The tensorial representation of the distributional stress–energy quadrupole and its dynamics

Gratus, Jonathan and Talaganis, Spyridon (2023) The tensorial representation of the distributional stress–energy quadrupole and its dynamics. Classical and Quantum Gravity, 40 (8). ISSN 0264-9381

Abstract

We investigate stress–energy tensors constructed from the covariant derivatives of delta functions on a worldline. Since covariant derivatives are used all the components transform as tensors. We derive the dynamical equations for the components, up to quadrupole order. The components do, however, depend in a non-tensorial way, on a choice of a vector along the worldline. We also derive a number of important results about general multipoles, including that their components are unique, and all multipoles can be written using covariant derivatives. We show how the components of a multipole are related to standard moments of a tensor field, by parallelly transporting that tensor field.