# Backreaction from quantum fluxes at the Kerr inner horizon

Tyler McMaken<EMAIL_ADDRESS>JILA and Department of Physics,
University of Colorado, Boulder, Colorado 80309, USA

###### Abstract

Black holes modeled by the Kerr metric are not semiclassically self-consistent
at or below the inner horizon. The renormalized stress-energy tensor (RSET) of
a scalar quantum field in the Unruh state has been found to diverge at the
Kerr inner horizon, causing the geometry to backreact in a non-trivial way. In
an effort to understand this backreaction, here the inner-horizon RSET is
computed for the full physically relevant parameter space of black hole spins
$a$ and polar angles $\theta$. Then, the backreaction is analyzed using a
framework for the dynamical behavior of mass inflation from continued
accretion. It is shown that the initial backreaction from the RSET does not
evolve the spacetime toward any known regular or extremal configuration, but
instead it brings the local interior geometry toward a chaotic, spacelike
singularity, classically stable over astrophysical timescales.

##  I Introduction

In general relativity, a rotating black hole is usually modeled by the Kerr
metric, an axisymmetric solution to Einstein’s field equations in an empty
spacetime. Within the Kerr black hole, an inner horizon marks the boundary of
Cauchy predictability, and beyond it lies an observable singularity, closed
timelike curves, and a wormhole to another external universe. But what if the
spacetime is not empty? Is it possible for these structures to form in a real
gravitational collapse? Put another way, what is the fate of the inner horizon
and its traversability if a Kerr black hole is subject to perturbations?

In the case of classical perturbations, the inner horizon is subject to the
mass inflation instability [1], an exponential divergence of the locally
measured stress-energy due to the presence of both ingoing and outgoing
streams of radiation or matter accreted onto the black hole. The backreaction
from this classical instability generically results in the geometry’s collapse
toward a strong, chaotic, oscillatory, spacelike singularity [2, 3, 4, 5, 6,
7, 8, 9], though if the black hole remains isolated from all external
influences aside from its own Price tail, the backreaction could result in a
weak, null singularity [10, 11].

In light of the classical divergence of stress-energy near the inner horizon,
one may wonder what role quantum effects play in the backreaction as the
curvature approaches the Planck scale. If a field in a vacuum state
${|\psi\rangle}$ is canonically quantized over a curved spacetime, its stress-
energy contribution to one-loop order can be renormalized and used as a source
for semiclassical field equations [12, 13] (throughout, assume geometrized
units where $G$=$c$=$\hbar$=1):

$G_{\mu\nu}=8\pi\langle\psi|\hat{T}_{\mu\nu}|\psi\rangle_{\text{ren}}.$ (1)

The computation of this renormalized stress-energy tensor (RSET) is a
difficult task for physically realistic vacuum states in even the most
symmetric spacetimes. But in recent years, novel computational methods have
led to a resurgence of interest, and most relevant to the present analysis,
Ref. [14] made use of state subtraction to calculate the mode-summed RSET of a
scalar field at the inner horizon of a four-dimensional (4D) Kerr black hole
(see [15] for a thorough derivation and Fig. 1 for the parameter space covered
by the study). The main conclusion of the study was that the ingoing double-
null flux component of the RSET (the $vv$-component, in Eddington-Finkelstein
coordinates) is generically nonvanishing, implying that the semiclassical
stress-energy will diverge when cast into coordinates that are regular across
the inner horizon (as seen, for example, in the local frame of an infalling,
outgoing111Infalling (outfalling) here means that the radial component of the
observer’s 4-velocity satisfies ${\dot{r}<0}$ (${\dot{r}>0}$), while ingoing
(outgoing) refers to a left-moving (right-moving) observer on a traditional
Penrose diagram. Ingoing observers are not considered here, since the Unruh
state differs substantially from the more realistic Minkowski in-state along
that portion of the inner horizon. observer reaching the inner horizon). The
implications are much stronger than the classical case—even in the complete
absence of external matter or radiation tails, the mere presence of an
initially empty quantum field is enough to interrupt the gravitational
collapse toward the full Kerr spacetime once the inner horizon is reached.

Figure 1: Parameter space of Kerr spin-to-mass ratios ${a/M}$ and polar angles
$\theta$ for which the dominant components of the RSET ${\langle
T_{\mu\nu}^{-}\rangle}$ have been calculated for a scalar field at the inner
horizon. Cyan plus signs indicate values calculated in Ref. [14] via point
splitting, solid blue triangles indicate values calculated in Ref. [14] via
state subtraction, and hollow orange circles show the values calculated in the
present study via state subtraction.

The divergence of the RSET at the inner horizon implies that the full geometry
of the Kerr metric is not semiclassically self-consistent and must be
substantially modified by the field’s backreaction. Ref. [14] suggested that
the backreaction should depend crucially on the signs of the RSET’s null flux
components, based on prior results from the spherically symmetric case [16]:
if the RSET’s $vv$-component at the inner horizon is positive (negative), an
observer approaching the inner horizon would experience an abrupt contraction
(expansion). Since this component was generically found to change signs for
different latitudes along the inner horizon and for different black hole
spins, the result to first order is a chaotic inner-horizon singularity with
local patches of abrupt contraction or expansion.

A more comprehensive analysis of the semiclassical backreaction in rotating
black holes was recently carried out in the context of the Kerr-de Sitter
spacetime, confirming the postulations of Ref. [14] that the RSET’s
$vv$-component dominates the evolution [17]. But if the RSET’s $vv$-component
changes sign across different latitudes, this would imply that some portions
of the inner horizon would necessarily remain unscathed by any quantum null
fluxes. Could a finely-tuned observer evade the semiclassical singularity at
the inner horizon? Not quite—Ref. [17] found that even in these regions, a
subdominant divergence in the RSET’s $v\varphi$-component causes the local
patch of geometry to experience a diverging amount of relative twist.

The goal of the present study is to provide independent validation of the
prior studies’ numerical work, extend the parameter space of known RSET values
(see Fig. 1), and understand as much as can be possibly gleaned about the Kerr
metric’s semiclassical evolution out of equilibrium. There is a hope that the
semiclassical gravity framework may be sufficient to describe the (meta)stable
endpoint of evolution from a gravitational collapse toward a black hole
[18]—though the black hole may evaporate on super-cosmological timescales, the
mass-inflation and semiclassical instabilities operate on light-crossing
timescales and may push the black hole toward a stable configuration with
extremal horizons, a regular core, a wormhole throat, or no horizon at all
(such as a gravastar). However, here evidence is given against the formation
of such objects from Kerr-like initial conditions. Instead, an analysis of the
backreaction using the mass-inflationary framework of Refs. [4, 5, 6] suggests
that the local geometry near the would-be inner horizon should evolve in a
Belinskii-Khalatnikov-Lifschitz (BKL)-like fashion [19] toward a strong,
spacelike singularity.

The derivation of the key details in the calculation of the RSET is presented
in Sec. II, the numerical results of the calculations are presented in Sec.
III, and the analysis of the backreaction is given in Sec. IV. In this
analysis, it should be noted here that one cannot actually conclude
definitively that a spacelike singularity is the generic outcome of the
semiclassical instability near the inner horizon. The backreaction of the Kerr
RSET is only valid while the geometry can still be well-approximated by the
vacuum Kerr metric, and as soon as the spacetime begins to evolve away from
equilibrium, a new, non-vacuum RSET would need to be found. Most prior studies
remain silent on the search for a path forward in our understanding other than
an admission to the necessity of a full theory of quantum gravity. But here,
another path is suggested: the local inner-horizon geometry under some mild
assumptions becomes elegantly simple during the initial stages of mass
inflation, and it thus may be possible in the future to compute the RSET in
this simplified spacetime to determine the full semiclassical evolution of the
near-inner-horizon geometry until the curvature reaches the Planck scale.

##  II Renormalization of the stress-energy tensor

The Kerr RSET at the inner horizon is here computed using the state
subtraction method employed in Refs. [15, 14]. The key details are outlined in
what follows.

The Kerr line element for a black hole with mass $M$ and angular momentum
${J\equiv aM}$ can be written in Boyer-Lindquist coordinates as

$\displaystyle ds^{2}=\rho^{2}\bigg{(}$
$\displaystyle\frac{dr^{2}}{(r^{2}+a^{2})^{2}\Delta_{r}}+\frac{\sin^{2}\\!\theta\
d\theta^{2}}{\Delta_{\theta}}$
$\displaystyle+\frac{\Delta_{\theta}(d\varphi-\Omega_{r}dt)^{2}-\Delta_{r}(dt-\Omega_{\theta}d\varphi)^{2}}{(1-\Omega_{r}\Omega_{\theta})^{2}}\bigg{)},$
(2)

with the conformal factor $\rho^{2}\equiv r^{2}+a^{2}\cos^{2}\\!\theta$, the
horizon function $\Delta_{r}\equiv(r^{2}+a^{2}-2Mr)/(r^{2}+a^{2})^{2}$ (with
two roots at the outer and inner horizon radii $r_{\pm}\equiv
M\pm\sqrt{M^{2}-a^{2}}$), the polar function
$\Delta_{\theta}\equiv\sin^{2}\\!\theta$, the angular velocity
$\Omega_{r}\equiv a/(r^{2}+a^{2})$ of the principal frame through the
coordinates, and the specific angular momentum $\Omega_{\theta}\equiv
a\sin^{2}\\!\theta$ of principal null congruence photons.

If a massless scalar test field $\Phi$ is minimally coupled to the Einstein-
Hilbert action and canonically quantized, the resulting Klein-Gordon wave
equation $\square$$\Phi=0$ is separable [20, 21] and lends itself to the field
mode decomposition

$\Phi_{\omega\ell m}(t,r,\theta,\varphi)=\text{e}^{im\varphi-i\omega t}S_{\ell
m}^{\omega}(\theta)R_{\omega\ell m}(r),$ (3)

where ${S_{\ell m}^{\omega}(\theta)}$ are prolate spheroidal harmonics,
normalized on the two-sphere according to the Meixner-Schäfke scheme (such a
normalization adds some extra factors to the calculation but is the scheme of
choice for the functions provided in Mathematica) [22].

The field $\Phi$ is then equipped with an Unruh-type vacuum state
${|U\rangle}$ [23, 24] to mimic the effects of a physically realistic
gravitational collapse, with positive frequencies along the past null
boundaries defined with respect to the proper time of an infalling observer
asymptotically approaching those radii [25].

In order to calculate the Unruh-state RSET at the inner horizon, Ref. [14]
constructed a bare mode sum composed of the appropriate differential operator
acting on the Hadamard two-point function. Since this sum is formally
divergent, the summand was then subtracted from that of another bare mode sum
equipped with a vacuum state known to lead to a vanishing RSET at the inner
horizon, in order to yield a finite RSET. This latter vacuum state,
constructed from a time-reversed, negative-mass Kerr spacetime, may appear
unconventional but nonetheless agrees with the Unruh-state RSET obtained from
a “comparison” state subtraction [17] as well as from a traditional point
splitting approach [14] (at least for two different values of the black hole
spin parameter on the polar axis; see Fig. 1).

Computing the two-point function at the inner horizon for a field initialized
at the past null boundaries necessitates solving a relativistic scattering
problem. Although the Unruh-state modes at the past horizon are not eigenmodes
of the radial Teukolsky equation, they can nonetheless be Fourier-decomposed
into an orthonormal set of eigenmodes $R^{\text{in}}_{\omega\ell m}$
(originating from past null infinity) and $R^{\text{up}}_{\omega\ell m}$
(originating from the past horizon) so that the two-point function and
therefore the RSET at the inner horizon can be expressed in terms of
numerically attainable 1D scattering coefficients [15]. These coefficients are
calculated using the Mano-Suzuki-Takasugi (MST) method [26, 27], using a
version of the Black Hole Perturbation Toolkit (BHPT) [28] adapted by the
author; see Appendix A of Ref. [25] for details. In the notation of MST222The
translation from the notation of MST, BHPT, and the present work to that of
Ref. [14] can be accomplished by setting $\rho^{\text{up}}_{\omega\ell
m}={(1-a^{2})^{-2i\omega_{+}(1-a^{2})^{-1/2}}C^{\text{ref}}_{\text{ext}}/C^{\text{inc}}_{\text{ext}}}$
and $B_{\omega\ell
m}={(1-a^{2})^{im/a}(r_{-}/r_{+})^{1/2}B^{\text{trans}}_{\text{int}}/B^{\text{trans}}_{\text{ext}}}$,
with the remaining coefficients obtained from the Wronskian conditions of Ref.
[25]., these scattering coefficients satisfy the asymptotic conditions

$\displaystyle R^{\text{in}}_{\omega\ell m}$
$\displaystyle\to\begin{cases}B^{\text{ref}}_{\text{ext}}r^{-1}\text{e}^{i\omega
r^{*}}+B^{\text{inc}}_{\text{ext}}r^{-1}\text{e}^{-i\omega
r^{*}},&r\to\infty\\\
B^{\text{trans}}_{\text{ext}}\text{e}^{-i\omega_{+}r^{*}},&r\to r_{+}\\\
B^{\text{ref}}_{\text{int}}\
\text{e}^{i\omega_{-}r^{*}}+B^{\text{trans}}_{\text{int}}\text{e}^{-i\omega_{-}r^{*}},&r\to
r_{-}\end{cases},$ (4a) $\displaystyle R^{\text{up}}_{\omega\ell m}$
$\displaystyle\to\begin{cases}C^{\text{trans}}_{\text{ext}}r^{-1}\text{e}^{i\omega
r^{*}},&r\to\infty\\\ C^{\text{inc}}_{\text{ext}}\
\text{e}^{i\omega_{+}r^{*}}+C^{\text{ref}}_{\text{ext}}\text{e}^{-i\omega_{+}r^{*}},&r\to
r_{+}\\\ C^{\text{trans}}_{\text{int}}\
\text{e}^{i\omega_{-}r^{*}}+C^{\text{ref}}_{\text{int}}\text{e}^{-i\omega_{-}r^{*}},&r\to
r_{-}\end{cases},$ (4b)

where ${\omega_{\pm}\equiv\omega-m\Omega_{r_{\pm}}}$ and the tortoise
coordinate $r^{*}$ is chosen to be

$r^{*}\equiv
r+\frac{1}{2\kappa_{+}}\ln\left|\frac{r-r_{+}}{2M}\right|+\frac{1}{2\kappa_{-}}\ln\left|\frac{r-r_{-}}{2M}\right|,$
(5)

with the surface gravity ${\kappa_{\pm}\equiv
Mr_{\pm}\Delta_{r}^{\prime}(r_{\pm})}$ (where a prime denotes differentiation
with respect to $r$), defined to be negative at the inner horizon.

In terms of these scattering coefficients, the state-subtracted mode sum
yielding the RSET at the inner horizon can be written explicitly [15, 17]. In
terms of the interior Eddington-Finkelstein coordinates ${u\equiv r^{*}-t}$
and ${v\equiv r^{*}+t}$, the three most relevant components of the RSET are
the $vv$-, $v\varphi$-, and $uu$-components, which are given by the equations
[14, 17]

$\displaystyle\langle T_{vv}^{-}\rangle$
$\displaystyle=\sumint\left(E_{vv(\omega\ell
m)}^{U-}-\coth\frac{\pi\omega_{-}}{-\kappa_{-}}\right),$ (6a)
$\displaystyle\langle T_{v\varphi}^{-}\rangle$
$\displaystyle=\sumint\frac{-m}{\omega_{-}}\left(E_{vv(\omega\ell
m)}^{U-}-\coth\frac{\pi\omega_{-}}{-\kappa_{-}}\right),$ (6b)
$\displaystyle\langle T_{uu}^{-}\rangle-\langle T_{vv}^{-}\rangle$
$\displaystyle=\sumint\left(\coth\frac{\pi\omega_{+}}{\kappa_{+}}-1\right)\left(1-\left|\frac{C^{\text{ref}}_{\text{ext}}}{C^{\text{inc}}_{\text{ext}}}\right|^{2}\right).$
(6c)

Here and throughout, each RSET component is written as ${\langle
T_{\mu\nu}^{\pm}\rangle\equiv\langle
U|\hat{T}_{\mu\nu}|U\rangle_{\text{ren}}|_{r_{\pm}}}$, and the symbols
$\sumint$ and ${E^{U-}_{vv(\omega\ell m)}}$ are defined by

$\displaystyle\sumint$ $\displaystyle(X)\equiv$
$\displaystyle\int_{0}^{\infty}\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\frac{(2\ell+1)(\ell-m)!}{(\ell+m)!}\frac{\omega_{-}|S_{\ell
m}^{\omega}(\theta)|^{2}}{32\pi^{2}Mr_{-}}(X)\ d\omega,$ (7) $\displaystyle
E_{vv(\omega\ell m)}^{U-}$ $\displaystyle=$
$\displaystyle\frac{r_{-}\omega_{-}}{r_{+}\omega_{+}}\Bigg{[}$
$\displaystyle\coth\frac{\pi\omega_{+}}{\kappa_{+}}\left(\left|\frac{B^{\text{ref}}_{\text{int}}}{B^{\text{trans}}_{\text{ext}}}\right|^{2}+\left|\frac{C^{\text{ref}}_{\text{ext}}}{C^{\text{inc}}_{\text{ext}}}\right|^{2}\left|\frac{B^{\text{trans}}_{\text{int}}}{B^{\text{trans}}_{\text{ext}}}\right|^{2}\right)$
$\displaystyle+2\text{csch}\frac{\pi\omega_{+}}{\kappa_{+}}\Re\left(\frac{C^{\text{ref}}_{\text{ext}}B^{\text{ref}}_{\text{int}}B^{\text{trans}}_{\text{int}}}{C^{\text{inc}}_{\text{ext}}(B^{\text{trans}}_{\text{ext}})^{2}}\right)$
$\displaystyle+\left(1-\left|\frac{C^{\text{ref}}_{\text{ext}}}{C^{\text{inc}}_{\text{ext}}}\right|^{2}\right)\left|\frac{B^{\text{trans}}_{\text{int}}}{B^{\text{trans}}_{\text{ext}}}\right|^{2}\Bigg{]}.$
(8)

Thus, the RSET at the Kerr inner horizon can be calculated in a
straightforward manner by numerically solving the radial and angular Teukolsky
equations and applying their eigenmode solutions to Eqs. (6a)–(6c).

##  III Numerical results

Figure 2: Spectrum of ${\langle T_{vv}^{-}\rangle}$ at the north or south pole
(${\theta=0^{\circ}}$) for a Kerr black hole with ${a/M=0.2}$. The sum over
the $\ell$\- and $m$-modes in the inset panel yields each point in main panel
(note that only the ${m=0}$ modes contribute when ${\theta=0^{\circ}}$), and
the integral over the $\omega$-modes in the main panel yields the total RSET
${\langle T_{vv}^{-}\rangle}$ (i.e., the point at ${a/M=0.2}$ in Fig. 3).
Positive (negative) values on the log plots are given by the red (blue)
points. The sampling and integration is performed using the adaptive
quadrature algorithm described in the text. Figure 3: Computed values of the
$vv$\- and $uu$-components of the RSET at the Kerr inner horizon along the
axis of rotation for different values of the black hole spin-to-mass ratio
${a/M}$. Positive (negative) values on the log plot are given by the red
(blue) points. Solid curves interpolate between the numerically-computed
points, and the dashed curve shows a 1/$a^{2}$ fit at low spins.  Figure 4:
Same setup as Fig. 3 but for black hole spins near extremality, with the near-
zero parameter ${\epsilon\equiv\sqrt{1-(a/M)^{2}}}$. The two lines show the
asymptotic behavior of the RSET’s double-null components near extremality,
given by the analytic expressions ${\langle
T_{vv}^{-}\rangle\to\epsilon^{4}/(7680\pi^{2})}$ and ${\langle
T_{uu}^{-}\rangle\to\epsilon^{5}/(960\pi^{2})}$. Figure 5: Parameter space for
the $vv$-component of the Kerr RSET at the inner horizon as a function of both
the black hole spin-to-mass ratio ${a/M}$ and the polar angle $\theta$. The
points at the locations given in Fig. 1 were computed explicitly using the
techniques described in the text, and the remaining portion of the parameter
space was filled in via interpolation.

In order to make the calculation of the RSET as efficient as possible,
integration is performed in the present work using an adaptive quadrature
algorithm, which has the advantages of both rapid convergence for preferably
sparse domain sampling and fine control over error estimation. An example of
the output of this scheme for a particular choice of parameters can be seen in
Fig. 2. The ultimate goal is to integrate and sum Eqs. (6a)–(6c) over all
possible $\omega$-, $\ell$-, and $m$-modes, which for brevity are written with
the notation

$\langle T_{\mu\nu}^{-}\rangle\equiv\int_{0}^{\infty}\langle
T_{\mu\nu}^{-}\rangle_{(\omega)}d\omega\equiv\int_{0}^{\infty}\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\langle
T_{\mu\nu}^{-}\rangle_{(\omega\ell m)}d\omega.$ (9)

Each point in the main panel of Fig. 2 corresponds to the integrand ${\langle
T_{vv}^{-}\rangle_{(\omega)}}$ for a particular value of the frequency
$\omega$, and each of these points is computed by summing over all possible
$\ell$\- and $m$-modes (see inset of Fig. 2). These sums converge
exponentially and usually require only modes with ${\ell\leq 2}$ for the error
threshold to be vastly dwarfed by the error in the $\omega$ integration,
though in some cases like the one shown in the figure, a larger number of
angular modes are needed.

Then, the algorithm proceeds by (i) computing four ${\langle
T_{\mu\nu}^{-}\rangle_{(\omega)}}$ points subdivided evenly in a closed
frequency domain, (ii) interpolating between these points with a cubic
B-spline, (iii) computing a fifth sample point in the center of the domain,
(iv) calculating the difference between this sample point and the center point
of the spline, and (v) if this difference lies above a set error threshold,
the domain is divided evenly in half and the algorithm is repeated from step
(i) for each new domain. This approach leads to a speedup of up to a factor of
10 in the total computational runtime compared to a fixed linear integration
method and additionally allows for more precise error control.

Adaptive quadrature approaches perform most poorly in regimes where the
integrand cannot be well-approximated by polynomial functions, which for the
integrands of Eqs. (6a)–(6c) occurs in the exponential decay regime at high
frequencies (notice in Fig. 2 the slight difference at ${\omega/M\sim 50}$
between the interpolating function and the dashed exponential fit). In these
cases, two options were tested, both yielding similar convergent results:
either the splines can be computed over log-frequency space and the domain
extended until convergence is reached, or enough points can be sampled so that
the remaining portion of the integrand can be fitted to an exponentially
decaying function and the integral extrapolated to infinity.

Two main sources of error are accounted for in the numerical calculations of
the RSET performed here. The first is truncation error, which is minimized in
the $\ell$\- and $m$-sums by cutting off the sum only when the next term
returns zero with the specified numerical precision, and which is controlled
in the $\omega$-integrals by the degree of confidence in the exponential decay
fit that is integrated to infinity. The second source of error is the global
discretization error from the numerical integration scheme, which is
controlled by specifying an error bound in the algorithm described above.
Accounting for both of these sources of uncertainty, the points in Figs. 2–4
are computed to a high enough precision that their error bars are smaller than
the points themselves in all but a couple edge cases.

First, consider the inner-horizon RSET at the north or south pole, where
${\theta=0^{\circ}}$. There only the modes with ${m=0}$ contribute to the
RSET, so that ${\langle T_{v\varphi}^{-}\rangle}$ vanishes and the calculation
of the other null components simplify considerably. The two double-null flux
components of the RSET are plotted as a function of the black hole’s angular
momentum $a$ in Figs. 3 and 4. Fig. 3 shows computed values for spins from
${a/M=0.15}$ to ${a/M\approx 0.997}$, yielding excellent quantitative
agreement with Fig. 2 of Ref. [14] and qualitative agreement with the Kerr-de
Sitter case in Fig. 4 of Ref. [17] (both of which only reach a minimum of
${a/M=0.55}$). The $vv$-component of the RSET is negative and vanishingly
small at high spins, and as the rotation of the black hole slows, this
component increases until reaching zero at ${a/M\approx 0.862}$ and continuing
to increase exponentially in ${a/M}$. However, at slow enough spins, ${\langle
T_{vv}^{-}\rangle}$ once again changes signs.

Both ${\langle T_{vv}^{-}\rangle}$ and ${\langle T_{uu}^{-}\rangle}$ are
expected to diverge as ${a\to 0}$, since in that limit the inner horizon
coincides with the ${r=0}$ singularity. In particular, the quantity ${\langle
T_{uu}^{-}\rangle-\langle T_{vv}^{-}\rangle}$, related to the Hawking outflux
per surface area ${4\pi r_{-}^{2}}$ [16], should diverge as $a^{-2}$ (compare
the dashed curve in Fig. 3) owing to the factor of $r_{-}$ in the denominator
of Eq. (II). An even stronger divergence is expected to be present in
${\langle T_{vv}^{-}\rangle}$ as ${a\to 0}$, since the scattering coefficients
${|B^{\text{ref}}_{\text{int}}|^{2}}$ and
${|B^{\text{trans}}_{\text{int}}|^{2}}$ from Eq. (4a) both diverge as
$a^{-2}$. However, the exact nature of the low-spin divergence of ${\langle
T_{vv}^{-}\rangle}$ is not easily found, since the spectrum includes non-
trivial contributions from successively higher frequencies $\omega$ as the
spin $a$ decreases, so that for ${a/M<0.15}$, ${\langle T_{vv}^{-}\rangle}$
could be either positive or negative (or even vanish in a highly fine-tuned
case).

While the double-null components of the RSET are expected to diverge at the
inner horizon as ${a/M\to 0}$, they will vanish as ${a/M\to 1}$, as shown in
Fig. 4 (compare Fig. 3 in the Supplemental Materials of Ref. [14]). In
particular, through an analysis of the asymptotic behavior of the near-
extremal scattering coefficients, it can be shown that in terms of a near-
extremal spin parameter ${\epsilon\equiv\sqrt{1-(a/M)^{2}}}$, the
$vv$-component of the RSET will vanish as ${\langle
T_{vv}^{-}\rangle\to\epsilon^{4}/(7680\pi^{2})}$, while the $uu$-component
will vanish as ${\langle T_{uu}^{-}\rangle\to\epsilon^{5}/(960\pi^{2})}$.

Off the axis of rotation, the computation of the inner-horizon RSET is less
clean. Even in the near-extremal limit, while the on-axis scattering
coefficients are dominated by low-frequency ${\ell=0}$ modes that create a
simple negative, exponentially decaying spectrum, the off-axis spectrum
generally contains both positive and negative peaks at low frequencies.
Nevertheless, in the limit as ${a/M\to 1}$ for all polar angles $\theta$, all
three components of the RSET analyzed here vanish.

The behavior of ${\langle T_{vv}^{-}\rangle}$ as a function of both spin $a$
and polar angle $\theta$ is shown in Fig. 5. As can be seen, while the sign of
${\langle T_{vv}^{-}\rangle}$ changes as the spin of the black hole is varied,
even for a single black hole with a given spin, ${\langle T_{vv}^{-}\rangle}$
will necessarily change signs at different latitudes along the inner horizon.
For ${a/M\gtrsim 0.862}$, ${\langle T_{vv}^{-}\rangle}$ is negative near the
pole and reaches zero exactly once between the pole and the equator. For
smaller spins, ${\langle T_{vv}^{-}\rangle}$ may change signs twice, though it
should be noted that the behavior of the RSET for large $\theta$ and very
small spins in Fig. 5 is only an extrapolated estimation.

Though not plotted here, the $v\varphi$-component of the RSET has also been
computed and is generically non-zero throughout the 2D parameter space, except
for at least one 1D zero-valued contour (just as ${\langle T^{-}_{vv}\rangle}$
in Fig. 5 contains two333Or three, if you count the $\theta$-parametrized
contour at ${a/M=1}$, though in this work the extremal case is not treated.
zero-valued contours given by the white regions). Though the divergence
associated with ${\langle T_{v\varphi}^{-}\rangle}$ is subdominant compared to
${\langle T_{vv}^{-}\rangle}$, the presence of a sign flip in ${\langle
T_{vv}^{-}\rangle}$ implies that ${\langle T_{v\varphi}^{-}\rangle}$ will
dominate the backreaction for at least one latitude, leading to a divergent
twisting of the geometry separating regions of local expansion and contraction
[17].

In conclusion, the renormalized stress-energy of a scalar quantum field in the
Unruh state has been calculated for most of the physically relevant parameter
space in the Kerr spacetime at the inner horizon. The results indicate that in
a locally inertial frame reaching the inner horizon, the field’s flux
generically diverges, and thus that the Kerr metric is not semiclassically
self-consistent at or beyond the inner horizon. To understand how the geometry
backreacts to this semiclassical instability, one must analyze the field
equations, as shown in the next section.

##  IV Backreaction

Analyses of semiclassical backreactions carry with them intricate subtleties.
Ideally, one would like to find a solution to the M$\o$ller-Rosenfeld
semiclassical field equations, Eq. (1), to see how the geometry and the
quantum field coevolve over time. However, as can be gleaned from the
calculations of the previous sections, the RSET is difficult to calculate and
currently is only known for a select handful of highly symmetric vacuum
spacetimes. Thus, the present calculations will apply only to a _weak
backreaction domain_ , where one is still far enough from the inner horizon
that the geometry can be well-approximated by the Kerr metric but close enough
that the semiclassical backreaction, of order $(\text{e}^{\kappa_{-}v}\
m_{P}/M)^{2}$ (where $m_{P}$ denotes the Planck mass), is not negligible.

###  IV.1 Spherical symmetry

The conclusions of any backreaction analysis are limited by the choice of
assumptions about how the metric in question should be generalized. In the
case of spherical symmetry (when ${a=0}$), such generalizations can be made
comprehensively. If double-null coordinates $u$ and $v$ are gauge-fixed to
match those of the vacuum RSET ${\langle T^{-}_{\mu\nu}\rangle}$, then the
remaining two functional degrees of freedom in the line element can be written
as $-\text{e}^{\sigma(u,v)}dudv+r(u,v)^{2}d\Omega^{2}$. The resulting field
equations imply that near the inner horizon,

$\partial_{v}r\approx\frac{4\pi r_{-}}{\kappa_{-}}\langle
T^{-}_{vv}\rangle+\mathcal{O}(\text{e}^{-\kappa_{-}v}).$ (10)

Recalling that the inner-horizon surface gravity $\kappa_{-}$ is defined here
to be negative, one would then conclude that an infalling observer near the
inner horizon would experience an abrupt contraction (expansion) in the
geometry’s area element when ${\langle T^{-}_{vv}\rangle}$ is positive
(negative) [16].

However, if instead the radial coordinate $r$ is gauge-fixed to match that of
the vacuum RSET and the remaining two metric degrees of freedom are encoded
by, for example, the positions of the black hole’s inner and outer horizons,
the same field equations then imply dynamical behavior in the horizon
structure rather than in the local area element. In particular, the outer
horizon will evaporate inward slowly over time, as per Hawking’s famous result
(as long as ${\langle T^{+}_{vv}\rangle<0}$, which is always true in the 2D
case) [29, 30, 31], while the inner horizon will evaporate rapidly outward
(inward) when ${\langle T^{-}_{vv}\rangle}$ is negative (positive). The inner
horizon in this case will move extremely quickly, at a timescale on the order
of the black hole’s light-crossing time [31].

While the two options described above for the spherically symmetric case might
seem distinct or even at odds with one another, they are actually completely
compatible. Extrapolating and reinterpreting those results as statements about
the dominant contributions to global, long-term dynamics, “which is
irresistible but not allowed” [32], would lead to the conclusion that black
holes with ${\langle T^{-}_{vv}\rangle>0}$ have interiors that contract until
a spacelike singularity is formed at ${r=0}$, while black holes with ${\langle
T^{-}_{vv}\rangle<0}$ would explode from the inside out to form extremal or
horizonless objects. However, since the RSET calculated here only remains
valid in the weak backreaction domain, absolutely no conclusions can be made
about long-term dynamics until the RSET is computed for the more general form
of the spacetime.

###  IV.2 Axisymmetry: initial tendencies

Now consider the case of black holes with non-zero rotation. Here it is clear
that the outcome must be more complicated than in the spherically symmetric
case, since the local interior geometry will no longer uniformly contract or
expand across the entire inner horizon; instead, ${\langle T^{-}_{vv}\rangle}$
is both positive and negative across different latitudes of the same black
hole (see Fig. 5). An excellent backreaction analysis for the case of Kerr-de
Sitter black holes was recently [17] carried out by decomposing the spacetime
into a set of double-null hypersurfaces gauge-fixed to match the null
coordinates in ${\langle T^{-}_{\mu\nu}\rangle}$.444The fixing comes in the
identification of the affine parameter $\lambda$, taken to be constant along
each hypersurface, with the Kruskal coordinate
$V_{-}\propto\text{e}^{-\kappa_{-}v}$. Along each hypersurface, the induced
metric can be written in a completely general form as

$g_{AB}dx^{A}dx^{B}=\gamma^{2}\left[\alpha^{-2}d\theta^{2}+\alpha^{2}\left(d\varphi+\tau
d\theta\right)^{2}\right],$ (11)

for the area element $\gamma^{2}$ and the arbitrary functions $\alpha$ and
$\tau$ that physically relate to notions of shear and twisting of null
geodesics as the hypersurface is moved toward the would-be inner horizon at
${\lambda\to 0}$, or equivalently, ${v\to\infty}$. The semiclassical Einstein
equations and Raychaudhuri equations can then be analyzed to yield

$\displaystyle\partial^{2}_{\lambda}\gamma=-\gamma\left[4\pi\langle
T^{-}_{\lambda\lambda}\rangle+4(\partial_{\lambda}\ln\alpha)^{2}+\alpha^{4}(\partial_{\lambda}\tau)^{2}\right],$
(12a)
$\displaystyle\partial_{\theta}(\gamma^{2}\alpha^{4}\partial_{\lambda}\tau)=8\pi\gamma^{2}\langle
T^{-}_{\lambda\varphi}\rangle+\partial_{\lambda}(\gamma^{2}\beta),$ (12b)

where $\beta$ measures the amount of twisting in the $\varphi$-direction for
light rays perpendicular to the surfaces of constant $\lambda$ as ${\lambda\to
0}$.

From Eq. (12a) it can be gleaned that for positive (negative) ${\langle
T^{-}_{vv}\rangle}$, the divergent RSET component ${\langle
T^{-}_{\lambda\lambda}\rangle\sim\lambda^{-2}}$ will also be positive
(negative) and will cause $\gamma$ to contract (expand) as ${\lambda\to 0}$.
In the inevitable intermediate cases where $\gamma$ neither contracts nor
expands, Eq. (12b) then predicts that a non-zero $\langle
T^{-}_{v\varphi}\rangle$ will cause an infinite local twisting. Thus, one
reproduces the same behaviors predicted in the spherically symmetric case
(i.e., the local blow-up or shrinking of the geometry near the inner horizon
dictated by the sign of ${\langle T^{-}_{vv}\rangle}$), with the addendum that
even in the cases where the geometry neither contracts nor expands, a
subdominant component of the RSET will still generically cause the local
geometry to diverge in a shearing manner.

Two comments concerning the above analysis are worth mentioning. First, while
the backreaction is performed with sufficient generality to reproduce the
effects predicted in the spherical case and predict additional rotational
effects, the restriction of the metric to null hypersurface cuts of constant
$v$ does not allow for a full global analysis of the spacetime’s evolution and
does not encompass the most general axisymmetric geometry possible (it for
example washes out any dynamical information related to the outgoing $u$
coordinate and its coupling with the other degrees of freedom). But more
importantly, the analysis does not invite any obvious path forward to
understand how the spacetime evolves beyond these initial tendencies. Does the
semiclassical inner-horizon instability remain confined to a (meta)stable
interior singularity (or even a regular configuration), or will inflationary
perturbations spread like a wildfire until they destroy the black hole from
the inside out? To address these problems at least partially, consider the
complementary analysis below.

Generalizations of the Kerr metric abound. Starting from the line element of
Eq. (II), one may, for example, promote the mass $M$ to an arbitrary function
${M(r,\theta)}$ [33], or allow dynamical behavior via ${M\to M(v)}$ and ${a\to
a(v)}$ [34], or leave all the functions $\Delta_{r}(r)$,
$\Delta_{\theta}(\theta)$, $\Omega_{r}(r)$, $\Omega_{\theta}(\theta)$, and
${\rho^{2}=\rho_{r}^{2}(r)+\rho_{\theta}^{2}(\theta)}$ arbitrary for full
Hamilton-Jacobi separability [20]. Additional generalizations beyond the form
of Eq. (II) also exist; imposing Klein-Gordon and timelike Hamilton-Jacobi
separability yields a 3-function class of metrics that reproduces a wide
variety of regular and singular spacetime candidates [35], and imposing only
asymptotic flatness and the preservation of the Carter constant leads to a
10-function class of general axisymmetric metrics [36]. Amidst all these
options, one thus desires a trade-off between generality and tractability in
capturing the most important physical behaviors to be modeled and understood.

The properties of the semiclassical radiation produced from a quantum scalar
field in the vacuum Kerr spacetime already seem to rule out a number of Kerr
generalizations. For example, the 3-function class of metrics put forward in
Ref. [35] claims to encompass a majority of the currently-proposed stable
endpoints of black hole evolution (aside from complete evaporation or
gravastar-like objects); however, the Einstein tensor for these metrics always
contains a vanishing $rt$-component, despite the fact that the semiclassical
RSET calculated here possesses a non-vanishing (and in fact diverging)
$rt$-component ${\langle T^{-}_{rt}\rangle}={\left(\langle
T^{-}_{vv}\rangle-\langle T^{-}_{uu}\rangle\right)R^{-2}\Delta_{r}^{-1}}$.
Presumably such black hole solutions that are regular and instability-free
cannot form from an initial Kerr-like collapse.

###  IV.3 Axisymmetry: mass inflationary approach

To generalize the Kerr metric in order to allow for the dynamics anticipated
from the vacuum RSET, the following assumptions will be made:

  1. (i)

The spacetime is axisymmetric, motivated both by simplicity and by the
preservation of this symmetry in the past Unruh state.

  2. (ii)

The spacetime is asymptotically flat at spatial infinity, as in the case of
the original Kerr metric.

  3. (iii)

The spacetime is initialized with a Kerr vacuum geometry, and the quantum
scalar field is initialized with an Unruh vacuum state mimicking the effects
of a gravitational collapse sufficiently far in the past.

  4. (iv)

The spacetime maintains conformal Hamilton-Jacobi separability [5]; i.e., the
equations of motion for massless particles are separable in spheroidal
coordinates (but not necessarily for massive particles, as in the case of
strict separability seen for Kerr).

Unfortunately, none of these assumptions are actually true for astrophysically
realistic black holes. The accretion flows observed by the GRAVITY instrument
[37] and the Event Horizon Telescope Collaboration [38] feature asymmetries in
$\varphi$, though in the stationary limit, rigidity theorems suggest that any
non-axisymmetric perturbations are part of the “hair” that a black hole will
eventually shed [39, 40]. In regard to Assumption (ii), the lack of asymptotic
flatness in the Universe has been measured with great precision [41], but any
effects from a cosmological horizon on the inner-horizon instability are
expected to be negligible, both due to the similarity in the RSET calculations
here to those of the Kerr-de Sitter case [17], and because the cosmological
constant is currently measured to be $\Lambda\sim 10^{-46}\
M_{\astrosun}^{-2}$ in geometrized units, much too small to have any practical
influence. As for Assumption (iii), black holes (and therefore near-inner-
horizon geometries) form under chaotic, rapidly evolving conditions, and even
after settling down, the most quiescent black holes are still non-vacuum and
accrete more than enough radiation to trigger the mass inflation instability
and destroy any vacuum inner horizon [8]. Additionally, physically relevant
fields in the Standard Model are those with spin 1 (electromagnetic), 1/2
(fermionic), or 2 (gravitational), so using a massless scalar field with spin
0 is a simplification seen as a proxy for, e.g., a single photonic degree of
freedom. Even regarding Assumption (iv), which provides a more general notion
than the strict separability that itself is justified by the observation of
long-lived accretion disks (lack of separability implies chaotic,
destabilizing particle orbits [35]), it will be seen in what follows that
conformal separability, while valid in the weak backreaction domain,
eventually breaks down once the vierbein of Eq. (16) ceases to be diagonal.

Under Assumptions (i)–(iv), the spacetime’s line element takes the form of Eq.
(II), except that the functions $\Delta_{r}(r)$, $\Delta_{\theta}(\theta)$,
$\Omega_{r}(r)$, $\Omega_{\theta}(\theta)$, and $\rho(r,\theta,t)$ are left
arbitrary. Assumption (ii) of asymptotic flatness, while not strictly required
for any aspect of this analysis (any cosmological contribution to the near-
inner-horizon geometry will be completely overwhelmed by the ignition of both
classical and semiclassical streams [5]), will impose the additional
conditions $\Delta_{r}\sim\Omega_{r}\sim r^{-2}$ and $\rho\sim r$.

The form of the line element of Eq. (II) naturally encodes an orthonormal
tetrad basis first written down by Carter [20]. This basis can be constructed
in the Newman-Penrose formalism by performing a null rotation on the
Kinnersley tetrad so that the resulting frame is symmetric with respect to the
ingoing and outgoing principal null directions [42]. The corresponding
vierbein $e^{\hat{m}}_{\ \mu}$, defined through

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=\eta_{\hat{m}\hat{n}}e^{\hat{m}}_{\
\mu}e^{\hat{n}}_{\ \nu}dx^{\mu}dx^{\nu}$ (13)

for the Minkowski metric tensor $\eta_{\hat{m}\hat{n}}$, can be written as the
product [7]

$e_{\hat{m}\mu}=(e_{\text{dyn}})_{\hat{m}\kappa}\ (e_{\text{fix}})^{\kappa}_{\
\mu},$ (14)

where $(e_{\text{dyn}})_{\hat{m}\kappa}$ is a dynamical vierbein that is only
a function of the variable $r$ (which is timelike in the interior), and
$(e_{\text{fix}})^{\kappa}_{\ \mu}$ is a fixed vierbein whose elements should
remain frozen at their inner-horizon values throughout the evolution of the
instability induced by the semiclassical backreaction. In terms of Boyer-
Lindquist coordinates, the fixed vierbein contains the basis vectors

$\displaystyle(e_{\text{fix}})^{0}$ $\displaystyle=\rho\ \partial_{r},$ (15a)
$\displaystyle(e_{\text{fix}})^{1}$
$\displaystyle=\frac{\rho}{1-\Omega_{r}\Omega_{\theta}}\left(\partial_{t}-\Omega_{\theta}\partial_{\varphi}\right),$
(15b) $\displaystyle(e_{\text{fix}})^{2}$ $\displaystyle=\rho\
\partial_{\theta},$ (15c) $\displaystyle(e_{\text{fix}})^{3}$
$\displaystyle=\frac{\rho\sqrt{\Delta_{\theta}}}{1-\Omega_{r}\Omega_{\theta}}\left(\partial_{\varphi}-\Omega_{r}\partial_{t}\right),$
(15d)

which serves to align the tetrad with the principal null directions of the
black hole. The dynamical vierbein then is purely diagonal:

$\displaystyle(e_{\text{dyn}})_{0}$
$\displaystyle=\frac{\text{e}^{-5\xi/2}}{R^{2}\sqrt{|\Delta_{r}|}}\
\partial_{0},$ (16a) $\displaystyle(e_{\text{dyn}})_{1}$
$\displaystyle=\text{e}^{\xi/2}\sqrt{|\Delta_{r}|}\ \partial_{1},$ (16b)
$\displaystyle(e_{\text{dyn}})_{2}$ $\displaystyle=\text{e}^{-\xi}\
\partial_{2},$ (16c) $\displaystyle(e_{\text{dyn}})_{3}$
$\displaystyle=\text{e}^{-\xi}\ \partial_{3},$ (16d)

where the redefinitions
${\rho(r,\theta,t)\to\text{e}^{-\xi(r)}\rho(r,\theta,t)}$ and
$\Delta_{r}(r)\to\text{e}^{3\xi(r)}\Delta_{r}(r)$ have been made for an
arbitrary radial function $\xi(r)$, which is brought to the dynamical vierbein
so that the vierbein may undergo a conformal expansion or contraction under
backreaction.

The Einstein tensor corresponding to this spacetime is far too intricate to be
presented here in any complete, meaningful way. However, at least in the
quasi-stationary limit (in particular, if the redefined conformal piece $\rho$
is written as
$\rho(r,\theta,t)=\text{e}^{\tilde{v}t}\sqrt{r^{2}+a^{2}\cos^{2}\\!\theta}$
for some vanishing small accretion parameter $\tilde{v}$), the field equations
have already been analyzed in detail in Refs. [4, 5, 6]. The resulting
behavior provides classical justification for the decomposition of Eq. (14),
since near the inner horizon, the functions $\Delta_{\theta}$, $\Omega_{r}$,
and $\Omega_{\theta}$ remain approximately fixed at their Kerr values near the
inner horizon, while the horizon function $\Delta_{r}$ and the inflationary
function $\xi$ become the dominant contributors to the evolution of the
Einstein tensor. The resulting tensor behaves at least initially like a
perfect fluid with equal ingoing and outgoing streams, with dominant inflating
tetrad-frame components $G_{00}\sim G_{11}\sim G_{01}\sim\Delta_{r}^{-1}$,
where the off-diagonal component $G_{01}$ is suppressed by a factor of
$\tilde{v}$. Classically, the evolution can then be continued to show that
$\Delta_{r}$ eventually stalls out at some tiny value [5], after which $\xi$
grows large and the spacetime undergoes a series of BKL-like bounces toward a
strong, spacelike singularity [7].

How might semiclassical effects modify this picture? The tetrad-frame
components of the RSET can be re-expressed in terms of the double-null
Eddington-Finkelstein coordinates used throughout this paper:

$\displaystyle\langle T_{00}\rangle$
$\displaystyle=\frac{\text{e}^{5\xi}}{\rho^{2}|\Delta_{r}|}\bigg{[}\langle
T_{uu}\rangle+\langle T_{vv}\rangle+2\langle T_{uv}\rangle\bigg{]},$ (17a)
$\displaystyle\langle T_{11}\rangle$
$\displaystyle=\frac{\text{e}^{-\xi}}{\rho^{2}|\Delta_{r}|}\bigg{[}\langle
T_{uu}\rangle+\langle T_{vv}\rangle-2\langle T_{uv}\rangle$
$\displaystyle-2\Omega_{r}\Big{(}\langle T_{u\varphi}\rangle-\langle
T_{v\varphi}\rangle\Big{)}+\Omega_{r}^{2}\langle
T_{\varphi\varphi}\rangle\bigg{]},$ (17b) $\displaystyle\langle T_{01}\rangle$
$\displaystyle=\frac{2\text{e}^{2\xi}}{\rho^{2}|\Delta_{r}|}\bigg{[}\langle
T_{uu}\rangle-\langle T_{vv}\rangle-\Omega_{r}\Big{(}\langle
T_{u\varphi}\rangle+\langle T_{v\varphi}\rangle\Big{)}\bigg{]},$ (17c)
$\displaystyle\langle T_{13}\rangle$
$\displaystyle=\frac{\text{e}^{\xi/2}}{\rho^{2}\sqrt{|\Delta_{r}|\Delta_{\theta}}}\bigg{[}\Omega_{\theta}\Big{(}\langle
T_{uu}\rangle+\langle T_{vv}\rangle-2\langle T_{uv}\rangle\Big{)}$
$\displaystyle+(1+\Omega_{r}\Omega_{\theta})\Big{(}\langle
T_{v\varphi}\rangle-\langle T_{u\varphi}\rangle\Big{)}+\Omega_{r}\langle
T_{\varphi\varphi}\rangle\bigg{]},$ (17d) $\displaystyle\langle T_{22}\rangle$
$\displaystyle=\frac{\text{e}^{2\xi}\Delta_{\theta}}{\rho^{2}\sin^{2}\\!\theta}\langle
T_{\theta\theta}\rangle,$ (17e) $\displaystyle\langle T_{33}\rangle$
$\displaystyle=\frac{\text{e}^{2\xi}}{\rho^{2}\Delta_{\theta}}\bigg{[}\Omega_{\theta}^{2}\Big{(}\langle
T_{uu}\rangle+\langle T_{vv}\rangle-2\langle T_{uv}\rangle\Big{)}$
$\displaystyle+2\Omega_{\theta}\Big{(}\langle T_{v\varphi}\rangle-\langle
T_{u\varphi}\rangle\Big{)}+\langle T_{\varphi\varphi}\rangle\bigg{]}.$ (17f)

Far enough away from the inner horizon, all components of the RSET should be
completely negligible, as they are suppressed by the Planck scale. Thus,
everywhere outside the black hole as well as inside when $v$ is not large, a
vacuum source should recover the standard Kerr solution with $\xi=0$ and
$\Delta_{r}=(r^{2}+a^{2}-2Mr)/(r^{2}+a^{2})^{2}$. However, in the weak
backreaction domain, once the null components of the RSET cease to be
vanishingly small, Eqs. (17) indicate that the radial- and time-components of
the tetrad-frame Einstein tensor will begin to diverge as $\Delta_{r}^{-1}$ as
one takes $\Delta_{r}\to 0$.

Though the RSET components ${\langle T^{-}_{uv}\rangle}$, ${\langle
T^{-}_{u\varphi}\rangle}$, and ${\langle T^{-}_{\varphi\varphi}\rangle}$ have
not been computed explicitly, one can make the assumption that their
contributions to Eqs. (17) will be subdominant compared to that of the double-
null components, and more importantly, that they will not contrive to cause
any of the specific combinations in the equations above to cancel exactly.
Even if they do, then the classical mass inflation phenomenon described in
Refs. [4, 5, 6] will take over the evolution and a spacelike singularity will
form. But based on the numerical results in Sec. III, it is apparent that
${\langle T^{-}_{00}\rangle}$ and ${\langle T^{-}_{11}\rangle}$ are the
dominant contributors to the evolution in the weak backreaction domain, while
the off-diagonal components ${\langle T^{-}_{01}\rangle}$ are also important
but initially much smaller than their diagonal counterparts (recall from Fig.
3 that the difference ${\langle T^{-}_{uu}\rangle}-{\langle
T^{-}_{vv}\rangle}$ is almost always several orders of magnitude smaller than
either individual component).

The Einstein tensor combination $G_{00}+G_{11}$ for the tetrad frame of Eq.
(14) can be written as

$\displaystyle G_{00}+G_{11}$
$\displaystyle=-\frac{2\Delta_{r}}{\rho^{2}}\bigg{(}\partial^{2}_{r_{*}}\xi+\left(\partial_{r_{*}}\xi\right)^{2}$
$\displaystyle-2\partial_{r_{*}}\ln\left(\frac{\rho}{1-\Omega_{r}\Omega_{\theta}}\right)\partial_{r_{*}}\xi\bigg{)}+F_{rt},$
(18a)

where the tortoise coordinate $r_{*}$ is defined by
$dr/dr_{*}=\text{e}^{3\xi}R^{2}\Delta_{r}$, and the function $F_{rt}$
encompasses subdominant terms related to the precise nature of any classical
accretion contributing to non-stationary conformal dynamics.

The dominant term in Eq. (18) is the one involving the second derivative of
the inflationary exponent $\xi$ [5]. Equating this term to the semiclassical
source from Eq. (17), which behaves as $\Delta_{r}^{-1}$ near the inner
horizon, yields after integration the approximate solution

$\displaystyle\partial_{r_{*}}\xi$
$\displaystyle\approx\frac{8\pi}{-\Delta_{r}}\Big{(}\langle
T^{-}_{uu}\rangle+\langle T^{-}_{vv}\rangle$
$\displaystyle-2\Omega_{r}(\langle T^{-}_{u\varphi}\rangle-\langle
T^{-}_{v\varphi}\rangle)+\Omega_{r}^{2}\langle
T^{-}_{\varphi\varphi}\rangle\Big{)},$ (19a)

as long as $\xi$ remains smaller than its derivatives. In the regime where the
double-null components of the RSET dominate over the shearing components, Eq.
(19) thus predicts that when the sum ${\langle T^{-}_{uu}\rangle+\langle
T^{-}_{vv}\rangle}$ is positive (negative), the spacetime’s conformal factor
$\text{e}^{-\xi}$ will abruptly contract (expand) as the inner horizon is
approached at $\Delta_{r}\to 0_{-}$. These initial tendencies align precisely
with that of the spherically symmetric case, since ${\langle
T^{-}_{uu}\rangle+\langle T^{-}_{vv}\rangle}$ is usually the same sign as
${\langle T^{-}_{vv}\rangle}$.

One can continue the evolution of the spacetime by examining the behavior of
the dynamical vierbein of Eqs. (16a)–(16d). Under the assumption that the
fixed vierbein of Eqs. (15a)–(15d) remains stable as the initial backreaction
ignites the inflationary tendencies described above, the geometry will be
governed solely by a diagonal, dynamical, homogeneous line element sourced by
inflating streams of semiclassical matter. The classical counterpart of this
analysis has already been carried out in detail in Ref. [8], yielding the so-
called inflationary Kasner metric: as one might anticipate, the streams of
matter are amplified in energy density as $\Delta_{r}$ plunges to zero and
$\xi$ increases rapidly. Then, the spacetime undergoes a bounce as
$\Delta_{r}$ freezes out at a small value and the $\text{e}^{\xi/2}$ term in
Eq. (16b) takes over and causes a radial expansion. The solution works for
both positive and negative sources of stress-energy, generically producing a
series of inflating Kasner-like bounces toward a spacelike singularity.

Eventually, the approximations made in this section will break down, and the
final evolution of the geometry near the spacelike singularity must be
relinquished to a higher-order theory of quantum gravity. However, the results
remain robust in the weak backreaction domain, and even in the presence of
additional non-negligible stress-energy sources not accounted for in the
analysis above, it has been shown [5] that the dominant double-null
contributors to the geometry’s initial inflation remain dominant to the next-
highest order, as long as one still has $|\Delta_{r}|\ll\partial_{r_{*}}\xi$
or $\partial_{r_{*}}\xi\ll 1$. If anything, the semiclassical contribution to
the classical mass inflation instability will cause the conformally separable
solution to break down faster than it otherwise would (due to the presence of
shearing terms that act to rotate the dynamical vierbein away from its initial
configuration). But numerical work in the classical case serves to indicate
that even with such perturbations, the generic result should still be a
chaotic, BKL-like collapse toward a spacelike singularity [7].

##  V Discussion

By now it is clear in the literature that quantum fields do not jive well with
vacuum black hole spacetimes. If the Kerr metric (the axisymmetric solution to
Einstein’s equations for a rotating black hole in an empty spacetime) is
immersed within a scalar quantum field, then even if that field begins in an
empty vacuum state, the gravitational collapse leading to the formation of the
black hole will cause a mixing of positive- and negative-frequency modes that
leads to the spontaneous production of particles. Usually, this semiclassical
flux of particles will be suppressed on the order of the Planck mass
$m_{P}^{2}$ divided by the black hole’s mass $M^{2}$, and the Kerr structure
will remain intact, but close enough to the inner horizon, this study confirms
the conclusion of previous studies [14, 17, 25] that semiclassical radiation
will diverge at the Kerr inner horizon.

To understand how the Kerr geometry will react to the diverging quantum field
near the inner horizon, it is natural to work in the framework of the
semiclassical Einstein field Eqs. (1), wherein the spacetime’s curvature is
sourced by the vacuum expectation value of the renormalized stress-energy
tensor (RSET) of the quantum field. The potential problems (mathematical,
physical, and philosophical) associated with such an approach have been
debated again and again over the years [43, 44, 45, 46], but it nonetheless
remains true that the semiclassical approach is perfectly valid as an
effective field theory of quantum gravity, as long as the RSET remains below
Planck-scale energies (at $10^{19}$ GeV, which is already orders of magnitude
above the grand unified scale, beyond the point where every atomic nucleus has
been dissociated and the very notion of particles and interactions is called
into question).

Here it is found that the RSET in double-null coordinates contains a non-zero
component ${\langle U|\hat{T}_{vv}|U\rangle_{\text{ren}}}$ near the Kerr inner
horizon, which, when converted to coordinates that are regular across that
horizon, yields an exponential divergence in the quantum flux. Such behavior
was also found in Refs. [14, 17]; the additional contributions here were to
compute more inner-horizon RSET components over the full parameter space ($a$,
$\theta$) (Fig. 5), find the asymptotic behavior at both high and low black
hole spins (Figs. 3, 4), and analyze the backreaction that the RSET elicits.

The most striking feature of the inner-horizon RSET is that, in contrast to
the classical case (or even the semiclassical 2D case), the double-null
components of the RSET can be either positive or negative at different points
along the inner horizon. The semiclassical Einstein equations then suggest
that as an initial tendency, the local geometry as one approaches the inner
horizon will rapidly contract (expand) wherever ${\langle T^{-}_{vv}\rangle}$
is positive (negative).

In semiclassical backreaction analyses, the above statement is usually the end
of the story. Ref. [17] takes it one step farther to show that in the
inevitable regions where the local geometry neither contracts nor expands, the
$v\varphi$-component of the RSET will cause an initial tendency for the
geometry to undergo an infinite twisting. However, the question of whether
anything more can be ascertained from the RSET about the geometry’s evolution
and the black hole’s final configuration is either left to speculation or
completely ignored, and for good reason—as soon as the inner horizon is pushed
even slightly away from its Kerr value, the Kerr vacuum RSET is no longer
valid and a new RSET in a more general spacetime would need to be found to
continue the evolution any further. Nonetheless, the goal of the current work
is to provide a path forward to do exactly that, so that one may determine
whether or not the semiclassical inner-horizon instability generically leads
to a spacelike singularity.

The backreaction analysis of Sec. IV.3 employed the conformally separable
framework of Refs. [4, 5, 6, 7] to determine how semiclassical fluxes can be
understood together with the classical fluxes that lead to mass inflation near
the inner horizon. One can immediately gather from the presence of a non-
vanishing RSET component ${\langle T^{-}_{rt}\rangle}={\left(\langle
T^{-}_{vv}\rangle-\langle T^{-}_{uu}\rangle\right)R^{-2}\Delta_{r}^{-1}}$ that
the semiclassical backreaction drives the evolution away from many of the
potential (meta)stable geometries proposed as potential endpoints of black
hole evolution, like the regular “eye of the storm” geometry or the Kerr
“black-bounce” geometry (see the end of Sec. IV.2). In contrast, the
conformally separable model generically predicts the presence of a strong,
spacelike singularity that is stable over astrophysical time scales.

In particular, in Sec. IV.3, the metric is generalized to allow for all the
functions in the Kerr line element of Eq. (II) to respond freely to the
presence of semiclassical stress-energy. The dominant evolution near the inner
horizon comes from the horizon function $\Delta_{r}(r)$ and the conformal
factor $\rho(r,\theta,t)$, and thus it becomes useful to analyze the spacetime
in a tetrad frame rotated to align with the principal null directions so that
the $\Delta_{r}$\- and $\rho$-dependent portions of the vierbein are purely
diagonal, Eqs. (16a)–(16d).

In this tetrad frame, the RSET’s dominant components are the diagonals
${\langle T^{-}_{00}\rangle}$ and ${\langle T^{-}_{11}\rangle}$, Eqs. (17),
while the off-diagonal ${\langle T^{-}_{01}\rangle}$ also diverges as
$\Delta_{r}^{-1}$ but is numerically seen to be usually several orders of
magnitude smaller than the diagonal components. As such, one is justified in
treating the initial semiclassical evolution with the inflationary Kasner
spacetime [8], which uses precisely the dynamical vierbein of Eqs. (16a)–(16d)
sourced by stress-energy components $T_{00}$ and $T_{11}$. Eventually (or even
initially, in the latitudes of infinite quantum twisting), this solution will
break down once the RSET’s angular components become non-negligible, but even
once the inflationary Kasner model breaks down, the full conformally separable
model can still be continued (albeit with less symmetry), as it has been shown
to be robust to a wide variety of stress-energy sources [5].

The advantage of the conformally separable model at describing the inner-
horizon backreaction is not only in its ability to encompass a wide range of
behaviors related to the mass inflation phenomenon, but also in its potential
to take the semiclassical evolution even deeper. It is almost miraculous that
such a chaotic instability leads at least initially to a spacetime (the
inflationary Kasner metric) that is so simple, elegant, and possesses a high
degree of symmetry. The RSET in this new homogeneous spacetime should be
relatively straightforward to compute, with the only difficulty coming from
mode-matching to the Kerr Unruh state (though the relevant mathematical
details have already been worked out in, e.g., Ref. [47]). In fact, the
renormalized vacuum polarization $\langle\hat{\phi}^{2}\rangle_{\text{ren}}$
has already been calculated in the inflationary Kasner spacetime in an
adiabatic vacuum state [48], with the conclusion that the semiclassical
contribution follows the same qualitative behavior as the classical stress-
energy.

It is time to take RSET calculations beyond vacuum black holes. From this work
it can be gleaned that the semiclassical inner-horizon instability causes the
local Kerr spacetime to be filled with diverging streams of radiation, and
while this work provides evidence that those streams should generically lead
to an astrophysically stable, chaotic, spacelike singularity, their exact
semiclassical evolution remains an open but tractable question.

###### Acknowledgements.

I would like to thank Andrew J. S. Hamilton, Adrian Ottewill, and Noa
Zilberman for many discussions that helped shape the ideas of this work. This
project is supported solely through teaching funds provided by the Department
of Physics at the University of Colorado Boulder.

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