Radiation coordinates in general relativity

Abstract

This paper introduces a system of coordinates (called radiation coordinates ) which may be useful in discussing problems of gravitational radiation. Starting with a general system of coordinates x i in curved space-time, an assigned timelike curve C , and an orthonormal tetrad assigned along C , the radiation coordinates x a of any event P are defined by a formula involving the world function. In radiation coordinates the equation of any null cone, drawn from an event x a' on C into the future, has the Minkowskian form This paper introduces a system of coordinates (called radiation coordinates ) which may be useful in discussing problems of gravitational radiation. Starting with a general system of coordinates x i in curved space-time, an assigned timelike curve C , and an orthonormal tetrad assigned along C , the radiation coordinates x a of any event P are defined by a formula involving the world function. In radiation coordinates the equation of any null cone, drawn from an event x a' on C into the future, has the Minkowskian form ƞ ab (x a - x a' ) (x b - x b’ ) = 0, ƞ ab = diag (1, 1, 1, - 1 ) . The metric tensor satisfies the coordinate conditions g ab (x b - x b' ) = ƞ ab (x b - x b' ), where x b' are regarded as functions of x b viz. the coordinates of the point of intersection of C with the null cone drawn into the past from x b . Continuity on C of the Jacobian matrix of the transformation x → x^ ~ is ensured by demanding the constancy of the components on the tetrad of the unit tangent vector to C . Continuity of the second derivatives of the transformation cannot be obtained except in very special circumstances. If space-time is flat, C a geodesic, and the orthonormal tetrad transported parallelly along C with the fourth vector tangent to C , then radiation coordinates reduce to the usual Minkowskian coordinates having C for time axis.