Foliation of time in quantum gravity

[exponent137]

Isham wrote:
http://arxiv.org/PS_cache/gr-qc/pdf/9310/9310031v1.pdf

When we come to a Diff(M)-invariant theory like classical general relativity the role of time is very different. If M is equipped with a Lorentzian metric g, and if its topology is appropriate, it can be foliated in many ways as a one-parameter family of space-like
surfaces, and each such parameter might be regarded as a possible definition of time.
However several problems arise with this way of looking at things:
• There are many such foliations, and there is no way of selecting a particular one, or special family of such, that is ‘natural’ within the context of the theory alone.
• Such a definition of time is rather non-physical since it provides no hint as to how it might be measured or registered.
• The possibility of defining time in this way is closely linked to a fixed choice of the metric g. It becomes untenable if g is subject to some type of quantum fluctuation.

Why different slicings are so important problem of quantum gravity. As I understand, because of quantum fluctuations spacelike distances modifies to timelike ones and vice versa.
Are there any other problems?

 

[tom.stoer]

Why different slicings are so important problem of quantum gravity.

One has to proof that the theory is invariant w.r.t. different foliations, i.e. that these (artificial) foliations do not introduce anomalies into the quatnum theory.

... and if its topology is appropriate

This may be a problem b/c a huge sector of classically allowed topologies must be excluded in order to define the quantum theory.

 

[exponent137]

One has to proof that the theory is invariant w.r.t. different foliations, i.e. that these (artificial) foliations do not introduce anomalies into the quatnum theory.This may be a problem b/c a huge sector of classically allowed topologies must be excluded in order to define the quantum theory.

What the problematic topologies are? I suppose that areas under horizon of black holes. Is anything else problematic?

 

[tom.stoer]

definition of a foliation requires global hyperbolicity http://en.wikipedia.org/wiki/Globally_hyperbolic_manifold

 

[exponent137]

definition of a foliation requires global hyperbolicity http://en.wikipedia.org/wiki/Globally_hyperbolic_manifold

globally hyperbolic if it satisfies a condition related to its causal structure. This is relevant to Einstein's theory of general relativity, and potentially to other metric gravitational theories.


I read, but it is very abstract. I please for a little clarification.

If we have a space-time without black hole, does this condition is valid? And what is with space-time with black holes.

Are here any simple, good examples?

 

[tom.stoer]

There is a well-known counter-example w/o any singularities, the Goedel spacetime with closed timelike curves

 

[exponent137]

Thus, black holes do not satisfy definition of a foliation which requires global hyperbolicity?

I please for the next answer:
It is known that special relativity is not problematic for quantization.
In special relativity two events, which are at the same time from one inertial system, are not at the same time from another inertial system.

When we calculate with wave function (WF), we calculate at the same time. Or it is enough that different events in wave function are timelike separated.
As I read Feynman's QED, amplitudes (or WF) are not calculated at the same time.
How it is with this that probabilities are calculated at the same time. Or they are enough to be calculated at timelike distances?

 

[tom.stoer]

It is known that special relativity is not problematic for quantization.

b/c you do not quantize SR, but you quantize some other theory XYZ on top of a flat, static and classical spacetime.

 

[exponent137]

Yes, this is true, but I wish to understand and to simplify time slicing as much as possible. I wish to see, why this quantisation in various slices is not problem in SR.

So, if we have observers from two inertial systems. Can be said that this is distinct time slicing?

 

[tom.stoer]

Different slicings in SR are not problematic b/c the spacetime is globally hyperbolic and different slicings are related via global Lorentz transformations. Canonical quantization does not only provide a Hamiltonian H but the full set of Lorentz group generators as Hilbert space operators. Via these generators one can explicitly prove Lorentz covariance and one can construct Lorentz (better: Poincare) group representations on the Hilbert space.

In GR global (rigid) Lorentz invariance is no longer a symmetry; instead one has local Lorentz invariance in tangent space + diffeomorphism invariance.

 

[exponent137]

Does different slicing (in SR or GR) mean passive of active diffeomorphism?

Is black hole's hyperbolicity cut at the horizon of the black hole (and it exists below horizon), or it does not exist below the horizon?