# Hamiltonian Formulation of GR for Elliptic Spacetime?

## Main Question or Discussion Point

This has bothered me for some time. In the ADM formulation, we foliate spacetime into 3+1 dimensions by creating 3 dimensional hypersurfaces via $T = constant$ along the worldline of some observer whose proper time is $T$. This allows us to write dynamical equations for the evolution of some initial 3D spacelike hypersurface $\Sigma$.

However, this only works for a general flat or hyperbolic manifold, because geodesics which are initially normal to $\Sigma$ will never cross. For an elliptic spacetime, those geodesics would eventually cross each other, such that the map $\Sigma_T \rightarrow \Sigma_{T'}$ would fail to be one-to-one. This means that solutions to Einstein's Equations found via the ADM formulation cannot have closed timelike curves, since every geodesic can only intersect each hypersurface once.

Therefore, what limitations does the ADM formulation of GR impose on the full set of solutions of Einstein's Equations? If we can't foliate an elliptic manifold, then doesn't that mean that we won't find any elliptic spacetime geometries using the ADM method?

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PAllen
2019 Award
So far as I know (and I'm not an expert in ADM), it assumes a-priori that the manifold is globally hyperbolic to achieve the required foliation. Then it could not be used to find non-hyperbolic manifold as a solution. Some other method would be required to find such a solution.

WannabeNewton
According to Wald (sec E.2 p. 459), in the Hamiltonian formulation of GR the space-like hypersurfaces in the one-parameter family foliating the space-time are taken to be Cauchy so the space-time would have to be globally hyperbolic (as PAllen noted). Hopefully this helps! Cheers.

Thanks a bunch, that's what I've read as well. I'm writing my undergrad thesis on the ADM formulation of GR and using it as a basis of a theory of Quantum Gravity, and this seems to be a major limitation for any QG theory built from the Hamiltonian formulation. I'll take it up with my advisor

PAllen
2019 Award
Thanks a bunch, that's what I've read as well. I'm writing my undergrad thesis on the ADM formulation of GR and using it as a basis of a theory of Quantum Gravity, and this seems to be a major limitation for any QG theory built from the Hamiltonian formulation. I'll take it up with my advisor
Why major? All it prevents are closed time like curves and some other dubious causal structures (e.g. intersection of a past light cone of e1 and future light cone of e2 that is not compact). Many physicists would say they expect QG to prohibit such things anyway.

atyy
Thanks a bunch, that's what I've read as well. I'm writing my undergrad thesis on the ADM formulation of GR and using it as a basis of a theory of Quantum Gravity, and this seems to be a major limitation for any QG theory built from the Hamiltonian formulation. I'll take it up with my advisor
There were comments on this in http://philosophyfaculty.ucsd.edu/faculty/wuthrich/pub/WuthrichChristianPhD2006Final.pdf [Broken]

"Thus, Hamiltonian formulations might apply to a larger class of models in GTR than does the initial value formulation, which only deals with globally hyperbolic spacetimes. There is a natural connection between these two formulations, but they need not coincide."

"The restriction to generally relativistic models with manifolds of topology R × Ʃ is far from innocent, for at least two reasons."

"In this sense, a Hamiltonian formulation of GTR does not a priori rule out the existence of closed timelike curves. It should be noted, however, in both known Hamiltonian formulations of GTR, spacetime is foliated using a global time function."

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PAllen
2019 Award
There were comments on this in http://philosophyfaculty.ucsd.edu/faculty/wuthrich/pub/WuthrichChristianPhD2006Final.pdf [Broken]

"Thus, Hamiltonian formulations might apply to a larger class of models in GTR than does the initial value formulation, which only deals with globally hyperbolic spacetimes. There is a natural connection between these two formulations, but they need not coincide."

"The restriction to generally relativistic models with manifolds of topology R × Ʃ is far from innocent, for at least two reasons."

"In this sense, a Hamiltonian formulation of GTR does not a priori rule out the existence of closed timelike curves. It should be noted, however, in both known Hamiltonian formulations of GTR, spacetime is foliated using a global time function."
ADM, being one of the [best] "known Hamiltonian formulations".

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atyy
ADM, being one of the [best] "known Hamiltonian formulations".
Actually, I don't quite get why Wuthrich says in principle any solution of GR which has topology R X Ʃ has a Hamiltonian formulation, even though the known cases have (I think) Ʃ = time.

PAllen
2019 Award
Actually, I don't quite get why he says in principle any solution of GR which has topology R X Ʃ has a Hamiltonian formulation, even though the known cases have (I think) Ʃ = time.
Hmm, I thought the R was time. A quick read of the section in your link suggests considering allowing initial conditions that are causally connected to support a slicing where R is not time. The author admits it is not clear how this would work.

In case the OP isn't aware, global hyperbolic restriction does not prevent spatially bounded solutions - each cauchy surface could be topologically a 3-sphere, for example.

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atyy