General Relativity: An Introduction for Physicists · Problem 4.10

Tensor calculus on manifolds — Hobson, Efstathiou & Lasenby

Physics · covariant-divergence-antisymmetric-tensor

Problem

For a general rank-2 tensor

T^{ab}

, show that the covariant divergence is given by

\nabla_a T^{ab} = \frac{1}{\sqrt{|g|}}\partial_a(\sqrt{|g|}\,T^{ab}) + \Gamma^b{}_{ca}T^{ac}.

Show further that if

A^{ab} = -A^{ba}

are the components of an antisymmetric rank-2 tensor then

\nabla_a A^{ab} = \frac{1}{\sqrt{|g|}}\partial_a(\sqrt{|g|}\,A^{ab}).

Hence show that if the antisymmetric tensor field

A^{ab}

vanishes on a hypersurface

S

that bounds a region

V

of an

N

-dimensional manifold then

\int_V (\nabla_a A^{ab})\sqrt{-g}\,d^N x = 0.