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A simple analytic example of the gravitational wave memory effect

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Abstract

We report an analytical example of the gravitational wave memory effect in exact plane wave spacetimes. A square pulse profile is chosen which gives rise to a curved wave region sandwiched between two flat Minkowski spacetimes. Working in the Brinkmann coordinate system, we solve the geodesic equations exactly in all three regions. Issues related to the continuity and differentiability of the solutions at the boundaries of the pulse are addressed. The evolution of the geodesic separation reveals displacement and velocity memory effects with quantitative estimates depending on initial values and the amplitude and width of the pulse. The deformation caused by the pulse on a ring of particles is then examined in detail. Formation of caustics is found in both scenarios, i.e. evolution of separation for a pair of geodesics and shape deformation of a ring of particles—a feature consistent with previous work on geodesic congruences in this spacetime. In summary, our analysis provides a useful illustration of memory effects involving closed-form exact expressions.

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Data Availability Statement

This manuscript has no associated data, or the data will not be deposited. [Author’s comment: This article is entirely theoretical. No data from any other source is used anywhere, in the paper. Hence, no extra data needs to be deposited.]

Notes

  1. For Maxwell fields, we have a third term in H(u, x, y) like \(B(u)(x^2+y^2)\).

  2. All the plots in this article are generated using Mathematica 12.

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Funding

I. C. is supported by University Grants Commission, Government of India through a Senior Research Fellowship with Reference ID: 523711.

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Authors and Affiliations

  1. Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur, 721 302, India

    Indranil Chakraborty & Sayan Kar

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  1. Indranil Chakraborty
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Correspondence to Indranil Chakraborty.

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Chakraborty, I., Kar, S. A simple analytic example of the gravitational wave memory effect. Eur. Phys. J. Plus 137, 418 (2022). https://doi.org/10.1140/epjp/s13360-022-02593-y

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