Understanding Proper Time and Arc-Length in GR Geodesic Curves

[TrickyDicky]

The proper time elapsed between two points in a timelike geodesic in GR is the arc-length of this curve segment.
Does this mean that we have an arc-length parameterization of the geodesic curve, with $\tau$ (proper time) as the parameter?

 

[bcrowell]

Does this mean that we have an arc-length parameterization of the geodesic curve, with $\tau$ (proper time) as the parameter?

This seems to be a question about terminology. I don't think it's common to hear people refer to this as "arc-length parametrization," but it is certainly closely analogous. I think it's more common for people to use phrases like "parametrized by proper time."

Note that this also works for spacelike curves, but in the case of lightlike curves it doesn't, so you have to use some other affine parameter rather than the one defined by the metric.

 

[TrickyDicky]

This seems to be a question about terminology. I don't think it's common to hear people refer to this as "arc-length parametrization," but it is certainly closely analogous. I think it's more common for people to use phrases like "parametrized by proper time."

I'm not sure if it is very commonly used, but it is in any (or most) calculus book(s).
What I wanted to stress is that timelike geodesics, if we want to abstract from an ambient higher dimension, can be parametrized by arc-length, which is a natural, intrinsic parametrization that all smooth curves admit. In this specific case (timelike geodesics) this parametrization by proper time, also reflects the intrinsic curvature of the manifold, right?

Note that this also works for spacelike curves, but in the case of lightlike curves it doesn't, so you have to use some other affine parameter rather than the one defined by the metric.

I should have specified I'm referring to physical particles paths, so it doesn't work for spacelike geodesics either.

 

[Ben Niehoff]

I'm not sure if it is very commonly used, but it is in any (or most) calculus book(s).
What I wanted to stress is that timelike geodesics, if we want to abstract from an ambient higher dimension, can be parametrized by arc-length, which is a natural, intrinsic parametrization that all smooth curves admit.

Careful here. As bcrowell mentioned, not all smooth curves admit an arc-length parametrization. Lightlike curves have zero arc-length, and so they must be parametrized by something else. This is a funny consequence of Lorentzian signature; in Euclidean signature, your statement is correct.

In this specific case (timelike geodesics) this parametrization by proper time, also reflects the intrinsic curvature of the manifold, right?

The proper time elapsed along a single curve is not enough information to give you the curvature, if that's what you mean.

If you know the proper time elapsed along the collection of all curves in some open region U, then this is equivalent to knowing the metric tensor in U. However, in order to get the curvature, you must also know the torsion...that is, you need to know about parallel transport; not just proper time. (If we assume the torsion is zero, then the metric is sufficient to give us the curvature).

I should have specified I'm referring to physical particles paths, so it doesn't work for spacelike geodesics either.

Are you interested in geodesics specifically, or general curves? Anyway, a spacelike curve doesn't have a proper time, but it does have a proper length, which is a perfectly fine parameter.

 

[bcrowell]

This seems to be a question about terminology. I don't think it's common to hear people refer to this as "arc-length parametrization," but it is certainly closely analogous. I think it's more common for people to use phrases like "parametrized by proper time."

I'm not sure if it is very commonly used, but it is in any (or most) calculus book(s).

I meant it's not common to hear people refer to it in relativity as "arc-length parametrization."

 

[TrickyDicky]

Careful here. As bcrowell mentioned, not all smooth curves admit an arc-length parametrization. Lightlike curves have zero arc-length, and so they must be parametrized by something else. This is a funny consequence of Lorentzian signature; in Euclidean signature, your statement is correct.

I'm trying to be as careful as I can, that is why I constrained my statement to material particle's paths.

The proper time elapsed along a single curve is not enough information to give you the curvature, if that's what you mean.

If you know the proper time elapsed along the collection of all curves in some open region U, then this is equivalent to knowing the metric tensor in U. However, in order to get the curvature, you must also know the torsion...that is, you need to know about parallel transport; not just proper time. (If we assume the torsion is zero, then the metric is sufficient to give us the curvature).

Since I'm trying to understand the GR scenario, torsion is assumed to be vanishing when I refer to the specific case of a timelike geodesic. In this case, being the single curve a geodesic, isn't it enough to have the geodesic parametric equations to know its metric and thus the curvature, given the fact that precisely the metric defines the geodesic?

Are you interested in geodesics specifically, or general curves? Anyway, a spacelike curve doesn't have a proper time, but it does have a proper length, which is a perfectly fine parameter.

For now, I'm interested just in understanding the timelike geodesics case.

 

[TrickyDicky]

I meant it's not common to hear people refer to it in relativity as "arc-length parametrization."

Agreed.

 

[Ben Niehoff]

OK, so assuming zero torsion: If you know the proper time elapsed along all timelike geodesics, I'm not sure if this is actually enough information to reconstruct the full metric tensor, since you don't know anything about spacelike or lightlike geodesics.

The answer depends on whether you can use your knowledge of timelike geodesics to work out the spacelike geodesics using some kind of clever construction (Schild's ladder, maybe?). I couldn't say off the top of my head whether it works.

 

[TrickyDicky]

The answer depends on whether you can use your knowledge of timelike geodesics to work out the spacelike geodesics using some kind of clever construction (Schild's ladder, maybe?). I couldn't say off the top of my head whether it works.

Do they need to be worked out? What particle's path corresponds to spacelike geodesics?

 

[TrickyDicky]

It seems to be possible.



http://arxiv.org/abs/0711.1217

 

[bcrowell]

OK, so assuming zero torsion: If you know the proper time elapsed along all timelike geodesics, I'm not sure if this is actually enough information to reconstruct the full metric tensor, since you don't know anything about spacelike or lightlike geodesics.

The answer depends on whether you can use your knowledge of timelike geodesics to work out the spacelike geodesics using some kind of clever construction (Schild's ladder, maybe?). I couldn't say off the top of my head whether it works.

Working in the other direction, knowing the lightlike geodesics isn't enough to determine the metric, because that just fixes everything up to a conformal factor. But if you know the lightlike geodesics and also some local proper times, then the proper times can be used to fix the conformal factor and you can find the metric. In other words, you need both light-cones and a clock in order to fix the metric.

So it seems to me that knowing all proper times along all timelike geodesics certainly suffices. If you know the set of all timelike geodesics, then you certainly know the set of lightlike geodesics, since those are in some sense the boundary of the set of timelike ones. Therefore you know the lightlike geodesics and you have information that fixes the conformal factor, and we know that's sufficient to fix the metric.

 

[TrickyDicky]

Working in the other direction, knowing the lightlike geodesics isn't enough to determine the metric, because that just fixes everything up to a conformal factor. But if you know the lightlike geodesics and also some local proper times, then the proper times can be used to fix the conformal factor and you can find the metric. In other words, you need both light-cones and a clock in order to fix the metric.

So it seems to me that knowing all proper times along all timelike geodesics certainly suffices. If you know the set of all timelike geodesics, then you certainly know the set of lightlike geodesics, since those are in some sense the boundary of the set of timelike ones. Therefore you know the lightlike geodesics and you have information that fixes the conformal factor, and we know that's sufficient to fix the metric.

This is pretty much along the lines of what I had in mind when I started this thread. I must say though, that I was not sure at all this is right (I'm not yet) and I wanted to make sure.

 

[TrickyDicky]

If you know the set of all timelike geodesics, then you certainly know the set of lightlike geodesics, since those are in some sense the boundary of the set of timelike ones.

But they are a boundary only if you consider an asymptotically flat universe, that's certainly not the case in FRW cosmologies for instance.

 

[TrickyDicky]

If we know the proper times, how many coordinates are needed to locate a point in a timelike geodesic (the paths of free massive particles)?

 

[TrickyDicky]

anybody?
..I thought this was an easy one.