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Register a new userNumerical study of the gravitational shock wave inside a spherical charged black hole
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We numerically investigate the interior of a four-dimensional, asymptotically flat, spherically symmetric charged black hole perturbed by a scalar field $Phi$. Previous study by Marolf and Ori indicated that late infalling observers will encounter an effective shock wave as they approach the left portion of the inner horizon. This shock manifests itself as a sudden change in the values of various fields, within a tremendously short interval of proper time $tau$ of the infalling observers. We confirm this prediction numerically for both test and self-gravitating scalar field perturbations. In both cases we demonstrate the effective shock in the scalar field by exploring $Phi(tau)$ along a family of infalling timelike geodesics. In the self-gravitating case we also demonstrate the shock in the area coordinate $r$ by exploring $r(tau)$. We confirm the theoretical prediction concerning the shock sharpening rate, which is exponential in the time of infall into the black hole. In addition we numerically probe the early stages of shock formation. We also employ a family of null (rather than timelike) ingoing geodesics to probe the shock in $r$. We use a finite-difference numerical code with double-null coordinates combined with a recently developed adaptive gauge method in order to solve the (Einstein + scalar) field equations and to evolve the spacetime (and scalar field) $ - $ from the region outside the black hole down to the vicinity of the Cauchy horizon and the spacelike $r=0$ singularity.
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We numerically compute the renormalized expectation value $langlehat{Phi}^{2}rangle_{ren}$ of a minimally-coupled massless quantum scalar field in the interior of a four-dimensional Reissner-Nordstrom black hole, in both the Hartle-Hawking and Unruh states. To this end we use a recently developed mode-sum renormalization scheme based on covariant point splitting. In both quantum states, $langlehat{Phi}^{2}rangle_{ren}$ is found to approach a emph{finite} value at the inner horizon (IH). The final approach to the IH asymptotic value is marked by an inverse-power tail $r_{*}^{-n}$, where $r_{*}$ is the Regge-Wheeler tortoise coordinate, and with $n=2$ for the Hartle-Hawking state and $n=3$ for the Unruh state. We also report here the results of an analytical computation of these inverse-power tails of $langlehat{Phi}^{2}rangle_{ren}$ near the IH. Our numerical results show very good agreement with this analytical derivation (for both the power index and the tail amplitude), in both quantum states. Finally, from this asymptotic behavior of $langlehat{Phi}^{2}rangle_{ren}$ we analytically compute the leading-order asymptotic behavior of the trace $langlehat{T}_{mu}^{mu}rangle_{ren}$ of the renormalized stress-energy tensor at the IH. In both quantum states this quantity is found to diverge like $b(r-r_{-})^{-1}r_{*}^{-n-2}$ (with $n$ specified above, and with a known parameter $b$). To the best of our knowledge, this is the first fully-quantitative derivation of the asymptotic behavior of these renormalized quantities at the inner horizon of a four-dimensional Reissner-Nordstrom black hole.
We numerically study the superradiant instability of charged massless scalar field in the background of charged stringy black hole with mirror-like boundary condition. We compare the numerical result with the previous analytical result and show the dependencies of this instability upon various parameters of black hole charge $Q$, scalar field charge $q$, and mirror radius $r_m$. Especially, we have observed that imaginary part of BQN frequencies grows with the scalar field charge $q$ rapidly.
In an ongoing effort to explore quantum effects on the interior geometry of black holes, we explicitly compute the semiclassical flux components $leftlangle T_{uu}rightrangle _{ren}$ and $leftlangle T_{vv}rightrangle _{ren}$ ($u$ and $v$ being the standard Eddington coordinates) of the renormalized stress-energy tensor for a minimally-coupled massless quantum scalar field, in the vicinity of the inner horizon (IH) of a Reissner-Nordstrom black hole. These two flux components seem to dominate the effect of backreaction in the IH vicinity; and furthermore, their regularization procedure reveals remarkable simplicity. We consider the Hartle-Hawking and Unruh quantum states, the latter corresponding to an evaporating black hole. In both quantum states, we compute $leftlangle T_{uu}rightrangle _{ren}$ and $leftlangle T_{vv}rightrangle _{ren}$ in the IH vicinity for a wide range of $Q/M$ values. We find that both $leftlangle T_{uu}rightrangle _{ren}$ and $leftlangle T_{vv}rightrangle _{ren}$ attain finite asymptotic values at the IH. Depending on $Q/M$, these asymptotic values are found to be either positive or negative (or vanishing in-between). Note that having a nonvanishing $leftlangle T_{vv}rightrangle _{ren}$ at the IH implies the formation of a curvature singularity on its ingoing section, the Cauchy horizon. Motivated by these findings, we also take initial steps in the exploration of the backreaction effect of these semiclassical fluxes on the near-IH geometry.
The horizon (the surface) of a black hole is a null surface, defined by those hypothetical outgoing light rays that just hover under the influence of the strong gravity at the surface. Because the light rays are orthogonal to the spatial 2-dimensional surface at one instant of time, the surface of the black hole is the same for all observers (i.e. the same for all coordinate definitions of instant of time). This value is 4*(pi)* (2Gm/c^2)^2 for nonspinning black holes, with G= Newtons constant, c= speed of light, and m= mass of the black hole. The 3-dimensional spatial volume inside a black hole, in contrast, depends explicitly on the definition of time, and can even be time dependent, or zero. We give examples of the volume found inside a standard, nonspinning spherical black hole, for several different standard time-coordinate definitions. Elucidating these results for the volume provides a new pedagogical resource of facts already known in principle to the relativity community, but rarely worked out.
We analyze and compute the semiclassical stress-energy flux components, the outflux $langle T_{uu}rangle$ and the influx $langle T_{vv}rangle$ ($u$ and $v$ being the standard null Eddington coordinates), at the inner horizon (IH) of a Reissner-Nordstrom black hole (BH) of mass $M$ and charge $Q$, in the near-extremal domain in which $Q/M$ approaches $1$. We consider a minimally-coupled massless quantum scalar field, in both Hartle-Hawking ($H$) and Unruh ($U$) states, the latter corresponding to an evaporating BH. The near-extremal domain lends itself to an analytical treatment which sheds light on the behavior of various quantities on approaching extremality. We explore the behavior of the three near-IH flux quantities $langle T_{uu}^-rangle^U$, $langle T_{vv}^-rangle^U$, and $langle T_{uu}^-rangle^H=langle T_{vv}^-rangle^H$, as a function of the small parameter $Deltaequivsqrt{1-(Q/M)^2}$ (where the superscript $-$ refers to the IH value). We find that in the near-extremal domain $langle T_{uu}^-rangle^Uconglangle T_{uu}^-rangle^H=langle T_{vv}^-rangle^H$ behaves as $proptoDelta^5$. In contrast, $langle T_{vv}^-rangle^U$ behaves as $proptoDelta^4$, and we calculate the prefactor analytically. It therefore follows that the semiclassical fluxes at the IH neighborhood of an evaporating near-extremal spherical charged BH are dominated by the influx $langle T_{vv}rangle^U$. In passing, we also find an analytical expression for the transmission coefficient outside a Reissner-Nordstrom BH to leading order in small frequencies (which turns out to be a crucial ingredient of our near-extremal analysis). Furthermore, we explicitly obtain the near-extremal Hawking-evaporation rate ($proptoDelta^4$), with an analytical expression for the prefactor (obtained here for the first time to the best of our knowledge). [Abridged]
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