General Relativity: An Introduction for Physicists · Problem 17.10

Linearised general relativity — Hobson, Efstathiou & Lasenby

Physics · greens-function-solution-linearised-field-equations

Problem

Show explicitly that

\bar{h}^{\mu\nu}(x^{\sigma}) = \bar{h}^{\mu\nu}_{(0)}(x^{\sigma}) - 2\kappa \int G(x^{\sigma} - y^{\sigma})T^{\mu\nu}(y^{\sigma})\, d^4y,

is a solution of the linearised Einstein field equations in the Lorenz gauge if

\bar{h}^{\mu\nu}_{(0)}(x^{\sigma})

is any solution of the linearised field equations in vacuo and

G(x^{\sigma} - y^{\sigma})

satisfies

\Box^2_x G(x^{\sigma} - y^{\sigma}) = \delta^{(4)}(x^{\sigma} - y^{\sigma}).