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Tensor Calculus vs. Differential forms

Despite its success in GR, for a deeper understanding the tensor calculus can not be fully recommended. Cartan used almost exclusively the differential forms language. Contrarily to other statements in the literature, it allows a compact desciption of all relevant operations on tensors, including contractions and so--called divergences (by means of the Hodge star operator). A brief and good introductory text which is

Harley Flanders, Differential forms with applications to the physical sciences, Dover 1989

It may be that the old-fashioned tensor calculus has guided Einstein to some wrong assumptions on the field equations. For example, the construction of eqn.(28) of the above paper (covariant derivative and contraction with respect to a value index) is quite unnatural in a differential forms context.

A criticism that has been given by Cartan (I am grateful to Jose G. Vargas for directing my attention to it) is that the field equations proposed by Einstein may be a weaker condition than teleparallelism in the sense that these equations dom not imply the first Bianchi identity. Also, they are a weaker condition that teleparallelism because, even if they implied the first Bianchi identity, this identity would not imply that the curvature is zero, as the same Bianchi identity as in teleparallelism can be obtained with less stringent conditions.