Strains in General Relativity

TL;DR

This paper refines the understanding of strains and relative accelerations in general relativity, emphasizing the importance of transport laws and observer dependence, with implications for gravity gradiometry and relativistic tests.

Contribution

It introduces a relativistically complete definition of strains considering transport laws and extends the concept of geodesic deviation to accelerated particles.

Findings

Generalized Szekeres' compass of inertia for accelerated particles

Emphasized observer-dependent formulation of strains

Contributed to the theory of relativistic gravity gradiometers

Abstract

The definition of relative accelerations and strains among a set of comoving particles is studied in connection with the geometric properties of the frame adapted to a "fiducial observer." We find that a relativistically complete and correct definition of strains must take into account the transport law of the chosen spatial triad along the observer's congruence. We use special congruences of (accelerated) test particles in some familiar spacetimes to elucidate such a point. The celebrated idea of Szekeres' compass of inertia, arising when studying geodesic deviation among a set of free-falling particles, is here generalized to the case of accelerated particles. In doing so we have naturally contributed to the theory of relativistic gravity gradiometer. Moreover, our analysis was made in an observer-dependent form, a fact that would be very useful when thinking about general relativistic tests on space stations orbiting compact objects like black holes and also in other interesting gravitational situations.