Every event is simultaneous with you at some time: proof?

  • Thread starter bcrowell
  • Start date
  • #26
bcrowell
Staff Emeritus
Science Advisor
Insights Author
Gold Member
6,723
423
The thing to look at here is the extensive literature on Fermi-Normal coordinates. The bounds on these are exactly that same as this problem.
Any chance you could point me to an online reference that's not paywalled and that that rigorously proves a bound? I didn't find anything in Wald, and in MTW there is a claim of a bound, but no proof that I was able to locate. At this point I think I have basically pieced together my own argument that works in 1+1 dimensions, as sketched in this thread, but I need to clean it up and polish it. I would like to see the generalization to 3+1 dimenions, but for pedagogical use I actually like the simplicity of 1+1. It would also be interesting to compare with other people's techniques and see if there's something slicker I could do.
 
  • #27
PAllen
Science Advisor
2019 Award
8,073
1,361
Any chance you could point me to an online reference that's not paywalled and that that rigorously proves a bound? I didn't find anything in Wald, and in MTW there is a claim of a bound, but no proof that I was able to locate. At this point I think I have basically pieced together my own argument that works in 1+1 dimensions, as sketched in this thread, but I need to clean it up and polish it. I would like to see the generalization to 3+1 dimenions, but for pedagogical use I actually like the simplicity of 1+1. It would also be interesting to compare with other people's techniques and see if there's something slicker I could do.
Interestingly, I find lots of papers on computation of such coordinates (both perturbatively, and exactly for special cases) with reference to the locality of validity, but I can't find any that explicitly compute the bounds on validity. Your 1+1 argument looks nice to me.
 
  • Like
Likes bcrowell
  • #28
bcrowell
Staff Emeritus
Science Advisor
Insights Author
Gold Member
6,723
423
I made a couple of mistakes in the post where I thought I'd proved a bound for 1+1 dimensions. Here I'm going to post a summary of what I've figured out, which ends up being inconclusive.

Statement of the problem: In Minkowski space, let W be a timelike, ##C^2## world-line parametrized by proper time ##\tau##. Fix a point P not on W. Suppose there exist distinct points M and N on W such that both PM and PN are orthogonal to W. That is, W considers both M and N to be simultaneous with P. MTW seems to claim that there is some sort of bound of the form (distance)(acceleration)##\ge##1, but they don't define it clearly anywhere that I can find. Is there a rigorous bound of this form, and if so, how should the relevant distance and acceleration be defined?

Define ##r(\tau)## as the vector from P to the event at W with that proper time, and let ##v## and ##a## be the first and second derivatives of ##r## with respect to ##\tau##.

My first attempt at a solution was to define ##f(\tau)=r\cdot v##, which gives ##f'=1+r\cdot a##. At M and N, we have ##f=0##, so by Rolle's theorem there is some intermediate point ##\iota## such that ##f'=0##. Since ##r## is spacelike at M and N, and W is timelike, ##r## is also spacelike at ##\iota##. Setting ##f'=0## gives ##r\cdot a=-1## at ##\iota##, which is sort of starting to look like the kind of bound MTW has in mind.

The obvious thing to try is to apply the Cauchy-Schwarz inequality ##|r\cdot a|\le |r||a|##, but in a semi-Riemannian space, this doesn't necessarily hold for spacelike vectors. If we restrict ourselves to 1+1 dimensions, then what we have for spacelike vectors is the reversed Cauchy-Schwarz inequality ##|r\cdot a|\ge ||r||\:||a||##, which doesn't lead to an estimate in the direction claimed by MTW. We could satisfy this inequality with an arbitrarily small proper acceleration or an arbitrarily small proper distance. If we had ##r\cdot v=0## at ##\iota##, then in 1+1 dimensions ##r## and ##a## would have to be collinear, which would provide an exact bound -- but we only have ##r\cdot v=0## at M and N, not at ##\iota##.

My second try was to use the proper distance from W to P. Define the projection ## P_v r = r-(r\cdot v) v ## (+--- metric) of vector r perpendicular to vector v, where ##v## is tangent to the observer's world-line W. In terms of this notation, we can define ## \ell^2 = -(P_v r)^2 ## (+--- metric). Differentiation of this with respect to proper time gives ##d(\ell^2)/d \tau=2(r\cdot v)(r\cdot a)##, which vanishes at points such as M and N where ##f=0##. If we specialize to 1+1 dimensions, then ##r\cdot a## can't vanish for spacelike ##r##, and then all places where ##d(\ell^2)/d \tau## vanishes must be places where ##f=0##. If M and N happen to be two isolated points at which ##f=0##, if we're in 1+1 dimensions, and if there are no other such points in between, then on the closed interval from M to N, ##\ell^2## achieves its maximum and minimum values at the endpoints M and N. At these points, ##\ell=||r||##. This may be a helpful characterization of what's going on, but doesn't seem to get us any closer to proving a bound of the form I'd been imagining.
 
  • #29
bcrowell
Staff Emeritus
Science Advisor
Insights Author
Gold Member
6,723
423
I think I finally understand this better, although, as often seems to be the case, this may have happened after I've exhausted the interest of others on PF.

In Euclidean space, there is a notion of a tubular neighborhood: https://en.wikipedia.org/wiki/Tubular_neighborhood . The tubular neighborhood has some radius. If I prescribe a smooth curve W, then the radius of the tubular neighborhood is that greatest possible radius of a non-self-intersecting piece of rope whose center coincides with W. There are two qualitatively different reasons why the rope could self-intersect. One is local: the radius of curvature of W is too small. The other is global: two points that are far apart as measured along W could be close together in the ambient Euclidean space. The former is obviously amenable to description in terms of differential geometry, while the latter is not.

In the Minkowski case, the local type of self-intersection occurs exactly when both ##r\cdot v## and its derivative vanish at the same proper time. This means that ##r\cdot a=-1## at a point where ##r\cdot v=0##. We always have ##a\cdot v=0##. Therefore in 1+1 dimensions, ##r## and ##a## are collinear. This means that we have ##\ell \alpha=1##, where ##\ell## is the proper distance to P and ##\alpha## is the magnitude of the proper acceleration. So if we only consider self-intersections of the local type, then we know that the radius of the tubular neighborhood is at least ##1/\alpha##, where ##\alpha## is a bound on the proper acceleration.
 
  • #30
bcrowell
Staff Emeritus
Science Advisor
Insights Author
Gold Member
6,723
423
BTW, the computation of a bound on the radius seems to be an open problem in GR. See Klein and Collas, General Transformation Formulas for Fermi-Walker Coordinates, http://arxiv.org/abs/0712.3838 , section 7.
 
  • #31
bcrowell
Staff Emeritus
Science Advisor
Insights Author
Gold Member
6,723
423
Actually the bound in #29 doesn't just apply in 1+1 dimensions. Since both r and a are perpendicular to W, they lie in a spacelike plane whose geometry is Euclidean, so you can apply the Euclidean Cauchy-Schwarz to them.
 

Related Threads on Every event is simultaneous with you at some time: proof?

Replies
12
Views
3K
Replies
21
Views
3K
  • Last Post
Replies
18
Views
1K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
18
Views
1K
  • Last Post
Replies
12
Views
3K
  • Last Post
3
Replies
64
Views
7K
Replies
47
Views
5K
Replies
4
Views
2K
Top