Some doubts concerning the mathematical bases of GR

[TrickyDicky]

The geometric object that makes sense for manifold is the parametrized path, P(s) which is a continuous function from reals to points in the manifold. If the path is timelike, then you can choose the parameter s to be proper time along the path, but that is not necessary to be able to talk about such things as parallel transport. s can be any monotonically increasing real-valued parameter. So there is no difficulty at all in doing parallel transport along lightlike paths. The only thing that is slightly annoying about it is that there is no natural best choice for parametrization in the case of null paths.

Both Riemannians and Pseudoriemannian manifolds use the Levi-Civita connection so in this particular case parallel transport does require s to be the parameter that locally extremizes the path. Even for null geodesics. As long as we want to respect the underlying manifold topology. A pseudosemimetric space for instance doesn't have this requirement, but PseudoRiemannian manifolds are not pseudosemimetric spaces.

 

[stevendaryl]

I mean precisely that, the topolog induced by a Lorentzian metric has to have several conditions added in order to agree with the smooth manifold topology, and I'm not referring only to the global topology you refer to with the "gluing of the paches". I mean the "local" topology that must be homeomorphic to R⁴ in order to agree with the smooth manifold natural topology. Why do you think coordinate patches don't have topological properties?

The local patches are just little sections of R⁴. There is really nothing much to distinguish one patch from another. In GR, locally everything looks like a little section of Minkowsky spacetime. The only issue really is how the patches are glued together. So I'm not sure what problem you're worried about.

 

[stevendaryl]

Both Riemannians and Pseudoriemannian manifolds use the Levi-Civita connection so in this particular case parallel transport does require s to be the parameter that locally extremizes the path. Even for null geodesics. As long as we want to respect the underlying manifold topology. A pseudosemimetric space for instance doesn't have this requirement, but PseudoRiemannian manifolds are not pseudosemimetric spaces.

I don't think what you are saying is correct. An extremal path (not necessarily the same thing as a geodesic, but it is in the case where the connection is compatible with the metric) from point A to point B is a parametrized path P(s) that extremizes the integral

\int \sqrt{g_{\mu \nu} \dfrac{dx^{\mu}}{ds}\dfrac{dx^{\mu}}{ds}}

The parameter s does not need to be proper time; it can be any monotonically increasing parameter. The above integral is invariant under reparametrization.

 

[stevendaryl]

What you call a simple case, the KS space, doesn't actually have the required added conditions to have the smooth manifolds topology at the overlaps, for instance the strong causality condition.

I think you might be mixing necessary and sufficient conditions. You can directly verify in the case of a Schwarzschild black hole that the metric is defined everywhere except at the singularity, and that it always has the signature (+ - - -). So you don't need a theorem telling you under what circumstances it is possible to do that. If you had a theorem giving necessary conditions, and those conditions weren't met in the Schwarzschild case, that would be an indication of an inconsistency.

So are the conditions you are talking about necessary or sufficient?

 

[TrickyDicky]

I think you might be mixing necessary and sufficient conditions. You can directly verify in the case of a Schwarzschild black hole that the metric is defined everywhere except at the singularity, and that it always has the signature (+ - - -). So you don't need a theorem telling you under what circumstances it is possible to do that. If you had a theorem giving necessary conditions, and those conditions weren't met in the Schwarzschild case, that would be an indication of an inconsistency.

So are the conditions you are talking about necessary or sufficient?

Necessary, and the putative inconsistence would be derived from the problem you've mentioned several times that starts with the metric rather than with the manifold and simply expects that a global topology exists that admits that metric.

 

[TrickyDicky]

I don't think what you are saying is correct. An extremal path (not necessarily the same thing as a geodesic, but it is in the case where the connection is compatible with the metric) from point A to point B is a parametrized path P(s) that extremizes the integral

\int \sqrt{g_{\mu \nu} \dfrac{dx^{\mu}}{ds}\dfrac{dx^{\mu}}{ds}}

The parameter s does not need to be proper time; it can be any monotonically increasing parameter.

Proper time τ in GR is precisely by definition the parameter that gives the extremal path.
Any Optics text will show that light doesn't follow paths parametrized by any monotonically increasing parameter.

The above integral is invariant under reparametrization.

Different metrics will give different paths.
We are talking about the variation of that integral: \deltaS=0

 

[Ilmrak]

Proper time τ in GR is precisely by definition the parameter that gives the extremal path.
Any Optics text will show that light doesn't follow paths parametrized by any monotonically increasing parameter.

Your book is lying xD
By definition a path is a continuos map \phi: \left[ 0, 1 \right] \rightarrow M\; .

Different metrics will give different paths.
We are talking about the variation of that integral: \deltaS=0

The integral is invariant under reparametrization, i.e. a change of the parameter s not of the metric.

Ilm

 

[stevendaryl]

Proper time τ in GR is precisely by definition the parameter that gives the extremal path.

No, that's not correct. Proper time is defined via

d\tau = \sqrt{g_{\mu \nu} \dfrac{dx^{\mu}}{ds} \dfrac{dx^{\nu}}{ds}} ds

The definition of proper time doesn't refer to extremal paths; if the path is non-extremal (which it will be if there are non-gravitational forces such as electromagnetism) the above still gives you proper time.

The definition of proper time is independent of your choice of the parameter ds. If you make a parameter change s' = f(s) then d\tau is unchanged.

Now, for time-like paths, you can actually use the freedom to change parameters to choose s = \tau, but that's really only a convenience. It doesn't have any consequences, other than making the mathematics simpler.

 

[stevendaryl]

Necessary, and the putative inconsistence would be derived from the problem you've mentioned several times that starts with the metric rather than with the manifold and simply expects that a global topology exists that admits that metric.

But in the case of the Schwarzschild solution, it's possible to explicitly calculate the metric, and see that it has signature (+ - - -) everywhere except at the singularity. This is clearest using Kruskal-Szekeres coordinates, as described here: http://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates

I'm not sure what problem you are worried about.

 

[stevendaryl]

Now, for time-like paths, you can actually use the freedom to change parameters to choose s = \tau, but that's really only a convenience. It doesn't have any consequences, other than making the mathematics simpler.

...and if the path is lightlike, then you can't use proper time as the parameter, because the parameter has to be monotonically increasing along the path. But you can certainly use other parameters. For example, in 2-D spacetime, the light-like path x=ct can be parametrized by t itself, giving \dfrac{dx^0}{dt} = 1 and \dfrac{dx^1}{dt} = c. Then the proper time \tau is given by:

d\tau = \sqrt{g_{\mu \nu} \dfrac{dx^\mu}{dt} \dfrac{dx^\nu}{dt}} dt
= 0

 

[TrickyDicky]

Your book is lying xD
By definition a path is a continuos map \phi: \left[ 0, 1 \right] \rightarrow M\; .



The integral is invariant under reparametrization, i.e. a change of the parameter s not of the metric.

Ilm

You are ignoring the context of my answers. Light in vacuum can only follow null geodesics which are not parametrized by just any monotonically increasing parameter, but the one that extremizes the path.

The invariance of the integral is derived from the fact it is integrating dtau.

 

[TrickyDicky]

No, that's not correct. Proper time is defined via

d\tau = \sqrt{g_{\mu \nu} \dfrac{dx^{\mu}}{ds} \dfrac{dx^{\nu}}{ds}} ds

The definition of proper time doesn't refer to extremal paths; if the path is non-extremal (which it will be if there are non-gravitational forces such as electromagnetism) the above still gives you proper time.

The definition of proper time is independent of your choice of the parameter ds. If you make a parameter change s' = f(s) then d\tau is unchanged.

Now, for time-like paths, you can actually use the freedom to change parameters to choose s = \tau, but that's really only a convenience. It doesn't have any consequences, other than making the mathematics simpler.

I thougt we were talking about geodesics, not just timelike paths, proper time parametrization is equivalent to the arc length parametrization, of course if the path is not geodesic the proper time is not defined like I wrote.

You guys seem to be discussing different things from the ones I'm talking about.

 

[stevendaryl]

You are ignoring the context of my answers. Light in vacuum can only follow null geodesics which are not parametrized by just any monotonically increasing parameter, but the one that extremizes the path.

Whether a path is extremal or not has nothing to do with the parametrization.

 

[stevendaryl]

I thougt we were talking about geodesics, not just timelike paths, proper time parametrization is equivalent to the arc length parametrization, of course if the path is not geodesic the proper time is not defined like I wrote.

You guys seem to be discussing different things from the ones I'm talking about.

I'm only responding to what you have said. You said that

Proper time τ in GR is precisely by definition the parameter that gives the extremal path.

That's false. Proper time has nothing to do with extremal paths. You can define proper time for non-extremal paths, as well. Whether a parametrized path is a geodesic is completely independent of whether you use proper time as the path parameter.

 

[TrickyDicky]

Really, you seem to be arguing for the sake of arguing about things that are not related to the OP and distract, in case you are not I apologize, and thanks for making my poor wording more understandable.

For your information there is a special circumstance that happens to coincide with the one at hand that relates parametrization with whether a path is extremal.
Natural parametrization, or unit speed (arc length) parametrization in the context of geodesics in a given metric and a Levi-Civita connection in a (pseudo)Riemannian manifold(see Morse theory of geodesics in pseudoRiemannian manifolds).
The extremal paths of the action functional coincide with the geodesics of the metric g in their natural (proper time in the Lorentzian case) parametrization.

 

[stevendaryl]

For your information there is a special circumstance that happens to coincide with the one at hand that relates parametrization with whether a path is extremal.
Natural parametrization, or unit speed (arc length) parametrization in the context of geodesics in a given metric and a Levi-Civita connection in a (pseudo)Riemannian manifold(see Morse theory of geodesics in pseudoRiemannian manifolds).
The extremal paths of the action functional coincide with the geodesics of the metric g in their natural (proper time in the Lorentzian case) parametrization.

Null geodesics cannot possibly be parametrized by proper time. But they certainly can be parametrized.

As I said, choosing proper time (or path length, in the case of Riemannian geometry) is convenient, but nothing depends on that choice, and you can't make that choice for null geodesics.

But extremal paths being the same as geodesics is independent of whether the parameter is proper time, or not.

The equation of a geodesic, for arbitrary parametrization is (if I haven't made a sign error):

\dfrac{d U^{\mu}}{d s} + \Gamma^{\mu}_{\nu \lambda} U^{\nu} U^{\lambda} - U^{\mu} \dfrac{d log(R)}{ds} = 0

where U^{\mu} is the tangent vector (\dfrac{d x^{\mu}}{d s}), and \Gamma^{\mu}_{\nu \lambda} is the connection coefficients (constructed from the metric tensor) and R is \dfrac{d \tau}{d s}, where \tau is proper time. If you have a null geodesic, or if you let the parameter s = \tau then the last term drops out, and you have the usual form of the geodesic equation:

\dfrac{d U^{\mu}}{d s} + \Gamma^{\mu}_{\nu \lambda} U^{\nu} U^{\lambda} = 0

 

[Ben Niehoff]

Really, you seem to be arguing for the sake of arguing about things that are not related to the OP and distract, in case you are not I apologize, and thanks for making my poor wording more understandable.

He is correcting your errors, which are not poor wording, but poor understanding. They are in fact relevant to the topological discussion in this thread. As I mentioned earlier, various points along a null curve are distinct precisely because they have different values of parameter. It doesn't matter which parameter you choose.

 

[stevendaryl]

I think that you might be talking about mathematical structures that are different from those considered in GR. In GR, you can have null geodesics, and you can have parametrized paths that are null geodesics that are not parametrized by proper time.

 

[TrickyDicky]

points along a null curve are distinct precisely because they have different values of parameter. It doesn't matter which parameter you choose.

Ok, if it doesn't matter which parameter you sure can choose tau (in fact it is only invariant for affine transformations of the parameter), that was all I was saying.
Now you tell stevendaryl, cause he says you can't.

 

[TrickyDicky]

The equation of a geodesic, for arbitrary parametrization is (if I haven't made a sign error):

\dfrac{d U^{\mu}}{d s} + \Gamma^{\mu}_{\nu \lambda} U^{\nu} U^{\lambda} - U^{\mu} \dfrac{d log(R)}{ds} = 0

where U^{\mu} is the tangent vector (\dfrac{d x^{\mu}}{d s}), and \Gamma^{\mu}_{\nu \lambda} is the connection coefficients (constructed from the metric tensor) and R is \dfrac{d \tau}{d s}, where \tau is proper time. If you have a null geodesic, or if you let the parameter s = \tau then the last term drops out, and you have the usual form of the geodesic equation:

\dfrac{d U^{\mu}}{d s} + \Gamma^{\mu}_{\nu \lambda} U^{\nu} U^{\lambda} = 0

Right, the first equation is for general geodesics with affine connection, the second is the equation used in (pseudo)riemannian manifolds and as you say you have s=tau.
I honestly don't know yet what you are arguing about. You seem to be saying one thing and its opposite in the same post.

 

[stevendaryl]

Ok, if it doesn't matter which parameter you sure can choose tau (in fact it is only invariant for affine transformations of the parameter), that was all I was saying.

If \tau is identically zero along the path (which it is for a null geodesic), then you can't use it as the parameter.

 

[stevendaryl]

Right, the first equation is for general geodesics with affine connection, the second is the equation used in (pseudo)riemannian manifolds and as you say you have s=tau.

No. I'm saying that IF it is a timelike geodesic, then you can choose s=\tau. If it is a NULL geodesic, then you CANNOT choose s=\tau. The original question was about null geodesics. In that case, the parameter s is NOT proper time.

 

[TrickyDicky]

Just to be clear, all this discussion started when I said that from the external observer the path of a light geodesic could be treated like a timelike geodesic , obviously in this case reparametrizing with t instead of tau since we are not adopting the frame of the photon where tau is zero.
I renamed the affine parameter as tau, which if one ignores the previous context could lead to confusion.

 

[stevendaryl]

Just to be clear, all this discussion started when I said that from the external observer the path of a light geodesic could be treated like a timelike geodesic , obviously in this case reparametrizing with t instead of tau since we are not adopting the frame of the photon where tau is zero.
I renamed the affine parameter as tau, which if one ignores the previous context could lead to confusion.

Okay. I feel like there must be some sense in which a lightlike geodesic is a limit of timelike geodesics.

 

[TrickyDicky]

Okay. I feel like there must be some sense in which a lightlike geodesic is a limit of timelike geodesics.

Exactly, that is the idea I wanted to convey, very clumsily indeed, I' ll see if I can make it mathematically precise.

 

[TrickyDicky]

Maybe we could say the affine parameter for light paths is the limit of the proper time parameter of a timelike geodesic as velocity tends to zero.

 

[PeterDonis]

Okay. I feel like there must be some sense in which a lightlike geodesic is a limit of timelike geodesics.

IMO this doesn't work. Timelike and null curves are two fundamentally distinct things, and I think they should be viewed that way.

I see the intuition that leads to the limit idea: in any particular inertial frame, if I look at timelike geodesics from the origin, (t, x) = (0, 0), to another surface of simultaneity, i.e., to various endpoints (t, x) with t always the same, their length gets shorter and shorter as their relative velocity approaches c (meaning x approaches t). So it's natural to think of a null geodesic lying on the light cone from the origin as a limit of those timelike geodesics as v -> c.

However, this set of geodesics only looks natural in that particular frame. Pick any of the timelike geodesics with nonzero x, and transform to the frame where that geodesic is the time axis. Then there will be *another* set of timelike geodesics, all going from the origin (which remains the same event) to the surface of simultaneity (t', x') with varying x', which *also* approach zero length as v -> c in this new frame. So now it will seem like we have a completely different set of timelike geodesics, but with the *same* limit--because the light cone from the origin is invariant.

There is, of course, an invariant "natural" set of timelike geodesics corresponding to any particular one we pick: the set whose endpoints lie on the hyperbola t^2 - x^2 = tau^2, where tau is the proper time along the geodesic we pick. But of course these geodesics all have the *same* length; they do *not* approach any null geodesic as a limit, though they do appear to "point" closer and closer to the light cone in the particular frame we picked. But again, we can transform to any other frame and change the way all of the geodesics appear to "point".

In sum, while the idea of null geodesics as a "limit" of timelike geodesics is intuitively appealing, I think it is best to resist this intuition, because it doesn't lead anywhere useful.

 

[Ben Niehoff]

Defining a limit on a space of paths requires you to define a topology on the space of paths. This is what topology is all about.

I think there are perfectly sensible topologies on the space of paths in which a null path is the limit of a sequence of timelike paths. Or spacelike paths, for that matter. These topologies, like the underlying manifold topology, are not induced from the pseudo-Riemannian metric.

However, as you point out, the proper time along the path will approach zero as the path approaches the limit. Therefore proper time is not a good parameter along null paths (which we already knew).

 

[TrickyDicky]

IMO this doesn't work. Timelike and null curves are two fundamentally distinct things, and I think they should be viewed that way.

I see the intuition that leads to the limit idea: in any particular inertial frame, if I look at timelike geodesics from the origin, (t, x) = (0, 0), to another surface of simultaneity, i.e., to various endpoints (t, x) with t always the same, their length gets shorter and shorter as their relative velocity approaches c (meaning x approaches t). So it's natural to think of a null geodesic lying on the light cone from the origin as a limit of those timelike geodesics as v -> c.

However, this set of geodesics only looks natural in that particular frame. Pick any of the timelike geodesics with nonzero x, and transform to the frame where that geodesic is the time axis. Then there will be *another* set of timelike geodesics, all going from the origin (which remains the same event) to the surface of simultaneity (t', x') with varying x', which *also* approach zero length as v -> c in this new frame. So now it will seem like we have a completely different set of timelike geodesics, but with the *same* limit--because the light cone from the origin is invariant.

There is, of course, an invariant "natural" set of timelike geodesics corresponding to any particular one we pick: the set whose endpoints lie on the hyperbola t^2 - x^2 = tau^2, where tau is the proper time along the geodesic we pick. But of course these geodesics all have the *same* length; they do *not* approach any null geodesic as a limit, though they do appear to "point" closer and closer to the light cone in the particular frame we picked. But again, we can transform to any other frame and change the way all of the geodesics appear to "point".

In sum, while the idea of null geodesics as a "limit" of timelike geodesics is intuitively appealing, I think it is best to resist this intuition, because it doesn't lead anywhere useful.

It is not only intuitive, I don't know why you would reject something only on the basis that you don't find it useful or convenient, when the arguments you offer basically go in the direction of confirming it.
The fact is that null geodesics in Minkowski spacetime are the asymptotical limit of timelike geodesics.

 

[PeterDonis]

I think there are perfectly sensible topologies on the space of paths in which a null path is the limit of a sequence of timelike paths. Or spacelike paths, for that matter.

Can you give a specific example? I'm not disputing what you say, in fact I said something similar in my recent post, but I wasn't thinking of the limit I described there in terms of a topology on the space of paths, and I'm not sure at first sight how to re-interpret it that way.