# Existence of quasi-euclidean spacelike hypersurfaces?

#### PAllen

I wonder if anyone knows or can figure out an answer to this question I've been thinking about:

In a smooth pseudo-riemannian manifold like those in GR, and given some arbitrarily long spacelike geodesic, is it always ( or almost always, e.g. except for passing through a singularity) possible to find some open spacelike 3-region containing the geodesic (possibly an exceedingly skinny region around the geodesic, as long as it is an open region) that is quasi-euclidean in geometry?

There may be a nicer defintion, but what I mean by a quasi-euclidean spacelike 3-region is that one can find some coordinate system on a spacetime (4) region containing the 3-region where:

1) one coordinate is zero throughout the 3-region
2) the 3x3 submetric corresponding to the other coordinates is positive definite throughout the 3-region.

Though phrased in coordinates, I believe this definition is coordinate independent; a given spacelike 3-region either has or does not have this property. Note, also, that even in Minkowski spacetime, it is trivial to construct spacelike 3-surfaces that are not quasi-euclidean (start with one that is and add sufficient bumps, but still keep it spacelike; you then can't achieve positive definite metric throughout - some areas will have negative metric components, brought in from the overall psuedo-riemmanion metric of the spacetime; if you squeeze them out in one area, you'll pick them up in another, no matter how you pick coordinates - as long as they are legitimate coordinates).

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#### bcrowell

Staff Emeritus
Gold Member
Re passing through a singularity, the usual description is that a singularity isn't a point in the spacetime, so you really can't have a geodesic that passes through a singularity. In fact, one way of defining a singularity is that it's a place that you can't extend geodesics through.

If the geodesic is a closed curve, then you could have issues with orientability.

I think the coordinate-based definition doesn't really work because we're on a manifold, where it may not be able to define a coordinate patch that covers the whole space.

Maybe a better way to define the conjecture is by asking whether, for given simply connected geodesic γ in 3+1 dimensions, there exists an open neighborhood O of γ and a spacelike orientable 3-surface S that is a subset of O containing γ, such that S is locally euclidean based on the metric of the original spacetime.

I think the answer is yes. The fact that it's a geodesic prevents you from running into kinks. If γ had a kink in it at point P, then I don't think there could be an S that was locally euclidean at P.

[Made a few edits to the above after posting.]

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#### PAllen

Note, I believe the answer in SR is that you can find a one parameter family of global, exactly euclidean, spacelike 3-planes containing any spacelike geodesic. These correspond to the inertial frames in which ends of the geodesic are simultaneous.

I also think, in GR, one can find quasi-euclidean 3-regions in a sufficiently small ball including a section of the geodesic; and that, in GR, you generally cannot find any globally quasi-eucliden 3-planes at all. My question amounts to whether restricted to an arbitrarily small open tube around an arbitrary geodesic, one can find quasi-euclidean 3-regions.

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#### PAllen

If the geodesic is a closed curve, then this would be impossible in a spacetime that wasn't orientable.
I meant to say a geodesic between two different events. Thus, for a closed geodesic you would have to cut out some tiny piece of it.

#### PAllen

I think the coordinate-based definition doesn't really work because we're on a manifold, where it may not be able to define a coordinate patch that covers the whole space.
That's a related question I had in mind. Looking some definitions of maximal atlases, but not have studied this material in detail, I was thinking you should always be able to find a coordinate patch containing some geodesic. Thinking about even extreme 2-surfaces, it seemed you could always find ribbon around a geodesic on which you could impose a single coordinate patch.

#### PAllen

Maybe a better way to define the conjecture is by asking whether, for given simply connected geodesic γ in 3+1 dimensions, there exists an open neighborhood O of γ and a spacelike orientable 3-surface S that is a subset of O containing γ, such that S is locally euclidean based on the metric of the original spacetime.
Independent of whether the coordinate definition could be made to work, I like this much better.

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