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arxiv: 2606.26152 · v1 · pith:T2TV7GURnew · submitted 2026-06-23 · 🌀 gr-qc · astro-ph.HE· hep-th

Observable strong field effects of extra spacetime dimension in the braneworld black hole

K. K. Nandi , R. N. Izmailov , R. Kh. Karimov , A. A. Potapov This is my paper

Pith reviewed 2026-06-26 01:30 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HEhep-th
keywords braneworldtidal chargestrong field lensingquasinormal modesblack hole shadowextra dimensions
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The pith

The tidal charge from an extra dimension alters strong-field lensing and quasinormal mode frequencies of black holes.

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

This paper examines how the tidal charge arising from an extra dimension in the braneworld scenario affects observable strong-field phenomena around black holes. It calculates changes to gravitational lensing in the strong deflection regime and to the quasinormal mode frequencies in the eikonal approximation. These modifications matter because they could be measured in high-resolution images of galactic center black holes or in the ringdown phase of gravitational wave events, potentially revealing signs of higher-dimensional spacetime.

Core claim

The DMPR braneworld black hole solution, which includes a tidal charge term ± Υ²/r² added to the Schwarzschild metric, leads to modified strong field lensing observables and quasinormal mode frequencies. In particular, the critical exponent γ for circular null orbits is lower in the DMPR- case, indicating stronger instability of photon orbits and thus brighter accretion disks. The shadow size of SgrA* can be used to constrain the value of Υ.

What carries the argument

The DMPR metric with its tidal charge Υ, which encodes the effect of the five-dimensional Weyl tensor projected onto the brane.

Load-bearing premise

The DMPR solution is assumed to be the valid effective metric on the four-dimensional brane.

What would settle it

Detection of a black hole shadow size for SgrA* that cannot be fit by any value of the tidal charge in the DMPR metric would falsify the predicted modifications.

Figures

Figures reproduced from arXiv: 2606.26152 by A. A. Potapov, K. K. Nandi, R. Kh. Karimov, R. N. Izmailov.

Figure 1
Figure 1. Figure 1: Lens geometry (Figure taken from J.B. Hartle, [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The angle of light deflection α− by DMPR−monotonically decreases with increase in Υ and is less than that by DMPR+, which shows that the tidal charge in DMPR− has some kind of a repulsive effect compared to the tidal charge in DMPR+, for which α+ monotonically increases. The Schwarzschild value occurs at Υ = 0. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The angular radius of the shadow θ − ∞ monotonically increases while θ + ∞ monotonically decreases. The benchmark is the Schwarzschild value that occurs at Υ = 0, which is θ ± ∞ ≃ 17 µarcsec [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The logarithm of the magnification ratio [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The image separation s − decreases with increasing Υ showing that the images become gradually unresolvable, contrary to the case of s + . 4 [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The Pretorius-Khurana exponent γ − shows that unstable null circular orbits leak out photons more profusely. Consequently, the accretion disk of DMPR− should be more luminous than that of DMPR+ [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The plot of u − m is seen to be monotonically increasing from the Schwarzschild value 2.6, while u + m decreases from that value. This behavior was used in Sec.5 to arrive at the upper limits on Υ. where r is the flux ratio r = µ1 P∞ n=2 µn (31) in which µ1 and µn are the magnifications of the first and nth image respectively. Converted to magnitudes, the difference be￾tween the first and all other images … view at source ↗
Figure 6
Figure 6. Figure 6: Fig.6 [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
read the original abstract

Inspired by the string theory, the braneworld picture introduces extra dimensions beyond the four that may have observable non-trivial effects in short distance (strong field) gravity experiments. A case in point is the Randall-Sundrum braneworld picture that projects the $5d$ bulk Weyl tensor onto the $3d$ brane providing a stress tensor in the effective Einstein field equations on the brane. Dadhich, Maartens, Papadopoulos and Rezania (DMPR) derived an exact braneworld black hole solution of the brane vacuum field equations. The solution formally resembles that of Reissner-Nordstr\"{o}m but is physically different from it since the "tidal charge" $\Upsilon$ in the solution is not the electric charge but an imprint from the fifth dimension allowing both signs in the power law modification $\pm \frac{\Upsilon ^{2}}{r^{2}}$ to the Schwarzschild metric $\left( \Upsilon = 0\right)$. The corresponding black holes are designated as DMPR$\pm$. We study here the effect of $\Upsilon$ on strong field lensing observables and compare in the eikonal limit the ring down quasinormal mode (QNM) frequencies of DMPR$-$ with those of DMPR$+$, the two variants of tidal charge modified Schwarzschild black hole ($\Upsilon = 0$). It turns out that the tidal charge can significantly modify the Schwarzschild lensing observables and QNM frequencies. In particular, we find that the Pretorius-Khurana critical exponent $\gamma$ of circular null orbits in the DMPR$-$ black hole has a lower value than that for the Schwarzschild black hole, which indicates a stronger Lyapunov instability suggesting that the accretion disks of DMPR$-$ black holes would appear brighter. The case of the SgrA* black hole is considered for a possible constraint on $\Upsilon$ from the EHT observation of its shadow size.

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

2 major / 1 minor

Summary. The paper claims that the tidal charge Υ in the DMPR braneworld black hole solutions (f(r) = 1 − 2M/r ± Υ²/r²) significantly modifies strong-field lensing observables and quasinormal mode (QNM) frequencies in the eikonal limit relative to Schwarzschild. In particular, the Pretorius-Khurana critical exponent γ is lower for DMPR− than for Schwarzschild, implying stronger Lyapunov instability and brighter accretion disks; the work also considers a possible constraint on Υ from the EHT shadow size of Sgr A*.

Significance. If the results hold after verification of the underlying approximations, the manuscript would provide concrete, observationally accessible signatures of extra dimensions in strong-field gravity, linking braneworld models to EHT shadow data and black-hole ringdown. The explicit comparison between DMPR+ and DMPR− cases and the direct tie to the Pretorius-Khurana exponent constitute a clear phenomenological advance.

major comments (2)
  1. [eikonal QNM analysis] The central claim that γ is lower for DMPR− (and hence that accretion disks appear brighter) rests on direct substitution of the DMPR metric into the leading eikonal formula ω ≈ l Ω − i (n + 1/2) λ; no verification is supplied that the reported difference survives 1/l corrections or a full numerical integration of the perturbation equations.
  2. [metric and field-equation setup] All reported modifications to lensing observables and QNM frequencies assume that the DMPR solution (derived from the projected 5d Weyl tensor) constitutes the complete effective 4D metric on the brane; the manuscript contains no check that additional bulk Weyl contributions are absent or negligible.
minor comments (1)
  1. The abstract states that calculations were performed and quotes a specific result (lower γ for DMPR−), yet the provided text does not include the explicit expressions for the lensing observables, the numerical values of γ, or the error estimates on the Sgr A* constraint.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below, indicating where the manuscript will be revised for clarity on the scope of the approximations used.

read point-by-point responses

  1. Referee: The central claim that γ is lower for DMPR− (and hence that accretion disks appear brighter) rests on direct substitution of the DMPR metric into the leading eikonal formula ω ≈ l Ω − i (n + 1/2) λ; no verification is supplied that the reported difference survives 1/l corrections or a full numerical integration of the perturbation equations.

    Authors: The comparison of the Pretorius-Khurana exponent γ is performed strictly within the leading eikonal limit, where γ is determined directly from the Lyapunov exponent λ of the unstable photon orbit. This is the standard approach for such phenomenological studies, and the lower value of γ for DMPR− arises from the metric modification at the photon sphere. While 1/l corrections or full numerical QNM computations could affect quantitative values, the qualitative indication of stronger instability is tied to the leading-order result. We will revise the manuscript to explicitly state the leading-order nature of the analysis and note that higher-order verification lies beyond the present scope. revision: partial

  2. Referee: All reported modifications to lensing observables and QNM frequencies assume that the DMPR solution (derived from the projected 5d Weyl tensor) constitutes the complete effective 4D metric on the brane; the manuscript contains no check that additional bulk Weyl contributions are absent or negligible.

    Authors: The DMPR metric is the exact vacuum solution to the effective 4D Einstein equations on the brane that incorporate the projected 5D Weyl tensor, as originally derived. In the effective 4D braneworld framework this is the complete metric employed for phenomenological calculations. A direct check that additional bulk Weyl terms are negligible would require a full 5D perturbation analysis, which is outside the scope of this work focused on observable 4D signatures. revision: no

standing simulated objections not resolved
  • Full verification that additional bulk Weyl contributions are absent or negligible would require a complete 5D analysis not performed in the manuscript.

Circularity Check

0 steps flagged

No circularity: direct substitution of external DMPR metric into standard formulas

full rationale

The paper adopts the DMPR metric (f(r) = 1 - 2M/r ± Υ²/r²) from prior independent work by Dadhich et al. and substitutes it into established lensing deflection-angle integrals and the eikonal QNM relation ω ≈ l Ω - i (n + 1/2) λ. No parameter is fitted to the target observables and then relabeled a prediction; no self-citation chain supplies the uniqueness or the functional form of the metric; the reported shifts in γ, shadow size, and Lyapunov exponent are computed outputs rather than identities. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

Review performed on abstract only; the ledger is therefore limited to elements explicitly named in the abstract. The tidal charge Υ is the sole free parameter; the braneworld projection is the key domain assumption; the extra dimension itself is an inherited entity from string theory.

free parameters (1)
  • Υ (tidal charge)
    Parameter appearing in the DMPR metric that can take either sign; its magnitude is to be constrained by observations rather than derived from first principles.
axioms (1)
  • domain assumption The Randall-Sundrum braneworld projects the 5d bulk Weyl tensor onto the 3d brane, supplying an effective stress tensor in the 4d Einstein equations.
    This is the foundational assumption that yields the DMPR solution referenced throughout the abstract.
invented entities (1)
  • Fifth spacetime dimension (braneworld bulk) no independent evidence
    purpose: Source of the tidal charge imprint on the 4d black-hole metric.
    Inherited from string theory; the paper provides no new falsifiable prediction for its existence beyond the existing DMPR construction.

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