Re: evolution of spacelike geodesics

On Mar 4, 9:45 am, dhillma...@xxxxxxxxx wrote:
In general relativity, how do spacelike geodesics evolve? E.g., if you
have a globally hyperbolic spacetime and choose a Cauchy surface, and
then evolve it a little bit forward in time, does a spacelike geodesic
in the first surface evolve into a spacelike geodesic in the second?
Or do the geodesics interact with one another, two combining into one
and so forth? Or what?

This is an interesting question. You must consider how you do the
evolution. A spacelike hypersurface will "evolve into" a spacelike
hypersurface, but it is similar to the evolution of a wave front in
optics; every point on the hypersurface "evolves into" a lot of
points in the future, and there are actually many ways to do the
slicing. There are unambiguous rules for how test masses evolve, but
in GR there are no meaningful "labels" on events (aka "points in
spacetime") - that is sort of the whole point. So a geodesic,
consisting of a one dimensional locus of events, will in some sense
evolve into a region, its domain of dependency, that is usually more
than one-dimensional.

However, you could put a trace particle on every point on a geodesic
and ask how they evolve, but you'd have to keep track of how they are
moving. I believe that if you use this point of view, a geodesic
evolves into itself. Consider a spacecraft in orbit that is
constantly shedding co-moving dust particles; they just move with the
spaceship and trace out the very same orbit, no?

Now, if you don't specify that our test particles are co-moving with
the geodesic, but have some alternative velocity, I think you get the
sort of thing you are looking for, but you have three degrees of
freedom at every point in how you specify that velocity: two angles
and a magnitude. That question is something like "What is the locus
of points at the forward tip of a light ray that you shine out of your
test object while it is in free fall?" if you use massless test
particles. Obviously, the tip of a light ray *needn't* form a
geodesic, nor would the tip of a massive stream of test particles
(particle beam etc). But... it might be an interesting question to
ask if it *can* form a geodesic, and if so, under what conditions. In
Minkowski space it certainly can.

Does anyone want to pursue this? Could the original poster re-
formulate the question?

.

Relevant Pages

  • Re: evolution of spacelike geodesics
    ... then evolve it a little bit forward in time, ... Or do the geodesics interact with one another, ... We have a Cauchy surface S inside a Lorentzian manifold. ... geodesics in the n-D spacetime. ...
    (sci.physics.research)
  • Re: evolution of spacelike geodesics
    ... then evolve it a little bit forward in time, ... We have a Cauchy surface S inside a Lorentzian manifold. ... Are we talking about geodesics in S with respect to h? ... Or about spacelike geodesics in M with respect to g which start tangential to S? ...
    (sci.physics.research)
  • evolution of spacelike geodesics
    ... how do spacelike geodesics evolve? ... have a globally hyperbolic spacetime and choose a Cauchy surface, ...
    (sci.physics.research)