Gravitational waves, black holes and cosmic strings in cylindrical symmetry

TL;DR

This paper explores the dynamics of gravitational waves, black holes, and cosmic strings within cylindrically symmetric Einstein gravity, deriving energy conservation laws and first laws of black-hole and string dynamics.

Contribution

It introduces an effective energy tensor for gravitational waves, formulates energy conservation and first laws for black holes and cosmic strings in cylindrical symmetry.

Findings

Energy tensor for gravitational waves matches massless Klein-Gordon field

Covariant conservation of energy-momentum with modified Thorne energy

Derived first laws for black-hole dynamics and cosmic strings

Abstract

Gravitational waves in cylindrically symmetric Einstein gravity are described by an effective energy tensor with the same form as that of a massless Klein- Gordon field, in terms of a gravitational potential generalizing the Newtonian potential. Energy-momentum vectors for the gravitational waves and matter are defined with respect to a canonical flow of time. The combined energy-momentum is covariantly conserved, the corresponding charge being the modified Thorne energy. Energy conservation is formulated as the first law expressing the gradient of the energy as work and energy-supply terms, including the energy flux of the gravitational waves. Projecting this equation along a trapping horizon yields a first law of black-hole dynamics containing the expected term involving area and surface gravity, where the dynamic surface gravity is defined with respect to the canonical flow of time. A first law for dynamic cosmic strings also follows. The Einstein equation is written as three wave equations plus the first law, each with sources determined by the combined energy tensor of the matter and gravitational waves.