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arxiv: 2606.30798 · v1 · pith:PPWX6CEFnew · submitted 2026-06-29 · ✦ hep-th · gr-qc

As Cold as a Black Hole: Extended Photon Spheres

Marcos Riojas This is my paper

Pith reviewed 2026-07-01 01:31 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords black hole thermodynamicsphoton sphereself-similar systemsIsrael junction conditionsTolman-Oppenheimer-Volkoff equationhillingar black holecoarse-grained entropyextended photon spheres
0
0 comments X

The pith

Extended photon spheres allow a one-parameter family of self-similar systems to mimic Schwarzschild black hole thermodynamics.

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

The paper argues that the photon sphere governs the Israel junction conditions and the Tolman-Oppenheimer-Volkoff equation in a way that makes them equivalent when radial pressure is zero. This equivalence permits computing coarse-grained entropies for certain systems without relying on the Euclidean path integral. A necessary and sufficient condition for thermodynamic mimicry is derived and shown to hold for a one-parameter family of self-similar systems. The hillingar black hole is part of this family and matches an ordinary black hole in temperature, photon ring, and entropy. Other members generally need massless walls at the boundaries of their extended photon spheres.

Core claim

The photon sphere controls central aspects of the Israel junction conditions, the Tolman-Oppenheimer-Volkoff equation, and finite-radius black hole thermodynamics. The Israel junction conditions and TOV equation are precisely equivalent at zero radial pressure. At fixed mass, adding shells in regions of positive specific heat lowers the asymptotic Hawking temperature, and the inverse specific heat at the photon sphere is proportional to negative Lambda. The exception to the belief that self-gravitating radiation cannot reach thermal equilibrium with a black hole arises from the hillingar black hole, which mimics a Schwarzschild black hole of mass M in Hawking temperature, photon ring, and co

What carries the argument

The extended photon sphere in self-similar systems, which enforces the necessary and sufficient condition for thermodynamic mimicry of black holes through control over junction conditions and equilibrium equations.

If this is right

  • At fixed mass, adding shells in regions of positive specific heat lowers the asymptotic Hawking temperature.
  • The inverse specific heat at the photon sphere is proportional to negative Lambda.
  • The Israel junction conditions and TOV equation are precisely equivalent at zero radial pressure.
  • Coarse-grained entropies can be computed without using the Euclidean path integral.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This condition might apply to other compact objects in general relativity that have self-similar structures.
  • Models of frozen stars or stiffest stars could be tested for similar mimicry properties in numerical relativity simulations.
  • Such systems could provide alternative explanations for observed phenomena that are usually attributed to black holes.
  • The requirement of massless walls suggests a way to stabilize these extended structures.

Load-bearing premise

The photon sphere controls central aspects of the Israel junction conditions, the TOV equation, and finite-radius black hole thermodynamics.

What would settle it

Finding a member of the one-parameter family that does not share the Hawking temperature with a Schwarzschild black hole of equal mass would disprove the necessary and sufficient condition.

read the original abstract

It is widely believed that self-gravitating radiation cannot reach thermal equilibrium with a black hole in asymptotically flat spacetime. The following observation is used to describe an exception to this rule. The photon sphere controls central aspects of the Israel junction conditions (IJCs), the Tolman-Oppenheimer-Volkoff (TOV) equation, and finite-radius black hole thermodynamics. Through these results, we will describe how to compute coarse-grained entropies without using the Euclidean path integral. For instance, we find the IJCs and TOV equation are precisely equivalent at zero radial pressure. At fixed mass, adding shells in regions of positive specific heat lowers the asymptotic Hawking temperature, and the inverse specific heat at the photon sphere is proportional to $-\Lambda$. The exception described here results from companion work with M.J. Strassler, where we found that a "hillingar black hole" (HBH) mimics an ordinary Schwarzschild black hole of mass $M$, sharing its Hawking temperature, photon ring, and, in equilibrium, its coarse-grained entropy $S = 4 \pi M^2$. Here, we show these features are not tuned; they follow uniquely from joint mechanical and thermodynamic constraints. A necessary and sufficient condition for thermodynamic mimicry is found that is satisfied by a one parameter family of self-similar systems, all of which, excepting the HBH, require massless walls at the edges of their extended photon spheres. This family includes "stiffest stars" and "frozen stars".

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper claims that the photon sphere controls central aspects of the Israel junction conditions (IJCs), the Tolman-Oppenheimer-Volkoff (TOV) equation, and finite-radius black hole thermodynamics, enabling computation of coarse-grained entropies without the Euclidean path integral. It identifies a necessary and sufficient condition for thermodynamic mimicry of a Schwarzschild black hole, satisfied by a one-parameter family of self-similar systems (including the hillingar black hole (HBH), stiffest stars, and frozen stars). These features follow uniquely from joint mechanical and thermodynamic constraints rather than tuning; non-HBH members of the family require massless walls at the edges of their extended photon spheres. The work presents an exception to the expectation that self-gravitating radiation cannot reach thermal equilibrium with a black hole in asymptotically flat spacetime.

Significance. If the central claims hold, the work would provide a concrete exception to standard expectations for thermal equilibrium in asymptotically flat spacetimes and a direct route to coarse-grained entropy via junction conditions and the TOV equation. The explicit construction of a one-parameter family satisfying the mimicry condition, together with the acknowledgment that non-HBH members require massless walls, supplies a falsifiable structure that could guide further analytic and numerical studies of self-similar configurations. The approach also offers a potential alternative to Euclidean methods for entropy calculations in finite-radius systems.

major comments (1)
  1. [Abstract (and any section deriving the mimicry condition)] The necessary and sufficient condition for thermodynamic mimicry and the claimed equivalence of the IJCs and TOV equation at zero radial pressure are presented as following uniquely from the constraints, yet the HBH properties and the photon-sphere control of these equations are stated to originate in the companion work with Strassler. The manuscript should supply an explicit, self-contained derivation of the condition (including the relevant equation relating inverse specific heat at the photon sphere to −Λ) so that the one-parameter family can be assessed independently of the companion paper.
minor comments (2)
  1. [Abstract] The abstract states that adding shells in regions of positive specific heat lowers the asymptotic Hawking temperature; a brief indication of the relevant equation or numerical check would clarify this step for readers.
  2. [Abstract and introduction] Notation for the extended photon sphere and the massless walls should be defined at first use with an explicit reference to the Israel junction conditions employed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading, positive evaluation of the work's significance, and recommendation for major revision. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract (and any section deriving the mimicry condition)] The necessary and sufficient condition for thermodynamic mimicry and the claimed equivalence of the IJCs and TOV equation at zero radial pressure are presented as following uniquely from the constraints, yet the HBH properties and the photon-sphere control of these equations are stated to originate in the companion work with Strassler. The manuscript should supply an explicit, self-contained derivation of the condition (including the relevant equation relating inverse specific heat at the photon sphere to −Λ) so that the one-parameter family can be assessed independently of the companion paper.

    Authors: We agree that the manuscript as written relies on the companion paper for the origin of the HBH properties and photon-sphere control of the IJCs and TOV equation. Although the current text states that the mimicry features follow uniquely from the joint mechanical and thermodynamic constraints, an explicit derivation of the necessary and sufficient condition (including the proportionality of inverse specific heat at the photon sphere to −Λ) is not fully self-contained. In the revised manuscript we will add this derivation directly in the main text (or a dedicated appendix), starting from the photon-sphere control of the junction conditions and TOV equation at zero radial pressure, deriving the mimicry condition step by step, and showing how it is satisfied by the one-parameter family. This will allow independent assessment without reference to the companion work. revision: yes

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 2 invented entities

The ledger is populated from the abstract alone; the central claims rest on the photon-sphere control assumption and the companion-work properties of the hillingar black hole.

free parameters (1)
  • one parameter of the self-similar family
    The family is described as one-parameter but the abstract gives no explicit value or fitting procedure.
axioms (1)
  • domain assumption The photon sphere controls central aspects of the Israel junction conditions, the TOV equation, and finite-radius black hole thermodynamics.
    Invoked in the abstract as the basis for equivalence at zero radial pressure and entropy calculations.
invented entities (2)
  • hillingar black hole (HBH) no independent evidence
    purpose: Mimics ordinary Schwarzschild black hole of mass M in temperature, photon ring, and coarse-grained entropy
    Introduced via companion work; no independent evidence supplied in the abstract.
  • extended photon spheres no independent evidence
    purpose: Enable thermodynamic mimicry for the family of systems
    New construct in the paper; no independent evidence supplied.

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discussion (0)

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Reference graph

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