⚛️ general relativity

Gravitational wave memory and geodesic congruence response in generalised Ellis-Bronnikov wormholes

This paper investigates gravitational wave memory in generalised Ellis-Bronnikov wormholes by analyzing the response of timelike geodesics and congruences to localized pulses, revealing that the resulting displacement, velocity, expansion, and shear signatures depend systematically on the wormhole's throat radius, steepness parameter, and the pulse's location relative to the throat.

Original authors: Soumya Bhattacharya, Suman Ghosh

Published 2026-09-10

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CC BY 4.0

Original authors: Soumya Bhattacharya, Suman Ghosh

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

Gravity is often imagined as a steady, invisible hand that holds the universe together, but in the most extreme environments, it behaves more like a dynamic, rippling fabric. When massive objects collide or accelerate, they send out gravitational waves—disturbances in the fabric of space and time that travel at the speed of light. For decades, scientists have focused on the immediate, rhythmic shaking these waves cause as they pass. However, a subtler, permanent effect has long been predicted: the "memory" of a gravitational wave. This is the idea that after a wave passes, the space between two objects does not simply return to its original state. Instead, the objects are left slightly closer together or slightly farther apart, and they may even be moving at a different speed relative to one another than they were before the wave arrived. This permanent shift is a direct signature of the violent, non-linear nature of gravity, offering a way to test our understanding of the universe in ways that standard wave detection cannot.

While we have detected these waves from colliding black holes and neutron stars, the universe may contain other, stranger objects that mimic black holes but are fundamentally different. One such theoretical possibility is a wormhole—a tunnel connecting two distant regions of space. Unlike black holes, which have a point of no return, a wormhole is a bridge that can be crossed. The most famous theoretical model for such a structure is the Ellis-Bronnikov wormhole, a smooth, spherical tunnel held open by a peculiar type of matter. Recently, physicists have developed a more flexible version of this model, called the generalised Ellis-Bronnikov wormhole, which allows for different shapes and sizes of the tunnel's narrowest point, known as the throat. The question remains: if a gravitational wave were to pass through such a tunnel, would the memory it leaves behind look different than the memory left by a wave passing through empty space or around a black hole?

In a new study, researchers set out to answer this by simulating how a burst of gravitational waves would interact with the geometry of these generalised wormholes. They did not look for the waves themselves, but rather for the permanent "scars" they would leave on the space around them. To do this, they imagined a pair of test particles floating freely near the wormhole, acting as a simple detector. As a short, localized pulse of gravitational waves swept through the scene, the researchers tracked how the distance between these particles changed. Crucially, they subtracted the natural movement caused by the wormhole's own gravity, isolating only the change induced by the wave pulse. They found that the particles did indeed end up in a new position and moving at a new speed compared to where they would have been without the wave. This confirmed that the memory effect exists in this environment, but its strength and character depend heavily on exactly where the wave hits the wormhole and how "steep" or "flat" the tunnel's walls are.

The study revealed that the location of the wave pulse relative to the wormhole's throat makes a dramatic difference. When the pulse passes directly through the center of the tunnel, the particles experience a change in their relative speed, but the total change in their separation distance averages out to zero over time. However, the shape of the space between them is permanently distorted. It is as if the space itself has been stretched or squeezed in a specific direction, even if the overall distance hasn't shifted. In contrast, when the wave passes through the tunnel away from the center, the symmetry breaks. The particles experience a permanent shift in both their distance and their speed, creating an asymmetric response that clearly distinguishes this event from a wave passing through empty space. This suggests that the geometry of the wormhole imprints a unique signature on the memory effect, acting like a fingerprint that reveals the structure of the tunnel the wave traveled through.

Beyond just tracking two particles, the researchers also examined the behavior of a continuous stream of particles, a concept known as a geodesic congruence. This approach allowed them to study how the entire volume of space expands or contracts as the wave passes. They found that when a wave hits the throat of the wormhole, the space expands and compresses in a balanced way, resulting in no net change in volume. However, the shape of that volume is permanently altered, a phenomenon they call "shear memory." When the wave hits the tunnel away from the center, this balance is broken, leading to a net change in the volume of space as well as its shape. The study also showed that the specific shape of the wormhole matters: as the tunnel becomes steeper and more localized, the memory effect becomes weaker and less sensitive to the passing wave. This indicates that the "hairs" or specific geometric details of a wormhole directly influence how it records the passage of gravitational waves.

These findings suggest that if we ever detect a gravitational wave signal that carries a memory effect, the specific pattern of that memory could tell us whether the wave passed near a black hole or through a wormhole. The permanent shift in position and the distortion of space are not just abstract mathematical curiosities; they are observable consequences of the universe's most extreme geometries. While this work remains a theoretical simulation based on the equations of general relativity, it provides a clear roadmap for what to look for. If future detectors can measure these subtle, permanent changes in the relative motion of objects, they may one day be able to distinguish between the familiar darkness of a black hole and the exotic, traversable tunnels of a wormhole, revealing a new chapter in our understanding of the cosmos.

Technical Summary: Gravitational-wave memory and geodesic-congruence response in generalised Ellis–Bronnikov wormholes

Problem Statement
Following the detection of gravitational waves (GWs) and observations of supermassive compact objects, there is a growing interest in testing General Relativity (GR) in the strong-field regime and distinguishing between black holes and "black hole mimickers," such as Exotic Compact Objects (ECOs). Among these mimickers, wormholes represent a speculative class of objects rooted in GR itself. While the four-dimensional Ellis–Bronnikov (EB) wormhole, sustained by a phantom scalar field, is a well-studied geometry, it requires negative energy density. To address energy condition violations, the Generalised Ellis–Bronnikov (GEB) model was introduced, which incorporates additional positive energy matter to satisfy the Weak Energy Condition at the throat.

Despite extensive study of GEB wormholes regarding lensing, quasinormal modes, and merger events, a comprehensive analysis of the gravitational-wave memory effect in this specific background has been absent from the literature. The memory effect refers to the permanent change in the relative distance between test particles (displacement memory) or their relative velocity (velocity memory) after a GW pulse passes. Furthermore, recent theoretical developments suggest that geodesic congruences experience a distinct kinematic memory known as B-memory, characterized by permanent changes in expansion and shear. This paper aims to fill the gap by investigating how the memory effect depends on the ambient GEB spacetime, specifically analyzing displacement, velocity, and B-memory in relation to wormhole parameters (throat radius $b_0$ and steepness $m$ ) and the pulse location.

Methodology
The authors adopt a theoretical and qualitative approach using the test-wave approximation. They do not solve the linearized Einstein equations for a self-consistent GW source but instead analyze the response of test particles and congruences to a localized, transient GW pulse propagating through a fixed GEB background.

Geometry and Setup: The GEB spacetime is described using a two-parameter family ( $m$ and $b_0$ ) in Schwarzschild-like coordinates, converted to a tortoise coordinate $l$ to handle the wormhole throat. The metric is perturbed by a GW pulse $H(u)$ in Bondi-Sachs-like coordinates, where $u = t - l$ is the outgoing null coordinate. The pulse profile is modeled as $H(u) = A \text{sech}^2(u/u_0)$ .
Operational Definition of Memory: Since GEB spacetimes possess two disconnected asymptotic regions and lack a single global null infinity, standard BMS-based memory definitions are not directly applicable. Instead, the authors define memory operationally by comparing the evolution of geodesics (or congruences) in the presence of the pulse versus the background evolution alone.
- Displacement Memory: Defined as the asymptotic difference in separation $\tilde{\Delta}X$ between two neighboring geodesics after the pulse passes, where $\tilde{\Delta}X = \Delta X_{GW} - \Delta X_{background}$ .
- Velocity Memory: Defined as the asymptotic difference in the rate of change of this separation.
- B-Memory: Analyzed using the Raychaudhuri equations for timelike geodesic congruences. The authors track the expansion scalar $\theta$ and shear tensor $\Sigma_{\alpha\beta}$ . B-memory is quantified by the integrated change in these quantities (specifically the area under the curve of $\theta$ and $\Sigma^2$ ) between asymptotic regions.
Numerical Analysis: The geodesic equations and Raychaudhuri equations are solved numerically using Mathematica. The analysis focuses on two scenarios:
- Throat-crossing geodesics: Geodesics passing through the wormhole throat ( $l=0$ ) where the pulse is located.
- Returning geodesics: Geodesics that approach the throat, reach a point of closest approach, and return to infinity, meeting the pulse away from the throat.

Key Contributions and Results

Geodesic Separation (Displacement and Velocity Memory):
- The study confirms that a GW pulse induces permanent changes in the separation of neighboring geodesics.
- Throat-crossing scenario: When the pulse is at the throat, the analysis reveals a finite displacement memory and a finite velocity memory in the radial ( $l$ ) direction. However, along the angular ( $\phi$ ) direction, while displacement memory is present, no significant velocity memory signature is observed asymptotically.
- Away-from-throat scenario: When the pulse meets geodesics away from the throat, both displacement and velocity memory are observed in both radial and angular coordinates. The response is asymmetric compared to the throat case.
- Parameter Dependence: The magnitude of the memory effect is systematically dependent on the wormhole parameters. Decreasing the throat radius $b_0$ (increasing the curvature relative to the pulse) or increasing the steepness parameter $m$ modifies the memory shift. Specifically, higher $m$ geometries (where the exotic matter is more localized away from the throat) result in a reduced pulse-induced deformation, indicating greater kinematic stability of the congruence.
Geodesic Congruence Response (B-Memory):
- Using the Raychaudhuri equations, the authors demonstrate that the memory effect is also visible in the kinematics of geodesic congruences (expansion and shear).
- Throat-crossing congruence: For a congruence crossing the throat with the pulse at $l=0$ , the net integrated expansion ( $\int \theta d\tau$ ) vanishes, meaning there is no net volume change. However, the integrated shear ( $\int \Sigma^2 d\tau$ ) is non-zero. This indicates that the pulse induces a permanent shape deformation of the congruence cross-section even without a net volume change.
- Asymmetry and Pulse Location: A critical finding is that when the pulse is located away from the throat (e.g., at $l = \pm 1/2$ ), the symmetry of the expansion profile is broken. The congruence experiences a net non-zero integrated expansion (either positive or negative depending on the side), whereas the throat-crossing case yields zero net expansion. This suggests the wormhole geometry imprints the pulse's location as an observable asymmetry in the congruence response.
- Parameter Sensitivity: Similar to the geodesic separation results, increasing the steepness parameter $m$ reduces the pulse-induced deformation in the expansion and shear profiles, suggesting that the B-memory observable is less sensitive to the pulse in "flatter" throat geometries.

Significance and Claims
The paper claims to provide the first comprehensive analysis of GW memory effects in the GEB wormhole background. The authors assert that their results demonstrate:

Distinct Signatures: Wormhole geometry imprints characteristic signatures on both detector-based (geodesic separation) and congruence-based (B-memory) observables.
Differentiation Mechanism: The memory response depends systematically on the throat radius, the steepness parameter, and crucially, the location of the pulse relative to the throat. This dependence offers a potential theoretical avenue to distinguish GEB wormholes from other compact objects or black holes.
B-Memory as a Diagnostic: The study highlights B-memory (specifically the integrated shear and expansion) as a complementary covariant diagnostic for GW memory, particularly useful in curved spacetimes where individual geodesic comparisons might be ambiguous.
Limitations: The authors explicitly state that their analysis is theoretical and qualitative, relying on the test-wave approximation. They do not claim to have modeled a realistic astrophysical GW source or solved the full perturbed Einstein equations. They note that a self-consistent perturbative treatment is a subject for future work.

The paper concludes that while the current analysis is limited to static, spherically symmetric solutions, the methodology establishes a framework for future investigations into rotating wormholes and more complex congruence classes.

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