Can SR be derived from causality alone?

[friend]

I previously mentioned the paper that tries to derive SR from causality alone:

http://arxiv.org/abs/1005.4172

But I had my doubts that it is valid. For it seems his projection procedure assumes a speed of light to begin with. For his "projection" procedure depends on the angle at which they are projected to a line of causality. Two unaffilitated events, depending on the angle they are projected onto a line could be seen as time-like, space-like, or light-like. This makes his break down into symmetric and anti-symmetric parts from which he derives the SR metric dependent on the angle of projection which is just another way of specifying the speed of light. So his derivation really depends on the speed of light, just like other derivations.

 

[bcrowell]

And I feel as though the "causality postulate" is also logically derived from Minkowski space, not vice versa.

You can do it either way around: https://www.physicsforums.com/showthread.php?t=534862

 

[bcrowell]

I previously mentioned the paper that tries to derive SR from causality alone:

http://arxiv.org/abs/1005.4172

But I had my doubts that it is valid. For it seems his projection procedure assumes a speed of light to begin with. For his "projection" procedure depends on the angle at which they are projected to a line of causality. Two unaffilitated events, depending on the angle they are projected onto a line could be seen as time-like, space-like, or light-like. This makes his break down into symmetric and anti-symmetric parts from which he derives the SR metric dependent on the angle of projection which is just another way of specifying the speed of light. So his derivation really depends on the speed of light, just like other derivations.

I don't see where they assume a speed of light. I don't think their use of the word "projection" implies any preexisting concept of angle. AFAICT the only notion of measure that they assume is that every observer has a clock that ticks once per event.

The whole treatment smells to me like a different presentation of what Geroch does in his cute and idiosyncratic popularization General Relativity from A to B. The five cases in figure 3 look to me like they probably map directly onto the cases that Geroch does.

If the whole thing isn't a complete swindle, then there has to be some point at which they rule out two possibilities: (1) that spacetime is Galilean, and (2) that spacetime isn't flat.

Given any pair of events, it is not necessarily true that one can be informed about the other. In this case, we say that the events are incomparable and write A||C.

In Galilean relativity, A||C never occurs. I suppose they must assume somewhere that A||C sometimes occurs, and clearly this should be a postulate, but they not only fail to state it explicitly as a postulate, they seem to make it very difficult to tell at what point they've made use of it.

Flatness likewise seems to be something that they either don't understand is a nontivial assumption or are aware of but want to sweep under the rug.

Well, these are the issues I'd have complained about if I'd been asked to referee the paper...and it doesn't seem to have been published in a peer-reviewed journal, even though they posted into arxiv in 2010...

 

[friend]

I don't see where they assume a speed of light. I don't think their use of the word "projection" implies any preexisting concept of angle. AFAICT the only notion of measure that they assume is that every observer has a clock that ticks once per event.

If you go to Fig 3, page 6, of http://arxiv.org/abs/1005.4172, the lower middle drawing has two events, yellow dots, 1 and 2, between two vertical lines which represent chains of events. They project a line from dot 1 through dot 2 to the right side chain and call these events light-like separated. However, it seems arbitrary to draw the line from dot 1 through dot 2. That depends on the angle of the line from dot 1. If the lines from the dots were drawn more horizontally, then event (dot) 1 would arrive at the right chain sooner than event 2. If the lines from the dots were drawn more vertically, event 2 would arrive at the right chain sooner than event 1. When the lines reach the chains depends on the vertical and horizontal distance of the dots to the chains and on the angle used to project them. It is not explained why these lines of projection have the same angle going left as going right, or why every dot has lines of projection of the same angle. It seems obvious that what is drawn are 2 dimensional light-cones from the dots, which already assumes the Minkowski-like metric and a speed of light. There is no new information in this paper that I can see.

 

[bcrowell]

They project a line from dot 1 through dot 2 to the right side chain and call these events light-like separated. However, it seems arbitrary to draw the line from dot 1 through dot 2.

I could be wrong -- I don't claim to have gone through every step of the paper -- but I think you're misinterpreting here. I don't think these lines are constructed, I think they're taken as some arbitrary initial structure that is given. The angles at which they're drawn aren't supposed to be significant. They're just a formless nestwork of points with no other initial structure besides the partial ordering.

 

[robphy]

This might also be helpful:
E. C. Zeeman
Causality Implies the Lorentz Group.
J. Math. Phys. April 1964 Volume 5, Issue 4, pp. 490-493

Along this line of reasoning...

A.A. Robb's (1914) axiomatic approach (using the notion of an ordering relation 'after') should be mentioned
http://en.wikipedia.org/wiki/Alfred_Robb

http://books.google.com/books?id=vp...hthUGyh&sig=csSzPceSNPGd4uoGmPMUKloDtU0&hl=en (comment by Torretti)


There are implicit assumptions of continuity.

I found the following paper on the arXiv:
http://arxiv.org/abs/1005.4172

This makes reference to Rafael Sorkin's (1987) "Causal Sets", a fundamentally-discrete approach to Quantum Gravity
http://en.wikipedia.org/wiki/Causal_sets

I remember hearing in a talk once upon a time that knowing the casual structure of spacetime was sufficient to give you the metric up to a (constant?) scale factor; however, I've been unable to track down a reference for that statement.

See Finkelstein's (1969) "Space-Time Code"
http://prola.aps.org/abstract/PR/v184/i5/p1261_1
http://streaming.ictp.trieste.it/preprints/P/68/019.pdf [preprint]

"The causal order C determines the conformal structure of space-time, or nine of the
ten components of the metric. The measure on spacetime fixes the tenth component."
[per spacetime event in 3+1 dimensions] (p. 1262, or p.3 in the preprint)