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In finite-lifetime black holes the quantum stress tensor stays finite at the inner horizon; only everlasting ones diverge, and inner-extremal ones grow much more slowly.

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2026-07-11 23:02 UTC pith:UN67G3FS

load-bearing objection Clean calculation showing that finite-lifetime trapped regions keep the RSET finite, with exponential growth for non-extremal inner horizons and only power-law growth for inner-extremal ones. the 2 major comments →

arxiv 2607.03916 v1 pith:UN67G3FS submitted 2026-07-04 gr-qc

Semiclassical regularity of compact trapped regions: From dynamical horizons to inner extremality

Ra\'ul Carballo-Rubio , Francesco Di Filippo , Stefano Liberati , Matt Visser This is my paper
classification gr-qc PACS 04.70.Dy04.62.+v04.20.Dw
keywords inner horizonsrenormalized stress-energy tensorPolyakov approximationmass inflationinner extremalitydynamical horizonsin-vacuumsemiclassical gravity
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Eternal black holes with inner horizons are known to be unstable both classically (mass inflation) and semiclassically (divergent renormalized stress-energy). The paper shows that the situation changes once the black hole is allowed to form and evaporate in finite time, so that the inner horizon never becomes a Cauchy horizon. Working in the s-wave Polyakov approximation, the authors compute the renormalized stress-energy tensor of a massless field in the natural “in” vacuum. The tensor remains finite everywhere. At a non-extremal inner horizon it still grows exponentially with advanced time, recovering the familiar divergence only in the everlasting limit; when the surface gravity vanishes (inner-extremal case) the growth is reduced to a mild power law. Inner-extremal geometries are therefore proposed as natural candidates for classically and semiclassically long-lived black-hole interiors.

Core claim

In dynamical spacetimes that form and evaporate a compact trapped region in finite time, the renormalized stress-energy tensor evaluated in the in-vacuum remains finite at every point, including the inner horizon. For non-extremal inner horizons the tensor grows exponentially with advanced time and diverges only when the lifetime is taken to infinity; for inner-extremal horizons the growth is only power-law.

What carries the argument

The in-vacuum of a massless scalar field in the s-wave Polyakov approximation, whose expectation value is obtained from the Boulware tensor plus Schwarzian derivatives of the coordinate maps that relate the asymptotic null coordinate to regular Kruskal-like coordinates on the dynamical horizons.

Load-bearing premise

The entire formation and evaporation history is replaced by two thin null shells, so continuous Hawking flux and continuous back-reaction never appear.

What would settle it

Compute the same RSET for a continuously evaporating geometry (or solve the semiclassical back-reaction equations) and check whether the exponential-versus-power-law distinction between non-extremal and inner-extremal horizons survives.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper computes the renormalized stress-energy tensor (RSET) of a massless scalar field in the s-wave Polyakov approximation for spherically symmetric geometries that contain a compact trapped region with both outer and inner horizons. For eternal (stationary) metrics—Reissner–Nordström, Hayward, and an inner-extremal example—it recovers the known result that no Hadamard state is regular on all horizon branches simultaneously, and that even vanishing inner surface gravity does not remove RSET singularities in the stationary setting. For dynamical geometries formed by an ingoing null shell and closed by a second (negative-energy) null shell after finite advanced time, the natural in-vacuum is shown to keep the RSET finite everywhere. Near a non-extremal dynamical inner horizon the relevant components grow exponentially with advanced time (recovering the stationary divergence only in the infinite-lifetime limit); for an inner-extremal horizon the growth is only power-law. The authors conclude that inner-extremal configurations are natural candidates for classically and semiclassically meta-stable black-hole interiors.

Significance. If the exponential-versus-power-law distinction survives continuous evaporation and back-reaction, the result supplies a concrete geometric criterion (vanishing inner surface gravity) that simultaneously evades classical mass inflation and suppresses the accumulation of semiclassical energy near the inner horizon. The calculation is parameter-free, rests on standard Polyakov formulae and Kruskal constructions, and cleanly separates the local peeling of null geodesics from the global presence or absence of a Cauchy horizon. The explicit expansions for three representative metrics and the transparent matching of the in-state to the appropriate Unruh states constitute a useful technical contribution that can be checked and extended by others.

major comments (2)
  1. The central claim that the RSET remains finite for any finite lifetime is established only on a fixed background; the paper itself notes (Sec. IV) that back-reaction must still be solved. For the non-extremal cases the exponential growth of ⟨T_V−V−⟩ (Eq. 112) already suggests that the fixed-background approximation will break down after a time of order 1/|κ−|. The claim that inner-extremal geometries are therefore “natural candidates for meta-stable interiors” therefore rests on an extrapolation that is not yet controlled. A short quantitative estimate of the timescale on which the integrated flux becomes order-one (or an explicit statement that this is left for future work) would strengthen the conclusion.
  2. The entire evaporation process is idealized as a single negative-energy null shell (Sec. III, Fig. 3). While the local peeling argument (tortoise expansions Eqs. 20, 50, 65 and the resulting V−–v relations) shows that the exponential-versus-power-law distinction is fixed by the instantaneous value of κ− rather than by the discontinuous jump, the paper never demonstrates that a continuous, slowly varying κ−(v) leaves the same qualitative growth rates intact. A brief remark or appendix sketch confirming that the leading asymptotic remains exponential (power-law) whenever κ− remains non-zero (zero) for a long interval would close this gap.
minor comments (4)
  1. Tables I–III and Fig. 2 summarize the eternal cases clearly, but Table IV for the dynamical case is terse; adding a short note that the growth rates of ⟨T_V−V−⟩ differ (exponential vs linear in v) would make the table self-contained.
  2. The constants A, B, C that appear in the inner-extremal tortoise (Eqs. 66–67) are given, yet their numerical values for a concrete choice of M, b, r− are never illustrated; a single numerical example would help the reader gauge the size of sub-leading terms.
  3. A few typographical inconsistencies remain (e.g., “T ransient” with a space in the section headings of Sec. III, and occasional missing spaces after commas in equations). A light copy-edit would remove them.
  4. Reference [41] (Hawking’s late remarks on information) is cited to support the claim that eternal Cauchy horizons are unphysical; a more precise pointer to the relevant passage would be helpful.

Circularity Check

0 steps flagged

No significant circularity: RSET finiteness and exponential-vs-power-law growth follow by direct evaluation of the Polyakov formula on explicitly prescribed metrics and coordinate maps.

full rationale

The paper's central results are obtained by applying the standard s-wave Polyakov expressions (Eqs. 7 and 10) for the renormalized stress-energy tensor, together with the Schwarzian derivative, to three families of metrics (Reissner–Nordström, Hayward, inner-extremal) that are written down explicitly. For the eternal cases the Kruskal-like coordinates are constructed from the surface gravities (Eqs. 13, 21, 51, 65–76) and the resulting components are expanded near each horizon branch (Tables I–III). For the dynamical cases the same procedure is repeated after matching null coordinates across the two null shells that open and close the trapped region (Eqs. 98–106, 114–122); the in-state is thereby shown to coincide locally with the appropriate Unruh state, remaining finite at every finite advanced time. The exponential growth for non-vanishing κ− versus the power-law growth for κ−=0 is read off from the asymptotic relation between the advanced-time coordinate and the regular inner-horizon coordinate (Eqs. 112 and 127–129). No free parameters are fitted to data, no uniqueness theorem is imported to force the form of the metric or the state, and the self-citations supply only the classical background geometries and the known mass-inflation literature; they do not enter the algebraic evaluation of the RSET. The double-shell idealization is an explicit modeling choice, not a circular reduction. Consequently the derivation chain is self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The central claim rests on standard 2-d conformal-field techniques applied to three families of spherically symmetric metrics. No free parameters are fitted; the only modeling choices are the Polyakov approximation, the null-shell idealization of collapse and evaporation, and the restriction to rational metric functions that admit explicit tortoise expansions. No new physical entities are postulated.

axioms (4)
  • domain assumption s-wave Polyakov approximation: the 4-d RSET is replaced by the 2-d conformal anomaly result multiplied by the spherical measure.
    Invoked throughout Secs. II–III; standard in the literature but uncontrolled for angular modes and near-horizon 4-d corrections.
  • ad hoc to paper Collapse and complete evaporation are modeled by two null shells that open and close a static intermediate geometry of finite advanced-time duration.
    Sec. III and Fig. 3; freezes continuous Hawking flux and continuous back-reaction into discontinuous jumps.
  • domain assumption Metric functions are rational of equal numerator/denominator degree so that tortoise coordinates admit elementary expansions near horizons.
    Stated in Sec. II A; covers RN, Hayward, and the inner-extremal example but excludes many other regular black-hole models.
  • standard math Fulling–Sweeny–Wald theorem guarantees that a Hadamard state remains Hadamard throughout the Cauchy development of a Cauchy surface.
    Cited in Sec. III to argue that the in-state is regular away from the idealized shells.

pith-pipeline@v1.1.0-grok45 · 27234 in / 2680 out tokens · 20973 ms · 2026-07-11T23:02:13.432927+00:00 · methodology

0 comments
read the original abstract

In eternal black-hole spacetimes, inner horizons are Cauchy horizons and are generically unstable. For non-extremal inner horizons, this includes both the classical mass-inflation instability and a semiclassical instability associated with divergences in the renormalized stress-energy tensor (RSET). Inner-extremal geometries, for which the inner-horizon surface gravity vanishes, evade classical mass inflation, but in stationary settings still suffer from singular behavior of the RSET. In this work, we show that the dynamical case is qualitatively different. Considering spacetimes describing the formation and evaporation of a compact trapped region in finite time, and working in the $s$-wave Polyakov approximation, we compute the expectation value of the stress-energy tensor in the in-vacuum state. Given that in this case the inner horizon is not a Cauchy horizon, the RSET remains finite everywhere. For generic non-extremal inner horizons, however, the RSET grows exponentially in time at the inner horizon, with a divergence emerging only in the asymptotic limit of an ever-lasting trapped region. For inner-extremal geometries this exponential growth is replaced by a considerably milder power-law growth. Such spacetimes may therefore be considered natural candidates for classically and semiclassically meta-stable black-hole interiors.

Figures

Figures reproduced from arXiv: 2607.03916 by Francesco Di Filippo, Matt Visser, Ra\'ul Carballo-Rubio, Stefano Liberati.

Figure 1
Figure 1. Figure 1: FIG. 1. (Partial) conformal diagram of an eternal black hole with two horizons, including the domains of outer communication [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Eternal black holes: Graphical summary of the (lack of) regularity of the different vacuum states defined — Boulware [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Conformal diagram of the dynamical spacetime under consideration. An incoming shell at [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

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Reference graph

Works this paper leans on

61 extracted references · 31 linked inside Pith · cited by 1 Pith paper

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    Unruh states (inner and outer) 10 E. Inner-extremal spacetime 11

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    Physical interpretation 20 D. Summary of the results for dynamical geometries 21 IV. Conclusions 21 Acknowledgments 22 References 22 3 I. INTRODUCTION Black-hole physics has traditionally focused on exterior regions and event horizons [1]. For non-extremal solutions, outer horizons are classically stable under perturbations, while semiclassically they und...

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    Boulware state The expectation value of the stress energy tensor in the Boulware state (5) can be easily obtained from Eq. (7) and reads ⟨B| ˆTuu |B⟩= −4r3(r− +r +) + 3r2 r2 − + 6r−r+ +r 2 + −12rr −r+(r− +r +) + 8r2 −r2 + 48πr3 ,(14) ⟨B| ˆTvv |B⟩=⟨B| ˆTuu |B⟩,(15) ⟨B| ˆTuv |B⟩= 1 48πr6 (r−r −)(r−r +) [3r−r+ −r(r − +r +)].(16) While these expectation value...

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    Unruh state (outer) We can define vacuum states that are regular at some of the horizons by splitting the positive and negative energy modes in a way different from Eq. (5). For instance, we can define the Unruh state|U +⟩regular on the outer horizon H + R by splitting the energy modes as ϕ= X ω e−iωv aL ω +e −iωU+ aR ω +h.c. ,(33) which correspond to the...

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    Boulware state The components of the stress-energy in the Boulware state read: ⟨B| ˆTuu |B⟩=− (r−r −)4 192πQ4(r)4 n [Q4(r)(4r−r − −3r +)−(r−r −)(r−r +)Q′ 4(r)]2 −2(r−r +) −Q4(r)(r−r −)2(r−r +)Q′′ 4(r) + 2(r−r −)2(r−r +)Q′ 4(r)2 −6Q 4(r)(r−r −)(r−r +)Q′ 4(r) −2Q 4(r)(r−r −)2Q′ 4(r) + 6Q4(r)2(r−r −) +6Q 4(r)2(r−r +) ,(62) ⟨B| ˆTvv |B⟩=⟨B| ˆTuu |B⟩,(63) ⟨B| ...

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