Einstein equations of gravitation are rewritten in terms of a certain symmetric connection $\Gamma^{\alpha}_{\beta\mu}$ which differs, in general, from the Levi--Civita metric connection $\oGamma^{\alpha}_{\beta\mu}$. The symmetric connection $\Gamma^{\alpha}_{\beta\mu}$ appears in a natural way and Einstein equations have a simple interpretation in terms of $\Gamma^{\alpha}_{\beta\mu}$. The equivalence between the so called \emph{purely metric}, \emph{purely affine} and \emph{metric--affine} formulations of General Relativity is proved. As an example, the theory of a charged gravitating barotropic fluid is discussed in detail. Other examples are also given.
On the equivalence between the purely metric and the purely affine theories of gravitation
FERRARIS, Marco;
1983-01-01
Abstract
Einstein equations of gravitation are rewritten in terms of a certain symmetric connection $\Gamma^{\alpha}_{\beta\mu}$ which differs, in general, from the Levi--Civita metric connection $\oGamma^{\alpha}_{\beta\mu}$. The symmetric connection $\Gamma^{\alpha}_{\beta\mu}$ appears in a natural way and Einstein equations have a simple interpretation in terms of $\Gamma^{\alpha}_{\beta\mu}$. The equivalence between the so called \emph{purely metric}, \emph{purely affine} and \emph{metric--affine} formulations of General Relativity is proved. As an example, the theory of a charged gravitating barotropic fluid is discussed in detail. Other examples are also given.
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