⚛️ general relativity

Displacement memory in regular black hole spacetimes

This paper numerically investigates the displacement memory effect induced by a wave pulse in regular black hole spacetimes, revealing that the resulting net displacement depends on the regularization parameter and pulse height, and differs significantly from that observed in singular black holes.

Original authors: Ritwik Acharyya, Sayan Kar

Published 2026-02-18

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CC BY 4.0

Original authors: Ritwik Acharyya, Sayan Kar

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 the universe as a giant, invisible trampoline made of spacetime. Usually, this trampoline is flat and calm. But when massive objects like black holes crash into each other, they send out ripples—gravitational waves—that shake the trampoline.

For decades, we've been able to detect the "shaking" of these waves. But there's a subtle, permanent side effect that we haven't seen yet, called Displacement Memory.

Think of it like this: If you give a trampoline a single, hard shove, it wobbles wildly. But once the shaking stops, the trampoline doesn't bounce back to exactly where it started. It settles into a slightly different position. That permanent shift is the "memory" of the wave.

This paper, written by Ritwik Acharyya and Sayan Kar, asks a fascinating question: Does this "memory" look different if the trampoline has a hole in the middle (a normal black hole) versus if the hole is patched up with a special, smooth material (a "regular" black hole)?

Here is a breakdown of their findings using everyday analogies:

1. The Setup: The "Pulse" and the "Test Particles"

The scientists created a mathematical model of a black hole. To test for memory, they imagined a "pulse" (a burst of gravitational waves) traveling through space. They placed two tiny "test particles" (like two marbles) floating near the black hole.

The Experiment: They watched how far apart the two marbles were before the wave hit, and how far apart they were after the wave passed.
The Goal: If the marbles end up in a different spot relative to each other than where they started, that's the Displacement Memory.

2. The Two Types of Black Holes

The team compared two scenarios:

The "Singular" Black Hole (The Classic): This is the standard black hole from Einstein's General Relativity. Imagine a funnel that gets infinitely deep and narrow until it hits a sharp, infinitely small point at the bottom (a singularity). It's like a drain with a sharp, jagged hole at the bottom.
The "Regular" Black Hole (The Patched One): This is a theoretical version where the sharp, jagged hole is replaced by a smooth, fuzzy core (like a deformed ball of dough). There is no sharp point; the center is "regular." This is what the authors call a "regularization" of the black hole.

3. The Discovery: The "Fingerprint" of the Black Hole

The researchers ran computer simulations to see how the marbles moved in both scenarios. Here is what they found:

The Wave Leaves a Mark: In both cases, the gravitational wave pulse pushed the marbles apart. When the wave stopped, the marbles didn't return to their original distance. They stayed slightly further apart. This confirmed the Displacement Memory exists.
The "Regular" vs. "Singular" Difference: This is the big news. The amount the marbles moved depended on the type of black hole.
- In the Singular (Classic) black hole, the marbles moved a certain amount.
- In the Regular (Patched) black hole, the marbles moved a different amount.
- The Analogy: Imagine hitting a drum with a sharp stick versus a soft mallet. Even if you hit them with the same force, the sound (the "memory" of the hit) is different. The "smoothness" of the black hole's center changes how the gravitational wave ripples through space.

4. The "Knobs" of the Universe

The scientists also turned various "knobs" in their simulation to see what changed the memory effect:

The "Smoothness" Knob ( $g$ ): The more "regular" (smooth) the black hole is, the smaller the memory effect becomes. The classic, jagged black hole leaves the biggest "bruise" on spacetime.
The "Spin" Knob ( $L_c$ ): If the marbles are spinning around the black hole faster, the memory effect gets stronger.
The "Pulse" Knob: A bigger, stronger wave creates a bigger permanent shift.

5. Why Does This Matter?

Right now, we can't tell the difference between a "smooth" regular black hole and a "jagged" singular one just by looking at them. They look the same from far away.

However, this paper suggests that gravitational wave memory could be the key to telling them apart.

If future detectors (like the next generation of LIGO or LISA) can measure this tiny, permanent shift in distance between particles, they might be able to say: "Hey, the black hole that caused this wave has a smooth center, not a sharp singularity!"

Summary

Gravitational Waves are ripples in the fabric of the universe.
Memory is the permanent "dent" left behind after the ripple passes.
Regular Black Holes (smooth centers) leave a different "dent" than Singular Black Holes (sharp centers).
By measuring this dent, we might finally prove whether the center of a black hole is a mathematical singularity or a smooth, physical object.

It's like trying to figure out if a pillow has a hidden rock inside by feeling how it bounces back after you punch it. The authors found that the "bounce back" (the memory) tells you exactly what's inside.

1. Problem Statement

The paper investigates the gravitational wave (GW) memory effect, specifically the displacement memory, within the context of regular black hole (RBH) spacetimes.

Context: While GW memory (a permanent displacement of test masses after a GW pulse passes) is a well-established prediction of General Relativity (GR), its behavior in spacetimes that resolve the central singularity (regular black holes) remains underexplored.
Objective: To determine if the displacement memory effect carries distinct signatures that can differentiate between:
1. Singular black holes (e.g., Schwarzschild).
2. Different classes of regular black holes (e.g., Bardeen, Hayward, and other parametric models).
3. The specific parameters governing the regularity of these spacetimes.

2. Methodology

The authors employ a theoretical and numerical approach combining geodesic analysis and geodesic deviation equations within a specific metric framework.

A. Spacetime Model

Metric: They utilize a restricted Bondi-Sachs type line element in outgoing Eddington-Finkelstein coordinates $(u, r, \theta, \phi)$ $(u, r, θ, ϕ)$ :
$ds^2 = -f(r)du^2 - 2dudr + (r^2 + rH(u))d\theta^2 + (r^2 - rH(u))\sin^2\theta d\phi^2$
- $f(r)$ : Represents the background static, spherically symmetric geometry (flat, Schwarzschild, or Regular Black Hole).
- $H(u)$ : Encodes the GW pulse profile, modeled as a sech-squared function: $H(u) = A \text{sech}^2(w(u-u_0))$ .
Regular Black Hole Families: The study analyzes several specific $f(r)$ $f (r)$ profiles derived from the Neves-Saa mass function and other models:
- Bardeen: $m(r) \propto (1 + (g/r)^2)^{-3/2}$ (where $b=2$ ).
- Hayward: $m(r) \propto (1 + (g/r)^3)^{-1}$ (where $b=3$ ).
- General Class: $m(r) \propto (1 + (g/r)^b)^{-a/b}$ .
- Alternative Class: A metric reducing to a deformed Reissner-Nordström geometry when the regularization parameter $g=0$ .
Regularization Parameter ( $g$ ): This parameter replaces the central singularity with a de Sitter core. $g=0$ recovers the singular Schwarzschild limit.

B. Analytical Framework

The authors use two complementary methods to quantify memory:

Geodesic Separation: Numerically integrating the geodesic equations for two nearby timelike test particles. The memory is defined as the permanent difference in their separation ( $\Delta r$ $Δ r$ and $\Delta \phi$ $Δ ϕ$ ) between the asymptotic past ( $u \to -\infty$ $u \to - \infty$ ) and future ( $u \to +\infty$ $u \to + \infty$ ).
- Key Constraint: The analysis requires non-zero conserved angular momentum ( $L_c$ ) to observe the memory effect, as the pulse derivative $H'(u)$ couples with $L_c$ in the equations of motion.
Geodesic Deviation: Solving the geodesic deviation equation in a tetrad basis to study the evolution of the deviation vector $\eta^\mu$ . This provides a linearized check on the separation results.

C. Numerical Implementation

The authors numerically solve the coupled differential equations for $r(u)$ , $\phi(u)$ , and their derivatives.
They compare scenarios with the pulse ( $H(u) \neq 0$ ) and without the pulse ( $H(u) = 0$ ) to isolate the memory contribution.
Parameters varied include the regularization parameter $g$ , angular momentum $L_c$ , pulse amplitude $A$ , and the profile exponent $b$ .

3. Key Contributions

Quantification of Memory in RBHs: The paper provides the first numerical quantification of displacement memory specifically for regular black hole spacetimes, moving beyond the standard Schwarzschild or Kerr backgrounds.
Parameter Dependence: It establishes a clear functional dependence of the memory magnitude on the regularization parameter $g$ .
Distinction Mechanism: It proposes that the magnitude of the displacement memory can serve as an observational signature to distinguish between singular and regular black holes, as well as between different regular black hole models.
Comparison of Methods: It validates the consistency between the non-linear geodesic separation method and the linear geodesic deviation method, noting that while they agree asymptotically, they differ in transient behavior (particularly in the angular component).

4. Key Results

Existence of Memory: A distinct displacement memory effect is confirmed in all regular black hole spacetimes studied. The separation between geodesics does not return to its initial value after the pulse passes.
Dependence on Regularization Parameter ( $g$ ):
- The magnitude of the displacement memory decreases as the regularization parameter $g$ increases.
- The Schwarzschild limit ( $g=0$ ) yields the largest memory effect.
- This creates a clear hierarchy: Singular Black Hole ( $g=0$ ) > Regular Black Hole (small $g$ ) > Regular Black Hole (large $g$ ).
Dependence on Angular Momentum ( $L_c$ ): The memory effect increases with higher conserved angular momentum $L_c$ .
Dependence on Profile Exponent ( $b$ ):
- Comparing Bardeen ( $b=2$ ) and Hayward ( $b=3$ ), the memory effect is more prominent for higher values of $b$ (Hayward) for the same set of parameters.
- The "Alternative Class" of regular black holes (Eq. 34 in the paper) exhibited even stronger memory effects than the Bardeen/Hayward families, suggesting curvature properties significantly influence the permanent displacement.
Asymptotic Behavior: At large $u$ , the difference in geodesic separation ( $\Delta r, \Delta \phi$ ) serves as a measurable "fingerprint" of the background geometry.
Bondi-Sachs Connection: The authors discuss that for specific parameter choices ( $b \leq 1$ ), the regularization parameter $g$ enters the subleading asymptotic terms of the Bondi mass, linking the memory effect to asymptotic flux-balance laws. However, for the standard Bardeen ( $b=2$ ) and Hayward ( $b=3$ ) cases, this dependence appears at higher orders.

5. Significance

Observational Prospects: As gravitational wave detectors (like LIGO, Virgo, and future space-based detectors like LISA) improve in sensitivity, the ability to detect the GW memory effect becomes realistic. This paper suggests that if memory is detected, its magnitude could theoretically reveal whether the source black hole is singular or regular.
Testing General Relativity: The results offer a new avenue to test GR against modified gravity theories or quantum gravity-inspired regular black hole models. If observations show a memory effect smaller than the Schwarzschild prediction for a given mass, it could imply the presence of a regular black hole core.
Theoretical Insight: The study bridges the gap between the mathematical construction of regular black holes and their dynamical response to gravitational radiation, highlighting how the "softening" of the central singularity (via parameter $g$ ) dampens the non-linear memory effect.

In conclusion, the paper demonstrates that displacement memory is a sensitive probe of spacetime regularity, with the magnitude of the effect acting as a discriminator between singular and regular black hole geometries.