Non causally linked events in special relativity

[Anaxagoras]

My question is about foundations of the special relativity theory. In Minkowski's way of presenting special relativity, with a signature "+ - - -", one associates to every couple of events, a spatio-temporal distance which is null on the light cone, positive if the two events are causally linked, and negative otherwise.

Now, in special relativity, the physical actions or physical trajectories must not get out of the null cone. In particular, a velocity vector has always a positive Minkowski length.

I know that, when we usually develop special relativity, we use as well negative as positive distances between events, and that it is a convenient mean to present special relativity. Nevertheless, from the point of view of the foundations of special relativity, I ask myself if it is absolutely necessary to attribute a (negative) length between two events that are not causally linked.

Wouldn't it be possible to reconstruct all the content of special relativity (at least what is really verifiable) without attributing any distance between non causally linked events? In other words, could we reconstruct the essence of the content of special relativity with only half of the Minkowski world, having only a metric in the interior of the null cone?

Thanks for any idea on the subject.

 

[ShayanJ]

That's just hiding our head in the sand. Why should anyone do that?

 

[Anaxagoras]

That's just hiding our head in the sand. Why should anyone do that?

Good question. As I said, my question is a not a question from the usual point of view of the physicist, who simply accepts a theory and then asks himself what can be deduced or not within the theory. Rather, my question is about foundations of a physical theory (namely of special relativity).

Now, special relativity, as any other physical theory, is based on several principles that you can decompose and for which you can ask: Is this particular element of the theory absolutely necessary or not for the development of theory? Is it independant from the other elements? Etc.

It is only from the positivistic point of view of someone who wants to accepts entirely a theory and simply inquire about its consequences, that my question could appear as "hiding his head in the sand". But if you want to question a theory and deeply understand its bases, I think that my question is totally meaningful and pertinent.

 

[Nugatory]

I ask myself if it is absolutely necessary to attribute a (negative) length between two events that are not causally linked.

The spacetime interval between two spacelike-separated events is physically significant; it's the (negative of the square of the) physical length of something. I'm not seeing how to build a physical theory without some notion of distance and length.

 

[Anaxagoras]

You say that the spacetime interval between two spacelike-separated event is physically significant as the physical length of something. Could you please develop this idea? I am not sure about it. Are you thinking about the "proper length" of a ruler that would join the two events A and B?

If I well understood you, you find the particular system of reference where your two events A and B are simultaneous. Then, you posit a motionless (in that particular frame) object at the place of the event A, and a motionless object at the place of the event B. Finally, you measure the physical distance between the objects A and B.

Is it what you meant by "it is the physical length of something"?

 

[Dale]

could we reconstruct the essence of the content of special relativity with only half of the Minkowski world, having only a metric in the interior of the null cone?

No. Every event has a different light cone. So even if we specify the light cone of some event and exclude all exterior events then we will still find pairs of interior events which are spacelike separated from each other.

 

[harrylin]

[..] my question is a not a question from the usual point of view of the physicist, who simply accepts a theory and then asks himself what can be deduced or not within the theory. Rather, my question is about foundations of a physical theory (namely of special relativity).

Now, special relativity, as any other physical theory, is based on several principles that you can decompose and for which you can ask: Is this particular element of the theory absolutely necessary or not for the development of theory? Is it independant from the other elements? Etc. [..].

The main principles are :
1. The relativity principle
2. The light "principle"*

See the intro of http://fourmilab.ch/etexts/einstein/specrel/www/

* It has been shown that the light principle can be reduced to the postulate that there is a limit speed with the value c.

 

[ShayanJ]

Good question. As I said, my question is a not a question from the usual point of view of the physicist, who simply accepts a theory and then asks himself what can be deduced or not within the theory. Rather, my question is about foundations of a physical theory (namely of special relativity).

Now, special relativity, as any other physical theory, is based on several principles that you can decompose and for which you can ask: Is this particular element of the theory absolutely necessary or not for the development of theory? Is it independant from the other elements? Etc.

It is only from the positivistic point of view of someone who wants to accepts entirely a theory and simply inquire about its consequences, that my question could appear as "hiding his head in the sand". But if you want to question a theory and deeply understand its bases, I think that my question is totally meaningful and pertinent.

You're looking at it the wrong way. In fact I doubt it that you understand SR properly.
The point is, the postulates underlying SR (mentioned by harrylin) imply (independent of the mathematical formulation) that not all pairs of events in the spacetime can influence each other, which is different from Newtonian concept of space and time(EDIT: Not that in Newtonian view of space and time all points in all times can influence all other points in all other times, but the limitations are of different nature). So in SR, two events of spacetime are either causally connected or disconnected. And from those pairs of events that are connected, some can only be connected using signals traveling at the speed of light and some can be connected by any speed less than the speed of light. So, independent from the mathematics, we know we have three kinds of spacetime intervals. Now when we want to develop a mathematical formulation of SR, we need some way to distinguish between these three kinds of intervals. For now, the most convenient and relevant formulation is using the Minkowski spacetime.You're welcome to find another formulation.

 

[Anaxagoras]

Thank you all for your different answers.

DaleSpam is right to say that it makes no sense in STR to "exclude events out of a null cone" because, as he says, we can always find two events, inside a given null cone, which are themselves spacelikely related. But it does not really constitute an argument to answer my question. Indeed, I was asking about the possibility to construct STR without attributing any measure of distance between two spacelikely separated events. I was not asking to "exclude all the events outside a light cone".

Now, coming to harrylin answer, I know that 1) the relativity principle and 2) the light principle are the most important and known principles that permit to construct STR. But they are not the only principles. There are other principles, that are often omitted because they are not specific to STR, but they exists anyway. And the supposition that there is an objective measure of (spatio-temporal) distance, which connect any two events (causally connected or not), is such an implicit principle that you usually accept, beside the two majors principles reported by harrylin. Therefore, it makes sense to ask if we can construct STR (and if we can in particular respect the two major principles) with a less restrictive hypothesis about the metric character of the universe of events.

The two answers for the moment that give the more interesting arguments to answer directly my question, I think, are the answer by Nugatory (but it would be clearer with more developments) and the last answer by Shyan.

But I don't see for the moment if it answer completely or not my question. I will have to come back to it later.

 

[Ibix]

Imagine a lamp equidistant between two mirrors. It emits a flash of light which reflects off the mirrors and returns to the lamp. The reflection events are space-like separated. If there is no invariant measure of their separation, why should the flashes' simultaneous return be invariant? Or, indeed, predictable? Or even defined?

 

[Dale]

But it does not really constitute an argument to answer my question. Indeed, I was asking about the possibility to construct STR without attributing any measure of distance between two spacelikely separated events.

I don't think that is possible. I certainly have never seen such a construction.

The metric is a symmetric bilinear form which maps every pair of vectors in the tangent space to a real number. I don't think that the math works if it only maps certain pairs of vectors in the tangent space. If you are not excluding them from the space then I think mathematically that the metric must provide a number.

 

[ShayanJ]

Thank you all for your different answers.

DaleSpam is right to say that it makes no sense in STR to "exclude events out of a null cone" because, as he says, we can always find two events, inside a given null cone, which are themselves spacelikely related. But it does not really constitute an argument to answer my question. Indeed, I was asking about the possibility to construct STR without attributing any measure of distance between two spacelikely separated events. I was not asking to "exclude all the events outside a light cone".

Now, coming to harrylin answer, I know that 1) the relativity principle and 2) the light principle are the most important and known principles that permit to construct STR. But they are not the only principles. There are other principles, that are often omitted because they are not specific to STR, but they exists anyway. And the supposition that there is an objective measure of (spatio-temporal) distance, which connect any two events (causally connected or not), is such an implicit principle that you usually accept, beside the two majors principles reported by harrylin. Therefore, it makes sense to ask if we can construct STR (and if we can in particular respect the two major principles) with a less restrictive hypothesis about the metric character of the universe of events.

The two answers for the moment that give the more interesting arguments to answer directly my question, I think, are the answer by Nugatory (but it would be clearer with more developments) and the last answer by Shyan.

But I don't see for the moment if it answer completely or not my question. I will have to come back to it later.

Your post doesn't make sense at all and I don't think you can propose a question about this that makes sense. The best advise I can give you is to put aside this question until you have a good understanding of SR. Then you can think about it and I promise you'll find out why this question doesn't make sense.
Actually its not a problem you have with SR. Your problem is with the way physical theories are constructed and also with the mathematics used here. So just continue learning physics and the related math!

 

[Anaxagoras]

DaleSpam: you are right that, usually, a bilinear form acts on the total space (but you need not consider the tangent space here, we are in special relativity, not in general relativity, so we can work directly on Minkowski space). Nevertheless, there is no mathematical problem to consider a metric that is only partly defined. The problem here is not mathematical but physical.

Ibix: I like very much your argument with the light and the mirrors. It goes straightly to the point. And I believe it shows that the spatial part of the metric is absolutely necessary to make even very basic predictions. Thanks.

 

[PeterDonis]

one associates to every couple of events, a spatio-temporal distance which is null on the light cone, positive if the two events are causally linked, and negative otherwise.

What you are talking about is the square of the "spacetime distance".

I ask myself if it is absolutely necessary to attribute a (negative) length between two events that are not causally linked.

A negative squared length. Yes, it is, because, as you said in your OP but have not apparently grasped the implications, the sign of the squared length depends on your metric signature convention. If you use the timelike convention, +---, then timelike intervals have positive squared length; but if you use the spacelike convention, -+++, then spacelike intervals have positive squared length. Since the metric signature convention cannot affect the physics, you cannot base any argument about physics on the sign of the squared length of spacetime intervals.

Wouldn't it be possible to reconstruct all the content of special relativity (at least what is really verifiable) without attributing any distance between non causally linked events?

No, because events being spacelike separated does not mean there is no physical meaning to the spacetime interval between them. The physical meaning of that interval is "proper distance"--the distance that would be measured between the two events by a ruler whose ends are located at the two events and which is moving in such a way as to make the two events simultaneous--just as the physical meaning of a timelike interval is "proper time"--the time that would be measured by a clock that moves inertially and passes through both events.

 

[robphy]

In some approaches to the foundations of relativity (like the "Ehlers-Pirani-Schild construction") and some attempts at quantum gravity (like "causal sets"),
there is emphasis on the causal structure of spacetime or spacetime-like structures, in which causality is primitive and then some sense of "space" or other structures are derived [in some limit]. The interest in this line of thinking is that one is looking for a physical motivation for a [macroscopically] smooth Lorentz-signature spacetime manifold, and how it might may be deconstructed (likely with some features demoted) in the search for a theory of quantum gravity.

An early approach is AA Robb's "after" partial-order relation.
(Here's a link to an old post of mine on causal structure.)

This might be enlightening to read:
http://www.mcps.umn.edu/assets/pdf/8.7_Winnie.pdf
http://www.mcps.umn.edu/assets/pdf/8.8_Sklar.pdf

"Causality Implies the Lorentz Group" (EC Zeeman)
http://dx.doi.org/10.1063/1.1704140

about the EPS construction...
http://link.springer.com/article/10.1007/s10714-012-1352-5 (free)
http://link.springer.com/article/10.1007/s10714-012-1353-4
http://projecteuclid.org/euclid.cmp/1103859104 (free)

 

[Anaxagoras]

Thank you Peter Donis for your message. My question was not about the negativity or rather positivity of spacelike events (which is of course pure convention). My question was about the possibility or not to forget totally, at a foundational level, the spacelike part of the metric. The following of your message confirm my interpretation of what Nugatory said in the very first answers to my questions. Thanks for it.

Robphy, I think that your message contains exactly the references I needed to deepen my investigations.

If it appears that some construction (together with some argument of "continuity", in a sense that is still necessary to make precise) permits to derive the spacelike part of the metric from the timelike part (the causal structure), then it would answer my foundational question.

Now I have to read the references you gave me and check it.

Best regards

 

[Dale]

Nevertheless, there is no mathematical problem to consider a metric that is only partly defined.

I don't believe this is correct. Do you have a reference supporting this claim? I think that a lot of important proofs and theorems would fail with this.

 

[stedwards]

It might be possible to eliminate any dependence upon $\sqrt{s^\mu g_{\mu\nu} s^\nu}$ altogether with the extremal of $\int s$, or $s \wedge \epsilon s$, where, to go further, the metric can be eliminated from the Christoffel tensor.

 

[Nugatory]

If I well understood you, you find the particular system of reference where your two events A and B are simultaneous. Then, you posit a motionless (in that particular frame) object at the place of the event A, and a motionless object at the place of the event B. Finally, you measure the physical distance between the objects A and B.

Is it what you meant by "it is the physical length of something"?

I mean that I have a single object, with one end at your point A and the other end at the point B. There is no causal relationship between these spacelike-separated events, but the separation between them is a direct measurement of the length of the object.

 

[PeterDonis]

My question was about the possibility or not to forget totally, at a foundational level, the spacelike part of the metric.

And the answer is, it isn't. Spacelike intervals are fundamentally different from timelike intervals (and null intervals). See below.

If it appears that some construction (together with some argument of "continuity", in a sense that is still necessary to make precise) permits to derive the spacelike part of the metric from the timelike part (the causal structure)

How could this be possible? Timelike and spacelike intervals are physically different things. You measure one with a clock and the other with a ruler. You can't make one out of the other.

 

[martinbn]

Take two space-like events q and p, and an observer whose world line contains q. Let A and B be the two events on his world line such that a light signal from A reaches p, reflects and goes back to B. Then the square of the space-time interval between q and p is the product of the space-time intervals between A and q and q and (in the more plus signs convention), which are time-like seperated. So you cannot have the interval between time-like events and not have it between space-like events, since it can be expressed by them.

 

[Dale]

This thread is closed. Mainstream SR certainly uses space like spacetime intervals. As such, that is the standard for discussion on PF.

If the OP has a professional scientific reference (See the forum rules) describing a different approach then we can discuss that specific reference in a new thread.