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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?

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# Foliation of time in quantum gravity

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