---
tag: JPE3A1SN7V
kind: concept
name: Einstein–Hilbert action
msc: [83C05, 49S05]
sources:
  - url: https://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_action
same_as:
  wikidata: Q239224
  wikipedia: Einstein–Hilbert action
---

## Summary

The Einstein–Hilbert action is the functional on Lorentzian metrics of a four-dimensional spacetime given by the integral of the [[scalar curvature|ZHX2VYQDZ3]] against the volume form.
Its critical points are exactly the solutions of the vacuum [[Einstein field equations|3MCZ97SW6V]].
It casts general relativity as a variational principle, like the other field theories of physics.

## Explanations

### L1 human

In general relativity, gravity is the curvature of spacetime.
The Einstein–Hilbert action adds up the curvature over all of spacetime into one number.
The shapes of spacetime that make this number stationary, under small changes of the metric, are those allowed by Einstein's equations without matter.
For example, flat Minkowski space, the spacetime of special relativity, has no curvature at all and is such a stationary shape.
Adding the action of matter to it gives the full Einstein equations, with matter as the source of curvature.