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I guess the answer is no.

Take an observer as a geodesic, seems its nature to choose his space as the space "spanned" by the basis vectors which are orthogonal to the time basis (the tangent vector of the geodesic).

However, for curved manifold, the previous chosen basis vector can only span a subspace of tangent space, but not a region on the manifold.

If we consider a region which is almost Minkowskian only, we can define the concept of space there, but what if we exclude it?

So whether there exists some nature definition of space in GR?

If the answer it no, how we build up our reference frame physically in GR? Or even the definition of reference frame just works in the Minkowskian-like region?

Qmechanic
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MakiseKurisu
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1 Answers1

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Not all spacetimes allow for a clear definition of "space", despite locally being Lorenzian. However, it is sometimes possible to define a foliation of spacetime that has space-like slices that together stack up into the full spacetime and makes it a globally hyperbolic spacetime. That is, you can define some kind of time coordinate that is constant on each slice, and these slices are Cauchy surfaces that only allow the observer to pass through them once. This produces a kind of definition of space as the content of the surfaces, although they can still be pretty arbitrary.

Anders Sandberg
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