GahQtc K 8ac7i

It will be convenient to take the contravariant form gah of the metric to be more fundamental and the covariant form gab as derived from it by (7.1). Using the alternating tensor tfabcd defined by the background metric, this relation can be expressed explicitly as

where (det g)"1 = 1 ff^W^^L/i is the determinant of the components of g^ in a basis which is orthonormal with respect to the metric

The difference between the connection T defined by g and the connection $ defined by § is a tensor, and can be expressed in terms of the covariant derivative of g with respect to T (cf § 3.3):

where we have used a stroke to denote covariant differentiation with respect to t1 and the symbol S to denote the difference between quantities defined from g and Then from (2.20),

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