Hamilton-Jacobi Equations for GR: Exploring Solutions

[nughret]

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.

 

[shoehorn]

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.

 

[Entropia]

There is no explicit form of the Hamilton-Jacobi equations for General Relativity (GR) as it is a highly complex theory involving ten components of the metric. However, there are various approaches that have been used to explore solutions to these equations.

One approach is to use numerical methods to solve the equations, which has been successful in obtaining solutions for some specific cases. Another approach is to use perturbation theory, where the equations are expanded in terms of a small parameter and solved iteratively.

Additionally, there have been attempts to find analytical solutions by using different coordinate systems or simplifying assumptions. However, these solutions may not accurately capture the full complexity of GR and are often limited to specific cases.

Overall, the Hamilton-Jacobi equations for GR are highly challenging to solve and require advanced mathematical techniques. It is an ongoing area of research and further developments in this field will likely lead to a better understanding of the theory and its solutions.