Simultaneity in General Relativity

[syra]

I had a much longer post typed with quotes and everything but I was auto-logged out, couldn't recover the text, and don't feel like typing it all in full. >:[

William Lane Craig, in "Einstein, Relativity, and Absolute Simultaneity" says that the Friedman metric as solution to Einstein's field equations (standard in cosmology) produces a unique hypersurface of simultaneity. Is this true, or is more than one hypersurface of simultaneity possible?

 

[Fredrik]

Short answer: Each coordinate system defines simultaneity differently, but there's one particular coordinate system that's particularly well suited to describe the large-scale behavior of the universe. That's the coordinate system he's talking about. (I doubt that he fully understands that though).

Longer answer: The Friedmann(-Lemaitre-Robertson-Walker) class of solutions of Einstein's equation describe those spacetimes that can be sliced into a one-parameter family of spacelike hypersurfaces that are homogeneous and isotropic in a specific technical sense. It's convenient to choose the parameter that labels the hypersurfaces to be the proper time (from the big bang to the event where it intersects the hypersurface) of a geodesic that's orthogonal to the hypersurfaces. This convention enables us to think of the hypersurface labeled by parameter value t as "space, at time t". The "preferred" coordinate system is defined by choosing the coordinate time of any event to be equal to the parameter that labels the hypersurface, and by choosing the coordinate distance between any two points in the same hypersurface to be the proper distance in that hypersurface.

There are many ways to slice one of these spacetimes into hypersurfaces that we can think of as "space" at different "times", but there's only one slicing with the nice properties mentioned above. That's why the "nicest" coordinate system we can associate with that slicing can be considered "preferred".

Next time, make sure that you check the "remember me" box when you log in.

 

[syra]

Ah man, thanks Fredrik. When I first read this from Craig, I instantly thought it was something like the case with the Levi-Cevita connection: it's the unique torsion-free one, but this doesn't mean that which tangent vectors are parallel or not on a manifold is uniquely determined (i.e. absolute), as there are many connections possible. Elegance and niceness isn't the same as absoluteness.