Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Foliation of time in quantum gravity

  1. Dec 13, 2011 #1
    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?
     
    Last edited: Dec 13, 2011
  2. jcsd
  3. Dec 13, 2011 #2

    tom.stoer

    User Avatar
    Science Advisor

    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.
     
    Last edited: Dec 14, 2011
  4. Dec 15, 2011 #3
    What the problematic topologies are? I suppose that areas under horizon of black holes. Is anything else problematic?
     
    Last edited: Dec 15, 2011
  5. Dec 15, 2011 #4

    tom.stoer

    User Avatar
    Science Advisor

  6. Dec 17, 2011 #5
    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?
     
  7. Dec 18, 2011 #6

    tom.stoer

    User Avatar
    Science Advisor

    There is a well-known counter-example w/o any singularities, the Goedel spacetime with closed timelike curves
     
  8. Dec 21, 2011 #7
    Thus, black holes do not satisfy definiton 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?
     
  9. Dec 21, 2011 #8

    tom.stoer

    User Avatar
    Science Advisor

    b/c you do not quantize SR, but you quantize some other theory XYZ on top of a flat, static and classical spacetime.
     
  10. Dec 22, 2011 #9
    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?
     
  11. Dec 22, 2011 #10

    tom.stoer

    User Avatar
    Science Advisor

    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.
     
  12. Dec 22, 2011 #11
    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?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Foliation of time in quantum gravity
  1. Quantum Gravity (Replies: 10)

  2. Gravity is Time? (Replies: 3)

  3. Gravity and time (Replies: 10)

Loading...