constraints
The field equations of general relativity can be separated into
hyperbolic evolution equations and elliptic constraints [ADM].
The evolution equations propagate an initial field configuration
'forward' in time, similar to other theories of classical fields.
However, due to the constraints one cannot choose the initial
configuration freely and this is very different from other classical
fields. In some sense the constraints 'connect' spacelike points and
thus one could call general relativity 'holistic' if this would
not be such an abused word.
We don't really know what the quantum theory of gravitation is,
but one would assume that the classical theory reflects the properties
of the underlying quantum theory and indeed the Wheeler-deWitt equation
is nothing but the operator version of one of the constraints.
I think one needs to keep this in mind when discussing thermodynamics
of general relativity, the information loss problem or the entropy of
black holes. E.g. if one specifies the metric near the horizon of a
(near spherically symmetric) black hole, the constraints already
determine the 3-geometry; Therefore I do not find it surprising that
counting microstates provides for a holographic result which differs
substantially from the naive expectation.
I would also think that an approach to the information
loss problem which emphasizes locality as 'conservative' is misguided.
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2 comments:
The constrained Hamiltonian formulation is not the only way to formulate general relativity. However, it does appear to be among the most promising routes to a quantum theory of gravity. These formulations often sneak in some assumption guaranteeing the existence of a Cauchy surface, on which to designate an initial configuration. This suggests an interesting argument that a large classes of spacetimes are not 'physically reasonable':
1. The physically reasonable spacetimes are those that satisfy our (yet-unconceived) best theory of quantum gravity.
2. Any formulation of quantum gravity will demand that its models admit a Cauchy surface.
3. Therefore, models that do not admit Cauchy surfaces are not physically reasonable.
Admitting a foliation of Cauchy surfaces is equivalent to the causality condition of global hyperbolicity. So, accepting this argument seems to rule out many of our non-globally hyperbolic friends: Klein bottles (not time-orientable), Godel's cosmology and many wormhole spacetimes (not chronal), and so on -- all on the basis of their causal structure alone.
It may feel good to be rid of this whackiness. But it seems hasty to me. In particular, I have yet to see a compelling argument for premise 2 -- why is it that a theory of quantum gravity will admit only globally hyperbolic models?
As for quantum gravity, an interesting question is what string theory has to say about this.
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