Compatibility of Determinism & CTC in General Relativity

[jk22]

GR makes use of the Riemann tensor itself using parellel transport along a closed curve.

Now GR says the 4th dimension were time, hence the vector comes to the same time and place it begun on the loop.

Now if determinism is admitted then at every time every quantity is defined uniquely at every place.

So if this determinism is admitted the end and start vector should be the same and hence there should be no curvature ?

 

[PeterDonis]

The thread title references CTCs, which are closed timelike curves, but that is not what your question is about. The closed curve around which we parallel transport a vector to see if it changes (a change shows a nonzero Riemann curvature tensor) is not the same thing as a closed timelike curve in particular spacetimes that have them (like the Godel universe). You can do the parallel transport test for curvature in any spacetime, whether it has CTCs or not.

 

[PeterDonis]

if this determinism is admitted the end and start vector should be the same

No. That is not what determinism says.

Determinism, as applied to GR, says that the entire 4-d spacetime geometry is determined by an appropriate set of initial data. (Note that there are quite a few technicalities here, some of which involve CTCs; but, as I noted in my previous post, your question actually has nothing to do with CTCs.) It also says that the worldline of any object in the spacetime is determined by an appropriate set of initial conditions. But the closed curve around which you do the parallel transport of a vector to test for curvature is not the worldline of any particle (although individual portions of it may be the worldlines of different particles). So determinism does not impose any requirement on what happens to the vector in the parallel transport test.

 

[PeterDonis]

if determinism is admitted then at every time every quantity is defined uniquely at every place

No, that is not what determinism says. Determinism says that every physical observable is uniquely defined at every event. But the two different values of the vector before and after being parallel transported around the loop in the parallel transport test are not two different values of the same physical observable. (They could be values of different physical observables.)