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Gio83
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Einstein's field equations (EFEs) describe the pointwise relation between the geometry of the spacetime and possible sources described by an energy-momentum tensor ##T^{ab}##. As well known, such equations can be derived from a variational principle applied to the following action: $$S=\int\, \left( L_{EH} + L_m \right)\, \sqrt{-g}\, d^4x\, ,$$ where ##L_{EH}=\frac{1}{2\kappa}R## is the Einstein-Hilbert Lagrangian and ##L_m## is the Lagrangian describing the matter/field sources. Thus EFEs are obtained by requiring the stationarity of ##S## with respect to variations of the metric, and they are given by $$R_{ab}−\frac{1}{2}g_{ab}\, R=\kappa\, T_{ab}\, .$$
Such equations are automatically compatible with energy-momentum conservation due to the validity of the doubly-contracted Bianchi identites, which imply essentially the vanishing of the covariant divergence of the left-hand side of EFEs.
However, EFEs take into account only the gravitational degrees of freedom that couple to matter: the "free part" of the gravitational field is said to be encoded in the Weyl tensor ##C_{abcd}## and its evolution/propagation is given by the trace-free part of the once-contracted (second) Bianchi identities, which are given by $$\nabla^d C_{abcd}=\nabla_{[a}\left( −R_{b]c}+\frac{1}{6}\, R\, g_{b]c} \right)\, .$$
In a 1+3 covariant splitting of the spacetime, such equations take a form which is strikingly similar to Maxwell's equations for electromagnetism (see https://arxiv.org/abs/gr-qc/9704059 for more details on such gravito-electromagnetic analogy).
Hence the question: would it be possible to obtain the equations of motion for the free gravitational field (the Bianchi identities above) from a variational principle? This, of course, implies the more detailed question: what would be the action and with respect to which quantity one has to vary such action in order to obtain the aforementioned equations of motion?
Such equations are automatically compatible with energy-momentum conservation due to the validity of the doubly-contracted Bianchi identites, which imply essentially the vanishing of the covariant divergence of the left-hand side of EFEs.
However, EFEs take into account only the gravitational degrees of freedom that couple to matter: the "free part" of the gravitational field is said to be encoded in the Weyl tensor ##C_{abcd}## and its evolution/propagation is given by the trace-free part of the once-contracted (second) Bianchi identities, which are given by $$\nabla^d C_{abcd}=\nabla_{[a}\left( −R_{b]c}+\frac{1}{6}\, R\, g_{b]c} \right)\, .$$
In a 1+3 covariant splitting of the spacetime, such equations take a form which is strikingly similar to Maxwell's equations for electromagnetism (see https://arxiv.org/abs/gr-qc/9704059 for more details on such gravito-electromagnetic analogy).
Hence the question: would it be possible to obtain the equations of motion for the free gravitational field (the Bianchi identities above) from a variational principle? This, of course, implies the more detailed question: what would be the action and with respect to which quantity one has to vary such action in order to obtain the aforementioned equations of motion?