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arXiv · 2504.06515

One-Dimensional Relativistic Self-Gravitating Systems

Robert B. Mann
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Abstract

One of the oldest problems in physics is that of calculating the motion of $N$ particles under a specified mutual force: the $N$-body problem. Much is known about this problem if the specified force is non-relativistic gravity, and considerable progress has been made by considering the problem in one spatial dimension. Here, I review what is known about the relativistic gravitational $N$-body problem. Reduction to one spatial dimension has the feature of the absence of gravitational radiation, thereby allowing for a clear comparison between the physics of one-dimensional relativistic and non-relativistic self-gravitating systems. After describing how to obtain a relativistic theory of gravity coupled to $N$ point particles, I discuss in turn the two-body, three-body, four-body, and $N$-body problems. Quite general exact solutions can be obtained for the two-body problem, unlike the situation in general relativity in three spatial dimensions for which only highly specified solutions exist. The three-body problem exhibits mild forms of chaos, and provides one of the first theoretical settings in which relativistic chaos can be studied. For $N\geq 4$, other interesting features emerge. Relativistic self-gravitating systems have a number of interesting problems awaiting further investigation, providing us with a new frontier for exploring relativistic many-body systems.

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Robert B. Mann. 2025-04-09. One-Dimensional Relativistic Self-Gravitating Systems. https://doi.org/10.3390/e26070612

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