Phase, Geodesics, and Space-Time Curvature

[jfy4]

Please read and critique this argument for me please, any help is appreciated.

Imagine a geodesic, and a matter wave that traverses this geodesic. The action of this matter wave determines the motion of the matter wave along this geodesic over a given space-time interval, and is specified by a metric and infinitesimal space-time lengths. The phase of the matter wave evolves along this geodesic, space-time interval.

Over a given space-time interval, there exist multiple geodesics along which a matter wave can traverse. This is done by holding the end-points of the space-time interval fixed, while varying the metric signature and space-time lengths. For each one of these geodesics, the matter wave's phase has the same initial value, but over the different geodesics evolves to different values with respect to each geodesic.

Therefore, a relationship between the phase of a matter wave, and its evolution along a geodesic should be derivable. A sum over all the phases (and hence the geodesics) would leave one with a prominent phase (or phase distribution) which could in turn be matched with a corresponding geodesic (over a fixed and given space-time interval, of which there should be a one-to-one correspondence between the phase and geodesic path). This geodesic can then be used to find the curvature of space-time (because of its intrinsic connection to curvature through the metric signature).

 

[Passionflower]

Please read and critique this argument for me please, any help is appreciated.

It is a very complicated argument.

Imagine a geodesic, and a matter wave that traverses this geodesic.

Here is the first complication. A wave is by definition non-local and is a four dimensional object (in spacetime). A geodesic however is a two dimensional object.

The phase of the matter wave evolves along this geodesic, space-time interval.

So, strictly speaking, this is not correct, since the phase evolves along a region (a group of geodesics) in spacetime. And if this spacetime is curved and non static you can readily see the enormous complications this will give to the phase of a wave. Wheeler, a long time ago, wrote some interesting stuff about that, but it never became mainstream science.

Over a given space-time interval, there exist multiple geodesics along which a matter wave can traverse.

Yes that is true, and phase coherence becomes even more problematic in such scenarios.

A sum over all the phases (and hence the geodesics) would leave one with a prominent phase (or phase distribution) which could in turn be matched with a corresponding geodesic (over a fixed and given space-time interval, of which there should be a one-to-one correspondence between the phase and geodesic path). This geodesic can then be used to find the curvature of space-time (because of its intrinsic connection to curvature through the metric signature).

I think I understand what you are getting at.

Let me rephrase it to see if I understand you:

You like to correlate the evolution of the phase of a matter wave with the vorticity, shear and expansion of a group of geodesics in order to determine curvature of spacetime?

 

[jfy4]

Thank you for responding!

I tried looking up vorticity but I could not find out what it means...

But I think you are interpreting my argument correctly. I imagine that a matter wave can evolve along many different geodesics which correspond to different curvatures and that the phase decoherence between these geodesics, like a slit experiment, would produce a prominent geodesic/phase which could be used to find curvature.

However, I did not know geodesics were 2d I thought they were path lengths. Could you explain this more to me?