Frame Fields in General Relativity - Specifying The Metric Using A Coframe

Specifying The Metric Using A Coframe

Alternatively, the metric tensor can be specified by writing down a coframe in terms of a coordinate basis and stipulating that the metric tensor is given by

This is just a fancy way of saying that the coframe is orthonormal. Whether this is used to obtain the metric tensor after writing down the frame (and passing to the dual coframe), or starting with the metric tensor and using it to verify that a frame has been obtained by other means, it must always hold true.