Accelerating Boundary Analog of a Kerr Black Hole

The Kerr metric is not spherically symmetric, but this is immaterial to the coordinates describing the collapsing star’s center. The regularity condition at the origin defines the behaviour of incoming modes, subsequently governing the character of outgoing particle production Wilczek (1993); Rothman (2000); Fabbri and Navarro-Salas (2005). The position of the center of the star is insensitive to the asymmetry of the surface event horizon and a (1+1)-dimensional model can be constructed by specializing to a single plane. Hence, the angular coordinates are not required in the calculation of the temperature of the radiation. This was discovered for the late-time extremal Kerr case by Rothman Rothman (2000). We will demonstrate this fact holds true for both the extremal and non-extremal Kerr spacetimes at all times.

The line element in spatial spherical coordinates is (throughout this paper we utilize natural units, $G=h=c=k_{B}=1$)

with

where

In this parametrization, $a$ is the ‘spin’ or mass-normalized angular momentum of the rotating black hole and $r_{s}=2M$ is the usual Schwarzschild radius.

The corresponding static coordinate metric of the (1+1)-dimensional Kerr metric can be obtained by setting $\phi=\theta=0$ in the (3+1)-dimensional case, Eq. (1). This yields the following line element in terms of the radial and time pieces,

with

Note that this is the same $a$ as that found in Eq. (1). The metric in (1+1) dimensions is introduced to find the associated radial trajectory (i.e. the mapping of inside to outside coordinates, see Sec. II.2) of the center of the black hole in (3+1) dimensions, which can be understood as the reflecting point of the incoming modes. Hence, the mirror trajectory in (1+1) is an appropriate model for describing the collapse and its subsequent effect on the modes, as far as an analysis of the radiation temperature is concerned. The angular coordinates $\theta$ and $\phi$ do not affect the temperature since they do not enter the equation for $f(r)$. This will illustrate an important 1-dimensional trait of the Kerr black hole, similar to single channel Bekenstein entropy flow Bekenstein (2003); Padmanabhan (2002). The resulting temperature is for the equivalent (3+1)-dimensional Kerr black hole.

Equation (4) contains two horizons at the radial coordinates $r_{p,m}=M\pm\sqrt{M^{2}-a^{2}}$ (where $r_{p},r_{m}$ correspond to the $\pm$ signs respectively), which reduce to the event horizon, $r=r_{s}$, and the curvature singularity, $r=0$, of the Schwarzschild metric in the limit $a\to 0$. Using Eq. (4) one can straightforwardly derive the analogous single moving mirror model Fulling and Davies (1976); Davies and Fulling (1977) trajectory, where the accelerating boundary plays the role of the black hole center. The resulting particle production emitted by the mirror can be analyzed through the calculation of Bogoliubov coefficients between the incoming and outgoing modes.

In Kerr spacetime, the temperature of emitted radiation observed by an inertial observer at infinity is

where $\beta=\sqrt{1-a^{2}/M^{2}}$ (see Appendix A for a derivation). We will show that the same Planck spectrum with an identical temperature holds for the accelerating mirror analog. Here $g=1/(4M)$ is the usual Schwarzschild surface gravity, and $k=M\Omega^{2}$ is the black hole spring constant Good and Ong (2015b). The parameter $\beta$ will allow us to present results for the continuous range from Schwarzschild ($\beta=1$) to extreme Kerr ($\beta=0$) solutions.

For a double null coordinate system $(u,v)$, with $u=t-r^{*}$ and $v=t+r^{*}$, the appropriate tortoise coordinate $r^{*}$ is found in the usual way, via

yielding

One then has the metric for the geometry describing the outside region $r>r_{p}$,

The matching condition (see e.g. Wilczek (1993); Fabbri and Navarro-Salas (2005)) with the flat interior geometry, described by the interior coordinates $U=T-r$ and $V=T+r$, is the trajectory of $r=0$, expressed in terms of the exterior function $u(U)$ with interior coordinate $U$. This matching is obtained via the association, $r^{*}=r$. We take $r^{*}(r=(v_{0}-U)/2)=(v_{0}-u)/2$, occurring along a light ray, $v_{0}$. We can choose either $v_{0}-2r_{p}\equiv v_{H}$ or $v_{0}-2r_{m}\equiv v_{H}$ because $u\rightarrow+\infty$ at $U=v_{H}$. Without loss of generality, we can set $v_{H}=0$, i.e. $v_{0}=2r_{p,m}$. Choosing $v_{0}=2r_{p}$ gives the correct Schwarzschild limit since for Schwarzschild, $r_{m}=0$. Another way of understanding this choice is that the modes that escape the incipient black hole necessarily have access to the $r=0$ center. Anticipating a transition to the mirror system, the outer radius is chosen for the shell position because $r_{p}>r_{m}$. That is, the modes from the shell $v_{0}=2r_{p}$ will reach the observer at $\mathscr{I}_{R}^{+}$ first in both the mirror and black hole system, already having passed through $r=0$ (reflecting off the mirror). Thus, we choose $v_{0}=2r_{p}$ rather than the inner radius which occurs at an earlier $v$.

Taking the outer horizon $v_{0}=2r_{p}$, the result for the exterior coordinate is expressed as

We can verify that the Schwarzschild expression is reproduced for $\beta=1$.

To ensure regularity of the modes, we require that they vanish at $r=0$ such that the origin acts like a moving mirror in the $(U,V)$ coordinates. Since there is no field behind $r<0$, the form of field modes can be determined, such that a $U\leftrightarrow v$ identification is made for the outgoing, Doppler-shifted right-movers Birrell and Davies (1984). We are now ready to analyze the analog mirror trajectory by making the identification $u(U)\leftrightarrow f(v)$, a known function of the advanced time $v$.

Using the standard moving mirror formalism Birrell and Davies (1984), we study the massless scalar field in $(1+1)$-dimensional Minkowski spacetime (following e.g.  Good and Ong (2015a)). From Eq. (10) the Kerr analog moving mirror trajectory is

Note that the prefactor of the first logarithm is simply $1/\kappa$ (the surface gravity at $r_{p}$) and the prefactor of the second logarithm is $1/\kappa_{m}$ (the surface gravity at $r_{m}$) and so for $\beta=1$ the reduction to the Schwarzschild case (where $g=\kappa$) is clear. This is now to be regarded as the trajectory of a perfectly reflecting boundary in flat spacetime rather than the origin as a function of coordinates in curved Kerr spacetime. We have reintroduced $g\equiv 1/(4M)$ to signal that we are now working in the moving mirror model with a background of flat spacetime, where $g$ is related to the acceleration parameter of the trajectory.

The rapidity, in advanced time, can be obtained via $-2\eta(v)=\ln f^{\prime}(v)$, where the prime denotes a derivative with respect to the argument Good and Linder (2018). This yields

The rapidity asymptotes at $v=0$, i.e. the mirror approaches the speed of light at the horizon, $u\rightarrow+\infty$. Again, this agrees with the Schwarzschild result for $\beta=1$. The proper acceleration $\alpha=e^{\eta(v)}\eta^{\prime}(v)$ is also easily found and diverges as $v\rightarrow 0^{-}$, with $\alpha(v\to 0^{-})=-\kappa/\sqrt{-4\kappa v}$. Thus, the late-time acceleration is related to the surface gravity $\kappa$, and both will be related to the temperature of the thermal spectrum of particles produced. The acceleration goes to zero at $v=-\infty$ as $-\beta/(g^{2}v^{2})$.

The trajectory in a conformal Penrose diagram is plotted in Fig. 1(a), showing the influence of the spin for different values of $\beta=\sqrt{1-a^{2}/M^{2}}$, while Fig. 1(b) displays the influence of the mass for different values of $g=1/(4M)$.

The extremal Kerr (EK) limit is defined by $J=M^{2}$, or $a=M$, corresponding to $\beta=0$. This means that $r_{p}=r_{m}=M$ and we have to redo our derivation in the $\beta=0$ limit to avoid division by zero in the last term of Eq. (8). The relevant radial and time pieces of the metric are given by

with

giving a horizon at $r=M$. The tortoise coordinate is found by the usual integration Eq. (7), giving

We perform the standard analysis by matching $r^{*}=r$, solving for the trajectory of the center and then applying the regularity condition of the modes. This gives the moving mirror trajectory,

Here we have set the shell to $v_{0}=2M$, so that the horizon is at $v_{H}=0$. Note that the constant term $2M$ in $f(v)$ has no effect on the acceleration or flux.

To signal that we are now in flat space with an accelerated boundary, we associate $M$ with $\mathcal{A}$, where $\mathcal{A}$ is the limiting uniform proper acceleration of the mirror at late times,

This gives $M=1/(2\sqrt{2}\mathcal{A})$. Thus the extremal Kerr case gives asymptotic uniform acceleration, and so the energy flux vanishes in this limit $v\to 0$. This is behavior in common with asymptotically inertial mirrors. The model still produces an infinite total particle count in contrast to asymptotic zero velocity (static) mirrors like that proposed by Walker and Davies Walker and Davies (1982), or the ‘Schwarzschild mirror with quantum purity’ model Good et al. (2020b, c); Good and Linder (2020), which yield finite total particle count $N$. An infinite total particle count is expected from the extremal Kerr case ($\beta=0$) because infinite soft particles (zero frequency) – an IR divergence – is present for uniform acceleration.

The energy flux is straightforward to derive from Eq. (13),

and is plotted as the blue curve in Fig. 4.

The total stress energy is found via substitution of Eq. (28) and Eq. (30) into

which gives

The particle mode-mode spectrum via substitution of Eq. (28) into Eq. (16) or Eq. (17) gives,

where $\omega_{p}\equiv\omega+\omega^{\prime}$, and $j\equiv 1+i\omega\sqrt{2}/\mathcal{A}$. The total energy carried by the particles is the same as that derived by the stress tensor radiation, Eq. (32),

We can compare Eq. (33) for the extreme Kerr (EK) case with the extreme Reissner-Nordström (ERN) case Good (2020),

where $q\equiv 1+2i\omega/\mathcal{A}_{\rm ERN}$. The forms are quite similar but differ in the details. For two equal mass ERN and EK black holes, the ERN emits more total energy than the EK,

Of course, the inverse relationship between acceleration and mass reverses the situation if instead we take two equal late-time acceleration ERN and EK mirrors. Then the EK emits more total energy than the ERN. Since $\mathcal{A}_{\text{ERN}}=1/(2M_{\text{ERN}})$ and $\mathcal{A}_{\text{EK}}=1/(2\sqrt{2}M_{\text{EK}})$, we have

Figure 5 compares the integrand of $E$ for the two extremal mirrors, $\omega|\beta_{\omega\omega^{\prime}}|^{2}$. Figure 6 shows the Penrose diagram of EK for various asymptotic accelerations, with a comparison to an ERN trajectory in black.