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

In summary, Hamiltonian mechanics can be performed in curved spacetime as long as a specific frame is chosen, with a specific time coordinate as the evolution parameter. This method works in flat spacetime and has been extended to curved spacetime by Arnowitt, Deser, and Misner in 1962.
  • #1
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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?
 
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  • #2
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.
 
  • #3
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.
 

1. What is a phase space in general relativity?

A phase space in general relativity is a mathematical concept that represents all possible states of a physical system. It is a space with multiple dimensions, where each dimension corresponds to a different physical quantity. In general relativity, the phase space is used to describe the position and momentum of particles in a gravitational field.

2. Why is a phase space important in general relativity?

A phase space is important in general relativity because it allows us to understand and predict the behavior of particles in a gravitational field. By using the phase space, we can calculate the trajectories of particles and determine how they will move and interact with each other under the influence of gravity.

3. How is a phase space different in general relativity compared to classical mechanics?

In classical mechanics, the phase space is a six-dimensional space, with three dimensions for position and three dimensions for momentum. However, in general relativity, the phase space is much more complex and can have an infinite number of dimensions, depending on the number of particles and the complexity of the gravitational field.

4. Can a phase space in general relativity be visualized?

It is difficult to visualize a phase space in general relativity due to its high dimensionality. However, there are ways to represent it mathematically using graphs and diagrams. Additionally, computer simulations can be used to visualize the behavior of particles in a phase space.

5. How does the concept of a phase space relate to the theory of relativity?

The concept of a phase space is an important aspect of the theory of relativity. It allows us to understand how particles move and interact in a gravitational field, which is a fundamental concept in the theory of relativity. Additionally, the phase space is used to calculate the effects of gravity on space and time, which are central concepts in the theory of relativity.

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