General Relativity: An Introduction for Physicists · Problem 17.11

Linearised general relativity — Hobson, Efstathiou & Lasenby

Physics · greens-function-fourier-transform

Problem

The Green's function

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

satisfies the equation

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

Show that the four-dimensional Dirac delta function can be written as

\delta^{(4)}(x^{\sigma} - y^{\sigma}) = \frac{1}{(2\pi)^4}\int \exp\left[ik_{\lambda}(x^{\lambda} - y^{\lambda})\right] d^4k.

Hence, by writing the Green's function in terms of its Fourier transform

\widetilde{G}(k^{\sigma})

, show that

G(x^{\sigma} - y^{\sigma}) = -\frac{1}{(2\pi)^4}\int \frac{1}{k^2}\exp\left[ik_{\lambda}(x^{\lambda} - y^{\lambda})\right] d^4k,

where

k^2 = k_{\lambda}k^{\lambda}