Rotating Stars in Relativity
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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 the equilibrium properties and on the nonaxisymmetric instabilities in fmodes and rmodes have been updated and several new sections have been added on analytic solutions for the exterior spacetime, rotating stars in LMXBs, rotating strange stars, and on rotating stars in numerical relativity.
Keywords
Neutron Star Gravitational Wave Strange Star Strange Quark Matter Relativistic Star1 Introduction
Rotating relativistic stars are of fundamental interest in physics. Their bulk properties constrain the proposed equations of state for densities greater than nuclear density. Accreted matter in their gravitational fields undergoes highfrequency 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 produce gravitational waves, the detection of which would initiate a new field of observational asteroseismology of relativistic stars.
There exist several independent numerical codes for obtaining accurate models of rotating neutron stars in full general relativity, including one that is freely available. One recent code achieves near machine accuracy even for uniform density models near the massshedding limit. The uncertainty in the highdensity equation of state still allows numerically constructed maximum mass models to differ by as much as a factor of two in mass, radius and angular velocity, and a factor of eight 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 0.28 ms.
In rotating stars, nonaxisymmetric perturbations have been studied in the Newtonian and postNewtonian approximations, in the slow rotation limit and in the Cowling approximation, but fully relativistic quasinormal modes (except for neutral modes) have yet to be obtained. A new method for obtaining such frequencies is the time evolution of the full set of nonlinear equations. Frequencies of quasiradial modes have already been obtained this way. Time evolutions of the linearized equations have also 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 (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 (rmodes) have received considerable attention since it was discovered that they are generically unstable to the emission of gravitational waves. The rmode instability could slow down newlyborn relativistic stars and limit their spin during accretioninduced spinup, which would explain the absence of millisecond pulsars with rotational periods less than ∼ 1:5 ms. Gravitational waves from the rmode instability could become detectable if the amplitude of rmodes is of order unity. Recent 3D simulations show that this is possible on dynamical timescales, but nonlinear effects seem to set a much smaller saturation amplitude on longer timescales. Still, if the signal persists for a long time (as has been found to be the case for strange stars) even a small amplitude could become detectable.
Recent advances in numerical relativity have enabled the longterm 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 current studies are limited to relativistic polytropes, but new 3D simulations with realistic equations of state should be expected in the near future.
The goal 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 their dynamical evolution. It focuses on the most recently available preprints, 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.
2 The Equilibrium Structure of Rotating Relativistic Stars
2.1 Assumptions
A relativistic star can have a complicated structure (such as a solid crust, magnetic field, possible superfluid interior, possible quark core, etc.). Still, its bulk properties can be computed with reasonable accuracy by making several simplifying assumptions.
The above arguments show that the bulk properties of an isolated rotating relativistic star can be modeled accurately by a uniformly rotating, zerotemperature 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.
2.2 Geometry of spacetime
One arrives at the above form of the metric assuming that i) the spacetime has a timelike Killing vector field t^{ a } and a second Killing vector field φ^{ a } corresponding to axial symmetry, ii) the spacetime is asymptotically flat, i.e. t_{ a }t^{ a }=1, φ_{ a }φ^{ a }=+∞ and t_{ a }φ^{ a }=0 at spatial infinity. According to a theorem by Carter [57], the two Killing vectors commute and one can choose coordinates x^{0}=t and x^{3}=φ (where x^{ a }, a=0, …3 are the coordinates of the spacetime), such that t^{ a } and φ^{ a } are coordinate vector fields. If, furthermore, the source of the gravitational field satisfies the circularity condition (absence of meridional convective currents), then another theorem [58] shows that the 2surfaces orthogonal to t^{ a } and φ^{ a } can be described by the remaining two coordinates x^{1} and x^{2}. A common choice for x^{1} and x^{2} are quasiisotropic coordinates, for which \({g_{r\theta }} = 0\) and \({g_{\theta \theta }} = {r^2}{g_{rr}}\) (in spherical polar coordinates), or and \({g_{\varpi z}} = 0\) and \({g_{rr}} = {r^2}{g_{\varpi \varpi }}\) (in cylindrical coordinates). In the slow rotation formalism by Hartle [143], a different form of the metric is used, requiring \({g_{\theta \theta }} = {g_{\phi \phi }}/{\sin ^2}\theta\).
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 ZeroAngularMomentumObservers (ZAMO) [23, 24]. These are observers whose worldlines are normal to the t=const: hypersurfaces, and they are also called Eulerian observers. Then, the metric function ω is the angular velocity 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 ψ has a geometrical meaning: \({e^\psi }\) is the proper circumferential radius of a circle around the axis of symmetry. In the nonrotating limit, the metric (5) reduces to the metric of a nonrotating relativistic star in isotropic coordinates (see [321] for the definition of these coordinates).
In rapidly rotating models, an ergosphere can appear, where g_{ tt }>0. In this region, the rotational framedragging is strong enough to prohibit counterrotating timelike 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 [108], but the associated growth time is comparable to the age of the universe [68].
2.3 The rotating fluid
2.4 Equations of structure
Thus, three of the four gravitational field equations are elliptic, while the fourth equation is a first order partial differential equation, relating only metric functions. The remaining nonzero components of the gravitational field equations yield two more elliptic equations and one first order partial differential equation, which are consistent with the above set of four equations.
2.5 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 massenergy of a configuration for a given baryon number and total angular momentum [49, 147]. In this case, the term involving F(Ω) in (20) vanishes.
The above rotation law is a simple choice that has proven to be computationally convenient. More physically plausible choices must be obtained through numerical simulations of the formation of relativistic stars.
Equilibrium properties.
circumferential radius  \(R = {e^\psi }\) 
gravitational mass  \(M = \int {\left( {{T_{ab}}  {\tfrac{1}{2}}{g_{ab}}T} \right){t^a}{{\hat n}^b}dV}\) 
baryon mass  \({M_0} = \int {\rho {u_a}{{\hat n}^a}dV}\) 
internal energy  \(U = \int {u{u_a}{{\hat n}^a}dV}\) 
proper mass  \({M_{\rm{p}}} = {M_0} + U\) 
gravitational binding energy  \(W = M  {M_{\rm{p}}}  T\) 
angular momentum  \(J = \int {{T_{ab}}{\phi ^a}{{\hat n}^b}dV}\) 
moment of inertia  I=J/Ω 
kinetic energy  T=1/2JΩ 
2.6 Equations of state
2.6.1 Relativistic polytropes
2.6.2 Hadronic equations of state
The true equation of state that describes the interior of compact stars is, still, largely unknown. This comes as a consequence of our inability to verify experimentally the different theories that describe the strong interactions between baryons and the manybody theories of dense matter, at densities larger than about twice the nuclear density (i.e. at densities larger than about 5×10^{14} g cm^{3}).
Many different socalled realistic EOSs have been proposed to date that all produce neutron star models that satisfy the currently available observational constraints. The two most accurate constraints are that the EOS must admit nonrotating neutron stars with gravitational mass of at least 1.44M_{⊙} and allow rotational periods at least as small as 1.56 ms (see [243, 187]). Recently, the first direct determination of the gravitational redshift of spectral lines produced in the neutron star photosphere has been obtained [74]. This determination (in the case of the lowmass Xray binary EXO 0748676) yielded a redshift of z=0:35 at the surface of the neutron star, corresponding to a mass to radius ratio of M/R=0.23 (in gravitational units), which is compatible with most normal nuclear matter EOSs and incompatible with some exotic matter EOSs.
The theoretically proposed EOSs are qualitatively and quantitatively very different from each other. Some are based on relativistic manybody theories while others use nonrelativistic theories with baryonbaryon interaction potentials. A classic collection of early proposed EOSs was compiled by Arnett and Bowers [20], while recent EOSs are used in Salgado et al. [261] and in [84]. A review of many modern EOSs can be found in a recent article by Haensel [138]. Detailed descriptions and tables of several modern EOSs, especially EOSs with phase transitions, can be found in Glendenning’s book [125].
High density equations of state with pion condensation have been proposed by Migdal [228] and Sawyer and Scalapino [264]. The possibility of kaon condensation is discussed by Brown and Bethe [51] (but see also Pandharipande et al. [241]). Properties of rotating Skyrmion stars have been computed in [237].
The realistic EOSs are supplied in the form of an energy density vs. pressure table and intermediate values are interpolated. This results in some loss of accuracy because the usual interpolation methods do not preserve thermodynamical consistency. Swesty [301] 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 in Nozawa et al. [236].
Usually, the interior of compact stars is modeled as a onecomponent ideal fluid. When neutron stars cool below the superfluid transition temperature, the part of the star that becomes superfluid can be described by a twofluid model and new effects arise. Andersson and Comer [9] have recently used such a description in a detailed study of slowly rotating superfluid neutron stars in general relativity, while the first rapidly rotating models are presented in [248].
2.6.3 Strange quark equations of state
Strange quark stars are likely to exist, if the ground state of matter at large atomic number is in the form of a quark fluid, which would then be composed of about equal numbers of up, down, and strange quarks together with electrons, which give overall charge neutrality [38, 98]. The strangeness per unit baryon number is ≃1. The first relativistic models of stars composed of quark matter were computed by Ipser, Kislinger, and Morley [157] and by Brecher and Caporaso [50], while the first extensive study of strange quark star properties is due to Witten [325].
2.7 Numerical schemes
All available methods for solving the system of equations describing the equilibrium of rotating relativistic stars are numerical, as no analytical selfconsistent solution for both the interior and exterior spacetime has been found. The first numerical solutions were obtained by Wilson [323] and by Bonazzola and Schneider [48]. Here, we will review the following methods: Hartle’s slow rotation formalism, the NewtonRaphson linearization scheme due to Butterworth and Ipser [55], a scheme using Green’s functions by Komatsu et al. [184, 185], a minimal surface scheme due to Neugebauer and Herold [235], and spectralmethod schemes by Bonazzola et al. [47, 46] and by Ansorg et al. [19]. Below we give a description of each method and its various implementations (codes).
2.7.1 Hartle
To order \({\mathcal O}({\Omega ^2})\), the structure of a star changes only by quadrupole terms and the equilibrium equations become a set of ordinary differential equations. Hartle’s [143, 148] method computes rotating stars in this slow rotation approximation, and a review of slowly rotating models has been compiled by Datta [82]. Weber et al. [317, 319] also implement Hartle’s formalism to explore the rotational properties of four new EOSs.
Weber and Glendenning [318] improve on Hartle’s formalism in order to obtain a more accurate estimate of the angular velocity at the massshedding limit, but their models still show large discrepancies compared to corresponding models computed without the slow rotation approximation [261]. 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 massshedding limit.
2.7.2 Butterworth and Ipser (BI)
The BI scheme [55] solves the four field equations following a NewtonRaphsonlike 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 highaccuracy 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 [55, 54]. Friedman et al. [113, 114] (FIP) extend the BI code to obtain a large number of rapidly rotating models based on a variety of realistic EOSs. Lattimer et al. [196] 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.
2.7.3 Komatsu, Eriguchi, and Hachisu (KEH)
In the KEH scheme [184, 185], the same set of field equations as in BI is used, but the three elliptictype 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 [184, 185, 95] 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 semiinfinite region [0, ∞) 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 [69] and polytropic and realistic models are computed in [71] and [70].
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 [295] and in [293]. It is available as a public domain code, named rns, and can be downloaded from [292].
2.7.4 Bonazzola et al. (BGSM)
In the BGSM scheme [47], 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 [261, 262] 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 [132, 43] (see Section 2.7.7).
While the field equations used in the BI and KEH schemes assume a perfect fluid, isotropic stressenergy 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.
2.7.5 Lorene/rotstar
Bonazzola et al. [46] 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 quasistationary models of binary neutron stars. The new method has been used in [133] for computing models of rapidly rotating strange stars and it has also been used in 3D computations of the onset of the viscositydriven instability to barmode formation [129].
2.7.6 Ansorg et al. (AKM)
A new multidomain spectral method has been introduced in [19, 18]. The method can use several domains inside the star, one for each possible phase transition. Surfaceadapted coordinates are used and approximated by a twodimensional Chebyshev expansion. Requiring transition conditions to be satisfied at the boundary of each domain, the field and fluid equations are solved as a free boundary value problem by a NewtonRaphson method, starting from an initial guess. The field equations 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, except for configurations at the massshedding limit (see Section 2.7.8)! The code has been used in a systematic study of uniformly rotating homogeneous stars in general relativity [265].
2.7.7 The virial identities
The two virial identities are an important tool for checking the accuracy of numerical models and have been repeatedly used by several authors [47, 261, 262, 236, 19].
2.7.8 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 by Stergioulas and Friedman in [295]. 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.
In [295], 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. [95], resulted from the fact that Eriguchi et al. and FIP used different versions of a tabulated EOS.
Nozawa et al. [236] 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 threepoint finitedifference 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.
Detailed comparison of the accuracy of different numerical codes in computing a rapidly rotating, uniform density model. The absolute value of the relative error in each quantity, compared to the AKM code, is shown for the numerical codes Lorene/rotstar, SF (at two resolutions), BGSM, and KEH (see text). The resolutions for the SF code are (angular × radial) grid points. See [236] for the definition of the various equilibrium quantities.
AKM  Lorene/  SF  SF  BGSM  KEH  

rotstar  (260×400)  (70×200)  
p̄ _{c}  1.0  
r_{P}/r_{e}  0.7  1×10^{3}  
Ω̄  1.41170848318  9×10^{6}  3×10^{4}  3×10^{3}  1×10^{2}  1×10^{2} 
M̄  0.135798178809  2×10^{4}  2×10^{5}  2×10^{3}  9×10^{3}  2×10^{2} 
M̄ _{0}  0.186338658186  2×10^{4}  2×10^{4}  3×10^{3}  1×10^{2}  2×10^{3} 
R̄ _{circ}  0.345476187602  5×10^{5}  3×10^{5}  5×10^{4}  3×10^{3}  1×10^{3} 
J̄  0.0140585992949  2×10^{5}  4×10^{4}  5×10^{4}  2×10^{2}  2×10^{2} 
Z _{p}  1.70735395213  1×10^{5}  4×10^{5}  1×10^{4}  2×10^{2}  6×10^{2} 
Z _{eq} ^{f}  0.162534082217  2×10^{4}  2×10^{3}  2×10^{2}  4×10^{2}  2×10^{2} 
Z _{eq} ^{b}  11.3539142587  7×10^{6}  7×10^{5}  1×10^{3}  8×10^{2}  2×10^{1} 
GRV3  4×10^{13}  3×10^{6}  3×10^{5}  1×10^{3}  4×10^{3}  1×10^{1} 
2.8 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 [149]. 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. An interesting solution has been found recently by Manko et al. [220, 221]. For nonmagnetized sources of zero net charge, the solution reduces to a 3parameter solution, involving the mass, specific angular momentum, and a parameter that depends on the quadrupole moment of the source. Although this solution depends explicitly only on the quadrupole moment, it approximates the gravitational field of a rapidly rotating star with higher nonzero multipole moments. It would be interesting to determine whether this analytic quadrupole solution approximates the exterior field of a rapidly rotating star more accurately than the quadrupole, \({\mathcal O}({\Omega ^2})\), slow rotation approximation.
The above analytic solution and an earlier one that was not represented in terms of rational functions [219] have been used in studies of energy release during disk accretion onto a rapidly rotating neutron star [279, 280]. In [276], a different approximation to the exterior spacetime, in the form of a multipole expansion far from the star, has been used to derive approximate analytic expressions for the location of the innermost stable circular orbit (ISCO). Even though the analytic solutions in [276] converge slowly to an exact numerical solution at the surface of the star, the analytic expressions for the location and angular velocity at the ISCO are in good agreement with numerical results.
2.9 Properties of equilibrium models
2.9.1 Bulk properties 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 gravitational mass, equatorial radius, and rotational period of the maximum mass model constructed with one of the softest EOSs (EOS B) (1.63M_{⊙}, 9.3 km, 0.4 ms) are a factor of two smaller than the mass, radius, and period of the corresponding model constructed by one of the stiffest EOSs (EOS L) (3.27M_{⊙}, 18.3 km, 0.8 ms). The two models differ by a factor of 5 in central energy density and by a factor of 8 in the moment of inertia!
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±0.02, a dimensionless angular momentum cJ/GM^{2} of 0.64±0.06, and an eccentricity of 0.66±0.04 [112]. 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 a factor of about two larger than for hadronic EOSs.
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 rotating model is roughly 15–20% higher than the mass of the maximum mass nonrotating model, for typical realistic hadronic EOSs. The corresponding increase in radius is 30–40%. The effect of rotation in increasing the mass and radius becomes more pronounced in the case of strange quark EOSs (see Section 2.9.8).
The deformed shape of a rapidly rotating star creates a distortion, away from spherical symmetry, in its gravitational field. Far from the star, the dominant multipole moment of the rotational distortion is measured by the quadrupolemoment tensor Q_{ ab }. For uniformly rotating, axisymmetric, and equatorially symmetric configurations, one can define a scalar quadrupole moment Q, which can be extracted from the asymptotic expansion of the metric function ν at large r, as in Equation (10).

Going further Although rotating relativistic stars are nearly perfectly axisymmetric, a small degree of asymmetry (e.g. frozen into the solid crust during its formation) can become a source of gravitational waves. A recent review of this can be found in [165].
2.9.2 Massshedding limit and the empirical formula
Weber and Glendenning [317, 318] derive analytically a similar empirical formula in the slow rotation approximation. However, the formula they obtain involves the mass and radius of the maximum mass rotating configuration, which is different from what is involved in (32).
2.9.3 Upper limits on mass and rotation: Theory vs. observation
The maximum mass and minimum period of rotating relativistic stars computed with realistic hadronic EOSs from the Arnett and Bowers collection [20] are about 3.3M_{⊙} (EOS L) and 0.4 ms (EOS B), while 1.4M_{⊙} neutron stars, rotating at the Kepler limit, have rotational periods between 0.53 ms (EOS B) and 1.7 ms (EOS M) [70]. The maximum, accurately measured, neutron star mass is currently still 1.44M_{⊙} (see e.g. [314]), but there are also indications for 2.0M_{⊙} neutron stars [167]. Core collapse simulations have yielded a bimodal mass distribution of the remnant, with peaks at about 1.3M_{⊙} and 1.7M_{⊙} [310] (the second peak depends on the assumption for the highdensity EOS — if a soft EOS is assumed, then black hole formation of this mass is implied). Compact stars of much higher mass, created in a neutron star binary merger, could be temporarily supported against collapse by strong differential rotation [30].
When magnetic field effects are ignored, conservation of angular momentum can yield very rapidly rotating neutron stars at birth. Recent simulations of the rotational core collapse of evolved rotating progenitors [151, 119] have demonstrated that rotational core collapse can easily 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 accretioninduced collapse of a white dwarf [212]). The existence of a magnetic field may complicate this picture. Spruit and Phinney [288] 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 [213] 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 [288]. A new investigation by Heger et al., yielding intermediate initial rotation rates, is presented in [152]. Clearly, more detailed computations are needed to resolve this important question.

Going further A review by J.L. Friedman concerning the upper limit on the rotation of relativistic stars can be found in [110].
2.9.4 The upper limit on mass and rotation set by causality
 1.
the high density EOS matches to the known low density EOS at some matching energy density ε_{m}, and
 2.
the matter at high densities satisfies the causality constraint (the speed of sound is less than the speed of light).
A first estimate of the absolute minimum period of uniformly rotating, gravitationally bound stars was computed by Glendenning [124] by constructing nonrotating models and using the empirical formula (32) to estimate the minimum period. Koranda, Stergioulas, and Friedman [186] improve on Glendenning’s results by constructing accurate, rapidly rotating models; they show that Glendenning’s results are accurate to within the accuracy of the empirical formula.
Furthermore, they show that the EOS satisfying the minimal set of constraints and yielding the minimum period star consists of a high density region at the causal limit (CL EOS), \(P = (\varepsilon  {\varepsilon _{\rm{c}}})\), (where ε_{c} is the lowest energy density of this region), which is matched to the known low density EOS through an intermediate constant pressure region (that would correspond to a first order phase transition). Thus, the EOS yielding absolute minimum period models is as stiff as possible at the central density of the star (to sustain a large enough mass) and as soft as possible in the crust, in order to have the smallest possible radius (and rotational period).
The above results have been confirmed in [139], where it is shown that the CL EOS has χ_{s}=0.7081, independent of ε_{c}, and the empirical formula (34) reproduces the numerical result (38) to within 2%.
2.9.5 Supramassive stars and spinup prior to collapse
Since rotation increases the mass that a neutron star of given central density can support, there exist sequences of neutron stars with constant baryon mass that have no nonrotating member. Such sequences are called supramassive, as opposed to normal sequences that do have a nonrotating member. A nonrotating star can become supramassive by accreting matter and spinningup to large rotation rates; in another scenario, neutron stars could be born supramassive after a core collapse. A supramassive star evolves along a sequence of constant baryon mass, slowly losing angular momentum. Eventually, the star reaches a point where it becomes unstable to axisymmetric perturbations and collapses to a black hole.
In a neutron star binary merger, prompt collapse to a black hole can be avoided if the equation of state is sufficiently stiff and/or the equilibrium is supported by strong differential rotation. The maximum mass of differentially rotating supramassive neutron stars can be significantly larger than in the case of uniform rotation. A detailed study of this massincrease has recently appeared in [215].
Cook et al. [69, 71, 70] have discovered that a supramassive relativistic star approaching the axisymmetric instability will actually spin up before collapse, even though it loses angular momentum. This potentially observable effect is independent of the equation of state and it is more pronounced for rapidly rotating massive stars. Similarly, stars can spin up by loss of angular momentum near the massshedding limit, if the equation of state is extremely stiff or extremely soft.
If the equation of state features a phase transition, e.g. to quark matter, then the spinup region is very large, and most millisecond pulsars (if supramassive) would need to be spinning up [289]; the absence of spinup in known millisecond pulsars indicates that either large phase transitions do not occur, or that the equation of state is sufficiently stiff so that millisecond pulsars are not supramassive.
2.9.6 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 observed surface dipole magnetic field strength of pulsars ranges between 10^{8} G and 2×10^{13} 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, one cannot exclude the existence of neutron stars with higher magnetic field strengths or the possibility that neutron stars are born with much stronger magnetic fields, which then decay to the observed values (of course, there are also many arguments against magnetic field decay in neutron stars [243]). 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 desirable and, in fact, Bocquet et al. [37] achieved the first numerical computation of such configurations. Following we give a brief summary of their work.
A magnetized relativistic star in equilibrium can be described by the coupled EinsteinMaxwell field equations for stationary, axisymmetric rotating objects with internal electric currents. The stressenergy tensor includes the electromagnetic energy density and is nonisotropic (in contrast to the isotropic perfect fluid stressenergy tensor). The equilibrium of the matter is given not only by the balance between the gravitational force, centrifugal force, and the pressure gradient; the Lorentz force due to the electric currents also enters the balance. For simplicity, Bocquet et al. consider only poloidal magnetic fields that preserve the circularity of the spacetime. Also, they only consider stationary configurations, which excludes magnetic dipole moments nonaligned with the rotation axis, since in that case the star emits electromagnetic and gravitational waves. The assumption of stationarity implies that the fluid is necessarily rigidly rotating (if the matter has infinite conductivity) [47]. Under these assumptions, the electromagnetic field tensor F^{ ab } is derived from a potential fourvector A_{ a } with only two nonvanishing components, A_{ t } and \({A_\phi }\), which are solutions of a scalar Poisson and a vector Poisson equation respectively. Thus, the two equations describing the electromagnetic field are of similar type as the four field equations that describe the gravitational field.
For magnetic field strengths larger than about 10^{14} G, one observes significant effects, such as a flattening of the equilibrium configuration. There exists a maximum value of the magnetic field strength of the order of 10^{18} G, for which the magnetic field pressure at the center of the star equals the fluid pressure. Above this value no stationary configuration can exist.
A strong magnetic field allows a maximum mass configuration with larger M_{max} than for the same EOS with no magnetic field and this is analogous to the increase of M_{max} induced by rotation. For nonrotating stars, the increase in M_{max} due to a strong magnetic field is 13–29%, depending on the EOS. Correspondingly, the maximum allowed angular velocity, for a given EOS, also increases in the presence of a strong magnetic field.

Going further An \(\mathcal{O}(\Omega )\) slow rotation approach for the construction of rotating magnetized relativistic stars is presented in [137].
2.9.7 Rapidly rotating protoneutron stars
Following the gravitational collapse of a massive stellar core, a protoneutron star (PNS) is born. Initially it has a large radius of about 100 km and a temperature of 50–100 MeV. The PNS may be born with a large rotational kinetic energy and initially it will be differentially rotating. Due to the violent nature of the gravitational collapse, the PNS pulsates heavily, emitting significant amounts of gravitational radiation. After a few hundred pulsational periods, bulk viscosity will damp the pulsations significantly. Rapid cooling due to deleptonization transforms the PNS, shortly after its formation, into a hot neutron star of T∼10 MeV. In addition, viscosity or other mechanisms (see Section 2.1) enforce uniform rotation and the neutron star becomes quasistationary. Since the details of the PNS evolution determine the properties of the resulting cold NSs, protoneutron stars need to be modeled realistically in order to understand the structure of cold neutron stars.
Hashimoto et al. [150] and Goussard et al. [134] construct fully relativistic models of rapidly rotating, hot protoneutron stars. The authors use finitetemperature EOSs [239, 195] to model the interior of PNSs. Important (but largely unknown) parameters that determine the local state of matter are the lepton fraction Y_{l} and the temperature profile. Hashimoto et al. consider only the limiting case of zero lepton fraction, Y_{l}=0, and classical isothermality, while Goussard et al. consider several nonzero values for Y_{l} and two different limiting temperature profiles — a constant entropy profile and a relativistic isothermal profile. In both [150] and [239], differential rotation is neglected to a first approximation.
The construction of numerical models with the above assumptions shows that, due to the high temperature and the presence of trapped neutrinos, PNSs have a significantly larger radius than cold NSs. These two effects give the PNS an extended envelope which, however, contains only roughly 0.1% of the total mass of the star. This outer layer cools more rapidly than the interior and becomes transparent to neutrinos, while the core of the star remains hot and neutrino opaque for a longer time. The two regions are separated by the “neutrino sphere”.
Compared to the T=0 case, an isothermal EOS with temperature of 25 MeV has a maximum mass model of only slightly larger mass. In contrast, an isentropic EOS with a nonzero trapped lepton number features a maximum mass model that has a considerably lower mass than the corresponding model in the T=0 case and, therefore, a stable PNS transforms to a stable neutron star. If, however, one considers the hypothetical case of a large amplitude phase transition that softens the cold EOS (such as a kaon condensate), then M_{max} of cold neutron stars is lower than the M_{max} of PNSs, and a stable PNS with maximum mass will collapse to a black hole after the initial cooling period. This scenario of delayed collapse of nascent neutron stars has been proposed by Brown and Bethe [51] and investigated by Baumgarte et al. [31].
An analysis of radial stability of PNSs [127] shows that, for hot PNSs, the maximum angular velocity model almost coincides with the maximum mass model, as is also the case for cold EOSs.
Because of their increased radius, PNSs have a different massshedding limit than cold NSs. For an isothermal profile, the massshedding limit proves to be sensitive to the exact location of the neutrino sphere. For the EOSs considered in [150] and [134], PNSs have a maximum angular velocity that is considerably less than the maximum angular velocity allowed by the cold EOSs. Stars that have nonrotating counterparts (i.e. that belong to a normal sequence) contract and speed up while they cool down. The final star with maximum rotation is thus closer to the massshedding limit of cold stars than was the hot PNS with maximum rotation. Surprisingly, stars belonging to a supramassive sequence exhibit the opposite behavior. If one assumes that a PNS evolves without losing angular momentum or accreting mass, then a cold neutron star produced by the cooling of a hot PNS has a smaller angular velocity than its progenitor. This purely relativistic effect was pointed out in [150] and confirmed in [134].

Going further The thermal history and evolutionary tracks of rotating PNSs (in the second order slow rotation approximation) have been studied recently in [300].
2.9.8 Rotating strange quark stars
As strange quark stars are very compact, the angular velocity at the ISCO can become very large. If the 1066 Hz upper QPO frequency in 4U 1820–30 (see [167] and references therein) is the frequency at the ISCO, then it rules out most models of slowly rotating strange stars in LMXBs. However, in [297] it is shown that rapidly rotating bare strange stars are still compatible with this observation, as they can have ISCO frequencies < 1 kHz even for 1.4 M_{⊙} models. On the other hand, if strange stars have a thin solid crust, the ISCO frequency at the massshedding limit increases by about 10% (compared to a bare strange star of the same mass), and the above observational requirement is only satisfied for slowly rotating models near the maximum nonrotating mass, assuming some specific values of the parameters in the strange star EOS [342, 340]. Moderately rotating strange stars, with spin frequencies around 300 Hz can also be accommodated for some values of the coupling constant α_{c} [338] (see also [131] for a detailed study of the ISCO frequency for rotating strange stars). The 1066 Hz requirement for the ISCO frequency depends, of course, on the adopted model of kHz QPOs in LMXBs, and other models exist (see next section).
2.10 Rotating relativistic stars in LMXBs
2.10.1 Particle orbits and kHz quasiperiodic oscillations
In the last few years, Xray 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 [192, 168, 249]). The most important feature of these sources is the observation of (in most cases) twin kHz quasiperiodic oscillations (QPOs). The high frequency of these variabilities and their quasiperiodic nature are evidence that they are produced in highvelocity 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 [72, 341].
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_{ISCO} for rapidly rotating stars have been used in [229] 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. For eccentric orbits, the periastron advances with a frequency ν_{pa} that is the difference between the Keplerian frequency of azimuthal motion ν_{K} and the radial epicyclic frequency ν_{r}. On the other hand, particles in slightly tilted orbits fail to return to the initial displacement ψ from the equatorial plane, after a full revolution around the star. This introduces a nodal precession frequency ν_{pa}, which is the difference between ν_{K} and the frequency of the motion out of the orbital plane (meridional frequency) ν_{ ψ }. Explicit expressions for the above frequencies, in the gravitational field of a rapidly rotating neutron star, have been derived recently by Marković [222], while in [223] highly eccentric orbits are considered. Morsink and Stella [231] compute the nodal precession frequency for a wide range of neutron star masses and equations of state and (in a postNewtonian analysis) separate the precession caused by the LenseThirring (framedragging) 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 [278]. These precession frequencies are relativistic effects and have been used in several models to explain the kHz QPO frequencies [291, 250, 2, 169, 5].
It is worth mentioning that it has recently been found that an ISCO also exists in Newtonian gravity, for models of rapidly rotating lowmass strange stars. The instability in the circular orbits is produced by the large oblateness of the star [170, 339, 5].
2.10.2 Angular momentum conservation during burst oscillations
Some sources in LMXBs show signatures of type I Xray bursts, which are thermonuclear flashes on the surface of the compact star [198]. Such bursts show nearlycoherent oscillations in the range 270–620 Hz (see [168, 299] for recent 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 antipodal hot spots in some sources that ignite at the same time. Alternative models also exist (see e.g. [249]).

Going further A very interesting topic is the modeling of the expected Xray spectrum of an accretion disk in the gravitational field of a rapidly rotating neutron star as it could lead to observational constraints on the source of the gravitational field. See e.g. [303, 279, 280, 34, 33], where work initiated by Kluzniak and Wilson [172] in the slow rotation limit is extended to rapidly rotating relativistic stars.
3 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 highdensity equation of state [13]. 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 oscillations of an equilibrium star are of small magnitude 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 [216, 118], but the Eulerian approach proved to be more suitable for numerical computations of mode frequencies and eigenfunctions [162, 218, 158, 160, 159]. Clement [64] 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. [156] and in the Eulerian approach by Managan [218] and Ipser and Lindblom [158]. While a lot has been learned from Newtonian studies, in the following we will focus on the relativistic treatment of oscillations of rotating stars.
3.1 Quasinormal modes of oscillation
The Eulerian perturbation in the fluid 4velocity δu^{ a } can be expressed in terms of vector harmonics, while the metric perturbation δg_{ 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 re ection in the equatorial plane and rotation by π. 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.
For a given pair (l, m), a solution exists for any value of the frequency ω_{i}, consisting of a mixture of ingoing and outgoing wave parts. Outgoing quasinormal 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.
 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}∼2 kHz, τ_{0}<1 s),

p(ressure)modes: nearly radial (f_{0}>4 kHz, τ_{0}>1 s),

g(ravity)modes: nearly tangential, only exist in stars that are nonisentropic or that have a composition gradient or first order phase transition (f_{0}<500 Hz, τ_{0}>5 s).

 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 rmodes in the Newtonian limit. Generically unstable to the CFS instability with growth times as short as a few seconds at high rotation rates.

 3.Polar and axial spacetime modes:

w(ave)modes: Analogous to the quasinormal modes of a black hole (very weak coupling to the fluid). High frequency, strongly damped modes (f_{0}>6 kHz, τ_{0}∼0:1 ms).

For a more detailed description of various types of oscillation modes, see [179, 178, 225, 56, 177].
3.2 Effect of rotation on quasinormal modes
 1.
The degeneracy in the index m is removed and a nonrotating mode of index l is split into 2l + 1 different (l, m) modes.
 2.
Prograde (m<0) modes are now different from retrograde (m>0) modes.
 3.A rotating “polar” lmode consists of a sum of purely polar and purely axial terms [293], e.g. for l=m,that is, rotation couples a polar lterm to an axial l±1 term (the coupling to the l+1 term is, however, strongly favoured over the coupling to the l−1 term [61]). Similarly, for a rotating “axial” mode with l=m,$$ P_l^{{\rm{rot}}} \sim \sum\limits_{l' = 0}^\infty {({P_{l + 2l'}} + {A_{l + 2l' \pm 1}}} ), $$(51)$$ A_l^{{\rm{rot}}}\sim \sum\limits_{l' = 0}^\infty {({A_{l + 2l'}} + {P_{l + 2l' \pm 1}}} ). $$(52)
 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.
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.

Going further The equations that describe oscillations of the solid crust of a rapidly rotating relativistic star are derived by Priou in [247]. The effects of superfluid hydrodynamics on the oscillations of neutron stars have been investigated by several authors, see e.g. [203, 67, 8, 10] and references therein.
3.3 Axisymmetric perturbations
3.3.1 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 maximummass 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 [286], Friedman, Ipser, and Sorkin [115] show that in the case of rotating stars a secular axisymmetric instability sets in when the mass becomes maximum along a sequence of constant angular momentum. An equivalent criterion (implied in [115]) is provided by Cook et al. [69]: 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 spinup time following a pulsar glitch. Eventually, the star encounters the onset of dynamical instability and collapses to a black hole (see [274] for recent numerical simulations). Thus, the onset of the secular instability to axisymmetric perturbations separates stable neutron stars from neutron stars that will collapse to a black hole.
Goussard et al. [134] extend the stability criterion to hot protoneutron 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. [127] compute frequencies and eigenfunctions of radial pulsations of hot protoneutron stars and verify that the secular instability sets in at the maximum mass turning point, as is the case for cold neutron stars.
3.3.2 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 spindown 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 quasiradial modes (with lowest order l=0 contribution) would, in principle, radiate gravitational waves.

Going further The stabilization, by an external gravitational field, of a relativistic star that is marginally stable to axisymmetric perturbations is discussed in [308].
3.4 Nonaxisymmetric perturbations
3.4.1 Nonrotating limit
Thorne, Campolattaro, and Price, in a series of papers [309, 245, 304], initiated the computation of nonradial modes by formulating the problem in the ReggeWheeler (RW) gauge [251] 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 [85]. Lindblom and Detweiler [202] 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 [86].
Chandrasekhar and Ferrari [61] 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 [163] 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 [246].
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.
Wmodes pose a more challenging numerical problem because they are strongly damped and the techniques used for f and pmodes fail to distinguish the outgoing wave part. The first accurate numerical solutions were obtained by Kokkotas and Schutz [181], followed by Leins, Nollert, and Soffel [197]. Andersson, Kokkotas, and Schutz [15] successfully combine a redefinition of variables with a complexcoordinate 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 complexcoordinate plane, where they have comparable amplitudes. Since this approach is purely numerical, it could prove to be suitable for the computation of quasinormal modes in rotating stars, where analytic solutions at infinity are not available.
The nonavailability 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, Mendell, and Ipser [206].
The authors obtain approximate nearzone 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 ReggeWheeler 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 zerofrequency 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 nearzone boundary condition yields fmode 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.
3.4.2 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 lmode of an initially nonrotating star couples to an axial l±1 mode in the presence of rotation. Conversely, an axial lmode couples to a polar l±1 mode as was first discussed by Chandrasekhar and Ferrari [61].
The equations of nonaxisymmetric perturbations in the slow rotation limit are derived in a diagonal gauge by Chandrasekhar and Ferrari [61], and in the ReggeWheeler gauge by Kojima [173, 175], where the complex frequencies \(\sigma = {\sigma _R} + i{\sigma _I}\) for the l=m modes of various polytropes are computed. For counterrotating modes, both σ_{ R } and σ_{ I } decrease, tending to zero, as the rotation rate increases (when σ passes through zero, the star becomes unstable to the CFS instability). Extrapolating σ_{ R } and σ_{ I } to higher rotation rates, Kojima finds a large discrepancy between the points where σ_{ R } and σ_{ 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 [174], it is shown that, for slowly rotating stars, the coupling between polar and axial modes affects the frequency of f and pmodes only to second order in rotation, so that, in the slow rotation approximation, to \({\mathcal O}(\Omega )\), the coupling can be neglected when computing frequencies.
The linear perturbation equations in the slow rotation approximation have recently been derived in a new gauge by Ruoff, Stavridis, and Kokkotas [257]. Using the ADM formalism, a first order hyperbolic evolution system is obtained, which is suitable for numerical integration without further manipulations (as was required in the ReggeWheeler gauge). In this gauge (which is related to a gauge introduced for nonrotating stars in [27]), 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 \({\mathcal O}(\Omega )\), and the presence of continuous bands in the spectrum (at this order in the rotation rate) has led to a series of detailed investigations of the properties of these modes (see [180] for a review). In a recent paper, Ruoff, Stavridis, and Kokkotas [258] finally show that the inclusion of both polar and axial parts in the computation of relativistic rmodes, at order \({\mathcal O}(\Omega )\), allows for discrete modes to be computed, in agreement with postNewtonian [214] and nonlinear, rapidrotation [294] calculations.
3.4.3 PostNewtonian approximation
A step toward the solution of the perturbation equations in full general relativity has been taken by Cutler and Lindblom [77, 79, 199], who obtain frequencies for the l=m fmodes in rotating stars in the first postNewtonian (1PN) approximation. The perturbation equations are derived in the postNewtonian formalism (see [36]), i.e. the equations are separated into equations of consistent order in 1/c.
Cutler and Lindblom show that in this scheme, the perturbation of the 1PN correction of the fourvelocity of the fluid can be obtained analytically in terms of other variables; this is similar to the perturbation in the threevelocity in the Newtonian IpserManagan scheme. The perturbation in the 1PN corrections are obtained by solving an eigenvalue problem, which consists of three second order equations, with the 1PN correction to the eigenfrequency of a mode Δω as the eigenvalue.
Cutler and Lindblom obtain a formula that yields Δω if one knows the 1PN stationary solution and the solution to the Newtonian perturbation equations. Thus, the frequency of a mode in the 1PN approximation can be obtained without actually solving the 1PN perturbation equations numerically. The 1PN code was checked in the nonrotating limit and it was found to reproduce the exact general relativistic frequencies for stars with M/R=0.2, obeying an N=1 polytropic EOS, with an accuracy of 3–8%.
Along a sequence of rotating stars, the frequency of a mode is commonly described by the ratio of the frequency of the mode in the comoving frame to the frequency of the mode in the nonrotating limit. For an N=1 polytrope and for M/R=0.2, this frequency ratio is reduced by as much as 12% in the 1PN approximation compared to its Newtonian counterpart (for the fundamental l=m modes) which is representative of the effect that general relativity has on the frequency of quasinormal modes in rotating stars.
3.4.4 Cowling approximation
In several situations, the frequency of pulsations in relativistic stars can be estimated even if one completely neglects the perturbation in the gravitational field, i.e. if one sets δg_{ ab }=0 in the perturbation equations [226]. In this approximation, the pulsations are described only by the perturbation in the fluid variables, and the scheme works quite well for f, p, and rmodes [209]. A different version of the Cowling approximation, in which δg_{ tr } is kept nonzero in the perturbation equations, has been suggested to be more suitable for gmodes [99], since these modes could have large fluid velocities, even though the variation in the gravitational field is weak.
Yoshida and Kojima [331] examine the accuracy of the relativistic Cowling approximation in slowly rotating stars. The first order correction to the frequency of a mode depends only on the eigenfrequency and eigenfunctions of the mode in the absence of rotation and on the angular velocity of the star. The eigenfrequencies of f, p_{1}, and p_{2} modes for slowly rotating stars with M/R between 0.05 and 0.2 are computed (assuming polytropic EOSs with N=1 and N=1.5) and compared to their counterparts in the slow rotation approximation.
For the l=2 fmode, the relative error in the eigenfrequency because of the Cowling approximation is 30% for weakly relativistic stars (M/R=0:05) and about 15% for stars with M/R=0.2; the error decreases for higher lmodes. For the p_{1} and p_{2} modes the relative error is similar in magnitude but it is smaller for less relativistic stars. Also, for pmodes, the Cowling approximation becomes more accurate for increasing radial mode number.
3.5 Nonaxisymmetric instabilities
3.5.1 Introduction
Rotating cold neutron stars, detected as pulsars, have a remarkably stable rotation period. But, at birth or during accretion, rapidly rotating neutron stars can be subject to various nonaxisymmetric instabilities, which will affect the evolution of their rotation rate.
If a protoneutron star has a sufficiently high rotation rate (so that, e.g. T/W>0.27 in the case of Maclaurin spheroids), it will be subject to a dynamical instability driven by hydrodynamics and gravity. Through the l=2 mode, the instability will deform the star into a bar shape. This highly nonaxisymmetric configuration will emit strong gravitational waves with frequencies in the kHz regime. The development of the instability and the resulting waveform have been computed numerically in the context of Newtonian gravity by Houser et al. [155] and in full general relativity by Shibata et al. [274] (see Section 4.1.3).
At lower rotation rates, the star can become unstable to secular nonaxisymmetric instabilities, driven by gravitational radiation or viscosity. Gravitational radiation drives a nonaxisymmetric instability when a mode that is retrograde in a frame corotating with the star appears as prograde to a distant inertial observer, via the ChandrasekharFriedmanSchutz (CFS) mechanism [60, 118]: A mode that is retrograde in the corotating frame has negative angular momentum, because the perturbed star has less angular momentum than the unperturbed one. If, for a distant observer, the mode is prograde, it removes positive angular momentum from the star, and thus the angular momentum of the mode becomes increasingly negative.
The instability evolves on a secular timescale, during which the star loses angular momentum via the emitted gravitational waves. When the star rotates more slowly than a critical value, the mode becomes stable and the instability proceeds on the longer timescale of the next unstable mode, unless it is suppressed by viscosity.
Neglecting viscosity, the CFS instability is generic in rotating stars for both polar and axial modes. For polar modes, the instability occurs only above some critical angular velocity, where the frequency of the mode goes through zero in the inertial frame. The critical angular velocity is smaller for increasing mode number l. Thus, there will always be a high enough mode number l for which a slowly rotating star will be unstable. Many of the hybrid inertial modes (and in particular the relativistic rmode) are generically unstable in all rotating stars, since the mode has zero frequency in the inertial frame when the star is nonrotating [6, 117].

Going further A numerical method for the analysis of the ergosphere instability in relativistic stars, which could be extended to nonaxisymmetric instabilities of fluid modes, is presented by Yoshida and Eriguchi in [327].
3.5.2 CFS instability of polar modes
The existence of the CFS instability in rotating stars was first demonstrated by Chandrasekhar [60] in the case of the l=2 mode in uniformly rotating, uniform density Maclaurin spheroids. Friedman and Schutz [118] show that this instability also appears in compressible stars and that all rotating, selfgravitating perfect fluid configurations are generically unstable to the emission of gravitational waves. In addition, they find that a nonaxisymmetric mode becomes unstable when its frequency vanishes in the inertial frame. Thus, zerofrequency outgoing modes in rotating stars are neutral (marginally stable).
In the Newtonian limit, neutral modes have been determined for several polytropic EOSs [156, 218, 158, 326]. The instability first sets in through l=m modes. Modes with larger l become unstable at lower rotation rates, but viscosity limits the interesting ones to l≤5. For an N=1 polytrope, the critical values of T/W for the l=3, 4, and 5 modes are 0.079, 0.058, and 0.045, respectively, and these values become smaller for softer polytropes. The l=m=2 “bar” mode has a critical T/W ratio of 0.14 that is almost independent of the polytropic index. Since soft EOSs cannot produce models with high T/W values, the bar mode instability appears only for stiff Newtonian polytropes of N≤0.808 [164, 283]. In addition, the viscositydriven bar mode appears at the same critical T/W ratio as the bar mode driven by gravitational radiation [162] (we will see later that this is no longer true in general relativity).
The postNewtonian computation of neutral modes by Cutler and Lindblom [79, 199] has shown that general relativity tends to strengthen the CFS instability. Compared to their Newtonian counterparts, critical angular velocity ratios ‖_{c}/Ω_{0} (where \({\Omega _0} = {(3{M_0}/4R_0^3)^{1/2}}\), and M_{0}, R_{0} are the mass and radius of the nonrotating star in the sequence) are lowered by as much as 10% for stars obeying the N=1 polytropic EOS (for which the instability occurs only for l=m≥3 modes in the postNewtonian approximation).
3.5.3 CFS instability of axial modes
Two independent computations in the Newtonian Cowling approximation [208, 16] showed that the usual shear and bulk viscosity assumed to exist for neutron star matter is not able to damp the rmode instability, even in slowly rotating stars. In a temperature window of 10^{5}K<T<10^{10} K, the growth time of the l=m=2 mode becomes shorter than the shear or bulk viscosity damping time at a critical rotation rate that is roughly one tenth the maximum allowed angular velocity of uniformly rotating stars. The gravitational radiation is dominated by the mass current quadrupole term. These results suggested that a rapidly rotating protoneutron star will spin down to Crablike rotation rates within one year of its birth, because of the rmode instability. Due to uncertainties in the actual viscous damping times and because of other dissipative mechanisms, this scenario also is consistent with somewhat higher initial spins, such as the suggested initial spin period of several milliseconds for the Xray pulsar in the supernova remnant N157B [224]. Millisecond pulsars with periods less than a few milliseconds can then only form after the accretioninduced spinup of old pulsars and not in the accretioninduced collapse of a white dwarf.
More recent studies of nonlinear couplings between the rmode and higher order inertial modes [21] and new 3D nonlinear Newtonian simulations [136] seem to suggest a different picture. The rmode could be saturated due to mode couplings or due to a hydrodynamical instability at amplitudes much smaller than the amplitude at which shock waves appeared in the simulations by Lindblom et al. Such a low amplitude, on the other hand, modifies the properties of the rmode instability as a gravitational wave source, but is not necessarily bad news for gravitational wave detection, as a lower spindown rate also implies a higher event rate for the rmode instability in LMXBs in our own Galaxy [11, 154]. The 3D simulations need to achieve significantly higher resolutions before definite conclusions can be reached, while the Arras et al. work could be extended to rapidly rotating relativistic stars (in which case the mode frequencies and eigenfunctions could change significantly, compared to the slowly rotating Newtonian case, which could affect the nonlinear coupling coefficients). Spectral methods can be used for achieving high accuracy in mode calculations; first results have been obtained by Villain and Bonazzolla [316] for inertial modes of slowly rotating stars in the relativistic Cowling approximation.

Going further If rotating stars with very high compactness exist, then wmodes can also become unstable, as was recently found by Kokkotas, Ruoff, and Andersson [183]. The possible astrophysical implications are still under investigation.
3.5.4 Effect of viscosity on the CFS instability
In normal neutron star matter, shear viscosity is dominated by neutronneutron scattering with a temperature dependence of T^{2} [101], and computations in the Newtonian limit and postNewtonian approximation show that the CFS instability is suppressed for T<10^{6} K10^{7} K [160, 159, 326, 199]. If neutrons become a superfluid below a transition temperature T_{s}, then mutual friction, which is caused by the scattering of electrons off the cores of neutron vortices could significantly suppress the fmode instability for T<T_{s} [204], but the rmode instability remains unaffected [205]. The superfluid transition temperature depends on the theoretical model for superfluidity and lies in the range 10^{8} K6×10^{9} K [240].
In a pulsating fluid that undergoes compression and expansion, the weak interaction requires a relatively long time to reestablish equilibrium. This creates a phase lag between density and pressure perturbations, which results in a large bulk viscosity [263]. The bulk viscosity due to this effect can suppress the CFS instability only for temperatures for which matter has become transparent to neutrinos [191, 41]. It has been proposed that for T>5×10^{9} K, matter will be opaque to neutrinos and the neutrino phase space could be blocked ([191]; see also [41]). In this case, bulk viscosity will be too weak to suppress the instability, but a more detailed study is needed.
In the neutrino transparent regime, the effect of bulk viscosity on the instability depends crucially on the proton fraction x_{p}. If x_{p} is lower than a critical value (∼1/9), only modified URCA processes are allowed. In this case bulk viscosity limits, but does not completely suppress, the instability [160, 159, 326]. For most modern EOSs, however, the proton fraction is larger than ∼1/9 at sufficiently high densities [194], allowing direct URCA processes to take place. In this case, depending on the EOS and the central density of the star, the bulk viscosity could almost completely suppress the CFS instability in the neutrino transparent regime [337]. At high temperatures, T>5×10^{9} K, even if the star is opaque to neutrinos, the direct URCA cooling timescale to T∼5×10^{9} K could be shorter than the growth timescale of the CFS instability.
3.5.5 Gravitational radiation from CFS instability
Conservation of angular momentum and the inferred initial period (assuming magnetic braking) of a few milliseconds for the Xray pulsar in the supernova remnant N157B [224] suggests that a fraction of neutron stars may be born with very large rotational energies. The fmode bar CFS instability thus appears as a promising source for the planned gravitational wave detectors [191]. It could also play a role in the rotational evolution of merged binary neutron stars, if the postmerger angular momentum exceeds the maximum allowed to form a Kerr black hole [28] or if differential rotation temporarily stabilizes the merged object.
3.5.6 Viscositydriven instability
A different type of nonaxisymmetric instability in rotating stars is the instability driven by viscosity, which breaks the circulation of the fluid [256, 164]. The instability is suppressed by gravitational radiation, so it cannot act in the temperature window in which the CFS instability is active. The instability sets in when the frequency of an l=m mode goes through zero in the rotating frame. In contrast to the CFS instability, the viscositydriven instability is not generic in rotating stars. The m=2 mode becomes unstable at a high rotation rate for very stiff stars, and higher mmodes become unstable at larger rotation rates.
In Newtonian polytropes, the instability occurs only for stiff polytropes of index N<0.808 [164, 283]. For relativistic models, the situation for the instability becomes worse, since relativistic effects tend to suppress the viscositydriven instability (while the CFS instability becomes stronger). According to recent results by Bonazzola et al. [39], for the most relativistic stars, the viscositydriven bar mode can become unstable only if N<0.55. For 1.4M_{⊙} stars, the instability is present for N<0.67.
These results are based on an approximate computation of the instability in which one perturbs an axisymmetric and stationary configuration, and studies its evolution by constructing a series of triaxial quasiequilibrium configurations. During the evolution only the dominant nonaxisymmetric terms are taken into account. The method presented in [39] is an improvement (taking into account nonaxisymmetric terms of higher order) of an earlier method by the same authors [41]. Although the method is approximate, its results indicate that the viscositydriven instability is likely to be absent in most relativistic stars, unless the EOS turns out to be unexpectedly stiff.
An investigation by Shapiro and Zane [269] of the viscositydriven bar mode instability, using incompressible, uniformly rotating triaxial ellipsoids in the postNewtonian approximation, finds that the relativistic effects increase the critical T/W ratio for the onset of the instability significantly. More recently, new postNewtonian [88] and fully relativistic calculations for uniform density stars [129] show that the viscositydriven instability is not as strongly suppressed by relativistic effects as suggested in [269]. The most promising case for the onset of the viscositydriven instability (in terms of the critical rotation rate) would be rapidly rotating strange stars [130], but the instability can only appear if its growth rate is larger than the damping rate due to the emission of gravitational radiation — a corresponding detailed comparison is still missing.
4 Rotating Stars in Numerical Relativity

The metric function ω in (5) describing the dragging of inertial frames by rotation is related to the shift vector through \({\beta ^\phi } =  \omega\). This shift vector satisfies the minimal distortion shift condition.

The metric satisfies the maximal slicing condition, while the lapse function is related to the metric function ν in (5) through \(\alpha = {e^\nu }\).

The quasiisotropic coordinates are suitable for numerical evolution, while the radialgauge coordinates [25] are not suitable for nonspherical sources (see [47] for details).

The ZAMOs are the Eulerian observers, whose worldlines are normal to the t=const: hypersurfaces.

Uniformly rotating stars have Ω=const: in the coordinate frame. This can be shown by requiring a vanishing rate of shear.

Normal modes of pulsation are discrete in the coordinate frame and their frequencies can be obtained by Fourier transforms (with respect to coordinate time t) of evolved variables at a fixed coordinate location [106].
Crucial ingredients for the successful longterm evolutions of rotating stars in numerical relativity are the conformal ADM schemes for the spacetime evolution (see [234, 277, 32, 4]) and hydrodynamical schemes that have been shown to preserve well the sharp rotational profile at the surface of the star [106, 294, 105].
4.1 Numerical evolution of equilibrium models
4.1.1 Stable equilibrium
New evolutions of uniformly and differentially rotating stars in 3D, using different gauges and coordinate systems, are presented in [93], while new 2D evolutions are presented in [273].
4.1.2 Instability to collapse
Shibata, Baumgarte, and Shapiro [275] study the stability of supramassive neutron stars rotating at the massshedding limit, for a Γ=2 polytropic EOS. Their 3D simulations in full general relativity show that stars on the massshedding sequence, with central energy density somewhat larger than that of the maximum mass model, are dynamically unstable to collapse. Thus, the dynamical instability of rotating neutron stars to axisymmetric perturbations is close to the corresponding secular instability. The initial data for these simulations are approximate, conformally flat axisymmetric solutions, but their properties are not very different from exact axisymmetric solutions even near the massshedding limit [73]. It should be noted that the approximate minimal distortion (AMD) shift condition does not prove useful in the numerical evolution, once a horizon forms. Instead, modified shift conditions are used in [275]. In the above simulations, no massive disk around the black hole is formed, as the equatorial radius of the initial model is inside the radius which becomes the ISCO of the final black hole. This could change if a different EOS is chosen.
4.1.3 Dynamical barmode instability
4.2 Pulsations of rotating stars
In the relativistic Cowling approximation, 2D time evolutions have yielded frequencies for the l=0 to l=3 axisymmetric modes of rapidly rotating relativistic polytropes with N=1.0 [104]. The higher order overtones of these modes show characteristic apparent crossings near massshedding (as was observed for the quasiradial modes in [330]).
Numerical relativity has also enabled the first study of nonlinear rmodes in rapidly rotating relativistic stars (in the Cowling approximation) by Stergioulas and Font [294]. For several dozen dynamical timescales, the study shows that nonlinear rmodes with amplitudes of order unity can exist in a star rotating near massshedding. However, on longer timescales, nonlinear effects may limit the rmode amplitude to smaller values (see Section 3.5.3).
4.3 Rotating core collapse
4.3.1 Collapse to a rotating black hole
More recently, Shibata [272] carried out axisymmetric simulations of rotating stellar collapse in full general relativity, using a Cartesian grid, in which axisymmetry is imposed by suitable boundary conditions. The details of the formalism (numerical evolution scheme and gauge) are given in [271]. It is found that rapid rotation can prevent prompt black hole formation. When \(q = {\mathcal O}(1)\), a prompt collapse to a black hole is prevented even for a rest mass that is 70–80% larger than the maximum allowed mass of spherical stars, and this depends weakly on the rotational profile of the initial configuration. The final configuration is supported against collapse by the induced differential rotation. In these axisymmetric simulations, shock formation for q<0.5 does not result in a significant heating of the core; shocks are formed at a spheroidal shell around the high density core. In contrast, when the initial configuration is rapidly rotating \((q = {\mathcal O}(1))\), shocks are formed in a highly nonspherical manner near high density regions, and the resultant shock heating contributes in preventing prompt collapse to a black hole. A qualitative analysis in [272] suggests that a disk can form around a black hole during core collapse, provided the progenitor is nearly rigidly rotating and \(q = {\mathcal O}(1)\) for a stiff progenitor EOS. On the other hand, q≪1 still allows for a disk formation if the progenitor EOS is soft. At present, it is not clear how much the above conclusions depend on the restriction to axisymmetry or on other assumptions — 3dimensional simulations of the core collapse of such initially axisymmetric configurations have still to be performed.
A new numerical code for axisymmetric gravitational collapse in the (2+1)+1 formalism is presented in [63].
4.3.2 Formation of rotating neutron stars
First attempts to study the formation of rotating neutron stars in axisymmetric collapse were initiated by Evans [96, 97]. Recently, Dimmelmeier, Font and Müller [90, 89] have successfully obtained detailed simulations of neutron star formation in rotating collapse. In the numerical scheme, HRSC methods are employed for the hydrodynamical evolution, while for the spacetime evolution the conformal flatness approximation [324] is used. Surprisingly, the gravitational waves obtained during the neutron star formation in rotating core collapse are weaker in general relativity than in Newtonian simulations. The reason for this result is that relativistic rotating cores bounce at larger central densities than in the Newtonian limit (for the same initial conditions). The gravitational waves are computed from the time derivatives of the quadrupole moment, which involves the volume integration of ρr^{4}. As the density profile of the formed neutron star is more centrally condensed than in the Newtonian case, the corresponding gravitational waves turn out to be weaker. Details of the numerical methods and of the gravitational wave extraction used in the above studies can be found in [91, 92].
New, fully relativistic axisymmetric simulations with coupled hydrodynamical and spacetime evolution in the lightcone approach, have been obtained by Siebel et al. [282, 281]. One of the advantages of the lightcone approach is that gravitational waves can be extracted accurately at null infinity, without spurious contamination by boundary conditions. The code by Siebel et al. combines the lightcone approach for the spacetime evolution with HRSC methods for the hydrodynamical evolution. In [281] it is found that gravitational waves are extracted more accurately using the Bondi news function than by a quadrupole formula on the null cone.
A new 2D code for axisymmetric core collapse, also using HRSC methods, has recently been introduced in [273].
Notes
Acknowledgements
I am grateful to Emanuele Berti, John L. Friedman, Wlodek Kluźniak, Kostas D. Kokkotas, and Luciano Rezzolla for a careful reading of the manuscript and for many valuable comments. Many thanks to Dorota GondekRosińska and Eric Gourgoulhon for comments and for supplying numerical results obtained with the Lorene/rotstar code, that were used in the comparison in Table 2. I am also grateful to Marcus Ansorg for discussions and to all authors of the included figures for granting permission for reproduction. This work was supported, in part, by the EU Programme “Improving the Human Research Potential and the SocioEconomic Knowledge Base” (Research Training Network Contract HPRNCT200000137), KBN5P03D01721, and the Greek GSRT Grant EPANM. 43/2013555.
Supplementary material
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