# Why does the Weyl tensor not show up in the Einstein field equations?

In the Einstein field equations, the only tensor that shows up is the Ricci tensor and the metric tensor, together with the Ricci scalar. The Weyl tensor though is a tensor that is a part of the Riemann curvature tensor as well (the off-diagonal part). It represents tidal forces, which are non-local.

Why doesn"t the Weyl tensor show itself in the Einstein field equations? Is it because gravity is based on the equivalence principle, which is a first-order equivalence?

• Have you learned how the Einstein equations guarantee the local conservation of energy and momentum via the vanishing covariant divergence of the Einstein tensor? Commented Sep 2, 2023 at 5:35
• TLDR; the Weyl tensor has a vanishing contraction with the metric i.e., $g^{\mu\nu}W_{\mu\nu\rho\sigma}=0$ and tells us nothing useful (especially with regards to gravitational fields in a vacuum). Commented Sep 2, 2023 at 6:42
• @Ghoster Is that comparable with the conservation of charge and currents in electromagnetism, as expressed by a vanishing divergence of the charge density and the current? Commented Sep 2, 2023 at 7:24
• @josephh Isn't it in fact the defining tensor for true gravitational fields? Commented Sep 2, 2023 at 7:25
• @josephh looks like that could be an answer (or at least the start of one) Commented Sep 2, 2023 at 13:49

Why does the Weyl tensor not show up in the Einstein field equations?

I think that the “moral” reason for that is that the Weyl tensor represents locally free, propagating part of the curvature and is thus not pointwise determined by matter fields.

However, Weyl is not completely independent of matter. By combining differential Bianchi identity, $$∇_{[α}R_{βγ]δ} {} ^\epsilon = 0$$, with Einstein equations we can arrive at the following constraint on Weyl tensor: $$∇^δC_{αβγδ} = 8πG \left( ∇_{[β}T_{α]γ} −\frac 1 3 g_{γ[α}∇_{β]}T \right).$$ Moreover, the linearized version of this equation can be “repackaged” as a wave equation describing propagation of a massless spin-2 field, thus making clear that these are the propagating degrees of freedom.

The Einstein field equations relate the curvature of spacetime to the distribution of matter and energy. While the Weyl tensor is important for understanding gravitational waves and certain aspects of the gravitational field's geometry, it does not directly involve matter and energy distribution. Therefore, it does not appear explicitly in the Einstein field equations.

My incomplete'' answer is that this can be interpreted also by Newtonian analogy. The Poisson equation is the Newtonian analogue to the Einstein field equations.