evolution of spacelike geodesics

Main Page | Report this Page

Science Forum Index » Physics - Research Forum » evolution of spacelike geodesics

Page 1 of 1

Author

Message

Guest

Posted: Sun Mar 04, 2007 4:45 am

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?

Thanks! (I'm interested in answers for 3d and 4d.)

paulaireilly

Posted: Wed Mar 07, 2007 5:11 pm

Guest

On Mar 4, 9:45 am, dhillma...@gmail.com wrote:

Quote:

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?

Marc Nardmann

Posted: Fri Mar 09, 2007 7:29 am

Guest

dhillman86@gmail.com wrote:

Quote:

Let's try to make this precise. First, what does "geodesic" mean here?
We have a Cauchy surface S inside a Lorentzian manifold (M,g). Let h
denote the Riemannian metric on S which is the restriction of g. 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? In the
latter case, the geodesics will in general not stay within S; you have
to assume that S is totally geodesic.

Second, what does "evolve" mean here? Let me suggest a meaning which
might or might not be the one you intended: We consider evolution by the
"normal exponential map" of S. That is, for each point x in S, we
consider the unique future-directed vector v[x] in T_xM which is normal
to T_xS and satisfies g(v[x],v[x])=-1 (timelike vectors v have negative
g(v,v) with respect to my convention), and we consider the g-geodesic
w[x] with w[x](0)=x and w[x]'(0)=v[x]. Let us assume that S is compact.
For suitable numbers t, we define S(t) to be the set of all points
w[x](t), where x is in S. Thus S(0)=S. Since S is compact, S(t) will be
a spacelike hypersurface for all t with sufficiently small absolute
value; it will even be a Cauchy surface. (When S is not compact, then
all sorts of problems arise in general: S(t) might not be well-defined
for any nonzero t because the geodesics exist only for small times,
where "small" depends on the start point x. Even if the set S(t) is
well-defined, it might not be a submanifold for any nonzero t, and so on.)

But whatever you mean precisely, the answer will probably be that
geodesics do in general *not* evolve into geodesics. What you mean by
"interact" and "two combining into one" is not clear to me.

-- Marc Nardmann

Guest

Posted: Sun Mar 11, 2007 10:18 am

On Mar 9, 9:29 am, Marc Nardmann <Marc.Nardm...@bigfoot.de> wrote:

Quote:

dhillma...@gmail.com 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?

Let's try to make this precise. First, what does "geodesic" mean here?
We have a Cauchy surface S inside a Lorentzian manifold (M,g). Let h
denote the Riemannian metric on S which is the restriction of g. 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? In the
latter case, the geodesics will in general not stay within S; you have
to assume that S is totally geodesic.

Second, what does "evolve" mean here? Let me suggest a meaning which
might or might not be the one you intended: We consider evolution by the
"normal exponential map" of S. That is, for each point x in S, we
consider the unique future-directed vector v[x] in T_xM which is normal
to T_xS and satisfies g(v[x],v[x])=-1 (timelike vectors v have negative
g(v,v) with respect to my convention), and we consider the g-geodesic
w[x] with w[x](0)=x and w[x]'(0)=v[x]. Let us assume that S is compact.
For suitable numbers t, we define S(t) to be the set of all points
w[x](t), where x is in S. Thus S(0)=S. Since S is compact, S(t) will be
a spacelike hypersurface for all t with sufficiently small absolute
value; it will even be a Cauchy surface. (When S is not compact, then
all sorts of problems arise in general: S(t) might not be well-defined
for any nonzero t because the geodesics exist only for small times,
where "small" depends on the start point x. Even if the set S(t) is
well-defined, it might not be a submanifold for any nonzero t, and so on.)

But whatever you mean precisely, the answer will probably be that
geodesics do in general *not* evolve into geodesics. What you mean by
"interact" and "two combining into one" is not clear to me.

-- Marc Nardmann

Thanks for the replies; yes, the question needs reformulating. What I
had in mind when I talked about Cauchy surfaces was an (n-1)-D Cauchy
surface whose intrinsic geodesics (via the induced metric) are also
(spacelike) geodesics in the n-D spacetime. I realized (actually just
before I read your message, Marc -- really! Smile

that these are rare
beasts. E.g., a curvy Cauchy surface in Minkowski space is not like
this. I take it these Cauchy surfaces are called "totally geodesic."
Obviously they exist in Minkowski space and you can foliate spacetime
with them. But what is the situation in general? Given a globally
hyperbolic spacetime, does there always exist a totally geodesic
Cauchy surface? Or almost never? And when can you foliate spacetime
with totally geodesic Cauchy surfaces?

Yes, I'm more interested in the compact case.

Now suppose you have a spacetime that is foliated with totally
geodesic compact Cauchy surfaces. Then I'd like to understand
something about how the geodesics evolve. Yes, this is a vaguer
question, but it would be interesting, for instance, if maps like your
exponential-map-generated evolutionary map (provided it is designed to
map a totally geodesic surface onto another such surface in the given
foliation) always send geodesics to geodesics.

Of course if foliating with totally geodesic surfaces only can happen
in Minkowski space, then the answer to the second question is much
less interesting. So just in case, let me try to formulate a more
general question. Consider a foliation of a spacetime by compact
Cauchy surfaces. Let the geodesics in question be those in a Cauchy
surface S given by the induced metric h. Look at very small evolutions
from one surface to another in the foliation. Is there any simple way
to describe how the geodesics evolve?

Marc Nardmann

Posted: Mon Mar 19, 2007 6:14 am

Guest

dhillman86@gmail.com wrote:

Quote:

What I
had in mind when I talked about Cauchy surfaces was an (n-1)-D Cauchy
surface whose intrinsic geodesics (via the induced metric) are also
(spacelike) geodesics in the n-D spacetime. I realized (actually just
before I read your message, Marc -- really! Smile

Almost never. (I could make this vague statement precise and prove it,
but I guess you do not need these tedious details.)

Quote:

And when can you foliate spacetime
with totally geodesic Cauchy surfaces?

Even less often.

Quote:

Now suppose you have a spacetime that is foliated with totally
geodesic compact Cauchy surfaces. Then I'd like to understand
something about how the geodesics evolve. Yes, this is a vaguer
question, but it would be interesting, for instance, if maps like your
exponential-map-generated evolutionary map (provided it is designed to
map a totally geodesic surface onto another such surface in the given
foliation) always send geodesics to geodesics.

Of course if foliating with totally geodesic surfaces only can happen
in Minkowski space,

Of course it happens for an arbitrary *product* metric -dt^2 +h on R x S
(where h is a fixed Riemannian metric on S), not only in Minkowski
space. But such metrics are very rare in the class of arbitrary
Lorentzian metrics on R x S.

Quote:

Consider a foliation of a spacetime by compact
Cauchy surfaces. Let the geodesics in question be those in a Cauchy
surface S given by the induced metric h. Look at very small evolutions
from one surface to another in the foliation. Is there any simple way
to describe how the geodesics evolve?

I don't know off the top of my head. Of course I could figure it out,
but, well, you can do it yourself! I assume you know all the relevant
formulae, so you just have to compute...

-- Marc Nardmann

Page 1 of 1 All times are GMT - 5 Hours The time now is Sun Sep 07, 2008 10:58 am Science Forum Index