⚛️ general relativity

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

This paper investigates how the tidal charge ( $\Upsilon$ ) arising from extra dimensions in the DMPR braneworld model significantly modifies strong-field gravitational lensing and quasinormal mode frequencies compared to standard Schwarzschild black holes, suggesting that DMPR$-$ variants exhibit stronger Lyapunov instability and could be constrained by Event Horizon Telescope observations of SgrA*.

Original authors: K. K. Nandi, R. N. Izmailov, R. Kh. Karimov, A. A. Potapov

Published 2026-06-26

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CC BY 4.0

Original authors: K. K. Nandi, R. N. Izmailov, R. Kh. Karimov, A. A. Potapov

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine our universe is like a giant, invisible sheet of fabric (a "brane") floating in a much larger, mysterious room (the "bulk"). In this story, gravity isn't just stuck to our sheet; it can leak out into the room and come back, leaving behind a faint "fingerprint" or "tidal charge" on our sheet. This paper explores what happens when a black hole forms on our sheet while carrying this extra-dimensional fingerprint.

Here is a breakdown of the paper's claims using simple analogies:

1. The Two Types of "Fingerprint" Black Holes

The scientists studied two versions of these black holes, which they call DMPR+ and DMPR-. Think of them as two different flavors of ice cream that look similar but taste very different.

The Standard Version (DMPR+): This is like a black hole that has been slightly "charged up" by the extra dimension. It behaves somewhat like a known type of black hole from standard physics (Reissner-Nordström), but the "charge" isn't electricity; it's a gravitational imprint from the 5th dimension.
The "Repulsive" Version (DMPR-): This is the more interesting one. Here, the extra-dimensional imprint acts like a repulsive force. Imagine trying to push two magnets together with the same poles facing each other; they resist. In this black hole, the extra dimension pushes back against gravity. This changes the black hole's structure significantly: it has a larger event horizon (the point of no return) and a "spacelike" singularity (the center) that behaves more like the classic black holes we know, rather than a weird time-traveling center.

2. The "Shadow" Test (The Event Horizon Telescope)

How can we tell if these extra-dimensional black holes exist? The paper suggests looking at the shadow of the black hole at the center of our galaxy, Sagittarius A* (SgrA*), using the Event Horizon Telescope (EHT).

The Analogy: Imagine a black hole is a hole in a piece of paper. If you shine a light behind it, the hole casts a shadow. The size of that shadow depends on how strong the gravity is.
The Finding: The "tidal charge" (the extra dimension's fingerprint) changes the size of this shadow.
- For the DMPR- (repulsive) version, the shadow gets bigger as the extra dimension's effect gets stronger.
- For the DMPR+ version, the shadow gets smaller.
The Result: By measuring the current size of SgrA*'s shadow, the authors calculated a limit. They say the extra-dimensional "charge" cannot be too strong, or the shadow would look different than what the EHT sees. They propose a specific upper limit for this charge based on current data.

3. The "Ring" of Light (Strong Lensing)

Black holes bend light so much that it can loop around them like a racetrack. This creates multiple images of the same background star, stacked on top of each other.

The Analogy: Think of a black hole as a funhouse mirror that stretches and bends light. The "strong field lensing" is when the light bends so hard it does a full loop around the mirror.
The Finding: The extra dimension changes how these light loops behave.
- In the DMPR- case, the light loops are slightly more unstable. It's like a tightrope walker who is wobbling more than usual. Because the light is less stable, it leaks out faster, making the accretion disk (the ring of hot gas around the black hole) appear brighter to us than a normal black hole would.
- The paper calculates specific numbers (called "exponents") that describe this wobbling. The "wobble" is stronger in the DMPR- version, suggesting these black holes would shine more brightly.

4. The "Vibration" of the Black Hole (Quasinormal Modes)

When a black hole is disturbed (like when two black holes smash together), it "rings" like a bell. These rings are called Quasinormal Modes (QNMs).

The Analogy: If you hit a bell, it makes a specific sound that fades away. The pitch (frequency) and how fast it fades (damping) tell you what the bell is made of.
The Finding: The extra dimension changes the "sound" of the black hole.
- In the DMPR- version, the "pitch" of the ring gets lower and the sound fades away faster as the extra dimension gets stronger.
- In the DMPR+ version, the opposite happens: the pitch gets higher.
- This means if we could listen to the "ringing" of a black hole, we could tell if it has an extra-dimensional fingerprint just by the tone.

Summary of the Paper's Main Claims

Extra dimensions leave a mark: They create a "tidal charge" that modifies how black holes behave, distinct from electric charge.
Two distinct behaviors: One version (DMPR-) acts like a repulsive force, making the black hole's shadow larger and its light ring brighter. The other (DMPR+) acts differently, shrinking the shadow.
Observable differences: These effects change the size of the black hole's shadow, the brightness of its surrounding light, and the "sound" (vibration frequency) it makes.
Current limits: Using the current images of SgrA* from the Event Horizon Telescope, the authors calculate that the extra-dimensional charge cannot be too large, or the shadow would look wrong. They provide a specific mathematical range for this charge based on today's observations.

The paper concludes that while we haven't definitively proven extra dimensions yet, the tools we have (like the EHT) are sensitive enough to start testing these ideas, and future, sharper telescopes might finally catch a glimpse of this "fifth dimension."

Technical Summary: Observable Strong Field Effects of Extra Spacetime Dimension in the Braneworld Black Hole

Problem Statement
The paper addresses the potential observable signatures of extra spatial dimensions within the context of the Randall-Sundrum (RS) braneworld scenario. In this framework, our universe is a 3-brane embedded in a 5-dimensional bulk. While gravity is assumed to propagate freely in the bulk, its projection onto the brane modifies the effective Einstein field equations. Specifically, the projection of the 5D bulk Weyl tensor introduces a "tidal charge" ( $\Upsilon$ ) term into the effective 4D metric. Unlike the electric charge in the Reissner-Nordström (RN) solution, this tidal charge arises from the fifth dimension and can theoretically possess both positive and negative signs. The authors aim to investigate how these two variants of the braneworld black hole—designated as DMPR $^+$ (positive tidal charge) and DMPR $^-$ (negative tidal charge)—differ from the standard Schwarzschild black hole ( $\Upsilon=0$ ) in the strong-field regime. The study focuses on distinguishing these models through strong-field gravitational lensing observables and quasinormal mode (QNM) frequencies, using the supermassive black hole SgrA* as a test case.

Methodology
The authors employ a combination of analytical general relativity and numerical evaluation of specific observables:

Metric Formulation: The study utilizes the exact vacuum black hole solution derived by Dadhich, Maartens, Papadopoulos, and Rezania (DMPR). The metric is defined by a tidal charge parameter $\Upsilon$ $Υ$ .
- DMPR $^+$ : Corresponds to a positive sign in the $\Upsilon^2/r^2$ term, formally resembling the RN metric but with a single horizon structure restricted by $\Upsilon \leq 0.5$ .
- DMPR $^-$ : Corresponds to a negative sign (achieved via $\Upsilon \to i\Upsilon$ ), resulting in a single horizon with a radius larger than the Schwarzschild radius and no upper bound on $\Upsilon$ .
Strong Field Lensing: The authors apply Bozza's analytical framework for strong-field gravitational lensing. They calculate the deflection angle $\alpha(\theta)$ $α (θ)$ near the photon sphere, deriving coefficients $a$ $a$ and $b$ $b$ that characterize the logarithmic divergence of the bending angle. From these, they compute key observables:
- Angular shadow radius ( $\theta_\infty$ ).
- Image separation ( $s$ ).
- Magnification ratio ( $r$ ) and its logarithmic magnitude ( $r_{mag}$ ).
Quasinormal Modes (QNMs): In the eikonal limit (high angular momentum $\ell \gg 1$ ), the authors relate QNM frequencies ( $\omega_{QNM}$ ) to the parameters of unstable circular null geodesics. They utilize the connection established by Stefanov et al., linking QNMs to the Lyapunov exponent ( $\lambda$ ), the angular velocity ( $\Omega_m$ ), and the lensing coefficient $a$ .
Stability Analysis: The study calculates the Pretorius-Khurana exponent ( $\gamma = \Omega_m / 2\pi\lambda$ ) to assess the Lyapunov instability of null orbits, which correlates with the brightness of accretion disks.
Observational Constraints: The authors compare their theoretical predictions for the angular shadow diameter of SgrA* against Event Horizon Telescope (EHT) observations (2017 data) to propose an upper limit on the tidal charge $\Upsilon$ .

Key Contributions and Results

Distinctive QNM Spectra: The paper demonstrates that the QNM frequencies for DMPR $^+$ $^{+}$ and DMPR $^-$ $^{-}$ exhibit opposite trends relative to the tidal charge $\Upsilon$ $Υ$ .
- For DMPR $^-$ , both the real and imaginary parts of the QNM frequencies decrease as $\Upsilon$ increases.
- For DMPR $^+$ , these parts increase as $\Upsilon$ increases.
- This provides a potential spectral signature to distinguish between the two braneworld variants and the standard Schwarzschild black hole.
Lensing Observables: The study reveals significant deviations in lensing observables:
- Shadow Size: The angular shadow radius $\theta_\infty$ for DMPR $^-$ increases monotonically with $\Upsilon$ , whereas for DMPR $^+$ , it decreases.
- Image Separation: The separation between relativistic images ( $s$ ) decreases with increasing $\Upsilon$ for DMPR $^-$ (making images harder to resolve), while it increases for DMPR $^+$ .
- Brightness: The magnification ratio indicates that sources lensed by DMPR $^-$ appear brighter than those lensed by DMPR $^+$ .
Lyapunov Instability and Accretion Disks: The Pretorius-Khurana exponent $\gamma$ is found to be lower for DMPR $^-$ compared to the Schwarzschild black hole. A lower $\gamma$ implies a stronger Lyapunov instability, meaning photons leak out from unstable circular orbits more rapidly. The authors conclude that accretion disks around DMPR $^-$ black holes would appear brighter than those around Schwarzschild or DMPR $^+$ black holes.
Constraints on SgrA:* Using the EHT measurement of SgrA*'s shadow diameter ( $2\theta_{obs} \approx 51.8 \pm 2.3$ $2 θ_{o b s} \approx 51.8 \pm 2.3$ $\mu$ $μ$ as), the authors derive an upper limit on the tidal charge.
- For the DMPR $^-$ metric (which they argue is the more physically consistent braneworld scenario as it preserves the spacelike nature of the singularity), the constraint is $0 \leq \Upsilon \leq 0.32467$ .
- The authors note that while SgrA* has a spin ( $a/M \approx 0.44$ ), the effect of spin on the shadow size is estimated to be $\leq 1$ $\mu$ as, suggesting the derived constraint on $\Upsilon$ remains robust despite the omission of spin in the static model.

Significance and Claims
The paper claims to be the first to systematically study the strong-field lensing observables and eikonal QNM frequencies for both variants of the DMPR braneworld black hole. The authors emphasize that while DMPR $^+$ is formally similar to the RN metric, it is physically distinct due to the origin of the charge (bulk Weyl projection vs. electromagnetic field). More importantly, the DMPR $^-$ solution represents a non-general relativistic spacetime with unique properties, such as a single horizon larger than the Schwarzschild radius and a spacelike singularity.

The significance of the work lies in identifying "tidal charge" as a potentially observable imprint of extra dimensions. The authors argue that the DMPR $^-$ model predicts a larger shadow and brighter accretion disks compared to the Schwarzschild case, offering a pathway to test the Randall-Sundrum braneworld scenario using current and future high-resolution observations (such as the Event Horizon Telescope and its next-generation counterpart, ngEHT). The paper concludes that while current EHT data allows for a non-zero tidal charge, future precision measurements of the shadow size and the detection of the photon ring substructure could further tighten these constraints or potentially rule out the braneworld hypothesis.