# What is the lagrangian of Einstein field equations

What is a lagrangian such that Euler-Lagrange equation (not sure if it's correct form for this case)

$$\frac{\partial \mathcal{L}}{\partial g_{\mu\nu}}=\partial_\lambda\frac{\partial \mathcal{L}}{\partial (\partial_\lambda g_{\mu\nu})}.$$

Gives us Einstein field equations?

This is almost certainly answered elsewhere, but the Hilbert Action, from which Einstein's equation can be derived, is:

$$S = \int d^{4}x\;\left(\sqrt{|g|}\frac{1}{16\pi G}R + \mathcal{L}_{m}\right)$$

taking the variation is pretty complicated (there are second derivatives of the metric in the action, and you have to deal with gauge invariance) and best looked up in a textbook, though. But note, that by this definition, we define $$T_{ab} = \frac{\delta \mathcal{L_m}}{\delta g^{ab}}$$

• The Wikipedia page does a pretty good job as an introduction of walking through the variation: en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_action. Also, if it is not clear, the action is an integral over $d^4x$. Jun 18, 2022 at 19:31
• Perhaps you'd like to add a general matter Lagrangian to that as well, since the Einstein field equations are usually formulated with the presence of a stress-energy tensor. The sign difference between them and the general arbitration in the overall choice could also be emphasized. Jun 18, 2022 at 19:34
• So the lagrangian is just $\mathcal{L}=\frac{\sqrt{|g|}R}{16\pi G}$? Jun 18, 2022 at 19:36
• @JavaGamesJAR yes Jun 18, 2022 at 19:41
• Really defining the volume integral means defining the theory of forms, which is usually beyond the scope of a first-year GR course. Jun 19, 2022 at 2:56

With the cosmological constant $$\Lambda$$ included, the Hilbert action for empty space is $$S = \frac{c^4}{16 \pi G} \int (R-2 \Lambda) \sqrt{-g} \, \mathrm{d}^4 x.$$

Wikipedia calls it the Einstein-Hilbert action, but this is wrong. The action is due to Hilbert, not to Einstein.

• Notice that this action does not yield the stress-energy tensor on the right-hand side of Einstein's Equations Jun 19, 2022 at 11:50
• Indeed, as stated, it is only for vacuum; matter can be added as above. Jun 19, 2022 at 11:51