Re: integrating Einstein's field eqns in a spacelike direction

Edward Green wrote:
On Mar 7, 11:32 am, Tom Roberts <tjroberts...@xxxxxxxxxxxxx> wrote:
It is not possible to have a purely timelike 3-d surface

I wonder if the OP is thinking of a 3-d surface in spacetime whose
normal is everywhere timelike.

Such a surface is spacelike, not timelike. A timelike surface would presumably be such that every vector tangent to the surface is timelike -- as I said that is not possible for a 3d surface.


On the other hand, maybe he is asking whether by taking purely local
measurments on a single world line -- where "local" admits of
infinitesimal excursions sufficient to establish differentials of all
orders on the line -- the structure of all of spacetime is
established.

This is also not possible -- the information along a single worldline is woefully inadequate to specify the geometry of the entire (3+1)-d manifold.


iuval wrote:
I meant a hypersurface whose normal is everywhere (or at least in
some regions) spacelike such as the 3D volume x=constant, while y, z
and t are allowed to vary.

Having a spacelike normal everywhere does not make a surface "timelike", in the sense of every tangent vector being timelike (a null surface also has spacelike normals). Indeed "timelike surface" is not a standard term, because it is confusing at best -- only some of the tangent vectors can be timelike. One can have a timelike worldline (i.e. every tangent vector is timelike), but that is a 1d locus.


Why is such a surface impossible in 4D?

It's not impossible, it's just not timelike, in the sense of every tangent vector being timelike.


Perhaps it is a matter of semantics--such a surface maybe shouldn't be
called timelike since one can move in spacelike directions on it?

Yes.


Anyway, the question still remains, with the clarification above.

The theorems I know about relating to the sufficiency of a Cauchy surface in GR require the surface to be achronal -- having only spacelike or null tangent vectors. Look in Hawking and Ellis for these theorems, and examine the proofs for how important the absence of a timelike tangent vector is in them.

My guess is that being achronal is important, and what you are asking about cannot be done.


Tom Roberts
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