Is it correct that based on the premise of foliation by Cauchy surfaces
- the ADM fomalism restricts the set of solutions for general relativity to causal manifolds
- and therefore excludes closed timelike and lightlike curves by construction?
Does that mean that based on a restriction of general relativity to spacetimes generated by time evolution in the ADM formalism the chronological protection conjecture is valid automatically?
Best Answer
I'm afraid the answer to this question is very long and technical, but here are some elements on how to solve this :
Depending on what you call the Hamiltonian formalism, you can always define it on any given spacetime. By which I mean, given a Lagrangian on some spacetime, it is always possible to perform a Legendre transform on it to obtain the appropriate phase space, which will involve the fields and their momenta rather than the jet extension of the field (ie you have the variables $\phi$ and $\pi$ rather than $\phi$ and $\partial_\mu \phi$). This is always possible and there are a variety of formalisms to do so, under various names such as the De Donder-Weyl, polysymplectic or multisymplectic formalism[1][2][3]. In these methods, you are roughly replacing the derivatives of the field by a phase space variable, we are not selecting any variable in particular to be our time parameter.
If you simply perform this, what you obtain from the Einstein-Hilbert action is quite simply equivalent to the Einstein field equations, simply with the appropriate quantities substituted [4].
In the case of the ADM formalism, we furthermore select a particular direction as the time direction. This is always more or less doable (assuming that our spacetime is time orientable), but on the other hand, this will not be particularly useful, since we in general don't have the corresponding orthogonal integrable distribution to this time flow (ie, a spacelike hypersurface), so that we don't really have a reasonable boundary to use as initial conditions. We could pick any boundary of a spacetime volume, as for any PDE, of course, but then, beyond purely practical issues (how do you define initial data on such a surface), you will get the same issue as in the following paragraph :
If we do have spacelike hypersurfaces, for instance if our spacetime is strongly causal, or if our spacetime isn't causal but a simple bundle with respect to time like the Minkowski torus (with time identified), things do become easier, but then from a PDE theory point of view, we don't necessarily have a well-posed problem. The ADM equations we have may not have any solutions or they may have more than one possible solutions. It's in fact quite generic for manifolds with closed timelike curves to have multiple developments, some of which are causal.
You can indeed use the ADM formalism on just about any spacetime (it has in fact been done a few times, such as in Carlip), but you do have to tread very carefully, as we can't assume most of the machinery of PDEs that we usually take for granted.
The benefit of globally hyperbolic spacetimes is of course that, given a nice enough spacetime (as described in the theorem of section 7.6 of Hawking Ellis), the development will be unique.