General Relativity: An Introduction for Physicists · Problem 17.16

Linearised general relativity — Hobson, Efstathiou & Lasenby

Physics · conservation-energy-momentum-integrated

Problem

From the conservation equation

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

, show that

\partial_0 T^{00} + \partial_k T^{0k} = 0 \qquad \text{and} \qquad \partial_0 T^{i0} + \partial_k T^{ik} = 0.

By integrating each equation over a spatial volume

V

whose bounding surface

S

encloses the energy–momentum source and using the three-dimensional divergence theorem, show that the quantities

Mc^2 \equiv \int_V T^{00}\, d^3\vec{y} \qquad \text{and} \qquad P^i \equiv \int T^{i0}\, d^3\vec{y}

are constants and give a physical interpretation of them.