What Are the Hamilton-Jacobi Equations for General Relativity?

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SUMMARY

The Hamilton-Jacobi equations for General Relativity (GR) can be derived using the Hamiltonian formulation outlined in Wald's text. The discussion highlights the challenges faced when attempting to use the 10 components of the metric as coordinates and R*sqrt(det(g)) as the Lagrangian density, which complicates the computation of canonical conjugates. A clear understanding of Wald's Appendix E is essential for successfully deriving the Hamilton-Jacobi equations in GR.

PREREQUISITES
  • Understanding of General Relativity principles and metrics
  • Familiarity with Hamiltonian mechanics
  • Knowledge of Lagrangian densities and their applications
  • Ability to compute canonical conjugates in theoretical physics
NEXT STEPS
  • Study Wald's "General Relativity" focusing on Appendix E for Hamiltonian formulation
  • Explore the derivation of Hamilton-Jacobi equations in classical mechanics
  • Research the application of Lagrangian densities in field theory
  • Examine canonical transformations and their role in theoretical physics
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The discussion is beneficial for theoretical physicists, graduate students in physics, and researchers focusing on the mathematical foundations of General Relativity and Hamiltonian mechanics.

nughret
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I was wondering if anyone had an explicit form of the hamilton-jacobi equations for GR.
I had a little attempt myself using the 10 components of the metric as the 'co-ordinates', R*sqrt(det(g)) as the Langrangian density, but the maths got a bit messy when trying to compute their canonical conjugates.
 
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nughret said:
I was wondering if anyone had an explicit form of the hamilton-jacobi equations for GR.
I had a little attempt myself using the 10 components of the metric as the 'co-ordinates', R*sqrt(det(g)) as the Langrangian density, but the maths got a bit messy when trying to compute their canonical conjugates.

The reason it was "messy" is because this approach obviously can't work. See Appendix E of Wald for a reasonably good treatment of the Hamiltonian formulation of GR. Once you've understood this material it should then be trivial to derive the Hamilton-Jacobi approach.
 

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