Hamiltonian Analysis in Curved Spacetime: Is it Possible in General Relativity?

[BWV]

can you construct (or if yes, is it regularly done) a Hamiltonian in curved spacetime? If you took a system and moved it into a strong gravitational field or accelerated it to relativistic speeds can you still do Hamiltonian mechanics?

[URL]http://upload.wikimedia.org/math/8/d/6/8d65ea399bf81fbc3c9ca911c44cd9f3.png[/URL]
(http://en.wikipedia.org/wiki/Canonical_coordinates)

would it then follow that this commutation relationship (with the addition of time) would give you the metric tensor?

 

[Ben Niehoff]

In any theory with a Lagrangian (i.e., most theories, including physics in curved spacetimes), you can define conjugate momenta, take a Legendre transformation, and thereby define a Hamiltonian.

However, in relativistic theories, you will run into a problem: The Hamiltonian constructed in this way vanishes identically! The crux of the problem is that time is now a coordinate rather than an evolution parameter.

The solution to the problem is to choose a specific frame; i.e. choose a specific time coordinate to be your evolution parameter, and define Hamiltonian mechanics in the usual way on the remaining spacelike coordinates. This works in flat spacetime and it can be shown that the equations of motion remain Lorentz-invariant (even though the Lorentz symmetry is no longer manifest, as it is in Lagrangian mechanics). I think it can be made to work in curved spacetime, too, although I haven't seen it specifically.

 

[dextercioby]

Flat spacetime allows you to perform a Hamiltonian analysis, for its background R^4 can be seen as an R-infinity of R^3. Thus picking a specific time slice, t_0, you can retrieve the classical R^3 as configurations space, hence R^3 x R^3 as its cotangent bundle.

A cumbersome topology of a curved spacetime introduces, I think, terrible complications to the flat space scenario. A Hamiltonian for GR was, however, written down in 1962 by Arnowitt, Deser & Misner.