Past exam of the mathematics course of the University of Cambridge 2009 iii Paper 56 1 a Solution Created 2026-10-03 Updated 2026-10-14
Work in geometrized units , with the stationary Killing vector field normalized by at infinity. Put , , and ; is the areal radius. The static spherical Komar mass in reciprocal radial gauge givesIndeed the future and outward unit normals to a round sphere are and , and . The Komar mass integral is therefore . Since and ,Taking the asymptotic limit yieldsThis agrees with . The finite-radius Komar integral need not equal its asymptotic value, because this solution is not vacuum. Restoring divides the mass expression by when has dimensions of length.