nLab Einstein equation

Idea

What are called Einstein’s equations are the equations of motion of gravity: the Euler-Lagrange equations induced by the Einstein-Hilbert action.

They say that the Einstein tensor $G$ of the metric/the field of gravity equals the energy-momentum tensor $T$ of the remaining force- and matter-fields:

$$G = T \,.$$

Properties

Existence and uniqueness

Given a choice of Cauchy surface $\Sigma$, the initial value problem for Einstein’s differential equations of motion is determined by a choice of Riemannian metric on $\Sigma$ and a second fundamental form along $\Sigma$.

With this data a solution to the equation exists and is unique. (Klainerman-Nicolo 03).

Related concepts

• equations of motion

  • Newton's laws of motion

  • Hamilton's equations, de Donder-Weyl-Hamilton equation

References

General

A general discssion is for instance in section 11 of

• Theodore Frankel, The Geometry of Physics - An Introduction

A discussion of the vacuum Einstein equations (only gravity, no other fields) in terms of synthetic differential geometry is in

• Gonzalo Reyes, A derivation of Einstein’s vacuum field equations (pdf)

PDE theory

Genuine PDE theory for Einstein’s equations goes back to local existence results by Yvonne Choquet-Bruhat in the 1950s. Global existence in the presence of a Cauchy surface was then shown in

• Sergiu Klainerman, Francesco Nicolo, The evolution problem in general relativity, Progress in Mathematical Physics, 25. Birkhäuser Boston, Inc., Boston, MA, 2003. xiv+385 pp. ISBN: 0-8176-4254-4

For further developments see

• H. Friedrich, A. D. Rendall, The Cauchy Problem for the Einstein Equations (arXiv:gr-qc/0002074)

• Alan D. Rendall, Partial differential equations in general relativity, Oxford University press 2008 (web)

• Hans Ringström, The Cauchy Problem in General Relativity, ESI Lectures in Mathematics and Physics 2009 (web)

• Yvonne Choquet-Bruhat, General relativity and the Einstein equations. Oxford University Press (2008) (publisher)

See also The Cauchy Problem in Classical Supergravity