Pulsation of black holes

Abstract

The Hawking–Penrose singularity theorem states that a singularity forms inside a black hole in general relativity. To remove this singularity one must resort to a more fundamental theory. Using a corrected dynamical equation arising in loop quantum cosmology and braneworld models, we study the gravitational collapse of a perfect fluid sphere with a rather general equation of state. In the frame of an observer comoving with this fluid, the sphere pulsates between a maximum and a minimum size, avoiding the singularity. The exterior geometry is also constructed. There are usually an outer and an inner apparent horizon, resembling the Reissner–Nordström situation. For a distant observer the horizon crossing occurs in an infinite time and the pulsations of the black hole quantum “beating heart” are completely unobservable. However, it may be observable if the black hole is not spherical symmetric and radiates gravitational wave due to the quadrupole moment, if any.

Appendices

Appendix A: From interior to exterior solution

Here we show how to obtain the exterior solution from the interior one using our “surface trick” in the context of general relativity.

The metric for a homogenous and isotropic fluid sphere is given by Eq. (1), where the scale factor a obeys Eq. (2). Following the procedure of Sect. 3, we rewrite Eq. (1) in the frame of the observer at infinity as

$$\begin{aligned} ds^2= & {} -F^2\left( 1-\frac{8\pi }{3}\rho x^2\right) d\bar{t}^2+x^2d\Omega _{(2)}^2\nonumber \\&+\left( 1-\frac{8\pi }{3}\rho x^2\right) ^{-1}dx^2, \end{aligned}$$

(69)

where the metric components must be understood as functions of $\bar{t}$ and x.

At the surface $r=r_0$ of the sphere ($r_0$ is a constant, since r is a comoving coordinate [46]), we have

$$\begin{aligned} ds^2= & {} -\left[ 1-\frac{8\pi }{3}\rho \left( \frac{x}{r_0}\right) x^2\right] d\tilde{t}^2 +x^2d\Omega _{(2)}^2\nonumber \\&+\left[ 1-\frac{8\pi }{3}\rho \left( \frac{x}{r_0}\right) x^2\right] ^{-1}dx^2. \end{aligned}$$

(70)

The metric (70) also represents the exterior metric at the surface $r=r_0$. Thus we can take this static spherically symmetric spacetime as the exterior solution of the fluid sphere. Similar to the proof in Sect. 4, we can show that the exterior geometry (70) matches continuously the interior one (1).

As an example, we consider the energy density

$$\begin{aligned} \rho =\lambda +\frac{\rho _m}{a^3}+\frac{\rho _r}{a^4}, \end{aligned}$$

(71)

with $\lambda$ for the cosmological constant energy density, ${\rho _m}/{a^3}$ for the dust density, and ${\rho _r}/{a^4}$ for the radiation density. Substituting Eq. (71) with $a=x/r_0$ into Eq. (70), one obtains

$$\begin{aligned} ds^2= & {} -\left( 1-\frac{2M}{x}-\frac{Q^2}{x^2} -\Lambda x^2\right) d\tilde{t}^2+x^2d\Omega _{(2)}^2\nonumber \\&+\left( 1-\frac{2M}{x}-\frac{Q^2}{x^2}-\Lambda x^2\right) ^{-1}dx^2, \end{aligned}$$

(72)

where

$$\begin{aligned} M\equiv \frac{4\pi }{3}\rho _m r_0^{3},\quad Q^2\equiv \frac{8\pi }{3}\rho _r r_0^{4},\quad \Lambda \equiv \frac{8\pi }{3}\lambda . \end{aligned}$$

(73)

The metric (72) is the Reissner–Nordström–de Sitter solution of the Einstein equations with the formal replacement $Q\rightarrow i Q$. While M plays the role of a mass, $Q^2$ also plays the role of the mass of radiation (not of electric charge) contained in a sphere. In fact, the thermal bath of radiation with density $\rho _r/a^4$ has nothing to do with free electric charges and the analogy with Reissner–Nordström–de Sitter is purely formal and not complete (because of the opposite sign of the term in $Q^2/x^2$).

Appendix B: From exterior to interior solution

Here we show how to obtain the interior geometry from the exterior one in the context of general relativity. We focus on the static spherically symmetric spacetime

$$\begin{aligned} ds^2= & {} -\left[ 1-\frac{8\pi }{3} \rho \left( x\right) x^2\right] d\tilde{t}^2+x^2d\Omega _{(2)}^2 \nonumber \\&+\left[ 1-\frac{8\pi }{3}\rho \left( x\right) x^2\right] ^{-1}dx^2. \end{aligned}$$

(74)

We assume that the spacetime exterior to a perfect fluid sphere is given by Eq. (74). Then our task is to look for the interior solution starting from this exterior. The normalization of the tangent $v^a$ to the trajectory described by the surface of the sphere in spacetime gives

$$\begin{aligned} \dot{x}^2=E^2-1+\frac{8\pi }{3}\rho x^2, \end{aligned}$$

(75)

with an overdot denoting differentiation with respect to the proper time t and where E is an integration constant. Equations (75) and (2) coincide provided that

$$\begin{aligned} x=a r_0\quad -k r_0^2=E^2-1, \end{aligned}$$

(76)

with $r_0$ a constant. This coincidence motivates us to consider simply the homogenous and isotropic perfect fluid sphere as the interior solution. Then $\rho$ plays the role of the energy density in the comoving frame. We have checked that the resulting interior solution does match the exterior solution continuously. In the following we consider, as an example, the Bardeen metric [54] as the exterior geometry outside a perfect fluid sphere.

The Bardeen line element describing a regular black hole is

$$\begin{aligned} ds^2= & {} -\left[ 1-\frac{2Mx^2}{ \left( x^2+q^2\right) ^{3/2}}\right] d\tilde{t}^2 +x^2d\Omega _{(2)}^2\nonumber \\&+\left[ 1-\frac{2Mx^2}{\left( x^2+q^2\right) ^{3/2}} \right] ^{-1}dx^2, \end{aligned}$$

(77)

where M is the mass and q is the monopole charge of a self-gravitating magnetic field described by a nonlinear electrodynamics [55]. This model has been revisited by Borde, who clarified the avoidance of singularities in this spacetime [56, 57]. For a certain range of the parameter q, the Bardeen metric describes a black hole. When $x\gg q$, it behaves as the Schwarzschild black hole but, when $x\ll q$, it behaves as de Sitter space, therefore, the spacetime in general has two apparent horizons. Explicitly, there are two such horizons when

$$\begin{aligned} |q|<\frac{4M}{3\sqrt{3}}. \end{aligned}$$

(78)

Given the exterior metric (77), one can read off the corresponding energy density of the perfect fluid sphere, which can be parameterized as

$$\begin{aligned} \rho= & {} \frac{\rho _m }{\left( a^2+a_0^2\right) ^{3/2}}, \end{aligned}$$

(79)

where $\rho _m$ and $a_0$ are two constants and a(t) is the time-dependent radius of the fluid sphere. Substituting Eq. (79) into Eq. (2) yields

$$\begin{aligned} {\dot{a}^2}=\frac{8\pi }{3}\frac{\rho _m a^2}{\left( a^2+a_0^2\right) ^{3/2}}-k, \end{aligned}$$

(80)

describing a pulsating sphere. However, the pulsation is again unobservable by an observer at infinity due to the unavoidable formation of an apparent horizon. On the other hand, if we regard the above equation as the Friedmann equation to study its cosmic evolution, we find it can give an exponential inflationary universe provided that $a\ll a_0$. Thus it would be interesting to investigate this inflationary model in great detail.