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arxiv: 2601.01451 · v3 · submitted 2026-01-04 · 🌀 gr-qc

Bounds on the photon sphere radius for spherically symmetric black holes in n-dimensional Einstein gravity

Yong Song , Jiaqi Fu , Yiting Cen This is my paper

Pith reviewed 2026-05-16 18:11 UTC · model grok-4.3

classification 🌀 gr-qc
keywords photon sphereblack holeshigher-dimensional gravityEinstein gravityenergy conditionsevent horizonADM massspherically symmetric
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The pith

The photon sphere radius around n-dimensional black holes is bounded above by [(n-1)M]^{1/(n-3)} and below by ((n-1)/2)^{1/(n-3)} times the horizon radius.

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

This paper establishes upper and lower bounds on the photon sphere radius for static spherically symmetric asymptotically flat black holes in n-dimensional Einstein gravity with n at least 4. The upper bound depends only on the ADM mass M and dimension n, while the lower bound involves the outer horizon radius under an extra monotonicity condition on the radial pressure. These constraints generalize the familiar four-dimensional results r_γ ≤ 3M and r_γ ≥ 1.5 r_H to higher dimensions for black holes sourced by anisotropic matter obeying the weak energy condition and having non-positive energy-momentum trace. The bounds limit possible geometries and therefore affect predictions for shadows, lensing, and quasinormal modes in higher-dimensional spacetimes.

Core claim

For static spherically symmetric asymptotically flat black holes in n-dimensional Einstein gravity (n ≥ 4) with anisotropic matter satisfying the weak energy condition and non-positive trace of the energy-momentum tensor, the photon sphere radius r_γ obeys the upper bound r_γ ≤ [(n-1)M]^{1/(n-3)}. When the additional assumption holds that |r^{n-1} p_r(r)| is monotonically decreasing, the lower bound r_γ ≥ ((n-1)/2)^{1/(n-3)} r_H also holds, where r_H is the outer event horizon radius.

What carries the argument

The photon sphere radius r_γ, defined as the radial coordinate of unstable null circular geodesics, bounded via integration of the Einstein equations under the stated energy conditions on the anisotropic stress-energy tensor.

If this is right

  • In four dimensions the inequalities recover the known bounds r_γ ≤ 3M and r_γ ≥ 3/2 r_H.
  • The results supply explicit dimension-dependent geometric constraints for Tangherlini-type black holes in n ≥ 5.
  • Black-hole shadow sizes and gravitational-lensing observables become bounded in terms of mass and horizon radius alone.
  • Quasinormal-mode frequencies tied to the photon sphere inherit the same dimension-dependent limits.

Where Pith is reading between the lines

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

  • If extra dimensions are large enough to affect astrophysical black holes, the upper bound would cap observable shadow diameters in terms of mass alone.
  • Relaxing the monotonicity assumption on radial pressure could extend the lower bound to a wider class of matter models.
  • Similar bounding techniques may apply to stationary axisymmetric black holes or to theories with higher-curvature corrections.

Load-bearing premise

The matter must be anisotropic, obey the weak energy condition, have non-positive energy-momentum trace, and for the lower bound the quantity |r^{n-1} p_r(r)| must decrease monotonically with r.

What would settle it

Observation of a static spherically symmetric black hole whose photon sphere lies outside these two inequalities while the matter still satisfies the weak energy condition and trace condition would contradict the bounds.

read the original abstract

The photon sphere, a hypersurface of null circular geodesics, plays a fundamental role in characterizing black hole spacetimes, influencing phenomena such as black hole shadows, gravitational lensing, and quasinormal modes. In this work, we derive both upper and lower bounds on the photon sphere radius for static, spherically symmetric, asymptotically flat black holes within $n$-dimensional Einstein gravity ($n\ge 4$), assuming an anisotropic matter field satisfying the weak energy condition and a non-positive trace of the energy-momentum tensor. For the upper bound, we obtain $r_\gamma\le [(n-1)M]^{\frac{1}{n-3}}$, where $M$ is the ADM mass. In the four-dimensional case ($n=4$), this reduces to $r_\gamma\le 3M$, in agreement with previous results. For the lower bound, under the additional assumption that $|r^{n-1}p_r(r)|$ is monotonically decreasing, we prove $r_\gamma\ge (\frac{n-1}{2})^{1/(n-3)}r_H$, where $r_H$ is the radius of the outer event horizon; for $n=4$ this gives $r_\gamma\ge \frac{3}{2}r_H$, also consistent with previous four-dimensional result. These results provide dimension-dependent geometric constraints that generalize well-known four-dimensional bounds to a specific class of higher-dimensional black holes (described by a Tangherlini-type metric) and deepen our understanding of spacetime structure in higher-dimensional gravitational theories.

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

0 major / 2 minor

Summary. The paper derives both upper and lower bounds on the photon sphere radius r_γ for static, spherically symmetric, asymptotically flat black holes in n-dimensional Einstein gravity (n≥4) described by a Tangherlini-type metric. Assuming an anisotropic matter field obeying the weak energy condition and with non-positive trace of the energy-momentum tensor, the upper bound is r_γ ≤ [(n-1)M]^{1/(n-3)}. Under the additional assumption that |r^{n-1} p_r(r)| is monotonically decreasing, the lower bound is r_γ ≥ ((n-1)/2)^{1/(n-3)} r_H. Both bounds recover the known four-dimensional results exactly.

Significance. If the derivations hold, the results generalize the standard four-dimensional photon-sphere bounds (r_γ ≤ 3M and r_γ ≥ 1.5 r_H) to higher dimensions under controlled energy conditions. This supplies dimension-dependent geometric constraints on black-hole spacetimes that can inform analyses of shadows, lensing, and quasinormal modes in higher-dimensional gravity. The explicit recovery of the n=4 limits and the use of standard energy conditions provide a consistent and falsifiable extension of existing results.

minor comments (2)
  1. The integration steps that produce the upper bound from the Einstein equations (starting from asymptotic flatness and the condition r Φ' = 1 at the photon sphere) would benefit from one or two additional intermediate inequalities to make the role of T ≤ 0 and the weak energy condition fully transparent.
  2. The monotonicity assumption on |r^{n-1} p_r(r)| is stated clearly but its physical motivation or range of applicability could be discussed briefly in the text to help readers assess how restrictive it is for realistic matter models.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The referee correctly notes that our bounds recover the standard four-dimensional results and provide a controlled generalization under the weak energy condition and non-positive trace. Since the report lists no specific major comments, we have no points requiring detailed rebuttal or revision at this stage.

Circularity Check

0 steps flagged

Derivation self-contained from Einstein equations plus energy conditions

full rationale

The central claims consist of explicit inequalities on r_γ obtained by integrating the Einstein-equation component for the metric function (or equivalent Φ') from asymptotic flatness to the photon-sphere locus rΦ'=1, using only the stated assumptions (weak energy condition, T≤0, and for the lower bound monotonicity of |r^{n-1}p_r|). These steps are direct consequences of the differential equations and inequalities; no parameter is fitted to data and then relabeled a prediction, no quantity is defined in terms of itself, and no load-bearing step reduces to a self-citation. The n=4 reductions are recovered as special cases but are not presupposed. The derivation is therefore independent of its target results.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

Central claim rests on the Einstein equations in n dimensions, the weak energy condition, non-positive trace of the stress-energy tensor, and one monotonicity condition on radial pressure; no free parameters or new entities are introduced.

axioms (3)
  • domain assumption Weak energy condition for anisotropic matter
    Invoked to obtain the upper bound on photon-sphere radius.
  • domain assumption Trace of energy-momentum tensor ≤ 0
    Used together with WEC to constrain the metric functions.
  • ad hoc to paper |r^{n-1} p_r(r)| is monotonically decreasing
    Additional assumption required only for the lower bound.

pith-pipeline@v0.9.0 · 5586 in / 1368 out tokens · 38016 ms · 2026-05-16T18:11:53.035091+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We derive both upper and lower bounds on the photon sphere radius for static, spherically symmetric, asymptotically flat black holes within n-dimensional Einstein gravity (n≥4), assuming an anisotropic matter field satisfying the weak energy condition and a non-positive trace of the energy-momentum tensor. For the upper bound, we obtain r_γ ≤ [(n-1)M]^{1/(n-3)}.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    R(r) = 3−n+μ(n−1)−16πr²p_r/(n−2) = 0 at the photon sphere; P'(r) = r^{n-1}/(2μ)[R(ρ+p_r)+2μT] with T≤0 and WEC used to sign the integrand.

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Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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