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arxiv: 2606.27795 · v1 · pith:UWEDRBKPnew · submitted 2026-06-26 · 🌀 gr-qc

Bounds on the radius of black hole shadows in n-dimensional Einstein gravity

Jiaqi Fu , Yong Song This is my paper

Pith reviewed 2026-06-29 04:07 UTC · model grok-4.3

classification 🌀 gr-qc
keywords black hole shadowhigher-dimensional gravityEinstein gravityenergy conditionsphoton sphereSchwarzschild-Tangherliniasymptotically flat spacetimesnull geodesics
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The pith

Black hole shadow radii in n dimensions obey lower and upper bounds fixed by horizon radius and ADM mass under energy conditions.

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

The paper proves model-independent bounds on the radius of the black hole shadow for static spherically symmetric asymptotically flat spacetimes in n-dimensional Einstein gravity with anisotropic matter. Under the weak energy condition the shadow radius cannot fall below a specific multiple of the horizon radius. Adding the strong energy condition and an asymptotic decay condition on the matter yields an upper bound in terms of the ADM mass. Both inequalities become equalities for the vacuum Schwarzschild-Tangherlini solution and recover the known four-dimensional results when n equals 4. A reader would care because the bounds supply geometric constraints on an observable quantity without committing to any particular matter model.

Core claim

For static spherically symmetric asymptotically flat black holes in n-dimensional Einstein gravity supported by anisotropic matter, the shadow radius r_sh satisfies r_sh ≥ ((n-1)/2)^{1/(n-3)} sqrt((n-1)/(n-3)) r_H under the weak energy condition, and r_sh ≤ sqrt((n-1)/(n-3)) [(n-1)M]^{1/(n-3)} under the weak and strong energy conditions plus asymptotic decay, with both bounds saturated by the vacuum Schwarzschild-Tangherlini solution.

What carries the argument

The photon sphere defined by the outermost unstable null circular geodesics, whose radius is determined from the effective potential of the null geodesic equation in the n-dimensional metric.

If this is right

  • The inequalities reduce exactly to the four-dimensional shadow bounds when n=4.
  • The vacuum Schwarzschild-Tangherlini solution achieves equality in both bounds for every n.
  • Any matter configuration obeying the stated energy conditions must produce a shadow inside the derived interval.
  • The bounds supply model-independent geometric restrictions on the observable shadow size in higher-dimensional spacetimes.

Where Pith is reading between the lines

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

  • The same geodesic analysis could be applied to rotating or non-asymptotically-flat cases to test whether analogous bounds survive.
  • Numerical construction of anisotropic matter solutions that approach but do not reach the bounds would quantify how much room the inequalities leave.
  • If extra dimensions are compactified, the bounds might serve as consistency checks on effective four-dimensional shadows.

Load-bearing premise

The matter fields obey the weak energy condition (and the strong energy condition plus decay at infinity for the upper bound).

What would settle it

An explicit static spherically symmetric asymptotically flat n-dimensional solution with anisotropic matter satisfying the weak energy condition whose computed shadow radius lies below the stated lower bound.

read the original abstract

The dark shadow cast by a black hole, determined by the outermost unstable null circular geodesics (the photon sphere), provides a direct probe of strong-field gravity. In this work, we derive model-independent lower and upper bounds on the shadow radius $r_{\mathrm{sh}}$ for static, spherically symmetric, asymptotically flat black holes in $n$-dimensional ($n\ge 4$) Einstein gravity, supported by an anisotropic matter field. For the lower bound, assuming the matter satisfies the Weak Energy Condition (WEC), we prove $r_{\mathrm{sh}}\geq \bigl(\frac{n-1}{2}\bigr)^{\frac{1}{n-3}}\sqrt{\frac{n-1}{n-3}}\,r_H$, where $r_H$ is the horizon radius. For the upper bound, under the WEC and the Strong Energy Condition (SEC), together with an asymptotic decay condition on the matter fields, we prove $r_{\mathrm{sh}}\leq\sqrt{\frac{n-1}{n-3}}\bigl[(n-1)M\bigr]^{\frac{1}{n-3}}$, where $M$ is the ADM mass. These results reduce to the known four-dimensional bounds and are saturated by the vacuum Schwarzschild-Tangherlini black hole. Our results generalize the four-dimensional shadow bounds to an arbitrary number of dimensions and provide model-independent geometric constraints on the observable shadow of higher-dimensional black hole spacetimes.

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 / 0 minor

Summary. The manuscript derives model-independent lower and upper bounds on the shadow radius r_sh for static, spherically symmetric, asymptotically flat black holes in n-dimensional (n≥4) Einstein gravity supported by anisotropic matter. Under the weak energy condition it proves r_sh ≥ ((n-1)/2)^{1/(n-3)} sqrt((n-1)/(n-3)) r_H; under the weak energy condition plus the strong energy condition and an asymptotic decay condition it proves r_sh ≤ sqrt((n-1)/(n-3)) [(n-1)M]^{1/(n-3)}. Both inequalities are saturated by the vacuum Schwarzschild-Tangherlini solution and reduce to the known four-dimensional bounds.

Significance. If the derivations hold, the work supplies the first general n-dimensional geometric constraints on black-hole shadows that follow directly from the Einstein equations and standard energy conditions. The saturation by the vacuum solution and the clean reduction to n=4 are strengths that confirm internal consistency. The results furnish falsifiable, parameter-free inequalities that can be used to test deviations from vacuum solutions in higher-dimensional gravity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of our manuscript, accurate summary of the results, and recommendation to accept. The report confirms the internal consistency of the bounds, their saturation by the Schwarzschild-Tangherlini solution, and their reduction to the known four-dimensional case. We have no major comments requiring response or revision.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from Einstein equations and energy conditions

full rationale

The paper derives the stated inequalities on r_sh directly from the n-dimensional Einstein equations for static spherically symmetric metrics, the null geodesic equation for the photon sphere, and the WEC/SEC plus asymptotic decay. The vacuum Schwarzschild-Tangherlini solution saturates the bounds by explicit calculation, but this is a consistency check rather than a definitional reduction. No load-bearing step invokes self-citation, fitted parameters renamed as predictions, or ansatze smuggled via prior work; the n=4 reduction is a special case of the same equations. The central claims therefore remain independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The claims rest on the standard energy conditions of general relativity and the geometric assumptions of the spacetime; no free parameters or new entities are introduced beyond those already present in nD Einstein gravity.

axioms (4)
  • domain assumption Weak Energy Condition holds for the anisotropic matter field
    Invoked to obtain the lower bound relating r_sh to r_H
  • domain assumption Strong Energy Condition holds for the anisotropic matter field
    Invoked together with asymptotic decay to obtain the upper bound in terms of ADM mass M
  • domain assumption Matter fields satisfy an asymptotic decay condition at infinity
    Required for the upper bound to involve the ADM mass
  • domain assumption Spacetime is static, spherically symmetric, and asymptotically flat in n-dimensional Einstein gravity (n≥4)
    Defines the metric ansatz and the null geodesic problem

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discussion (0)

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Reference graph

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