World-Line Perturbation Theory

Abstract

The motion of a compact body in space and time is commonly described by the world line of a point representing the instantaneous position of the body. In General Relativity such a world-line formalism is not quite straightforward because of the strict impossibility to accommodate point masses and rigid bodies. In many situations of practical interest it can still be made to work using an effective hamiltonian or energy-momentum tensor for a finite number of collective degrees of freedom of the compact object. Even so exact solutions of the equations of motion are often not available. In such cases families of world lines of compact bodies in curved space-times can be constructed by a perturbative procedure based on generalized geodesic deviation equations. Examples for simple test masses and for spinning test bodies are presented. *

Curvature Dynamics in General Relativity

“…Taking traces of the Bianchi identity (7) and its divergence (8), it follows that the Ricci tensor has a divergence-free extension, the Einstein tensor:…”

“…The Bianchi identity (7) can be rewritten as an identity for the Weyl tensor; it then takes the form…”

“…In the case of gravity, it is simpler to derive the equation for the overall curvature in the presence of energy densities (the Riemann curvature) than for pure gravity (the Weyl curvature). The derivation starts from the Bianchi identity (7), taking a divergence:…”

“…[5] 2 For a detailed and thorough introduction with references see [6]. For a derivation and discussion with references see [7].…”