I Need to resort to spherical wavefront to derive the LTs?

[PeterDonis]

I would start with the spacetime interval

If one wants to derive the interval formula, as the OP says he does, of course one cannot assume it.

 

[robphy]

No. I would start with the spacetime interval: $$ds^2=-c^2 dt^2+dx^2+dy^2+dz^2$$ This single formula contains all of special relativity.

Since you have to assume something I like to assume the fewest and most powerful things. So for special relativity it would be this.

For time dilation I would start with analyzing the muon experiment of Bailey.

https://www.nature.com/articles/268301a0

The idea is to focus on things that are actual frame invariant experimental outcomes.

To me, the muon experiment (and for similar particles with different speeds in the lab frame) is the experimental discovery of a “circle” in a position vs time diagram drawn the lab frame.

If one could repeat the experiments measured in another internal frame, one would obtain the same circle with the data points “shifted” along the circle, which could then be reconciled by an appropriate transformation (to be uncovered).

One of the implications of such a “circle” on a position-vs-time diagram is the invariance of the asymptotes (corresponding to what appear to be maximum signal speeds).

Once one has a “circle” on a plane (plus additional assumptions that suggest the displacements on a position vs time graph form a vector space), probably all of Minkowskian spacetime geometry can be recovered.

A similar story could have been applied to Euclidean geometry and to Gailiean Spacetime geometry.

 

[Dale]

If one wants to derive the interval formula, as the OP says he does, of course one cannot assume it.

Agreed. One also cannot derive the interval formula from the invariance of c alone.

 

[pervect]

To be shown. x^2 = a^2 if, and only if , either x=a or x=-a

The first phase of the proof.

Assume x^2 = a^2, then show x=a or x=-a. Proof:
(x^2 - a^2) = 0. Factoring (x+a)*(x-a) = 0. Thus, either x=a or x=-a, so the first section of the proof has been completed.

The second phase of the proof

Assume x=a or x=-a. Show that x^2=a^2. This can be done by direct substitution.

QED.

A quick search for counter-examples should also be convincing - a counterexample would show the two statements are not equivalent. (But they are).

Thus, the only difference between the assumptions x^2=a^2 and the assumption that x=a is that the later doesn't include the possibility that x=-a. If there is a difference between the proofs starting with the original assumptions x^2 =a^2 and a proof involving x=a, the place to look for the difference is when x=-a.

 

[Saw]

You interpreted incorrectly. The fact that the equation for a light ray happens to look like the interval equation does not mean that a derivation using the former must be a derivation of the latter. It isn't. The Wikipedia article says quite clearly that it is about derivations of the Lorentz transform (by which it really means the Lorentz boost), not derivations of the interval. And, as I have already said, none of its derivations are derivations of the interval.

Again, do you have a reference for a derivation of the interval? Because if not, this thread should be closed as we have no valid basis for any discussion of what you now say you want to talk about.

Well, if I interpreted it incorrectly, of which I am not sure, I say in my discharge that it was no big blunder, because I would have been misguided by the wording of the Wiki text, when they take a formula that exactly -as you euphemistically point out- "looks like the interval" (I would say that it "is" the interval) and then they say that it is "invariant" (literally, it "takes the same form in both frames") and finally they mention a derivation, albeit rudimentary, when they state that this is "because of relativity postulates" (that was precisely my complaint, lack of sufficient motivation)!

Again, do you have a reference for a derivation of the interval? Because if not, this thread should be closed as we have no valid basis for any discussion of what you now say you want to talk about.

Take Spacetime physics, Wheeler and Taylor, section 3.7, "Invariance of the interval proven", where they use as reference a timelike interval, when a light is flashed transversely with regard to the relative motion between the two frames, in the line of the light clock thought experiment. But I am fine with their approach.

 

[Dale]

If there is a difference between the proofs starting with the original assumptions x^2 =a^2 and a proof involving x=a, the place to look for the difference is when x=-a.

Yes, exactly.

…

 

[PeterDonis]

I would have been misguided by the wording of the Wiki text

Only if you ignore the fact that the "interval" given was specifically stated in the article to be for a spherical wave front of light, not a general formula that would apply to any interval whatever. And since you explicitly referred to that fact in your OP, I don't think you can blame your misunderstanding on Wikipedia.

if I interpreted it incorrectly, of which I am not sure

The fact I referred to above should make it obvious that your claimed intepretation cannot be correct.

 

[PeterDonis]

Take Spacetime physics, Wheeler and Taylor, section 3.7, "Invariance of the interval proven", where they use as reference a timelike interval, when a light is flashed transversely with regard to the relative motion between the two frames, in the line of the light clock thought experiment. But I am fine with their approach.

Ok, good. Then, once we have the invariant interval, as has already been stated, we can derive the Lorentz boost simply by finding the group of transformations that leaves this interval invariant, which will mean leaving the spacetime geometry invariant. In other words, we are looking for a group of isometries of the spacetime. This would be a derivation of the Lorentz boost that did not rely on any properties of spherical wave fronts of light.

 

[Saw]

This would be a derivation of the Lorentz boost that did not rely on any properties of spherical wave fronts of light.

Are you implying that Wiki’s commented derivation is faulty because it does rely on the the properties of a spherical wavefront of light?

 

[PeterDonis]

Are you implying that Wiki’s commented derivation is faulty because it does rely on the the properties of a spherical wavefront of light?

Not for that reason alone, no. We eventually agreed in this thread, I think, that the derivation your referenced in the OP was faulty, but not for that reason.

 

[strangerep]

I concede that the title of the thread may be misleading, because I mentioned LTs, but I think that the OP is clear in that my concern is only about the way to derive the *ST interval*.

Then take a few steps back, a deep breath, and listen carefully...

View attachment 319299

This is not the usual, most physically fundamental, meaning of the Relativity Postulate.

From Rindler** :

Principle of Relativity:
The laws of physics are identical in all inertial frames,
or, equivalently,
the outcome of any physical experiment is the same when performed with identical initial conditions relative to any inertial frame.

One then searches for the maximal group of coordinate transformations $(t,x^i) \to (t',x'^i)$ which ensures that every inertial (i.e., non-accelerating) frame is transformed into another inertial frame, i.e., $$ \frac{d^2 x^i}{dt^2} ~=~ 0 ~~~~ \Leftrightarrow ~~~~ \frac{d^2 x'^i}{dt'^2} ~=~ 0 ~. $$ This determining equation already has spatial isotropy inherent within it, therefore we may use this (i.e., without an extra separate assumption) when searching for the coordinate transformations. Similarly for spatiotemporal translation invariance.

From this (long, tedious) analysis, looking especially for a subgroup corresponding to velocity boosts such that velocity boosts in along any given spatial direction form a 1-parameter Lie group, one finds the Lorentz group, together with a universal constant with dimensions of inverse velocity squared that emerges unbidden from the analysis. By comparison with experiment, one realizes that this constant corresponds to $c^{-2}$. But one does not mention anything about light at the beginning.

Having the Lorentz transformations, one then notices that it preserves a particular quadratic form in spacetime. I.e., by this route, the invariant spacetime interval is derived from the Relativity Principle.

Importantly, the usual Light Postulate (i.e., assume invariance of the speed of light) is not actually necessary. Similarly, anything to with spherical wavefronts of light is not needed up front. All such things are simply shortcuts to make the LT derivation quicker, hence more palatable to the average student. But if you seek the deepest foundation that we know of, then study the so-called "1-postulate" derivation sketched above.

Here are some older PF threads where I talked about this stuff:

https://www.physicsforums.com/threa...-1-spacetime-only.1000831/page-2#post-6468652
Post #44.

https://www.physicsforums.com/threads/linearity-of-the-lorentz-transformations.975920/#post-6219004
Post #6.

https://www.physicsforums.com/threads/derivation-of-the-lorentz-transformations.974098/
Post #26.

Other PF members have also talked about it. Try searching for "1-postulate" and/or for ways of deriving the LTs.

Textbook Reference:

**) Rindler, Introduction to Special Relativity,
Oxford University Press, 1991 (2nd Ed.), ISBN 0-19-85395-2-5.

 

[Saw]

Then take a few steps back, a deep breath, and listen carefully...

From this (long, tedious) analysis

(...)

Having the Lorentz transformations, one then notices that it preserves a particular quadratic form in spacetime. I.e., by this route, the invariant spacetime interval is derived from the Relativity Principle.

(...)

Other PF members have also talked about it. Try searching for "1-postulate" and/or for ways of deriving the LTs.

Thanks, but as stated in the text from me that you have quoted I am not interested (in this thread) in deriving the LTs and then noticing that they "preserve a particular quadratic form in spacetime", but acting the other way round: deriving the invariant ST interval and then looking for what transformations preserve it. In particular, I am interested in what assumptions are needed to make this ST interval derivation.

I recognize that doing it the way you mention may have a sound reason and bring high benefits, but my interest lies in the other route. If that requires taking as postulate light invariance, that is not a problem for me. If that requires taking as postulate something more than relativity postulate, that is not a problem either. So the question ultimately boils down to "what assumptions do you think are necessary to derive the ST interval"? For example, take as reference the one that has been accepted by PeterDonis as a valid derivation, from section 3.7 of Taylor-Wheeler. Can you spell out all its assumptions?

 

[Saw]

Ok, good. Then, once we have the invariant interval, as has already been stated, we can derive the Lorentz boost simply by finding the group of transformations that leaves this interval invariant, which will mean leaving the spacetime geometry invariant. In other words, we are looking for a group of isometries of the spacetime.

But the thread is not about what goes after the derivation of ST interval. The thread is about what assumptions must be made for deriving the ST interval. Reference about what goes after may be interesting only inasmuch as we realize that some assumption that is introduced ex post was actually made ex ante because it was implicit in the ST derivation.

So you take Wheeler-Taylor derivation of the ST interval as a valid reference, as I do myself. But the question is: what assumptions or postulates are, in your opinion, implicit in this derivation?

 

[strangerep]

Thanks, but as stated in the text from me that you have quoted I am not interested (in this thread) in deriving the LTs and then noticing that they "preserve a particular quadratic form in spacetime", but acting the other way round: deriving the invariant ST interval and then looking for what transformations preserve it. In particular, I am interested in what assumptions are needed to make this ST interval derivation.

(Sigh.) This is becoming ridiculous.

So the question ultimately boils down to "what assumptions do you think are necessary to derive the ST interval"?

"Necessary"? I already told you.

For example, take as reference the one that has been accepted by PeterDonis as a valid derivation, from section 3.7 of Taylor-Wheeler. Can you spell out all its assumptions?

They use a 2-postulate method and a lot of hand wavy diagrams. I don't think it qualifies as a "derivation", but rather a plausibility argument.

Anyway, I'm outta here.

 

[Ibix]

But the question is: what assumptions or postulates are, in your opinion, implicit in this derivation?

Write down the time taken for light to travel up and down the light clock, $\Delta t(v)$. Write down the distance travelled by the lower mirror, $\Delta x(v)$. Notice that $c^2\Delta t^2-\Delta x^2$ is independent of $v$. Note that similar constructions are possible for spacelike and null separated pairs of events and the same formula is always independent of $v$.

Used Pythagoras' Theorem (so assuming homogeneity and isotropy), the principle of relativity, and the invariance of the speed of light. The same as Einstein's original paper.

 

[Sagittarius A-Star]

Note that similar constructions are possible for spacelike and null separated pairs of events and the same formula is always independent of $v$.

This can be found in chapter 2 of
http://www.faculty.luther.edu/~macdonal/Interval/Interval.html

 

[Dale]

Thanks, but as stated in the text from me that you have quoted I am not interested (in this thread) in deriving the LTs and then noticing that they "preserve a particular quadratic form in spacetime", but acting the other way round: deriving the invariant ST interval and then looking for what transformations preserve it. In particular, I am interested in what assumptions are needed to make this ST interval derivation

I don’t think that your objection to @strangerep is valid. A derivation of the Lorentz transform and then a derivation of the spacetime interval is still a derivation of the spacetime interval.

At a maximum Einstein’s two postulates are necessary for deriving the ST interval. This is done by deriving the Lorentz transform and then showing the invariance of the interval. That forms a valid proof of the ST interval.

There are several valid “one postulate” derivations, but they all require one postulate plus experimental evidence to determine the value of the unknown parameter, c. So that is still two assumptions. I don’t think it can be derived in less.

 

[Ibix]

I don’t think that your objection to @strangerep is valid. A derivation of the Lorentz transform and then a derivation of the spacetime interval is still a derivation of the spacetime interval.

I think @Saw wants to be able to derive the LTs, $\Lambda$, from $\eta=\Lambda\eta\Lambda^T$ given $\eta$, rather than deriving $\eta$ given $\Lambda$. Strangerep's approach is the latter. So what's needed is a thought experiment that leads to the interval/$\eta$ without explicit dependence on the Lorentz transforms.

 

[Dale]

I think @Saw wants to be able to derive the LTs, $\Lambda$, from $\eta=\Lambda\eta\Lambda^T$ given $\eta$, rather than deriving $\eta$ given $\Lambda$. Strangerep's approach is the latter. So what's needed is a thought experiment that leads to the interval/$\eta$ without explicit dependence on the Lorentz transforms.

I think it is fairly pointless for you and I to argue about @Saw’s intentions. To me he seems to be focused on the number of assumptions, particularly implicit or hidden assumptions, required for deriving the spacetime interval. But it isn’t a good sign that more than 100 posts in this is a matter of confusion

 

[Saw]

I think @Saw wants to be able to derive the LTs, $\Lambda$, from $\eta=\Lambda\eta\Lambda^T$ given $\eta$, rather than deriving $\eta$ given $\Lambda$. Strangerep's approach is the latter. So what's needed is a thought experiment that leads to the interval/$\eta$ without explicit dependence on the Lorentz transforms.

Thanks, that is exactly what I want. And then calmly reflect on the assumptions inherent to such derivation of the ST interval. As simple as that. Shall we do it on the basis of Wheeler and Taylor's derivation, as you started to do? @strangerep suggested that Taylor and Wheeler's derivation may not be good enough, but I myself cannot think of a better one, so I would work on the basis of that one.

If so, would you agree, as a first step, that the steps of such derivation are as follows?

 

[vanhees71]

I don’t think that your objection to @strangerep is valid. A derivation of the Lorentz transform and then a derivation of the spacetime interval is still a derivation of the spacetime interval.

Talking about the didactically convincing "derivation" of some particular result in physics is of course always thinking about, which assumptions should be made in this derivation.

Einstein's choice was to assume the validity of Newton's Lex Prima, the special principle of relativity (existence of a (global) inertial frame) and the independence of the speed of light in vacuo on the velocity of the light source relative to the observer. Then, using (not so explicitly stated) also the assumption that space for any inertial observer is described as a Euclidean affine manifold (as in Newtonian mechanics). He derived the Lorentz transformation from using these postulates to synchronize clocks, at rest relative to each other and relative to an inertial frame, and how the description of space and time change for an observer moving with constant velocity relative to this inertial frame, thus defining also an inertial frame. Then he proved that also the Maxwell equations are Lorentz invariant, and the problem with the incompatibility of Maxwell theory with Galilei invariance and the observed absence of an absolute inertial frame, defined as the rest frame of some "aether". That's a physicist's derivation, i.e., giving a instrumental definition of how to synchronize clocks in such a way as to make the two postulates, which took from the features of Maxwell's equations under the assumption that they must obey the special principle of relativity, the very ones that are necessary to establish a new mathematical description of space and time given the absence of an absolute inertial reference frame.

Another derivation, that is much simpler mathematically, is to use Einstein's postulates together with the Euclidicity of space for any inertial observer and just use the invariance of the Minkowski quadratic form (which implies the invariance of the Minkowski bilinear form) to derive the Lorentz group (or rather the Poincare group by assuming space-time-translation symmetry too) as the corresponding symmetry group of the implied space-time model, i.e., Minkowski space as a 4D affine manifold with an indefinite fundamental from (Minkowski bilinear form) of signature (1,3) or equivalently (3,1). This has the advantage of great mathematical elegance but the disadvantage of being pretty abstract rather than physical as is Einstein's derivation.

Another variant of this more mathematically inclined pedagogics is to only use the special principle of relativity and the Euclidicity of space for any inertial observer to figure out what the possible spacetime symmetries are, and I found the answer pretty appealing when I first read about this approach: It comes out that there are only two possible spacetime symmetries (and thus spacetime models, because these can be derived from the symmetries), i.e., Galilei-Newton or Einstein-Minkowski spacetime.

It think, in fact, it's best to use both, the physics and mathematics approach to the "derivation of the Lorentz/Poincare transformation", because it has a higher chance of gaining a really deep understanding or the transformation from both the physical and the mathematical meaning.

At a maximum Einstein’s two postulates are necessary for deriving the ST interval. This is done by deriving the Lorentz transform and then showing the invariance of the interval. That forms a valid proof of the ST interval.

There are several valid “one postulate” derivations, but they all require one postulate plus experimental evidence to determine the value of the unknown parameter, c. So that is still two assumptions. I don’t think it can be derived in less.

Indeed, you need pretty constraining symmetry assumptions about time and space to arrive at the Minkowski spacetime of special relativity. You can, however, leave out the 2nd of Einstein's postulate and derive that there's only Galilei-Newton or Einstein-Minkowski spacetime left. Then you can leave it to the experimentalists figuring out which model is more accurate in describing Nature. The answer is well known since Maxwell: It's Einstein-Minkowski spacetime, and to a very high accuracy the limiting speed of the Minkowski space-time model is indeed the speed of em. waves in vacuo. The latter is an empirical conclusion from the determination of the upper limit of the photon mass, which is at $m_{\gamma} < 10^{-18} \text{eV}$.

 

[Dale]

that is exactly what I want.

Post 110. Just saying…

 

[Saw]

Another variant of this more mathematically inclined pedagogics is to only use the special principle of relativity and the Euclidicity of space for any inertial observer to figure out what the possible spacetime symmetries are, and I found the answer pretty appealing when I first read about this approach: It comes out that there are only two possible spacetime symmetries (and thus spacetime models, because these can be derived from the symmetries), i.e., Galilei-Newton or Einstein-Minkowski spacetime.

Adding to what I said in post 110, my intention is precisely to zoom on the assumptions that make us prefer the Einstein-Minkowski spacetime through the adoption of the so-called ST interval (the invariant Minkowski bilinear form, as you refer to it in the previous paragraph).

 

[Dale]

my intention is precisely to zoom on the assumptions that make us prefer the Einstein-Minkowski spacetime through the adoption of the so-called ST interval

But that in no way prevents you from doing the derivation through the Lorentz transform. I think that you need to decide what you really want. More than 100 posts in and it seems to change every time we hear from you.

If you want to minimize assumptions then any of the two postulate derivations of the Lorentz transform or any of the one postulate derivations plus an experimentally determined invariant speed will get you to the spacetime interval.

If you want to derive the spacetime interval without the Lorentz transform then you may not be able to minimize the assumptions.

 

[PeroK]

Adding to what I said in post 110, my intention is precisely to zoom on the assumptions that make us prefer the Einstein-Minkowski spacetime through the adoption of the so-called ST interval (the invariant Minkowski bilinear form, as you refer to it in the previous paragraph).

I've just looked at this thread. I've no idea how you dragged it out to 110+ posts. It's crazy!

 

[robphy]

I've just looked at this thread. I've no idea how you dragged it out to 110+ posts. It's crazy!

Post dilation?

 

[Saw]

If you want to derive the spacetime interval without the Lorentz transform then you may not be able to minimize the assumptions.

I never said that I want to minimize the assumptions!!!! I just said that I want to have a clear idea about which are needed and which are not and my concern is precisely that I fear that often too few are declared, as I have repeated plenty of times!!!. Please stop questioning what the object of the OP is, It is what Ibix said in his post 108 and I confirmed in post 110: deriving the ST interval (full stop, not LT) and, yes, of course, being clear on how (on the basis of which assumption this has been done). Is that so difficult to understand? It seems so, given how often I have to repeat it, but it is not so.

I've just looked at this thread. I've no idea how you dragged it out to 110+ posts. It's crazy!

I am also quite tired that the object of the thread is constantly put into question. If anybody wants to discuss this simple thing, he/she is invited to stay. Whoever does not like it, thanks for your comments so far and goodbye. If anybody wants to close the thread, fair enough. Yet the truth is that the object is clear and I am making a good-faith attempt to discuss it.

 

[Ibix]

Shall we do it on the basis of Wheeler and Taylor's derivation, as you started to do?

There's nothing left to do after post #105 and the reference in @Sagittarius A-Star's #106 that fills in the setup for spacelike and null separated events, except fill in the value of the flight time of the light, which tells you that $\Delta t=2l/\sqrt{c^2-v^2}$ (where $l$ is the separation of the mirrors) and $\Delta x=v\Delta t$. Those are general expressions valid in all frames moving perpendicular to the line between mirrors, and $c^2\Delta t^2-\Delta x^2$ is manifestly invariant. I already listed the assumptions used.

 

[PeterDonis]

I am also quite tired that the object of the thread is constantly put into question.

It keeps getting put into question because nobody but you seems to know what it is.

It is what Ibix said in his post 108 and I confirmed in post 110

So it took 110 posts to finally get what you say is a clear statement of what you want. If you are tired after all that, imagine how the rest of us feel.

But even then you throw us another change:

my intention is precisely to zoom on the assumptions that make us prefer the Einstein-Minkowski spacetime through the adoption of the so-called ST interval

What is this "make us prefer"? Where did "prefer" come into it? Prefer over what?

If you want to know what assumptions are required to derive the Minkowski interval (I have no idea why you keep saying "ST"--what does that stand for?), that can be done without any "preference" at all. It's just a straightforward question of logic, which has been answered.

If you want to know why we "prefer" the Minkowski interval over something else, the answer should be obvious: because it makes correct predictions. But that's a different question from the one you said you wanted answered in post #110.

 

[Ibix]

I have no idea why you keep saying "ST"--what does that stand for?

Space Time, I assumed