Einstein equation




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Equality and Equivalence



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 GG of the metric/the field of gravity equals the energy-momentum tensor TT of the remaining force- and matter-fields:

G=T. G = T \,.


Existence and uniqueness

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

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



A general discssion is for instance in section 11 of

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

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

Last revised on September 14, 2016 at 11:23:43. See the history of this page for a list of all contributions to it.