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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.