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Tetrads in General Relativity

XI. Field Equations

Einstein-Palatini Action

   The canonical Levi-Civita connection is characterized by the fact that the connection is compatible with the metric and has no torsion. However, a priori, metric and connection are separate concepts, and the notion of curvature can be defined for arbitrary connections. To take metric and connection as independent variables in the action principle, was suggested by work of Palatini (1919) and given its variational form by Einstein (1923-25) in his search for the unification of gravity and electrodynamics.

   In the Einstein-Palatini framework, the Einstein-Hilbert action integral (11.06) is written in terms of the Ricci tensor (10.17) defined by contraction of the Riemann (10.3), so that the Ricci tensor depends on the connection but not explicitly on the metric:

11.7

\[{S_{{\rm{EP}}}}[g,\Gamma ] := \frac{1}{{2\kappa }}\int {{d^4}x} \sqrt {\left| g \right|} \left( {{g^{\mu \nu }}{R_{\mu \nu }}[\Gamma ] + 2\Lambda } \right)\]

The pair of metric and connection $\left\{ {{g^{\mu \nu }},\Gamma _{\mu \nu }^\lambda } \right\}$ constitute the basic gravitational variables, without the restriction that the connection is torsion free, i.e. symmetric in its lower indices. The Einstein equations are then obtained by varying the (inverse) metric, and the metric compatible and torsion-free Levi-Civita connection by varying the connection. It is called a first-order formulation as the variables to vary over involve only up to first derivatives in the action.

   In the actual variational calculation it is convenient to introduce the contortion tensor $C_{\mu \nu }^\lambda $, i.e. the deviation from the Levi-Civita connection, as the variable to be varied. Solving the Euler-Lagrange equations of motion, one is led to the conclusion that the connection is the Levi-Civita one, up to an arbitrary vector ${A_\mu }$:

11.8

\[\Gamma _{\mu \nu }^\lambda = {\left. {\Gamma _{\mu \nu }^\lambda } \right|_{{\rm{LC}}}} + {A_\mu }\delta _\nu ^\lambda \qquad{A_\mu }: = C_{\sigma \mu }^\sigma \]

Since ${A_\mu }$ can have any value, one may set this vector equal to zero. In fact, it was already stated by Einstein in 1925 that, in order to obtain the correct equations of motion, the trace of the torsion tensor (1.23) must vanish: $T_{\mu \nu }^\nu = 3 {A_\mu } = 0$.

   However, there is altogether no need to impose any condition because the arbitrariness of ${A_\mu }$ corresponds to a (Noether) gauge symmetry of the action, as is easily verified. As a consequence, the equations of motion are invariant. Therefore, one may conclude that the theory described by the Einstein-Palatini action (11.7) is indeed Einstein's GRT, without any ambiguity.