Black hole inner horizon evaporation in semiclassical gravity

Abstract

In this work we analyse the backreaction of a quantum field on a spherically symmetric black hole geometry with an inner horizon, i.e. an internal boundary of the trapped region. We start with a black hole background with an inner horizon which remains static after its formation. We quantise a massless scalar field on it and calculate its renormalised stress-energy tensor in the Polyakov approximation. We use this tensor as a source of perturbation on top of the background spacetime. We find that the inner horizon has a tendency to evaporate outward much more quickly than the outer one evaporates inward through the Hawking effect. This suggests a revised picture of a semiclassically selfconsistent evaporation in which the dominant dynamical effect comes from the inner horizon, the cause of which can be seen as an interplay between the wellknown unstable nature of this horizon and a locally negative energy contribution from the quantum vacuum. We also look at backreaction on backgrounds which resemble gravitational collapse, where the inner horizon moves towards the origin. There we find that, depending on the nature of the background dynamics, horizon-related semiclassical effects can become dominant and invert the collapse.

Classical mass inflation vs semiclassical inner horizon inflation

“…Additionally, for a dynamically formed black hole, if one takes into account Hawking evaporation of the outer horizon [20,21], then the trapped region should disappear long before a Cauchy horizon or a null singularity forms (assuming no stable extremal remnant is generated). Furthermore, as shown in a previous work by the present authors [22], the backreaction from the quantum vacuum around the inner apparent horizon (i.e. the inner boundary of a dynamically formed trapped region) should also be taken into account long before any consideration of the Cauchy horizon and its divergences.…”

“…The main goal of this work is to see whether and how semiclassical physics can have an influence on the evolution of the inner horizon inside a black hole undergoing mass inflation. A recent work by the present authors [22] analyses the backreaction around static and dynamical inner horizons in simple geometries which do not incorporate mass inflation. The result was that the inner horizon tends to be pushed outward due to semiclassical effects.…”

“…As in [22], we will use the RSET of a massless scalar field in the Polyakov approximation as the source of backreaction. Although this approximation is far from being the exact RSET of a 3+1 dimensional geometry, it is sufficient to describe Hawking evaporation at the outer horizon, and is therefore a good candidate to give us a first glimpse at horizon-related semiclassical effects around the inner apparent horizon.…”

“…We construct the "in" vacuum [33,34] of the 1+1 dimensional radial-temporal sector of the spacetime, used to calculate the RSET in this approximation, by following the movement of light-like geodesics from past null infinity up to the region of interest, which in this case is the vicinity of the inner apparent horizon inside the mass-inflated region of a dynamically formed black hole. As discussed in [22], the accumulative effect that the inner horizon has on null geodesics results in a sensitivity of the RSET to the past of a large part of the collapse geometry, unlike what occurs for the outer horizon. We therefore employ the same tactic as in that work to simplify the initial conditions of the quantum modes in this region: we consider their propagation as being in a flat spacetime up to an advanced time v = v 0 , from where it continues inside a black hole with an inner horizon, as shown in fig.…”

Renormalization of $\langleϕ^2\rangle$ at the inner horizon of rotating, accreting black holes

“…In the charged, spherically symmetric case, early work indicated that T µν would also likely diverge at the inner horizon to produce a spacelike singularity, but could also remain regular in certain cases [26,27]. In the past few years, an explosion of works studying semiclassical Reissner-Nordström mass inflation have shown that the ingoing null component T vv yields a non-zero value at the inner horizon, and switching to a time coordinate which is regular through the inner horizon shows that the renormalized stress-energy tensor does physically diverge there [28][29][30][31][32][33][34]. Back-reaction from T vv alone is generally believed to cause a strong curvature singularity [31], though there is no complete semiclassically consistent solution to verify this yet.…”