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Peter Michor
Peter Michor
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It has found application. The main problem is that Einstein's equations have no type due to the large symmetry group (the whole diffeomorphism group). So Firstfirst one has to fix a gauge; this is done by fixing a space like hypersurface $\Sigma$.

Added in edit: As Deane Yang mentioned, after fixing a gauge, Einsteins equations become hyperbolic.

Then the initial conditions are a Riemannian metric on $\Sigma$ and a second funcdamental form along $\Sigma$. It is one of the great advances in nonlinear PDE-theory that then there exists a solution and it is unique and it is smooth, if all initial data are small and smooth. This was proved by Christodoulou and Klainerman. See, e.g.,

  • MR1946854 (2004f:58036) Reviewed Klainerman, Sergiu(1-PRIN); Nicolò, Francesco(I-ROME2M) 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 an account of this.

It has found application. The main problem is that Einstein's equations have no type due to the large symmetry group (the whole diffeomorphism group). So First one has to fix a gauge; this is done by fixing a space like hypersurface $\Sigma$. Then the initial conditions are a Riemannian metric on $\Sigma$ and a second funcdamental form along $\Sigma$. It is one of the great advances in nonlinear PDE-theory that then there exists a solution and it is unique and it is smooth, if all initial data are small and smooth. This was proved by Christodoulou and Klainerman. See, e.g.,

  • MR1946854 (2004f:58036) Reviewed Klainerman, Sergiu(1-PRIN); Nicolò, Francesco(I-ROME2M) 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 an account of this.

It has found application. The main problem is that Einstein's equations have no type due to the large symmetry group (the whole diffeomorphism group). So first one has to fix a gauge; this is done by fixing a space like hypersurface $\Sigma$.

Added in edit: As Deane Yang mentioned, after fixing a gauge, Einsteins equations become hyperbolic.

Then the initial conditions are a Riemannian metric on $\Sigma$ and a second funcdamental form along $\Sigma$. It is one of the great advances in nonlinear PDE-theory that then there exists a solution and it is unique and it is smooth, if all initial data are small and smooth. This was proved by Christodoulou and Klainerman. See, e.g.,

  • MR1946854 (2004f:58036) Reviewed Klainerman, Sergiu(1-PRIN); Nicolò, Francesco(I-ROME2M) 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 an account of this.

Source Link
Peter Michor
Peter Michor
  • 25.3k
  • 2
  • 64
  • 112

It has found application. The main problem is that Einstein's equations have no type due to the large symmetry group (the whole diffeomorphism group). So First one has to fix a gauge; this is done by fixing a space like hypersurface $\Sigma$. Then the initial conditions are a Riemannian metric on $\Sigma$ and a second funcdamental form along $\Sigma$. It is one of the great advances in nonlinear PDE-theory that then there exists a solution and it is unique and it is smooth, if all initial data are small and smooth. This was proved by Christodoulou and Klainerman. See, e.g.,

  • MR1946854 (2004f:58036) Reviewed Klainerman, Sergiu(1-PRIN); Nicolò, Francesco(I-ROME2M) 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 an account of this.