On the Ultrarelativistic Limit of General Relativity

TL;DR

This paper explores the ultrarelativistic limit of General Relativity, revealing a simplified theory characterized by a null hypersurface structure and the Carroll group, as an alternative to the classical Newtonian limit.

Contribution

It introduces a novel ultrarelativistic limit of Einstein's theory, leading to a new geometric structure and symmetry group, expanding understanding of extreme relativistic regimes.

Findings

Ultrarelativistic limit results in a null hypersurface structure.

The theory simplifies to ordinary differential equations along singular congruences.

The symmetry group transitions from the Galilei to the Carroll group.

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 L\'evy-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.