3+1 split

[Sukratu Barve]

To frame my quaestion, I need to refer to
Wald's book on GR. In the chapter on 3+1 formalism,
we learn that the geometry along with the field
equations allows one to think of a non-invertible
4-tensor h(representing the 3-metric on the leaf)
and another symmetric 4-tensor K for extrinsic
curvature
of the leaf in the foliation. We derive equations
using the projected T mu nu and also using Gauss-
Codacci relations. All along we keep in mind that
the h and K are defined already and indeed we use
that
to derive their equations (and apply the quantization
procedure). Now if one begins with their
equations assuming that h and K are not defined as 3
projection and extrinsic curvature resp., but just as
SOME symmetric tensors, I believe one should be able
to do the further quantization procedure just as well.
I do not know how to interprete this. My h and K are
not necessarily the 3- metric and extrinsic curvature
now. NEither is it clear that they are related by any
diffeomorphism to the 3-metric and extrinsic
curvature.
Comments?



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