A. Zee, Quantum Field Theory in a Nutshell · Problem VIII.1.8

Field Theory Redux

Physics — QFT

Problem

The vielbeins for a spacetime with Minkowski metric is defined by

g_{\mu\nu}(x) = e^a{}_\mu(x)\eta_{ab}e^b{}_\nu(x)

, where the Minkowski metric

\eta_{ab}

replaces the Euclidean metric

\delta_{ab}

. The indices

a

and

b

are to be contracted with

\eta_{ab}

. For example,

R^{ab} = d\omega^{ab} + \omega^a{}_c\eta^{cd}\omega_d{}^b

. Show that everything goes through as expected.