

Approximate universal relationships

Yagi and Yunes (2013b, 2013a) recently discovered a set of universal relationships that relate the moment of inertia, the tidal love number and the (spin-induced) quadrupole moment for slowly rotating neutron stars and quark stars (for another review of the I-Love-Q relations see also Yagi and Yunes 2017 where applications of these relations are also presented). The word “universal” in this context means within the framework of a particular theory of gravitation, but independent of the equation of state, provided the equation of state belongs to the class of cold, realistic equations of state, i.e., those that for the most part agree below the nuclear saturation density where our knowledge of nuclear physics is robust. More specifically, the universal relationships were established numerically between properly defined non-dimensional versions of the moment of inertia, the tidal Love number and the quadrupole moment. In particular, if M is the gravitational mass of the star, Yagi and Yunes introduced the following dimensionless quantities: \(\bar{I} \equiv I/M^3\), \(\bar{Q} \equiv -Q/(M^3\chi ^2)\), where \(\chi \equiv J/M^2\) is the dimensionless NS spin parameter, and \(\bar{\lambda }^\mathrm{(tid)}\equiv \lambda ^\mathrm{(tid)}/M^5\). Here \(\lambda ^\mathrm{(tid)}\) is the tidal Love number, which determines the magnitude of the quadrupole moment tensor, \(Q_{ij}\), induced on the star by an external quadrupole tidal tensor field \(\mathcal {E}_{ij}\) through the relation \(Q_{ij}=-\, \lambda ^\mathrm{(tid)}\mathcal {E}_{ij}\). The universal relations can be expressed through the following fitting formulae (Yagi and Yunes 2013b) (see also Lattimer and Lim 2013)

$$\begin{aligned} \ln y_i = a_i+b_i\ln x_i+c_i(\ln x_i)^2 + d_i(\ln x_i)^3+e_i(\ln x_i)^4, \end{aligned}$$
(109)

where \(y_i\) and \(x_i\) are a pair of two variables from the trio \(\bar{I}\), \(\bar{\lambda }^\mathrm{(tid)}\) and \(\bar{Q}\), and the values of the coefficients \(a_i,b_i,c_i,d_i,e_i\) are given in Table 3.

Table 3 Coefficients for the fitting formulae of the neutron star and quark star I-Love, \(I{-}Q\) and Love-Q relations

As pointed out in Yagi and Yunes (2013a) these relations could have been anticipated because in the Newtonian limit \(\bar{I} \propto C^{-2}\), \(\bar{Q} \propto C^{-1}\) and \(\bar{\lambda }^\mathrm{(tid)} \propto C^{-5}\), indicating the existence of one-parameter relation between the trio \(\bar{I}\), \(\bar{\lambda }^\mathrm{(tid)}\), \(\bar{Q}\). Here, C is the compactness of the star. The advantage of the existence of such universal relations is that in principle the measurement of one of the I-Love-Q parameters determines the other two, and one can use these relations to lift quadrupole moment and spin–spin degeneracies that arise in parameter estimation from future gravitational wave observations of compact binaries involving neutron stars (Yagi and Yunes 2013b, a). These relations could also help constrain modified theories of gravity (Yagi and Yunes 2013b, a) (but see below).

Shortly after the discovery of these relations, several works attempted to test the limits of the universality of these relations. Maselli et al. (2013) relaxed the small tidal deformation approximation assumed in Yagi and Yunes (2013b, 2013a) and derived universal relations for the different phases during a neutron star inspiral, concluding that these relations do not deviate significantly from those reported in Yagi and Yunes (2013b, 2013a). On the other hand, Haskell et al. (2014), considered neutron star quadrupole deformations that are induced by the presence of a magnetic field. They built self-consistent magnetized equilibria with the LORENE libraries and concluded that the I-Love-Q universal relations break down for slowly rotating neutron stars (spin periods \(> 10\) s), and for polar magnetic field strengths \(B_p > 10^{12}\,\mathrm {G}\). Doneva et al. (2014b) considered self-consistent, equilibrium models of spinning neutron stars beyond the slow-rotation approximation adopted in Yagi and Yunes (2013b, 2013a). They use the RNS code to built rapidly rotating stars, and find that with increasing rotation rate, the \(\bar{I}\)-\(\bar{Q}\) relation departs significantly from its slow-rotation limit deviating up to 40% for neutron stars and up to 75% for quark stars. Moreover, they find that the deviation is EOS dependent and for a broad set of hadronic and strange matter EOS the spread due to rotation is comparable to the spread due to the EOS, if one considers sequences with fixed rotational frequency. For a restricted set of EOSs, that do not include models with extremely small or large radii, they were still able to find relations that are roughly EOS-independent at fixed rotational frequencies. However, Pappas and Apostolatos (2014) using the RNS code, showed that even for rapidly rotating neutron stars universality is again recovered, if instead of the \(\bar{I}\)-\(\bar{Q}\) and angular frequency parameters, one focuses on the 3 dimensional parameter space spanned by the dimensionless spin angular momentum \(\chi \), the dimensionless mass quadrupole \(\bar{Q}\) and the dimensionless spin octupole moment \(\beta _2\equiv -s_3/\chi ^3\), where \(s_3 \equiv -S_3/M^4\), and where, again, \(S_3\) is the spin octupole moment of the Hansen–Geroch moments (Geroch 1970a; Hansen 1974). Moreover, Pappas and Apostolatos (2014) show that if one considers the parameter space \((\chi ,\bar{I},\bar{Q})\), then the \(I{-}Q\) EOS universality is recovered, in the sense that for each value \(\chi \) there exists a unique universal \(\bar{I}\)-\(\bar{Q}\) relation.

It should be pointed out (Yagi and Yunes 2013b, a; Pappas and Apostolatos 2014) that these “universal” relations hold not among the moments themselves, but among the rescaled, dimensionless moments, where the mass scale is factored out. Thus, the introduction of a scale will lift the apparent degeneracy among different EOSs.

Given the existence of such universal relations relating moments of neutron and quark stars, a fundamental question then arises: what is the origin of the universality? Yagi et al. (2014) performed a thorough study to answer this question and concluded that universality arises as an emergent approximate symmetry in that relativistic stars have an approximate self-similarity in their isodensity contours, which leads to the universal behavior observed in their exterior multipole moments. Work by Chan et al. (2015) has explored the origin of the I-Love relation through a post-Minkowskian expansion for the moment of inertia and the tidal deformability of incompressible stars.

Another way deviations from universality can take place are in protoneutron stars for which a cold, nuclear EOS in not applicable. Martinon et al. (2014) find that the I-Love-Q relations do not apply following one second after the birth of a protoneutron star, but that they are satisfied as soon as the entropy gradients are smoothed out typically within a few seconds. See also Marques et al. (2017), where a new finite temperature hyperonic equation of state is constructed and finds a similar conclusion as Martinon et al. (2014) regarding thermal effects.

Pani and Berti (2014) have extended the Hartle–Thorne formalism for slowly rotating stars to the case of scalar tensor theories of gravity and explored the validity of the I-Love-Q relations in scalar–tensor theories of gravity focusing on theories exhibiting the phenomenon of spontaneous scalarization (Damour and Esposito-Farèse 1993, 1996). Pani and Berti find that I-Love-Q relations exist in scalar-tensor gravity and interestingly also for spontaneously scalarized stars. Most remarkably, the relations in scalar–tensor theories coincide with their general relativity counterparts to within less than a few percent. This result implies that the I-Love-Q relations may not be used to distinguish between general relativity and scalar–tensor theories. We note that a similar conclusion was drawn by Sham et al. (2014) in the context of Eddington-inspired Born–Infeld gravity where the I-Love-Q relations were found to be indistinguishable than those of GR—an anticipated result (Pani and Berti 2014). More recently, Sakstein et al. (2017) have found the the I-C (C for compactness) in beyond Hordenski theories are clearly distinct from those in GR.

The effects of anisotropic pressure have been explored by Yagi and Yunes (2015). They find that anisotropy breaks the universality, but that the I-Love-Q relations remain approximately universal to within 10%. Finally, Yagi and Yunes (2016) considered anisotropic pressure to build slowly rotating, very high compactness stars that approach the black hole compactness limit, in order to answer the question of how the approximate I-Love-Q relations become exact in the BH limit. While the adopted methodology provides some hints into how the BH limit is approached, an interesting, and perhaps, definitive way to probe this is to consider unstable rotating neutron stars and perform dynamical simulations of neutron star collapse to black hole with full GR simulations.

In addition to the I-Love-Q relations, Pappas and Apostolatos (2014) find that for realistic equations of state there exists a universal relation between \(\alpha _2\) and \(\beta _3\), i.e., \(\beta _3=\beta _3(\alpha _2)\), while Yagi et al. (2014) discover a similar universal relation between \(\gamma _4\) and \(\alpha _2\), i.e., \(\gamma _4=\gamma _4(\alpha _2)\). These new approximate universal relations provide a type of “no-hair” relations among the multipole moments for neutron stars and quark stars. Motivated by these studies, Manko and Ruiz (2016b), show that there exists an infinite hierarchy of universal relations for neutron star multipole moments, assuming that neutron star exterior field can be described by four arbitrary parameters as in Manko et al. (1995).

Rapidly rotating equilibrium configurations in modified theories of gravitation

With the arrival of “multimessenger” astronomy, gravitational wave and electromagnetic signatures of compact objects will soon offer a unique probe to test the limits of general relativity. Neutron stars are an ideal astrophysical laboratory for testing gravity in the strong field regime, because of their high compactness and because of the coupling of possible extra mediator fields with the matter.

Testing for deviations from general relativity would preferably require a generalized framework that parametrizes such deviations in an agnostic way as in the spirit of the parameterized post-Newtonian approach (Will 2014) (which systematically models post-Newtonian deviations from GR), or in the spirit of the parametrized post-Einsteinian approach (Yunes and Pretorius 2009; Cornish et al. 2011), which parametrizes a class of deviations from general relativistic waveforms within a certain regime. In the absence of such a complete parameterized framework, the existence of alternative theories of gravity are welcome not only as a means for testing for such deviations, but also for gaining a better understanding of how to develop a generalized, theory-agnostic framework of deviations from general relativity. Motivated by these ideas and by observations that can be interpreted as an accelerated expansion of the Universe (Riess et al. 1998) a number of extended theories of gravitation have been proposed as alternatives to a cosmological constant in order to explain dark energy (see e.g., Tsujikawa 2010; Paschalidis et al. 2011; Bloomfield et al. 2013; de Rham 2014; Joyce et al. 2016; Koyama 2016 for reviews and multiple aspects of such theories). The “infrared” predictions of modified gravity theories have been investigated extensively, and recently their strong-field predictions have attracted considerable attention (see e.g., Berti et al. 2015 for a review). Studies of spherically symmetric and slowly rotating neutron stars in modified gravity are reviewed in Berti et al. (2015), thus we focus here on the bulk properties of equilibrium, rapidly rotating neutron stars in modified theories of gravity.

Doneva et al. (2013b) presented a study of rapidly rotating neutron stars in scalar-tensor theories of gravity, by extending the RNS code to treat these theories in the Einstein frame, while computing physical quantities in the Jordan frame. The Jordan frame action considered in Doneva et al. (2013b) is given by

$$\begin{aligned} S = \frac{1}{16\pi }\int d^4 x\sqrt{-\tilde{g}}\left[ F(\varPhi )\tilde{R}-Z(\varPhi )g^{\mu \nu }\partial _\mu \varPhi \partial _\nu \varPhi - 2 U(\varPhi )\right] + S_m(\varPsi _m;\tilde{g}_{\mu \nu }),\nonumber \\ \end{aligned}$$
(110)

where \(\tilde{g}_{\mu \nu }\) is the Jordan frame metric, \(\tilde{R}\) the Ricci scalar accociated with \(\tilde{g}_{\mu \nu }\), \(\varPhi \) the scalar field, \(U(\varPhi )\) the potential and \(S_m\) denotes the matter action and \(\varPsi _m\) denotes the matter fields. The functions \(U(\varPhi )\), \(F(\varPhi )\) and \(Z(\varPhi )\) control the dynamics of the scalar field. However, requiring that the gravitons carry positive energy implies \(Z(\varPhi ) > 0\), and non-negativity of the scalar field kinetic energy requires \(2F(\varPhi )Z(\varPhi )+3(dF/d\varPhi )^2 \ge 0\). Note that the matter action does not involve \(\varPhi \). Via a conformal transformation

$$\begin{aligned} g_{\mu \nu } = F(\varPhi ) \tilde{g}_{\mu \nu } \end{aligned}$$
(111)

and a scalar field redefinition via

$$\begin{aligned} \left( \frac{d\phi }{d\varPhi }\right) ^2 = \frac{3}{4}\left( \frac{d\ln F(\varPhi )}{d\varPhi }\right) ^2+ \frac{Z(\varPhi )}{2F(\varPhi )} \end{aligned}$$
(112)

and letting

$$\begin{aligned} \mathcal {A}(\phi )=1/\sqrt{F(\varPhi )}, 2V(\phi )=U(\varPhi )/F(\varPhi )^2, \end{aligned}$$
(113)

one recovers the Einstein frame action

$$\begin{aligned} S = \frac{1}{16\pi }\int d^4 x\sqrt{- g}\left[ R-2g^{\mu \nu }\partial _\mu \phi \partial _\nu \phi - 4 V(\phi )\right] + S_m(\varPsi _m;A(\phi )^2 g_{\mu \nu }) ,\qquad \end{aligned}$$
(114)

where R is the Ricci scalar associated with the Einstein frame metric \(g_{\mu \nu }\). Variation of this action with respect to \(g_{\mu \nu }\) and \(\phi \) yields the equations of motion for the metric and for the scalar field, which are then cast in a convenient form and coupled to the equilibrium fluid equations that make it straightforward to extend the general relativistic equations solved by the RNS code. In their study, Doneva et al. (2013b) set \(V(\phi )=0\) and consider two choices for the function \(\mathcal {A}(\phi )\), namely \(\ln \mathcal {A}(\phi )=k_0\phi \) and \(\ln \mathcal {A}(\phi )=\beta \phi ^2/2\), while setting \(\lim _{r\rightarrow \infty } \phi = 0\) and focusing on rigidly rotating equilibria. The former choice for \(\mathcal {A}(\phi )\) is equivalent to Brans–Dicke theory, but the latter choice while it is indistinguishable from general relativity in the weak field regime, leads to the emergence of new phenomenology, such as a bifurcation due to non-uniqueness of solutions (Damour and Esposito-Farèse 1993, 1996). Observations currently constrain \(k_0\) and \(\beta \) to values \(k_0 < 4\times 10^{-3}\) and \(\beta \gtrsim -\,4.5\) (Will 2014; Freire et al. 2012; Antoniadis et al. 2013; Shibata et al. 2014). However, as pointed out in Popchev (2015); Ramazanoǧlu and Pretorius (2016), a massive scalar field naturally circumvents these observational bounds if the Compton wavelength of the scalar field is small compared to the binary orbital separation. The equation of state adopted in Doneva et al. (2013b) is a polytrope \(P=k\rho ^{1+1/n}\), with \(n=0.7463\) and \(k=1186\) in units where \(G=c=M_{\odot }=1\).

For \(\ln \mathcal {A}(\phi )=k_0\phi \) with the largest allowed value \(k_0 = 4\times 10^{-3}\), Doneva et al. (2013b) find that even for stars rotating at the mass shedding limit, their the total mass, radius and angular momentum are practically indistinguishable from their counterparts in general relativity. However, for \(\ln \mathcal {A}(\phi )=\beta \phi ^2/2\), while all general relativity solutions are also solutions of the scalar tensor theory with \(\phi =0\), for certain values of \(\beta \) and a certain range of neutron star densities new solutions emerge with non-trivial scalar field values that are also energetically favored (Damour and Esposito-Farèse 1993, 1996). This phenomenon is known as spontaneous scalarization and for the equation of state adopted in Doneva et al. (2013b), Harada has argued that the phenomenon occurs only for \(\beta \lesssim -\,4.35\) (Harada 1998). One of the important findings in Doneva et al. (2013b) is that rapid rotation extends the range of \(\beta \) values for which spontaneous scalarization can take place, and in particular that along the mass-shedding limit the bound becomes \(\beta < -\,3.9\). In addition, it is found that rapid rotation changes significantly several bulk properties from their GR counterparts. Examples of such properties include the mass, radius, angular momentum, and moment of inertia as can be seen in Fig. 6. Of all bulk quantities those affected the most by the scalar field are the angular momentum and the moment of inertia of the star, which can differ up to a factor of two from their corresponding values in general relativity. It is also worth noting that the deviation of the bulk properties from their GR values, increases further if one considers smaller values of \(\beta \), that are still in agreement with the observations. Based on the sensitivity of the moment of inertia (even at slow rotation rates), Doneva et al. suggested that the moment of inertia could be an astrophysical probe of theories exhibiting spontaneous scalarization.

Fig. 6
figure6

From left to right, top to bottom the plots correspond to mass versus radius, mass vs central energy density, angular frequency versus dimensionless angular momentum, and moment of inertia versus angular frequency. For the plots in the top row, solid lines correspond to nonrotating stars, while dotted lines correspond to the mass-shedding sequence. The angular frequency versus dimensionless angular momentum has only the mass-shedding sequence, while the moment of inertia plot corresponds to models for which the central energy density is fixed at \(\sim \epsilon _c/c^2 = 1.5 \times 10^{15}\,\mathrm g/cm^3\). (Image reproduced with permission from Doneva et al. 2013b, copyright by APS)

In a subsequent paper, Doneva et al. (2014c) extended the equilibrium solutions of rapidly rotating compact stars for the spontaneous scalarization model \(\ln \mathcal {A}(\phi )=\beta \phi ^2/2\) with \(V(\phi )=0\), for tabulated equations of state. For the cases when scalarization occurs, they find results similar to those reported in Doneva et al. (2013b). In addition, they compute orbital and epicyclic frequencies for particles orbiting these neutron star models and find considerable differences of these frequencies between the scalar tensor theory and general relativity for the maximum-mass rotating models (but not so for models with spin frequency of \(\sim 700\,\mathrm {Hz}\) or less, with the exception of very stiff equations of state).

The \(I{-}Q\) relation for rapidly rotating stars in the model \(\ln \mathcal {A}(\phi )=\beta \phi ^2/2\) was considered by Doneva et al. (2014a). The authors find that the \(I{-}Q\) relation is nearly EOS independent for scalarized rapidly rotating stars, and that the spread of the relationship for higher rotation rates increases compared to general relativity. They also find that smaller negative values of \(\beta \) lead to larger deviations from the general relativitivstic \(I{-}Q\) relation, but the deviations (at most 5% for \(\beta =-4.5\)) are less than the anticipated accuracy of the observations. These results provide, yet, another example where the \(I{-}Q\) relation may not be able to provide strong constraints on deviations from general relativity. We note that similar conclusions hold for rapidly rotating stars in Einstein–Gauss–Bonnet-dilaton gravity (Kleihaus et al. 2014).

In a recent paper, Doneva and Yazadjiev (2016) studied rapidly rotating stars for the model \(\ln \mathcal {A}(\phi )=\beta \phi ^2/2\), but this time extending it to the case of a massive scalar field by adding a potential \(V(\phi )=m_\phi ^2 \phi ^2/2\). In this case, In this case, the scalar field is short-range and observations practically leave the value of \(\beta \) unconstrained. However, for the spontaneous scalarization of the neutron star one must have \(10^{-16}\,\mathrm {eV} \lesssim m_\phi \lesssim 10^{-9}\,\mathrm {eV}\). Adopting the value \(\beta = -\,6\), Doneva and Yazadjiev find that the \(I{-}Q\) relation remains universal, but they deviate substantially (up to \(\sim 20\%\)) from those in general relativity. Thus, the \(I{-}Q\) relation could be used to infer deviations from general relativity.

Another modified gravity theory that has been considered in the context of rapidly rotating stars is a particular model of f(R) gravity (Sotiriou and Faraoni 2010; de Felice and Tsujikawa 2010) with \(f(R)=R+a R^2\) sometimes referred to as \(R^2\) gravity. It can be shown that the Einstein frame action of this particular model of f(R) gravity can be cast in the form (114) with \(\ln A(\phi )=-\phi /\sqrt{3}\), but with a non-zero potential \(V(\phi )=(1-\exp (-2\phi /\sqrt{3}))^2/16a\) (Yazadjiev et al. 2015). Motivated by the results found for static and slowly rotating stars in \(R^2\) gravity (Yazadjiev et al. 2014; Staykov et al. 2014), Yazadjiev et al. (2015) modified the RNS code to allow for the construction of rapidly rotating neutron star models in \(R^2\) gravity. Adopting different equations of state, they find that rapid rotation enhances the discrepancy in global quantities such as mass, radius, and angular momentum between \(R^2\)-gravity and general relativistic stars. Also, the differences become larger as the coupling constant a increases. Generically, the \(R^2\)-gravity maximum neutron star mass is larger than the corresponding limit in general relativity. Yazadjiev et al. adopted \(a/M_{\odot }^2\in [0,10^4]\), which is within the Gravity Probe B constraint \(a\lesssim 5\times 10^{5}\,\mathrm {km^2}\), but much larger than the Eöt-Wash experiment constraint \(a\lesssim 10^{-16}\,\mathrm {km^2}\) (Näf and Jetzer 2010). However, the bound from Gravity Probe B is still relevant because the chameleon nature of f(R) gravity can give rise to different effective values at different scales (Näf and Jetzer 2010). For the mass-shedding sequences and with \(a=10^4\,M_{\odot }^2\), they find that for the equations of state considered, the maximum fractional differences between general relativity and \(R^2\)-gravity in maximum mass and maximum moment of inertia are 16.6 and 65.6%, respectively.

Armed with the \(R^2\)-gravity code, Doneva et al. (2015b) studied the universality of the \(I{-}Q\) relation. They find that \(R^2\) gravity exhibits an EOS-independent \(I{-}Q\) relation, but that the differences with the Einstein gravity can be as large as \(\sim 20\%\) for \(a=10^4\,M_{\odot }^2\), similar to the deviations found in Doneva and Yazadjiev (2016) for a scalar–tensor model \(\ln \mathcal {A}(\phi )=\beta \phi ^2/2\) and a massive scalar field. Thus, while it would be difficult to use the \(I{-}Q\) relation in order to single out a specific extended theory of gravity, this relation could potentially be used to infer deviations from general relativity and to exclude some theories of extended gravity.

Fig. 7
figure7

The physical regime on the mass-circumferential equatorial radius plane for neutron star solutions in EdGB theory for the FPS (left) and DI-II (right) EOSs. The left boundary in each panel designates the static sequence and the upper and right boundary the mass-shedding sequence. The values of the coupling constant \(\alpha =0,1,2\) are in units of \(M_{\odot }^2\). (Image reproduced with permission from Kleihaus et al. 2016, copyright by APS)

In addition to theories mentioned that can be cast in the usual form of scalar-tensor theories of gravity, the Einstein-dilaton-Gauss–Bonet (EdGB) theory is another example that has received attention in the context of rapidly rotating neutron stars. EdGB is inspired by heterotic string theory (Gross and Sloan 1987; Metsaev and Tseytlin 1987), and the effective action is given by

$$\begin{aligned} S = \frac{1}{16\pi }\int d^4 x\sqrt{- g}\left[ R-\frac{1}{2}g^{\mu \nu }\partial _\mu \varPhi \partial _\nu \varPhi + \alpha e^{-\beta \varPhi }R_{GB}^2\right] + S_m(\varPsi _m; g_{\mu \nu }),\qquad \end{aligned}$$
(115)

where \(\varPhi \) is the dilaton field, \(\gamma \) is a coupling constant, \(\alpha \) is a positive coefficient and \(R_{GB}^2=R_{\mu \nu \rho \sigma }R^{\mu \nu \rho \sigma }-4R_{\mu \nu }R^{\mu \nu }+R^2\) is the Gauss–Bonet term. The equations of motion for this theory are given by (see, e.g., Kleihaus et al. 2016)

$$\begin{aligned} \Box \varPhi= & {} \alpha \gamma e^{-\beta \varPhi }R_{GB}^2 \end{aligned}$$
(116)
$$\begin{aligned} G_{\mu \nu }= & {} 8\pi T_{\mu \nu }+\frac{1}{2}\left[ \nabla _\mu \varPhi \nabla _\nu \varPhi -\frac{1}{2}\nabla _\rho \varPhi \nabla ^\rho \varPhi \right] \nonumber \\&-\,\alpha e^{\beta \varPhi }\left[ H_{\mu \nu }+4(\beta ^2\nabla ^\rho \varPhi \nabla ^\sigma \varPhi -\beta \nabla ^\rho \nabla ^\sigma \varPhi )P_{\mu \rho \nu \sigma }\right] , \end{aligned}$$
(117)

where

$$\begin{aligned} H_{\mu \nu }= & {} 2\left[ RR_{\mu \nu }-2R_{\mu \rho }R^{\rho }{}_{\nu }-2R_{\mu \rho \nu \sigma }R^{\rho \sigma }+R_{\mu \rho \sigma \lambda }R_{\nu }{}^{\rho \sigma \lambda }\right] \nonumber \\&-\,\frac{1}{2}g_{\mu \nu }R^2_{GB}, \end{aligned}$$
(118)
$$\begin{aligned} P_{\mu \nu \rho \sigma }= & {} R_{\mu \nu \rho \sigma }+2g_{\mu [\sigma } R_{\rho ]\nu }+2g_{\nu [\rho } R_{\sigma ]\mu }+Rg_{\mu [\rho }g_{\sigma ]\nu }, \end{aligned}$$
(119)

are second-order partial differential equations because of the particular form of the Gauss–Bonet term. In this theory black hole solutions exist only for up to a maximum value of \(|\alpha |\) (Kanti et al. 1996), hence rotating neutron star solutions are interesting to find only in this regime. Pani et al. (2011) build models of slowly rotating compact stars in this theory and find that only the product \(\alpha \beta \) matters for the structure of compact stars in EdGB theory, whereas the larger the value of this product the smaller the maximum neutron star mass that can be supported in this theory. They also find that stellar solutions do not exist for arbitrarily large values of \(\alpha \beta \) (this was already known about the existence of black hole solutions, Kanti et al. 1996, in this theory). As a result, the maximum observed mass could be used to place constraints on \(\alpha \beta \).

Kleihaus et al. (2016) develop a code for building rapidly rotating neutron stars in EdGB theory. The authors consider two different equations of state, FPS and DI-II (Diaz Alonso and Ibanez Cabanell 1985). They confirm the results of Pani et al. (2011) and in addition find that rotation enhances the effects of deviations from GR (see Fig. 7). Furthermore, the authors find that the quadrupole moment depends on the value of the EdGB coupling constant and that the dependence is enhanced for larger value of the angular velocity. Finally, Kleihaus et al. discover that the GR \(I{-}Q\) relation extends to EdGB theory with weak dependence on the value of the coupling parameter \(\alpha \) when the NS dimensionless spin is 0.4. Therefore, EdGB theory provides yet another example where the \(I{-}Q\) relations cannot be utilized to constrain deviations from GR.

Differentially rotating neutron stars

The non-uniformity of rotation in the early stages of the life of a compact object (or right after the merger of a binary system) opens another dimension in the allowed parameter space of equilibrium models. The simplest description appropriate for neutron stars is the 1-parameter law (45), introduced in Komatsu et al. (1989a, b); Eriguchi et al. (1994). Relativistic models of differentially rotating stars were constructed numerically by Baumgarte et al. (2000), where it was pointed out that these configurations can support more mass than uniformly rotating stars. The authors coined the term “hypermassive” neutron stars for these compact objects whose mass exceeds the supramassive limit.

Examples of equilibrium sequences of differentially rotating polytropic models, using the above rotation law, were constructed by Stergioulas et al. (2004). Table 4 shows the detailed properties of a fixed rest mass sequence (A), in which the central density decreases as the star rotates more rapidly and of a sequence of fixed central density (B), in which the mass increases significantly with increasing rotation. \(\varOmega _c\) and \(\varOmega _e\) are the values of the angular velocity at the center and at the equator, respectively while \(r_p\) and \(r_e\) is the coordinate radius at the pole and at the equator (other quantities shown are defined as in Table 1). While most models along these sequences are quasi-spherical (meaning that the maximum density appears at the center), the fastest rotating members are quasi-toroidal, with an off-center maximum density. An example is shown in Fig. 8.

Table 4 Properties of two sequences of differentially rotating equilibrium models
Fig. 8
figure8

Density stratification for model A11 of Table 4, displaying an off-center density maximum. In comparison, the shape of the nonrotating star of same rest mass is shown, scaled by the equatorial radius of the rotating model (dashed line). (Image reproduced with permission from Stergioulas et al. 2004, copyright by RAS)

Fig. 9
figure9

Different types of sequences of differentially rotating equilibrium models (see text for a detailed description). Here \(\tilde{A} = \hat{A}^{-1}\). (Image reproduced with permission from Ansorg et al. 2009, copyright by the authors)

Ansorg et al. (2009) found 4 different types of differentially rotating models (which they label as type A, B, C and D) for the same 1-parameter law (45) which exists in parts of the allowed parameter space. For a sufficiently weak degree of differential rotation, sequences with increasing rotation terminate at the mass-shedding limit, but for moderate and strong rates of differential rotation equilibrium sequences can exhibit a continuous transition to a regime of toroidal fluid bodies. Figure 9 displays sequences of \(N=1\) polytropes with various values of the parameter \(\tilde{A}=\hat{A}^{-1}=r_e/A\) and a fixed central density. In the vertical axis, the parameter \(\tilde{\beta }\) is related to the shape of the surface of the star and ranges between 0 (when rotation is limited by mass shedding at the equator) and 1 (when the radius on the polar axis becomes 0, indicating the transition to a toroidal configuration). As the axis ratio \(r_p/r_e\) is varied, type A sequences start at a nonrotating model and terminate at the mass-shedding limit. Type B sequences have no nonrotating member, but connect models at the mass-shedding limit to toroidal configurations. Type C sequences connect the nonrotating limit to toroidal configurations, while type D sequences connect models at the mass-shedding limit to other models at the mass-shedding limit. It will be interesting to study the stability properties of models of these different types. One should keep in mind that these different types arise for the simple 1-parameter law (45) and a more complicated picture may arise for multi-parameter rotation laws. More recently, Studzińska et al. (2016) thoroughly explored the parameter space for the rotation law (45) and determined how the maximum mass depends on the stiffness, on the degree of differential rotation and on the maximum density, taking into account all types of solutions that were shown to exist in Ansorg et al. (2009).

A well know fact about differentially rotating neutron stars is that they can support more mass than the supramassive limit—the maximum mass when allowing for maximal uniform rotation. Neutron stars with mass larger than the supramassive limit are known as hypermassive neutron stars. Equilibrium sequences of differentially rotating models with polytropic equations of state, and using the same differential rotation law have been constructed by Lyford et al. (2003). There, the focus was on the effects of differential rotation on the maximum mass configuration. The authors find that differential rotation can support about 50% more mass than the TOV limit mass, as opposed to uniform rotation that typically increases the TOV limit by about 20%. In a subsequent paper, Morrison et al. (2004) extended this result to realistic equations of state. However, recent calculations by Gondek-Rosinska et al. (2017) focussing on \(n=1\) polytropes, discover that the maximum mass depends not only on the degree of differential rotation, but also on the type of solution identified in Ansorg et al. (2009), i.e., A, B, C or D. The authors find that different classes have different maximum mass limits and even for moderate degrees of differential rotation \(\hat{A}^{-1} \sim 1\), the maximum rest-mass configuration can be significantly higher than 2.0 times the TOV limit. Although, masses greater than two times the TOV limit can never be achieved in hypermassive neutron stars formed following a binary neutron star merger, it would be interesting to investigated the dynamical stability of the maximum mass configurations constructed in Gondek-Rosinska et al. (2017).

Proto-neutron stars

Following the gravitational collapse of a massive stellar core, a proto-neutron star (PNS) is born with an initially large temperature of order 50 MeV and a correspondingly large radius of up to 100 km. If the PNS is slowly rotating, one can study its evolution assuming spherical symmetry (see, for example Burrows and Lattimer 1986; Burrows et al. 1995; Bombaci et al. 1995; Keil and Janka 1995; Keil et al. 1996; Prakash et al. 1997; Pons et al. 1999, 2000; Prakash et al. 2001; Strobel et al. 1999). Up to a time of about 100 ms after core bounce, the PNS is lepton rich and consists of an unshocked core at densities \(n>0.1 \,\mathrm {fm}^{-3}\), with entropy per baryon \(s\sim 1\), surrounded by a transition region and a low-density but high-entropy, shocked envelope with \(s \sim \) 4–10, which extends to large radii. The lepton number is roughly \(Y_l\sim 0.4\) and neutrinos in the core and in the transition region are trapped (the PNS is opaque to neutrinos), while at densities less than \(n\sim 6\times 10^{-4}\,\mathrm {fm}^{-3}\) the outer envelope becomes transparent to neutrinos. Within about 0.5 s, the outer envelope cools and contracts with the entropy per baryon becoming roughly \(s\sim 2\) throughout the star (the lepton number in the outer envelope drops to \(Y_l\sim 0.3\)). Further cooling results in a fully deleptonized, hot neutron star at several tens of seconds after core bounce, with a roughly constant entropy per baryon of \(s\sim \) 1–2. After several minutes, when the neutron star has cooled to \(T<1\,\mathrm {MeV}\), the thermal effects are negligibly small in the bulk of the star and a zero-temperature EOS can be used to describe its main properties.

The structure of hot PNSs is described by finite-temperature EOSs, such as those presented in Lattimer and Swesty (1991); Sugahara and Toki (1994); Toki et al. (1995); Lalazissis et al. (1997); Strobel et al. (1999b); Pons et al. (1999); Shen et al. (1998); Hempel and Schaffner-Bielich (2010); Typel et al. (2010); Fattoyev et al. (2010); Shen et al. (2011b, 2011a); Hempel et al. (2012); Steiner et al. (2013) (see also Oertel et al. 2017 for a review). These candidate EOSs differ in several respects (for example in the thermal pressure at high densities). The sample of cold EOSs that has been extended, so far, is quite limited and does not correspond to the wide range of possibilities allowed by current observational constraints. Therefore, PNS models that have been constructed only cover a small region of the allowed parameter space. Understanding the detailed evolution of a PNS is significant, as the star could undergo transformations that could be associated with direct or indirect observational evidence, such as the delayed collapse of a hypermassive PNS (see Brown and Bethe 1994; Baumgarte et al. 1996, 2000).

If the PNS is born rapidly rotating, its evolution will sensitively depend on the rotation rate and other factors, such as the development of the magnetorotational instability (MRI). Some partial understanding has emerged by studying quasi-equilibrium sequences of rotating models (Hashimoto et al. 1994; Strobel et al. 1999; Sumiyoshi et al. 1999; Villain et al. 2004). Exact equilibria can be found in the case that the model is considered to be barotropic, where all thermodynamical quantities (energy density, pressure, entropy, temperature) depend only on the baryon number density. Special cases, such as homentropic or isothermal stars have also been considered. In Villain et al. (2004) a barotropic EOS was constructed by rescaling temperature, entropy and lepton number profiles that were obtained from detailed, one-dimensional simulations of PNS evolutions, while the rotational properties of the models were taken from two-dimensional core-collapse simulations.

The main conclusion from the studies of sequences of quasi-equilibrium models is that PNSs that are born with moderate rotation, will contract and spin up during the cooling phase (see e.g., Goussard et al. 1998; Strobel et al. 1999). This could lead to a PNS rotating with large enough rotation rate for secular or dynamical instabilities to become interesting. It is not clear, however, whether the quasi-stationary approximation is valid when the stars reach the mass-shedding limit, as, upon further thermal contraction, the outer envelope could actually be shed from the star, resulting in an equatorial stellar wind. It should be noted here that a small amount of differential rotation significantly affects the mass-shedding limit, allowing more massive stars to exist than uniform rotation allows.

Studies of PNSs are being extended to include additional effects, such as entropy and lepton-driven convective instabilities and hydromagnetic instabilities (Epstein 1979; Livio et al. 1980; Burrows and Lattimer 1986; Burrows and Fryxell 1992; Miralles et al. 2000, 2002, 2004; Dessart et al. 2006; Lasky et al. 2012), meridional flows (Eriguchi and Müller 1991), local and mean-field magnetic dynamos (Thompson and Duncan 1993; Xu and Busse 2001; Bonanno et al. 2003; Reinhardt and Geppert 2005; Naso et al. 2008), magnetic braking and viscous damping of differential rotation (Shapiro 2000; Liu and Shapiro 2004; Duez et al. 2004; Thompson et al. 2005; Duez et al. 2006b), and the MRI and Tayler instabilities (Akiyama et al. 2003; Kotake et al. 2004; Thompson et al. 2005; Ardeljan et al. 2005; Masada et al. 2006; Shibata et al. 2006; Cerdá-Durán et al. 2007; Masada et al. 2007; Stephens et al. 2007; Bisnovatyi-Kogan and Moiseenko 2008; Spruit 2008; Kiuchi et al. 2008; Obergaulinger et al. 2009; Siegel et al. 2013; Guilet et al. 2016). These effects will be important for the evolution of both PNSs formed after core collapse, as well as for hypermassive or supramassive neutron stars possibly formed after a binary neutron star merger.

Rotating relativistic stars in LMXBs

Particle orbits and kHz quasi-periodic oscillations

X-ray observations of accreting sources in LMXBs have revealed a rich phenomenology that is waiting to be interpreted correctly and could lead to significant advances in our understanding of compact objects (see Lamb et al. 1998; van der Klis 2000; Psaltis 2001). The most important feature of these sources is the observation (in most cases) of twin kHz quasi-periodic oscillations (QPOs) (see van der Klis 2006; Abramowicz and Fragile 2013 for reviews on QPOs). The high frequency of these variabilities and their quasi-periodic nature are evidence that they are produced in high-velocity flows near the surface of the compact star. To date, there exist a large number of different theoretical models that attempt to explain the origin of these oscillations. No consensus has been reached, yet, but once a credible explanation is found, it will lead to important constraints on the properties of the compact object that is the source of the gravitational field in which the kHz oscillations take place. The compact stars in LMXBs are spun up by accretion, so that many of them may be rotating rapidly; therefore, the correct inclusion of rotational effects in the theoretical models for kHz QPOs is important. Under simplifying assumptions for the angular momentum and mass evolution during accretion, one can use accurate rapidly rotating relativistic models to follow the possible evolutionary tracks of compact stars in LMXBs (Cook et al. 1994c; Zdunik et al. 2002).

In most theoretical models, one or both kHz QPO frequencies are associated with the orbital motion of inhomogeneities or blobs in a thin accretion disk. In the actual calculations, the frequencies are computed in the approximation of an orbiting test particle, neglecting pressure terms. For most equations of state, stars that are massive enough possess an ISCO, and the orbital frequency at the ISCO has been proposed to be one of the two observed frequencies. To first order in the rotation rate, the orbital frequency at the prograde ISCO is given by (see Kluźniak et al. 1990)

$$\begin{aligned} f_{\mathrm {ISCO}} \simeq 2210 \, (1+0.75\chi ) \left( \frac{1\,M_{\odot }}{M} \right) \,\mathrm {Hz}, \end{aligned}$$
(120)

where \(\chi :=J/M^2\). At larger rotation rates, higher order contributions of \(\chi \) as well as contributions from the quadrupole moment Q become important and an approximate expression has been derived by Shibata and Sasaki (1998), which, when written as above and truncated to the lowest order contribution of Q and to \(\mathcal{O}(\chi ^2)\), becomes

$$\begin{aligned} f_{\mathrm {ISCO}} \simeq 2210 \, (1+0.75\chi +0.78\chi ^2-0.23Q_2) \left( \frac{1\,M_{\odot }}{M} \right) \,\mathrm {Hz}, \end{aligned}$$
(121)

where \(Q_2=-Q/M^3\).

Notice that, while rotation increases the orbital frequency at the ISCO, the quadrupole moment has the opposite effect, which can become important for rapidly rotating models. Numerical evaluations of \(f_{\mathrm {ISCO}}\) for rapidly rotating stars have been used in Miller et al. (1998) to arrive at constraints on the properties of the accreting compact object.

In other models, orbits of particles that are eccentric and slightly tilted with respect to the equatorial plane are involved (the relativistic precession model). For eccentric orbits, the periastron advances with a frequency \(\nu _{\mathrm {pa}}\) that is the difference between the Keplerian frequency of azimuthal motion \(\nu _{\mathrm {K}}\) and the radial epicyclic frequency \(\nu _{\mathrm {r}}\). On the other hand, particles in slightly tilted orbits fail to return to the initial displacement \(\psi \) from the equatorial plane, after a full revolution around the star. This introduces a nodal precession frequency \(\nu _{\mathrm {pa}}\), which is the difference between \(\nu _{\mathrm {K}}\) and the frequency of the motion out of the orbital plane (meridional frequency) \(\nu _{\psi }\). Explicit expressions for the above frequencies, in the gravitational field of a rapidly rotating neutron star, have been derived recently by Marković (2000), while in Marković and Lamb (2000) highly eccentric orbits are considered. Morsink and Stella (1999) compute the nodal precession frequency for a wide range of neutron star masses and equations of state and (in a post-Newtonian analysis) separate the precession caused by the Lense–Thirring (frame-dragging) effect from the precession caused by the quadrupole moment of the star. The nodal and periastron precession of inclined orbits have also been studied using an approximate analytic solution for the exterior gravitational field of rapidly rotating stars (Sibgatullin 2002). These precession frequencies are relativistic effects and have been used in several models to explain the kHz QPO frequencies (Stella et al. 1999; Psaltis and Norman 2000; Abramowicz and Kluźniak 2001; Kluźniak and Abramowicz 2002; Amsterdamski et al. 2002).

It is worth mentioning that it has recently been found that an ISCO also exists in Newtonian gravity, for models of rapidly rotating low-mass strange stars. The instability in the circular orbits is produced by the large oblateness of the star (Kluźniak et al. 2001; Zdunik and Gourgoulhon 2001; Amsterdamski et al. 2002) (see also Török et al. 2014 for a more recent study). Epicyclic frequencies for Maclaurin spheroids in Newtonian gravity have also been computed by Kluźniak and Rosińska (2013).

Epicyclic frequencies for rapidly rotating strange stars have been computed by Gondek-Rosińska et al. (2014) adopting the MIT bag model for the equation of state of quark matter. They find that the orbits around rapidly rotating strange quark stars are mostly affected by the stellar oblateness, rather than by the effects of general relativity.

For reviews on applications of current QPO models and what one can learn about the properties of NSs see Bhattacharyya (2010) , Török et al. (2010), Pappas (2012), Miller and Lamb (2016) and Özel and Freire (2016).

Going further:   Observations of some LMXBs finding that the difference in the frequencies of the peak QPOs is equal to half the spin frequency of the star raise some questions regarding the validity of the popular beat-frequency model (Miller et al. 1998) (but see Lamb and Miller 2003). Motivated by this tension, another model for QPOs is suggested by Li and Narayan (2004) in which it is argued that a strong magnetic field may truncate the inner parts of the disk and at the interface between the accretion disk and the magnetosphere surrounding the accreting star that gas becomes Rayleigh–Taylor and, possible also, Kelvin–Helmholtz unstable, leading to nonaxisymmetric structures which result in the high-frequency QPOs that can explain observations. For other studies considering the impact of magnetic fields see also Kluźniak and Rappaport (2007), Kulkarni and Romanova (2008), Tsang and Lai (2009), Lovelace et al. (2009), Bakala et al. (2010), Bakala et al. (2012), Fu and Lai (2012), Romanova et al. (2012), Long et al. (2012) and references therein. Another model that can explain the observations where the difference in the frequencies of the twin peaks is equal to half the spin frequency of the star is the so-called non-linear resonance model (Kluźniak et al. 2004) (see also Lee et al. 2004; Horák et al. 2009; Urbanec et al. 2010). For a recent work investigating the compatibility of realistic neutron star equations of state with several QPO modes see Török (2016). Constants of motion in stationary, axisymmetric spacetimes have been investigated recently in Markakis (2014).

Angular momentum conservation during burst oscillations

Some sources in LMXBs show signatures of type I X-ray bursts, which are thermonuclear bursts on the surface of the compact star (Lewin et al. 1995). Such bursts show nearly-coherent oscillations in the range 270–620 Hz (see van der Klis 2000; Strohmayer 2001; Strohmayer and Bildsten 2006; Watts 2012 for reviews). One interpretation of the burst oscillations is that they are the result of rotational modulation of surface asymmetries during the burst. In such a case, the oscillation frequency should be nearly equal to the spin frequency of the star. This model currently has difficulties in explaining some observed properties, such as the oscillations seen in the tail of the burst, the frequency increase during the burst, and the need for two anti-podal hot spots in some sources that ignite at the same time. Alternative models also exist (see, e.g., Psaltis 2001).

Changes in the oscillation frequency by a few Hz during bursts have been associated with expansion and contraction of the burning shell. Cumming et al. (2002) compute the expected spin changes in general relativity taking into account rapid rotation. Assuming that the angular momentum per unit mass is conserved, the change in angular velocity with radius is given by

$$\begin{aligned} \frac{d\ln \varOmega }{d\ln r} = - 2 \left[ \left( 1-\frac{v^2}{2}-\frac{R}{2} \frac{\partial \nu }{\partial r} \right) \left( 1-\frac{\omega }{\varOmega }\right) - \frac{R}{2\varOmega }\frac{\partial \omega }{\partial r}\right] , \end{aligned}$$
(122)

where R is the equatorial radius of the star and all quantities are evaluated at the equator. The slow rotation limit of the above result was derived previously by Abramowicz et al. (2001). The fractional change in angular velocity can then be estimated as

$$\begin{aligned} \frac{\varDelta \varOmega }{\varOmega } = \frac{d\ln \varOmega }{d\ln r}\left( \frac{\varDelta r}{R}\right) , \end{aligned}$$
(123)

where \(\varDelta r\) is the coordinate expansion of the burning shell, a quantity that depends on the shell’s composition. Cumming et al. find that in the expansion phase the expected spin down is a factor of two to three times smaller than observed values, if the atmosphere rotates rigidly. More detailed modeling is needed to fully explain the origin and properties of burst oscillations (see Watts 2012 for a recent review on theoretical models of thermonuclear bursts).

A very interesting topic is modeling the expected X-ray spectrum of an accretion disk in the gravitational field of a rapidly rotating neutron star or of the “hot spot” on its surface as it could lead to observational constraints on the source of the gravitational field. See, e.g., Thampan and Datta (1998), Sibgatullin and Sunyaev (1998), Sibgatullin and Sunyaev (2000), Bhattacharyya (2002), Bhattacharyya (2001), where work initiated by Kluźniak and Wilson (1991) in the slow rotation limit is extended to rapidly rotating relativistic stars.

Following an earlier work which uses approximate spacetimes (Cadeau et al. 2005), light curves from ray-tracing on spacetimes corresponding to realistic models of rapidly rotating neutron stars (generated with the RNS code) are obtained by Cadeau et al. (2007) assuming that the X-ray photons arise from a hot spot on the NS. There it was shown that the dominant effect due to rotation comes from the stellar oblateness, and that approximating a rapidly rotating star as a sphere results in large errors if one is trying to fit for the radius and mass. However, for cases with stellar spin frequencies \(<\,\sim 300\,\mathrm {Hz}\) rapidly rotating spacetime models are not necessary and only the stellar oblateness has to be taken into consideration. As a result, Morsink et al. (2007) develop the Oblate Schwarzschild (OS) model in which photons emerge from a hot spot in the NS oblate surface, and they reach the observer following the geodesics of a corresponding Schwarzschild spacetime, while doppler effects due to rotation are taken into consideration as in the standard model of Miller and Lamb (1998). Morsink et al. demonstrate that the OS model suffices to describe the effects due to the NS rotation. An approximate analytic model for pulse profiles taking into account gravitational light bending, doppler effect, anisotropic emission and time delays is presented by Poutanen and Beloborodov (2006). Another simple model adopting the Hartle–Thorne approximation for generating pulse profiles from rotating neutron stars is developed by Psaltis and Özel (2014). For related studies see also Lee and Strohmayer (2005), Bauböck et al. (2012), Ck et al. (2013), Miller and Lamb (2015) and references therein.

Going further: A number of theoretical works whose aim to model atomic lines in NS atmospheres in order to infer the NS properties from the atomic line redshift see, e.g., (Özel and Psaltis 2003; Bildsten et al. 2003; Chang et al. 2005, 2006; Bhattacharyya et al. 2006; Bauböck et al. 2013; Özel 2013; Heinke 2013; Bauböck et al. 2015).

Oscillations and stability

The study of oscillations of relativistic stars is motivated by the prospect of detecting such oscillations in electromagnetic or gravitational wave signals. In the same way that helioseismology is providing us with information about the interior of the Sun, the observational identification of oscillation frequencies of relativistic stars could constrain the high-density equation of state (Andersson and Kokkotas 1996). The oscillations could be excited after a core collapse or during the final stages of a neutron star binary merger. Rapidly rotating relativistic stars can become unstable to the emission of gravitational waves.

When the displacement due to the oscillations of an equilibrium star are small compared to its radius, it will suffice to approximate them as linear perturbations. Such perturbations can be described in two equivalent ways. In the Lagrangian approach, one studies the changes in a given fluid element as it oscillates about its equilibrium position. In the Eulerian approach, one studies the change in fluid variables at a fixed point in space. Both approaches have their strengths and weaknesses.

In the Newtonian limit, the Lagrangian approach has been used to develop variational principles (Lynden-Bell and Ostriker 1967; Friedman and Schutz 1978), but the Eulerian approach proved to be more suitable for numerical computations of mode frequencies and eigenfunctions (Ipser and Managan 1985; Managan 1985; Ipser and Lindblom 1990, 1991b, a). Clement (1981) used the Lagrangian approach to obtain axisymmetric normal modes of rotating stars, while nonaxisymmetric solutions were obtained in the Lagrangian approach by Imamura et al. (1985) and in the Eulerian approach by Managan (1985) and Ipser and Lindblom (1990). While a lot has been learned from Newtonian studies, in the following we will focus on the relativistic treatment of oscillations of rotating stars.

Quasi-normal modes of oscillation

A general linear perturbation of the energy density in a static and spherically symmetric relativistic star can be written as a sum of quasi-normal modes that are characterized by the indices (lm) of the spherical harmonic functions \(Y_l^m\) and have angular and time dependence of the form

$$\begin{aligned} \delta \varepsilon \sim f(r) P_l^m(\cos \theta ) e^{i(m\phi +\omega _{\mathrm {i}} t)}, \end{aligned}$$
(124)

where \(\delta \) indicates the Eulerian perturbation of a quantity, \(\omega _{\mathrm {i}}\) is the angular frequency of the mode as measured by a distant inertial observer, f(r) represents the radial dependence of the perturbation, and \(P_l^m(\cos \theta )\) are the associated Legendre polynomials. Normal modes of nonrotating stars are degenerate in m and it suffices to study the axisymmetric \((m=0)\) case.

The Eulerian perturbation in the fluid 4-velocity \(\delta u^a\) can be expressed in terms of vector harmonics, while the metric perturbation \(\delta g_{ab}\) can be expressed in terms of spherical, vector, and tensor harmonics. These are either of “polar” or “axial” parity. Here, parity is defined to be the change in sign under a combination of reflection in the equatorial plane and rotation by \(\pi \). A polar perturbation has parity \((-\,1)^l\), while an axial perturbation has parity \((-\,1)^{l+1}\). Because of the spherical background, the polar and axial perturbations of a nonrotating star are completely decoupled.

A normal mode solution satisfies the perturbed gravitational field equation,

$$\begin{aligned} \delta (G^{ab}-8 \pi T^{ab})=0, \end{aligned}$$
(125)

and the perturbation of the conservation of the stress-energy tensor,

$$\begin{aligned} \delta (\nabla _aT^{ab})=0, \end{aligned}$$
(126)

with suitable boundary conditions at the center of the star and at infinity. The latter equation is decomposed into an equation for the perturbation in the energy density \(\delta \varepsilon \) and into equations for the three spatial components of the perturbation in the 4-velocity \(\delta u^a\). As linear perturbations have a gauge freedom, at most six components of the perturbed field equation (125) need to be considered.

For a given pair (lm), a solution exists for any value of the frequency \(\omega _{\mathrm {i}}\), consisting of a mixture of ingoing and outgoing wave parts. Outgoing quasi-normal modes are defined by the discrete set of eigenfrequencies for which there are no incoming waves at infinity. These are the modes that will be excited in various astrophysical situations.

The main modes of pulsation that are known to exist in relativistic stars have been classified as follows (\(f_0\) and \(\tau _0\) are typical frequencies and damping times of the most important modes in the nonrotating limit):

  1. 1.

    Polar fluid modes are slowly damped modes analogous to the Newtonian fluid pulsations:

    • f(undamental)-modes: surface modes due to the interface between the star and its surroundings (\(f_0 \sim 2 \,\mathrm {kHz}\), \(\tau _0<1 \,\mathrm {s}\)),

    • p(ressure)-modes: nearly radial (\(f_0 > 4 \,\mathrm {kHz}\), \(\tau _0 > 1 \,\mathrm {s}\)),

    • g(ravity)-modes: nearly tangential, degenerate at zero frequency in nonrotating isentropic stars; they have nonzero frequencies in stars that are non-isentropic or that have a composition gradient or a first order phase transition (\(f_0<500 \,\mathrm {Hz}\), \(\tau _0 > 5 \,\mathrm {s}\)).

  2. 2.

    Axial and hybrid fluid modes:

    • inertial modes: degenerate at zero frequency in nonrotating stars. In a rotating star, some inertial modes are generically unstable to the CFS instability; they have frequencies from zero to kHz and growth times inversely proportional to a high power of the star’s angular velocity. Hybrid inertial modes have both axial and polar parts even in the limit of no rotation.

    • r(otation)-modes: a special case of inertial modes that reduce to the classical axial r-modes in the Newtonian limit. Generically unstable to the CFS instability with growth times as short as a few seconds at high rotation rates.

  3. 3.

    Polar and axial spacetime modes:

    • w(ave)-modes: Analogous to the quasi-normal modes of a black hole (very weak coupling to the fluid). High frequency, strongly damped modes (\(f_0>6 \,\mathrm {kHz}\), \(\tau _0 \sim 0.1 \,\mathrm {ms}\)).

For a more detailed description of various types of oscillation modes, see Kokkotas (1997b, 1997a), McDermott et al. (1988), Carroll et al. (1986) and Kokkotas and Schmidt (1999).

Effect of rotation on quasi-normal modes

In a continuous sequence of rotating stars that includes a nonrotating member, a quasi-normal mode of index l is defined as the mode which, in the nonrotating limit, reduces to the quasi-normal mode of the same index l. Rotation has several effects on the modes of a corresponding nonrotating star:

  1. 1.

    The degeneracy in the index m is removed and a nonrotating mode of index l is split into \(2l+1\) different (lm) modes.

  2. 2.

    Prograde (\(m<0\)) modes are now different from retrograde (\(m>0\)) modes.

  3. 3.

    A rotating “polar” l-mode consists of a sum of purely polar and purely axial terms (Stergioulas 1996), e.g., for \(l=m\),

    $$\begin{aligned} P_l^{\mathrm {rot}} \sim \sum _{l'=0}^\infty (P_{l+2l'} +A_{l+2l' \pm 1}), \end{aligned}$$
    (127)

    that is, rotation couples a polar l-term to an axial \(l \pm 1\) term (the coupling to the \(l+1\) term is, however, strongly favoured over the coupling to the \(l-1\) term, Chandrasekhar and Ferrari 1991). Similarly, for a rotating “axial” mode with \(l=m\),

    $$\begin{aligned} A_l^{\mathrm {rot}} \sim \sum _{l'=0}^\infty (A_{l+2l'} +P_{l+2l' \pm 1}). \end{aligned}$$
    (128)
  4. 4.

    Frequencies and damping times are shifted. In general, frequencies (in the inertial frame) of prograde modes increase, while those of retrograde modes decrease with increasing rate of rotation.

  5. 5.

    In rapidly rotating stars, apparent intersections between higher order modes of different l can occur. In such cases, the shape of the eigenfunction is used in the mode classification.

In rotating stars, quasi-normal modes of oscillation have been studied only in the slow rotation limit, in the post-Newtonian, and in the Cowling approximations. The solution of the fully relativistic perturbation equations for a rapidly rotating star is still a very challenging task and only recently have they been solved for zero-frequency (neutral) modes (Stergioulas 1996; Stergioulas and Friedman 1998). First frequencies of quasi-radial modes have now been obtained through 3D numerical time evolutions of the nonlinear equations (Font et al. 2002).

Going further:   The equations that describe oscillations of the solid crust of a rapidly rotating relativistic star are derived by Priou (1992). The effects of superfluid hydrodynamics on the oscillations of neutron stars have been investigated by several authors, see, e.g., Lindblom and Mendell (1994); Comer et al. (1999), Andersson and Comer (2001), Andersson et al. (2002), Andersson et al. (2004), Prix et al. (2004), Sidery et al. (2008), Samuelsson and Andersson (2009), Passamonti et al. (2009a), Andersson et al. (2011), Passamonti and Andersson (2012), Passamonti et al. (2016) and references therein.

Axisymmetric perturbations

Secular and dynamical axisymmetric instability

Along a sequence of nonrotating relativistic stars with increasing central energy density, there is always a model for which the mass becomes maximum. The maximum-mass turning point marks the onset of an instability in the fundamental radial pulsation mode of the star.

Applying the turning point theorem provided by Sorkin (1982), Friedman et al. (1988) provide a sufficient condition for a secular axisymmetric instability of rotating stars, when the mass becomes maximum along a sequence of constant angular momentum. An equivalent criterion (implied in Friedman et al. 1988) is provided by Cook et al. (1992): The secular axisymmetric instability sets in when the angular momentum becomes minimum along a sequence of constant rest mass. The instability first develops on a secular timescale that is set by the time required for viscosity to redistribute the star’s angular momentum. This timescale is long compared to the dynamical timescale and comparable to the spin-up time following a pulsar glitch. Eventually, the star encounters the onset of dynamical instability and collapses to a black hole (see Shibata et al. 2000a for recent numerical simulations). Thus, the onset of the secular instability to axisymmetric perturbations separates stable neutron stars from neutron stars that will undergo collapse to a black hole. More recently, Takami et al. (2011) investigated the dynamical stability of rotating stars, computing numerically the neutral point of the fundamental, quasi-radial F-mode frequency, which signals the onset of the dynamical stability. In their simulations, they found that the F-mode frequency can go through zero before a star reaches turning point. Prabhu et al. (2016) investigate the axisymmetric stability of rotating relativistic stars through a variational principle and show that the sign of the canonical energy gives a necessary and sufficient condition for dynamical instability. In addition, they determine a lower bound for exponential growth.

Goussard et al. (1997) extend the stability criterion to hot proto-neutron stars with nonzero total entropy. In this case, the loss of stability is marked by the configuration with minimum angular momentum along a sequence of both constant rest mass and total entropy. In the nonrotating limit, Gondek et al. (1997) compute frequencies and eigenfunctions of radial pulsations of hot proto-neutron stars and verify that the secular instability sets in at the maximum mass turning point, as is the case for cold neutron stars.

Fig. 10
figure10

Apparent intersection (due to avoided crossing) of the axisymmetric first quasi-radial overtone (\(H_1\)) and the first overtone of the \(l=4\) p-mode (in the Cowling approximation). Frequencies are normalized by \(\sqrt{\rho _{\mathrm {c}}/4\pi }\), where \(\rho _{\mathrm {c}}\) is the central energy density of the star. The rotational frequency \(f_{\mathrm {rot}} \) at the mass-shedding limit is 0.597 (in the above units). Along continuous sequences of computed frequencies, mode eigenfunctions are exchanged at the avoided crossing. Defining quasi-normal mode sequences by the shape of their eigenfunction, the \(H_1\) sequence (filled boxes) appears to intersect with the \({}^4p_1\) sequence (triangle), but each sequence shows a discontinuity, when the region of apparent intersection is well resolved. In the notation \({}^l\mathrm{mode}_n\), the superscript indicates the l of the perturbation, while the subscript indicates the harmonic overtone. (Image reproduced with permission from Yoshida and Eriguchi 2001, copyright by RAS)

Axisymmetric pulsation modes

Axisymmetric (\(m=0\)) pulsations in rotating relativistic stars could be excited in a number of different astrophysical scenarios, such as during core collapse, in star quakes induced by the secular spin-down of a pulsar or during a large phase transition, or in the merger of two relativistic stars in a binary system, among others. Due to rotational couplings, the eigenfunction of any axisymmetric mode will involve a sum of various spherical harmonics \(Y_l^0\), so that even the quasi-radial modes (with lowest order \(l=0\) contribution) would, in principle, radiate gravitational waves.

Quasi-radial modes in rotating relativistic stars have been studied by Hartle and Friedman (1975) and by Datta et al. (1998) in the slow rotation approximation. Yoshida and Eriguchi (2001) study quasi-radial modes of rapidly rotating stars in the relativistic Cowling approximation and find that apparent intersections between quasi-radial and other axisymmetric modes can appear near the mass-shedding limit (see Fig. 10). These apparent intersections are due to avoided crossings between mode sequences, which are also known to occur for axisymmetric modes of rotating Newtonian stars. Along a continuous sequence of computed mode frequencies an avoided crossing occurs when another sequence is encountered. In the region of the avoided crossing, the eigenfunctions of the two modes become of mixed character. Away from the avoided crossing and along the continuous sequences of computed mode frequencies, the eigenfunctions are exchanged. However, each “quasi-normal mode” is characterized by the shape of its eigenfunction and thus, the sequences of computed frequencies that belong to particular quasi-normal modes are discontinuous at avoided crossings (see Fig. 10 for more details). The discontinuities can be found in numerical calculations, when quasi-normal mode sequences are well resolved in the region of avoided crossings. Otherwise, quasi-normal mode sequences will appear as intersecting.

Fig. 11
figure11

Frequencies of several axisymmetric modes along a sequence of rapidly rotating relativistic polytropes of \(N=1.0\), in the Cowling approximation. On the horizontal axis, the angular velocity of each model is scaled to the angular velocity of the model at the mass-shedding limit. Lower order modes are weakly affected by rapid rotation, while higher order modes show apparent mode intersections. (Image reproduced with permission from Font et al. 2001, copyright by RAS)

Several axisymmetric modes have recently been computed for rapidly rotating relativistic stars in the Cowling approximation, using time evolutions of the nonlinear hydrodynamical equations (Font et al. 2001) (see Font et al. 2000 for a description of the 2D numerical evolution scheme). As in Yoshida and Eriguchi (2001), Font et al. (2001) find that apparent mode intersections are common for various higher order axisymmetric modes (see Fig. 11). Axisymmetric inertial modes also appear in the numerical evolutions.

The first fully relativistic frequencies of quasi-radial modes for rapidly rotating stars (without assuming the Cowling approximation) have been obtained recently, again through nonlinear time evolutions (Font et al. 2002) (see Sect. 5.2).

Going further:   The stabilization, by an external gravitational field, of a relativistic star that is marginally stable to axisymmetric perturbations is discussed in Thorne (1998).

Nonaxisymmetric perturbations

Nonrotating limit

Thorne, Campolattaro, and Price, in a series of papers (Thorne and Campolattaro 1967; Price and Thorne 1969; Thorne 1969), initiated the computation of nonradial modes by formulating the problem in the Regge–Wheeler (RW) gauge (Regge and Wheeler 1957) and numerically computing nonradial modes for a number of neutron star models. A variational method for obtaining eigenfrequencies and eigenfunctions has been constructed by Detweiler and Ipser (1973). Lindblom and Detweiler (1983) explicitly reduced the system of equations to four first order ordinary differential equations and obtained more accurate eigenfrequencies and damping times for a larger set of neutron star models. They later realized that their system of equations is sometimes singular inside the star and obtained an improved set of equations, which is free of this singularity (Detweiler and Lindblom 1985).

Chandrasekhar and Ferrari (1991) expressed the nonradial pulsation problem in terms of a fifth order system in a diagonal gauge, which is formally independent of fluid variables. Thus, they reformulate the problem in a way analogous to the scattering of gravitational waves off a black hole. Ipser and Price (1991) show that in the RW gauge, nonradial pulsations can be described by a system of two second order differential equations, which can also be independent of fluid variables. In addition, they find that the diagonal gauge of Chandrasekhar and Ferrari has a remaining gauge freedom which, when removed, also leads to a fourth order system of equations (Price and Ipser 1991).

In order to locate purely outgoing wave modes, one has to be able to distinguish the outgoing wave part from the ingoing wave part at infinity. This is typically achieved using analytic approximations of the solution at infinity.

W-modes pose a more challenging numerical problem because they are strongly damped and the techniques used for f- and p-modes fail to distinguish the outgoing wave part. The first accurate numerical solutions were obtained by Kokkotas and Schutz (1992), followed by Leins et al. (1993). Andersson et al. (1995) successfully combine a redefinition of variables with a complex-coordinate integration method, obtaining highly accurate complex frequencies for w-modes. In this method, the ingoing and outgoing solutions are separated by numerically calculating their analytic continuations to a place in the complex-coordinate plane, where they have comparable amplitudes. Since this approach is purely numerical, it could prove to be suitable for the computation of quasi-normal modes in rotating stars, where analytic solutions at infinity are not available.

The non-availability of asymptotic solutions at infinity in the case of rotating stars is one of the major difficulties for computing outgoing modes in rapidly rotating relativistic stars. A method that may help to overcome this problem, at least to an acceptable approximation, has been found by Lindblom et al. (1997). The authors obtain approximate near-zone boundary conditions for the outgoing modes that replace the outgoing wave condition at infinity and that enable one to compute the eigenfrequencies with very satisfactory accuracy. First, the pulsation equations of polar modes in the Regge–Wheeler gauge are reformulated as a set of two second order radial equations for two potentials—one corresponding to fluid perturbations and the other to the perturbations of the spacetime. The equation for the spacetime perturbation reduces to a scalar wave equation at infinity and to Laplace’s equation for zero-frequency solutions. From these, an approximate boundary condition for outgoing modes is constructed and imposed in the near zone of the star (in fact, on its surface) instead of at infinity. For polytropic models, the near-zone boundary condition yields f-mode eigenfrequencies with real parts accurate to 0.01–0.1% and imaginary parts with accuracy at the 10–20% level, for the most relativistic stars. If the near zone boundary condition can be applied to the oscillations of rapidly rotating stars, the resulting frequencies and damping times should have comparable accuracy.

Slow rotation approximation

The slow rotation approximation is useful for obtaining a first estimate of the effect of rotation on the pulsations of relativistic stars. To lowest order in rotation, a polar l-mode of an initially nonrotating star couples to an axial \(l \pm 1\) mode in the presence of rotation. Conversely, an axial l-mode couples to a polar \(l \pm 1\) mode as was first discussed by Chandrasekhar and Ferrari (1991).

The equations of nonaxisymmetric perturbations in the slow rotation limit are derived in a diagonal gauge by Chandrasekhar and Ferrari (1991), and in the Regge–Wheeler gauge by Kojima (1992, 1993b), where the complex frequencies \(\sigma = \sigma _R + i \sigma _I\) for the \(l=m\) modes of various polytropes are computed. For counterrotating modes, both \(\sigma _R\) and \(\sigma _I\) decrease, tending to zero, as the rotation rate increases (when \(\sigma \) passes through zero, the star becomes unstable to the CFS instability). Extrapolating \(\sigma _R\) and \(\sigma _I\) to higher rotation rates, Kojima finds a large discrepancy between the points where \(\sigma _R\) and \(\sigma _I\) go through zero. This shows that the slow rotation formalism cannot accurately determine the onset of the CFS instability of polar modes in rapidly rotating neutron stars.

In Kojima (1993a), it is shown that, for slowly rotating stars, the coupling between polar and axial modes affects the frequency of f- and p-modes only to second order in rotation, so that, in the slow rotation approximation, to \(\mathcal{O}( \varOmega )\), the coupling can be neglected when computing frequencies. This result was already known from the original work of Hartle et al. (1972), where it was noted that a reversal of the direction of rotation cannot change the shape of the mode or its frequency.

The linear perturbation equations in the slow rotation approximation have been derived in a new gauge by Ruoff et al. (2002). Using the Arnowitt–Deser–Misner (ADM) formalism (Arnowitt et al. 2008), a first order hyperbolic evolution system is obtained, which is suitable for numerical integration without further manipulations (as was required in the Regge–Wheeler gauge). In this gauge (which is related to a gauge introduced for nonrotating stars in Battiston et al. 1971), the symmetry between the polar and axial equations becomes directly apparent.

The case of relativistic inertial modes is different, as these modes have both axial and polar parts at order \(\mathcal{O}(\varOmega )\), and the presence of continuous bands in the spectrum (at this order in the rotation rate) has led to a serie