---
tag: 3MCZ97SW6V
kind: concept
name: Einstein field equations
msc: [83C05, 35Q76]
sources:
  - url: https://en.wikipedia.org/wiki/Einstein_field_equations
same_as:
  wikidata: Q273711
  wikipedia: Einstein field equations
---

## Summary

The Einstein field equations are the equations of general relativity.
They relate the curvature of a four-dimensional [[Lorentzian manifold|BY4QYYS788]], the spacetime, to the energy and momentum of matter in it.
For the metric they form a system of nonlinear second-order partial differential equations, which is hyperbolic after a suitable choice of coordinates.
Their solutions include the flat Minkowski space, the Schwarzschild black hole and the expanding cosmological models.

## Explanations

### L1 human

Gravity in Einstein's theory is not a force but the curvature of spacetime.
Matter and energy tell spacetime how to curve, and the curvature tells matter how to move.
The field equations are the precise form of the first half of this slogan.
Empty, flat spacetime with no matter is the simplest solution, and the field around a single star or black hole is the next.