ASYMPTOTIC PROPERTIES OF THE WEYL TENSOR IN HIGHER DIMENSIONS

Abstract

The Riemann tensor is the most important geometric quantity that one can compute at any given point on a manifold of interest. It measures the extent to which the metric at the given point deviates from flatness. The Riemann tensor can be decomposed in two parts: a tracefree part that is given by the Weyl tensor and a part that is related to the Ricci curvature and the scalar associated with this. Of course, in Einstein’s theory of gravity, the Ricci part of the Riemann tensor is constrained to be equal to the energy-momentum tensor of the matter fields via the Einstein equation. Thus, in general relativity, the Weyl tensor is the geometric quantity that encodes the propagating degrees of freedom in a solution of the theory. The existence of gravitational radiation is an important prediction of general relativity and, as explained above, it is the Weyl tensor that contains information about the specific characteristics of this radiation. In general, a source of gravitational radiation has non-trivial dynamics, Thus, the emitted gravitational radiation, and by implication, the Weyl tensor will be complicated. However, in four spacetime dimensions, the Weyl tensor exhibits a “peeling” property, which ensures that at distances far enough away from the isolated source, complicated components of the radiation decay away. More precisely, in 4d, the Weyl tensor of an asymptotically flat spacetime is of the form: