General Relativity: An Introduction for Physicists · Problem 16.2

Inflationary cosmology — Hobson, Efstathiou & Lasenby

Physics · scalar-field-energy-momentum-tensor

Problem

Show that the energy–momentum tensor of a scalar field,

T_{\mu\nu} = (\partial_\mu\phi)(\partial_\nu\phi) - g_{\mu\nu}\left[\tfrac{1}{2}(\partial_\sigma\phi)(\partial^\sigma\phi) - V(\phi)\right],

satisfies the condition

\nabla_\mu T^{\mu\nu} = 0