On the Ultrarelativistic Limit of General Relativity

TL;DR

This work investigates the ultrarelativistic limit of General Relativity (c->0), where light cones shrink to a null congruence and spacetime becomes a degenerate four-dimensional null hypersurface $V^{(1)}_4$, with Carroll symmetry. Using a degenerate metric framework and adapted tetrads, the authors derive ultrarelativistic field equations that are simpler than Einstein's equations and evolve along the generators of the singular congruence; matter can be included via an expansion in epsilon=c^2, yielding a hierarchy of equations. The paper demonstrates the approach with vacuum and dust solutions, including gravitational-wave-like dynamics and isotropic expansion, highlighting the method's potential for near-ultralrelativistic regimes and for developing post-ultrarelativistic corrections. This provides a tractable, geometrically transparent avenue to study strong gravity limits and the interplay between causality, degeneracy, and dynamics in GR.

Abstract

As is well-known, Newton's gravitational theory can be formulated as a four-dimensional space-time theory and follows as singular limit from Einstein's theory, if the velocity of light tends to the infinity. Here 'singular' stands for the fact, that the limiting geometrical structure differs from a regular Riemannian space-time. Geometrically, the transition Einstein to Newton can be viewed as an 'opening' of the light cones. This picture suggests that there might be other singular limits of Einstein's theory: Let all light cones shrink and ultimately become part of a congruence of singular world lines. The limiting structure may be considered as a nullhypersurface embedded in a five-dimensional spacetime. While the velocity of light tends to zero here, all other velocities tend to the velocity of light. Thus one may speak of an ultrarelativistic limit of General Relativity. The resulting theory is as simple as Newton's gravitational theory, with the basic difference, that Newton's elliptic differential equation is replaced by essentially ordinary differential equations, with derivatives tangent to the generators of the singular congruence. The Galilei group is replaced by the Carroll group introduced by Lévy-Leblond. We suggest to study near ultrarelativistic situations with a perturbational approach starting from the singular structure, similar to post-Newtonian expansions in the $c \to \infty$ case.