The discussion revolves around the relationship between proper time and arc-length in the context of geodesic curves in General Relativity (GR). Participants explore the implications of proper time as a parameter for timelike geodesics, the terminology used to describe this relationship, and the conditions under which arc-length parametrization is applicable to different types of curves.
Participants generally express differing views on the terminology and implications of proper time and arc-length parametrization. There is no consensus on whether proper time alone is sufficient to reconstruct the metric tensor or the relationship between timelike and spacelike geodesics.
Participants note limitations regarding the assumptions made about torsion and the types of curves being discussed, particularly in relation to the intrinsic properties of timelike, spacelike, and lightlike geodesics.
TrickyDicky said:Does this mean that we have an arc-length parameterization of the geodesic curve, with [itex]\tau[/itex] (proper time) as the parameter?
I'm not sure if it is very commonly used, but it is in any (or most) calculus book(s).bcrowell said: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 should have specified I'm referring to physical particles paths, so it doesn't work for spacelike geodesics either.bcrowell said: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 said: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?
I should have specified I'm referring to physical particles paths, so it doesn't work for spacelike geodesics either.
I meant it's not common to hear people refer to it in relativity as "arc-length parametrization."TrickyDicky said:I'm not sure if it is very commonly used, but it is in any (or most) calculus book(s).bcrowell said: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 trying to be as careful as I can, that is why I constrained my statement to material particle's paths.Ben Niehoff said: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.
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?Ben Niehoff said: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).
For now, I'm interested just in understanding the timelike geodesics case.Ben Niehoff said: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 said:I meant it's not common to hear people refer to it in relativity as "arc-length parametrization."
Do they need to be worked out? What particle's path corresponds to spacelike geodesics?Ben Niehoff said: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.
Ben Niehoff said: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.
bcrowell said: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.
But they are a boundary only if you consider an asymptotically flat universe, that's certainly not the case in FRW cosmologies for instance.bcrowell said: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.