Quantum fluxes at the inner horizon of a spherical charged black hole

TL;DR

This paper calculates quantum flux components near the inner horizon of a charged black hole, revealing finite asymptotic values that depend on charge-to-mass ratio and exploring their potential backreaction effects.

Contribution

It provides explicit semiclassical flux calculations at the inner horizon of Reissner-Nordström black holes for different quantum states and charge ratios, highlighting their regularization and physical implications.

Findings

Flux components attain finite asymptotic values at the IH.

Asymptotic flux values depend on the charge-to-mass ratio.

Nonvanishing fluxes imply possible curvature singularities on the Cauchy horizon.

Abstract

In an ongoing effort to explore quantum effects on the interior geometry of black holes, we explicitly compute the semiclassical flux components $\left\langle T_{uu}\right\rangle _{ren}$ and $\left\langle T_{vv}\right\rangle _{ren}$ ($u$ 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-Nordstr\"om 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 $\left\langle T_{uu}\right\rangle _{ren}$ and $\left\langle T_{vv}\right\rangle _{ren}$ in the IH vicinity for a wide range of $Q/M$ values. We find that both $\left\langle T_{uu}\right\rangle _{ren}$ and $\left\langle T_{vv}\right\rangle _{ren}$ 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 $\left\langle T_{vv}\right\rangle _{ren}$ 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.

Equations

$$G_{\alpha \beta }=8\pi {⟨T_{\alpha \beta }⟩}_{ren}.$$

$$d{s}^2=-f(r)d{t}^2+\frac{1}{f(r)}d{r}^2+r^{2}d{\Omega }^2,$$

$$\frac{d^2{\psi }_{\omega l}}{d{r}_{\ast }^2}+\left[{\omega }^2-V_l\left(r\right)\right]{\psi }_{\omega l}=0,$$

$$V_l\left(r\right)=f(r)\left[\frac{l\left(l+1\right)}{r^2}+\frac{df/dr}{r}\right].$$

$${\psi }_{\omega l}\cong e^{-i\omega r_{\ast }},r_{\ast }\to -\infty .$$

$${⟨T_{yy}⟩}_{ren}\left(x\right)=\frac{1}{2}\underset{x^{\prime }\to x}{\lim }{G}^{\left(1\right)}{\left(x,x^{\prime }\right)}_{,y{y}^{\prime }},$$

$$G^{\left(1\right)}\left(x,x^{\prime }\right)=\hslash \sum _{l,m}{\int }_0^{\infty }d\omega E_{\omega lm}\left(x,x^{\prime }\right),$$

$$E_{\omega lm}^H\left(x,x^{\prime }\right)$$

$$J^R=\left\{f_{\omega lm}^R\left(x\right),f_{\omega lm}^{R\ast }\left(x^{\prime }\right)\right\},J^L=\left\{f_{\omega lm}^L\left(x\right),f_{\omega lm}^{L\ast }\left(x^{\prime }\right)\right\}$$

$$J^{RL}=2\Re \left[{\rho }_{\omega l}^{up}\left\{f_{\omega lm}^R\left(x\right),f_{(-\omega )lm}^{L\ast }\left(x^{\prime }\right)\right\}\right].$$

$$f_{\omega lm}^{R,L}(x)=\frac{1}{r\sqrt{4\pi \left|\omega \right|}}{Y}_{lm}(\theta ,\phi ){\tilde{f}}_{\omega l}^{R,L}\left(t,r\right)$$

$${\tilde{f}}_{\omega l}^R\cong A_{\omega l}{e}^{i\omega u}+B_{\omega l}{e}^{-i\omega v},{\tilde{f}}_{\omega l}^L\cong A_{\omega l}{e}^{i\omega v}+B_{\omega l}{e}^{-i\omega u}.$$

$${⟨T_{yy}⟩}_{ren}^H\left(x\right)=\frac{\hslash }{2}\underset{x^{\prime }\to x}{\lim }\sum _{l,m}{\int }_0^{\infty }d\omega E_{\omega lm}^H{\left(x,x^{\prime }\right)}_{,y{y}^{\prime }}.$$

$$J_{,u{u}^{\prime }}^R\to \left\{Y_{lm}(\theta ,\phi ),Y_{lm}^{\ast }({\theta }^{\prime },\phi )\right\}{\left|A_{\omega l}\right|}^2.$$

$${⟨T_{uu}⟩}_{ren}^H$$

$${\hat{E}}_{\omega l}^H=\frac{\omega \coth \tilde{\omega }}{\pi r_{-}^2}\left[{\left|A_{\omega l}\right|}^2+{\text{cosh}}^{-1}\tilde{\omega }\Re \left({\rho }_{\omega l}^{up}{A}_{\omega l}{B}_{\omega l}\right)\right]$$

$${⟨T_{uu}^{-}⟩}_{ren}^H={⟨T_{vv}^{-}⟩}_{ren}^H$$

$${⟨T_{yy}^{-}⟩}_{ren}^U$$

$$\Delta {\hat{E}}_{\omega l\left(yy\right)}^U=\frac{\omega }{2\pi r_{-}^2}\left(1-\coth \tilde{\omega }\right){\left|{\tau }_{\omega l}^{up}\right|}^2\left({\left|A_{\omega l}\right|}^2+{\delta }_y^v\right).$$

$$r_{,yy}$$

$$r_{,y}$$

Full text

Introduction.

The analytically extended Kerr and Reissner-Nordström (RN) metrics, describing respectively spinning and spherical charged isolated black holes (BHs), reveal a traversable passage through an inner horizon (IH) to another external universe Carter:1966 ; GravesBrill:1960 .

Consider a traveler intending to access this other universe. To do so, she must pass through the BH interior, and in particular, through the IH. What will she encounter along her way? Is her mission doomed to fail? Does this external universe actually exist? Answering these questions requires understanding how quantum fields change the internal geometry of BHs. The most renowned phenomenon in which quantum effects profoundly transform the classical spacetime picture is the process of BH evaporation due to Hawking radiation Hawking:1974 ; Hawking:1975 . In fact, already at the classical level, it was demonstrated that introducing matter (or perturbation) fields on BH backgrounds may affect their regularity. A notable example is the null weak Tipler curvature singularity that forms along the Cauchy horizon (CH, the IH ingoing section) in both spinning Ori:1992 ; Ori:1999 ; BradyDrozMorsnik:1998 ; Dafermos:2017 and spherical charged Hiscock:1981 ; PoissonIsrael:1990 ; OriMassInflation:1991 ; BradySmith:1995 ; Piran ; Burko:1997 ; Dafermos:2005 BHs. The analogous effect of quantum perturbations is often expected to be significantly stronger BirrellDavies:1978 ; Hiscock:1980 ; Ottewill:2000 , but this issue remains inconclusive, making it the main motivation for this work.

A theoretical framework that lends itself to this problem is the semiclassical formulation of general relativity, considering matter fields as quantum fields propagating in a classical curved spacetime, obeying the semiclassical Einstein field equation, given (in units $G=c=1$) by:

[TABLE]

Here $G_{\alpha\beta}$ is the Einstein tensor, and the source term $\left\langle T_{\alpha\beta}\right\rangle_{ren}$ is the renormalized expectation value of the stress-energy tensor (RSET) associated with the quantum field. Note the emergent requirement for self-consistency: spacetime curvature induces a non-trivial stress-energy in the quantum fields, which in turn deforms the spacetime metric — an effect known as backreaction. A possible way to handle this complexity is to break the problem into steps of increasing order in the mutual effect, initially computing $\left\langle T_{\alpha\beta}\right\rangle_{ren}$ for a fixed, classical background metric. But already at this level, one faces a serious challenge: the computation of the RSET on curved backgrounds.

Recall that already in flat spacetime the stress-energy tensor of a quantum field formally diverges, but this is usually handled through the normal-ordering scheme, which is ill-defined in curved spacetime. The intricate regularization procedure required in curved spacetime, along with its inevitable numerical implementation, has made this computation a decades-lasting hurdle in the study of semiclassical problems. However, the recently developed pragmatic mode-sum regularization (PMR) method AAt:2015 ; AAtheta:2016 ; AARSET:2016 ; LeviRSET:2017 , rooted in covariant point-splitting Christensen:1976 ; Christensen:1978 , has made this task more accessible. (See, however, earlier works employing other methods, e.g. Candelas:1980 ; Frolov:1982 ; Fawcett:1983 ; Candelas-Howard:1984 ; Howard-Candelas:1984 ; HowardRSET:1984 ; Candelas_Jensen:1986 ; Anderson:1989 ; Jen_Otte:1989 ; Anderson:1990 ; McLau_Jen_Otte:1992 ; Ander_His_Sam:1995 ; Otte_Taylor:2011 ; Taylor:2019 ).

The PMR method overcomes the main difficulty in the numerical implementation of point splitting by treating the coincidence limit analytically, through construction of “modewise” counter-terms. It has been successfully used in recent years to compute both the vacuum expectation value $\left\langle\Phi^{2}\right\rangle_{ren}$ and the RSET for a quantum scalar field $\Phi$ on various BH exteriors AAt:2015 ; AAtheta:2016 ; AARSET:2016 ; LeviRSET:2017 ; AAKerr:2017 . On BH interiors, however, only $\left\langle\Phi^{2}\right\rangle_{ren}$ has been computed in that method so far (initially for Schwarzschild SchAssaf:2018 , reproducing previous results Candelas_Jensen:1986 ). Although $\left\langle\Phi^{2}\right\rangle_{ren}$ is not the quantity most relevant for backreaction, it nevertheless provides valuable insights for the computation of the more divergent RSET. In particular, in a recent paper GroupPhiRN:2019 , $\left\langle\Phi^{2}\right\rangle_{ren}$ was investigated both numerically and analytically inside RN, with extensive study of the IH vicinity. The RSET trace (for a minimally-coupled scalar field) was consequently found to diverge at the IH. The following work is a natural continuation of previous ones, providing novel results for certain key components of the RSET inside a BH — which directly demonstrate the divergence of semiclassical energy-momentum fluxes at the CH. 111See also Taylor:2019 ; but note that the unusual quantum state constructed there does not allow investigating the anticipated semiclassical CH divergency.

We hereby consider a spherically-symmetric charged BH, whose geometry is described by the RN metric:

[TABLE]

where $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\varphi^{2}$, and $f\left(r\right)\equiv 1-2M/r+Q^{2}/r^{2}$ with mass $M$ and charge $Q$. We consider a non-extremal BH, with $0<Q/M<1$. The event horizon (EH) and the IH are located at $r=r_{+}$ and $r=r_{-}$ respectively, with $r_{\pm}\equiv M\pm\sqrt{M^{2}-Q^{2}}$. For later use, we define the two surface gravity parameters, $\kappa_{\pm}=\left(r_{+}-r_{-}\right)/2r_{\pm}^{2}$.

Upon this background we introduce an (uncharged) minimally-coupled massless scalar quantum field $\Phi\left(x\right)$, obeying the (covariant) d’Alembertian equation, $\square\Phi=0$. We decompose the field into modes, which, owing to the metric symmetries, may be separated into $e^{-i\omega t}$, spherical harmonics $Y_{lm}\left(\theta,\varphi\right)$, and a function of $r$ Group:2018 . The latter is encoded in the radial function $\psi_{\omega l}\left(r\right)$, satisfying:

[TABLE]

with the effective potential

[TABLE]

$r_{*}$ is the standard tortoise coordinate defined through $dr/dr_{*}=f(r)$, varying from $r_{*}\rightarrow-\infty$ at the EH to $r_{*}\to\infty$ at the IH.

In the BH interior $f\left(r\right)<0$, meaning the coordinate $r$ is now timelike. Then, assuming a free incoming wave at the EH, Eq. (2) is endowed with the initial condition

[TABLE]

We consider our field in two quantum states: the Hartle-Hawking (HH) state HH:1976 ; Israel:1976 , corresponding to a BH in thermal equilibrium, and the more physically realistic Unruh state Unruh:1976 , describing an evaporating BH.

We introduce the null Eddington coordinates inside the BH, $u=r_{*}-t$ and $v=r_{*}+t$. The flux components of the RSET, $\left\langle T_{uu}\right\rangle_{ren}$ and $\left\langle T_{vv}\right\rangle_{ren}$, are of particular interest 222Note that the fluxes do not contribute to the RSET trace, which was shown to diverge in GroupPhiRN:2019 .. The reason is threefold. First and foremost, as we shall see, it is these components that seem to be the most significant for backreaction near the CH, with a remarkable accumulating effect on the form of the metric (as opposed to minor local distortions associated with other RSET components). In addition, note that although the classical RN background contains a non-zero stress-energy tensor (of the sourceless electromagnetic field), its $T_{uu}$ and $T_{vv}$ components vanish identically, leaving quantum contributions to prevail. Finally, their regularization procedure turns out to be especially manageable. Accordingly, aiming for the “heart” of the RSET in the context of backreaction, this work focuses on the flux components $\left\langle T_{uu}\right\rangle_{ren}$ and $\left\langle T_{vv}\right\rangle_{ren}$ in the IH vicinity.

In the next section we implement the PMR $\theta$-splitting variant AAtheta:2016 ; LeviThetaRSET to obtain expressions for the renormalized semiclassical flux components in both quantum states, revealing notable simplicity when taking the IH limit. We then provide numerical results for various $Q/M$ values, noting various issues that arise. Finally, we present a preliminary analysis of backreaction and implications to the fate of our traveler.

Developing the near-IH flux expressions.

In what follows, indices $U$ and $H$ correspond to the Unruh and HH states, respectively. As mentioned, we shall only consider the two flux components $\left\langle T_{uu}\right\rangle_{ren}$ and $\left\langle T_{vv}\right\rangle_{ren}$, and for their uniform treatment we introduce the symbol $y$, representing either $u$ or $v$.

The basic PMR expression for the trace-reversed RSET is given in Eq. (2.6) of Ref. LeviRSET:2017 . In the case of interest (i.e. the flux components $\left\langle T_{yy}\right\rangle_{ren}$ evaluated at $r\to r_{-}$ using $\theta$-splitting), two remarkable simplifications occur: (i) the PMR counter-term $\tilde{L}_{yy}\left(x,x^{\prime}\right)$ vanishes LeviThetaRSET ; Sup ; and (ii) since $g_{yy}=0$, $T_{yy}$ coincides with its trace-reversed counterpart. The expression then simplifies to

[TABLE]

where $G^{\left(1\right)}\left(x,x^{\prime}\right)=\left\langle\left\{\Phi(x),\Phi(x^{\prime})\right\}\right\rangle$, and $\left\{p(x),q(x^{\prime})\right\}$ denotes $p(x)q(x^{\prime})+p(x^{\prime})q(x)$. We can also express $G^{\left(1\right)}$ as

[TABLE]

where the mode contributions $E_{\omega lm}\left(x,x^{\prime}\right)$ inside a RN BH, in the HH state, are given by

[TABLE]

(cf. Eq. (4.3) in Group:2018 ) where

[TABLE]

and

[TABLE]

Here $\tilde{\omega}\equiv\pi\omega/\kappa_{+}$, the star denotes complex conjugation, and $\Re$ marks the real part. Hereafter, $\rho_{\omega l}^{up}\,\left(\tau_{\omega l}^{up}\right)$ represents the reflection (transmission) coefficient for the “up” modes outside the BH Group:2018 . The mode functions $f_{\omega lm}^{R,L}\left(x\right)$ are given by

[TABLE]

where $\tilde{f}_{\omega l}^{R}=e^{-i\omega t}\psi_{\omega l}(r)$ and $\tilde{f}_{\omega l}^{L}=e^{i\omega t}\psi_{\omega l}(r)$, and $\psi_{\omega l}\left(r\right)$ is the aforementioned radial function solving Eq. (2) with the initial condition (4). (For more details see Group:2018 .)

A similar expression exists for the Unruh-state counterpart, $E_{\omega lm}^{U}$. In what follows, we shall describe the analysis for the HH state solely. For the Unruh state the analysis is similar and we shall merely quote final results below (with the more detailed derivation deferred to Sup ). Note that due to time-inversion symmetry of the HH state (unlike the Unruh state), $\left\langle T_{uu}\right\rangle_{ren}^{H}=\left\langle T_{vv}\right\rangle_{ren}^{H}$ everywhere.

We are interested in the asymptotic behavior at the IH, where the effective potential $V_{l}\left(r\right)$ vanishes like $f\propto r-r_{-}$. Hence the radial equation (2) for $\psi_{\omega l}$ admits the general asymptotic solution $A_{\omega l}e^{i\omega r_{*}}+B_{\omega l}e^{-i\omega r_{*}}$ (with constant coefficients $A_{\omega l},\,B_{\omega l}$ ), which in turn implies

[TABLE]

Equations (5,6) yield

[TABLE]

It is interesting to inspect $E_{\omega lm}^{H}\left(x,x^{\prime}\right)_{,yy^{\prime}}$ within the near-IH approximation (7). Consider, for example, the contribution coming from the $J^{R}$ term. Focusing for concreteness on $y=u$, we readily see that the $\partial_{uu^{\prime}}$ operator annihilates the terms depending on $v$ in Eq. (7). Also, $r_{,u}=f/2\propto r-r_{-}$ vanishes at $r\to r_{-}$, altogether yielding at the limit $\left(u^{\prime},v^{\prime},\varphi^{\prime}\right)\to\left(u,v,\varphi\right)$ (corresponding to $\theta$-splitting) and $r\to r_{-}$:

[TABLE]

Remarkably, although $J^{R}$ itself does contain terms like $\propto e^{i\omega(v+u)}=e^{2i\omega r_{*}}$ at the IH limit, $J_{,uu^{\prime}}^{R}$ is free of such oscillatory terms — and is in fact entirely independent of $r_{*}$ (and $t$). This simplification occurs for all three “$J$” terms in the expression for $E_{\omega lm}^{H}\left(x,x^{\prime}\right)_{,uu^{\prime}}$. Combining their contributions and summing over $m$, one readily obtains at the IH

[TABLE]

where $\delta\theta\equiv\theta^{\prime}-\theta$, and $F_{l}^{H}\equiv\int_{0}^{\infty}d\omega\,\hat{E}_{\omega l}^{H}$ where

[TABLE]

(see fuller derivation in Sup ).

The sequence $F_{l}^{H}$ appearing in Eq. (9) approaches a non-vanishing constant $\beta\equiv F_{l\to\infty}^{H}$. One can show Sup , analytically, that $\beta=\left(\kappa_{-}^{2}-\kappa_{+}^{2}\right)/24\pi r_{-}^{2}$. Taking the $\delta\theta\to 0$ limit (using the methods of Ref. AAtheta:2016 ; see also Sup ), we obtain the final result

[TABLE]

Here, the upper $"-"$ index indicates the IH limit.

The analogous Unruh-state expression is Sup :

[TABLE]

where $\Delta F_{l\left(yy\right)}^{U}\equiv\int_{0}^{\infty}d\omega\,\Delta\hat{E}_{\omega l\left(yy\right)}^{U}$ and

[TABLE]

Note that the two Unruh-state flux components are not independent: From energy-momentum conservation, $4\pi r^{2}\left(\left\langle T_{uu}(x)\right\rangle_{ren}^{U}-\left\langle T_{vv}(x)\right\rangle_{ren}^{U}\right)$ is constant (it is actually the Hawking outflux; see Sup ).

Numerical results.

Recalling the Wronskian relation $\left|\tau_{\omega l}^{up}\right|^{2}=1-\left|\rho_{\omega l}^{up}\right|^{2}$, the final expressions (11,12) for the near-IH fluxes in both quantum states reveal simple dependence on $A_{\omega l},\,B_{\omega l}$ and $\rho_{\omega l}^{up}$. We numerically compute $A_{\omega l}$ and $B_{\omega l}$ by integrating the radial equation (2) from $r_{+}$ to $r_{-}$ (and $\rho_{\omega l}^{up}$ likewise, by solving the radial equation outside the BH). We then compute the three flux quantities $\left\langle T_{yy}^{-}\right\rangle_{ren}$ (that is $\left\langle T_{yy}^{-}\right\rangle_{ren}^{H},\,\left\langle T_{uu}^{-}\right\rangle_{ren}^{U}$ and $\left\langle T_{vv}^{-}\right\rangle_{ren}^{U}$) at the IH, as prescribed in Eqs. (11,12). For further numerical details, see Sup . We find exponential convergence of both the integral over $\omega$ (entailed in $F_{l}^{H},\Delta F_{l}^{U}$) and the sum over $l$, for all three quantities $\left\langle T_{yy}^{-}\right\rangle_{ren}$, as they attain well-defined finite values. Note that a finite non-vanishing $\left\langle T_{vv}^{-}\right\rangle_{ren}$ implies a curvature singularity at the CH, since transforming to a regular Kruskal-like coordinate $V=-e^{-\kappa_{-}v}$ yields $\left\langle T_{VV}^{-}\right\rangle_{ren}\propto e^{2\kappa_{-}v}\to\infty$.

Remarkably, the three quantities $\left\langle T_{yy}^{-}\right\rangle_{ren}$ may be either positive or negative, depending on $Q/M$. We find that sufficiently close to extremality all three flux components become negative, whereas further away from extremality they are all positive. Whether the diverging $\left\langle T_{VV}^{-}\right\rangle$ is positive or negative is crucial for the nature of tidal deformation (contraction vs. expansion), a point expanded hereafter. Figure 1 displays the three flux quantities $\left\langle T_{yy}^{-}\right\rangle_{ren}$ in the range $0.96<Q/M<1$, exhibiting the transition from positive to negative values at around $Q/M\sim 0.97$. More precisely, the sign change occurs at $Q/M$ values of $q_{v}^{U}\cong 0.9650,\,q_{u}^{U}\cong 0.9671$ and $q_{y}^{H}\cong 0.9675$ for $\left\langle T_{vv}^{-}\right\rangle_{ren}^{U},\,\left\langle T_{uu}^{-}\right\rangle_{ren}^{U}$ and $\left\langle T_{yy}^{-}\right\rangle_{ren}^{H}$, respectively.

Figure 2 displays the three flux quantities in a wider range $0.1\leq Q/M<1$. Note the very rapid increase in the fluxes as $Q/M$ decreases. This is perhaps not too surprising, since a decrease in $Q/M$ implies an (even faster) decrease in $r_{-}/M$, and correspondingly an increasing curvature at the IH.

Another notable feature is the decay of the fluxes as $Q/M\to 1$. Remarkably, in the near-extremal domain (characterized by $\left|Q/M-1\right|\ll 1$), the flux computation lends itself to analytical treatment (which we defer to a future paper Near-extremality ), leading to excellent agreement with the numerical data illustrated on the rightmost part of Fig. 1.

Backreaction near the CH.

The semiclassical backreaction, being of order $\propto\hbar/M^{2}=(m_{p}/M)^{2}$ (where $m_{p}$ denotes the Planck mass), is basically an extremely weak effect for macroscopic BHs. For instance, for astrophysical BHs it is typically $<10^{-75}$. However, these effects accumulate along the EH, causing its area to drastically shrink upon evaporation. Likewise, as we shall shortly see, semiclassical effects may also accumulate near the CH (and in addition, they become singular there). Thus, semiclassical backreaction is presumably negligible — and hence the actual backreacted geometry should be well approximated by the original RN metric — as long as (i) the BH hasn’t had the chance yet to significantly evaporate (that is, the $v$ interval since the BH formation is much smaller than the evaporation timescale), and (ii) we are not too close to the CH.

To address backreaction, we write the general spherically-symmetric metric in double-null coordinates as $-e^{\sigma}dudv+r^{2}d\Omega^{2}$. The two unknown metric functions, $r\left(u,v\right)$ and $\sigma\left(u,v\right)$, are to be determined from the semiclassical Einstein equation (1). This system contains constraint equations, which are two independent ODEs (one along each null direction) that involve the flux components $\left\langle T_{yy}^{-}\right\rangle_{ren}$ only; and evolution equations, which are two coupled PDEs involving $\left\langle T_{uv}\right\rangle_{ren}$ and $\left\langle T_{\theta\theta}\right\rangle_{ren}$. Our analysis will mainly rely on the two constraint equations, which we write uniformly as

[TABLE]

To proceed, we shall now restrict the analysis to the weak-backreaction domain, in which $r,\,\sigma_{,y}$ and $\left\langle T_{yy}\right\rangle_{ren}$ (but not necessarily $r_{,y}$) are still well approximated by their original RN background values. 333By its very definition, this weak-backreaction domain must satisfy restrictions (i,ii) mentioned above. Correspondingly, in what follows we consider the RN-background RSET and explore its backreaction effect via the semiclassical Einstein equation.

Furthermore, we shall focus on the near-CH portion of this weak-backreaction domain 444This is the region in which $e^{-\kappa_{-}(u+v)}$ is already $\ll 1$ (hence in the RN background $r,\,\sigma_{,y}$ and \bigl{\langle}T_{yy}\bigr{\rangle}_{ren} are well approximated by their near-CH values); and correspondingly, the drift effect in $r$ is already present — but still hasn’t accumulated much.. In this region, we may replace the right hand side of Eq. (14) by the constant $-4\pi r_{-}\left\langle T_{yy}^{-}\right\rangle_{ren}$, and $\sigma_{,y}$ by $-\kappa_{-}$ (its near-CH value in RN). We obtain a trivial linear ODE for $r_{,y}$, which is easily solved. After an exponentially decaying term ($\propto e^{\sigma}$) is dropped, we are left with

[TABLE]

This result expresses a small but steady asymptotic drift of $r\left(u,v\right)$ in both null directions. In the long run (i.e. at sufficiently large $u$ and/or $v$) this drift would result in a major deviation of $r$ from its RN value — which would eventually lead us away from the weak-backreaction domain.

From Eq. (15) it becomes clear that this remarkable accumulative effect is dictated solely by the flux components, namely, it is independent of the other RSET components.

To discuss the physical implications of this result, let us assume our infalling traveler moves towards the IH ingoing section, and approaches the near-IH domain where the semiclassical drift is present. We shall consider now the effect of the drift in the $v$ direction. 555Both $\left\langle T_{uu}^{-}\right\rangle_{ren}$ and $\left\langle T_{vv}^{-}\right\rangle_{ren}$ are associated with a drift effect at the CH vicinity, but as mentioned, only $\left\langle T_{vv}^{-}\right\rangle_{ren}$ induces a singular effect there. The effect of $\left\langle T_{uu}^{-}\right\rangle_{ren}$ may be associated with a steady drift of $r$ along the CH. We emphasize that although the near-CH drift in $r$ is very “slow” in terms of $v$ (i.e. $r_{,v}\ll 1$), it actually happens at an exceedingly fast rate for our infalling traveler — which (in the fiducial RN geometry) would arrive the CH at a finite proper time 666In particular, recall that $dr/d\tau\propto dv/d\tau\propto e^{\kappa_{-}v}$.. The nature of this physical effect may crucially depend on the sign of $\left\langle T_{vv}^{-}\right\rangle_{ren}$ — and hence on the value of $Q/M$. For $Q/M<q_{v}^{U,H}$, $\left\langle T_{vv}^{-}\right\rangle_{ren}>0$ and correspondingly our traveler will undergo sudden contraction. However, for $Q/M>q_{v}^{U,H}$, $\left\langle T_{vv}^{-}\right\rangle_{ren}$ is negative — implying an abrupt expansion.

This analysis still needs to be extended to the domain of strong backreaction, which actually entails two types of extensions: (i) to the domain of very late time (i.e. very large $v$), in which significant evaporation has already occurred 777Obviously this extension is needed in Unruh state only., and (ii) to the region very close to the CH.

Discussion.

Motivated by long-standing expectations that semiclassical effects may drastically influence the interior geometry of spinning or charged BHs, this work focused on the RSET flux components (for a minimally-coupled massless scalar field), in the IH vicinity, on a fixed RN background. We presented novel results for the flux components in the Unruh and HH states for various $Q/M$ values. Both flux components $\left\langle T_{uu}\right\rangle_{ren}$ and $\left\langle T_{vv}\right\rangle_{ren}$ — in both quantum states — exhibit finite asymptotic values at the IH. Recall, however, that a non-vanishing finite $\left\langle T_{vv}\right\rangle_{ren}$ implies unbounded curvature (and unbounded tidal force) at the CH ($v\to\infty$), because the corresponding Kruskal-like component $\left\langle T_{VV}\right\rangle_{ren}$ then diverges as $e^{2\kappa_{-}v}$.

Hiscock Hiscock:1980 previously demonstrated that in the Unruh state in a Kerr-Newman BH either $\left\langle T_{uu}^{-}\right\rangle_{ren}$ or $\left\langle T_{vv}^{-}\right\rangle_{ren}$ (or possibly both) are non-vanishing — indicating that the corresponding Kruskal fluxes diverge on at least one of the two IH sections. Still, this result left the semiclassical CH singularity inconclusive: Note that it is exclusively the ingoing section of the IH which maintains the causal and physical role of a CH in an astrophysical BH. 888In particular, a semiclassical divergence that occurs at the outgoing IH section of the Kerr-Newman background is not expected to realize in a realistic BH produced by gravitational collapse. Our results show that both $\left\langle T_{uu}^{-}\right\rangle_{ren}$ and $\left\langle T_{vv}^{-}\right\rangle_{ren}$ are generically nonvanishing — demonstrating for the first time the divergence of the Kruskal flux component $\left\langle T_{VV}\right\rangle_{ren}\propto e^{2\kappa_{-}v}$ at the CH.

It is also worth comparing the semiclassical RSET divergence $\propto e^{2\kappa_{-}v}$ found here with its classical counterpart. Classical perturbations are known to give rise to curvature divergence at the CH, typically like $v^{-n}e^{2\kappa_{-}v}$ (with $n$ a positive integer depending on the type of perturbation) Hiscock:1981 ; OriMassInflation:1991 ; OriLinear:1999 . In this sense, the aforementioned semiclassical divergence at the CH is stronger than the one associated with classical perturbations.

Our numerical results indicate that all flux components change their signs at around $Q/M\sim 0.97$, being negative for larger $Q/M$ and positive (and typically much larger) for smaller $Q/M$ values. The sign may have crucial implications to the nature of the tidal effect: catastrophic contraction (for $\left\langle T_{vv}^{-}\right\rangle_{ren}>0$) vs. expansion (for $\left\langle T_{vv}^{-}\right\rangle_{ren}<0$).

We also made initial steps towards analyzing the semiclassical backreaction effects of the fluxes on the near-CH geometry (in both the Unruh and HH states). The result expressed in Eq. (15) hints for drastic deformation of the area coordinate $r$ on approaching the CH. However, the analysis provided here was rather preliminary. It should be extended, as mentioned, beyond the domain of weak backreaction. In particular, this picture may change in the next iteration, in which the RSET is re-evaluated with respect to the *backreacted *geometry.

Other obvious extensions are in order. First, it would be worthwhile to generalize the analysis to all RSET components, and also to the entire interior domain $r_{-}<r<r_{+}$. More importantly, this investigation should be extended from the scalar to the more realistic electromagnetic quantum field — and in addition, from the spherical RN background to the astrophysically much more relevant background of a spinning BH.

Acknowledgements.

This work was supported by the Asher Fund for Space Research at the Technion, and by the Israel Science Foundation under Grant No. 600/18.