Need to resort to spherical wavefront to derive the LTs?

[Sagittarius A-Star]

unless I messed up somewhere

After you will have corrected the starting point, as @Dale pointed out, the next thing to correct is, that you check an equation, that you intend to work only for a null-interval, with a test interval of ##2s \cdot c##. That cannot work.

 

[Saw]

You messed up at the beginning. As I already pointed out before your starting point is wrong. ##\Delta t## can be negative and your starting point does not reflect that.

If you use a correct starting point and correct algebra then you will get a valid result.

Also, we are currently working only with null spacetime intervals. We have not as yet made any assumption that non-null spacetime intervals are invariant. That is a separate assumption.

Now my starting point does reflect that time cannot be negative (this has also been pointed out many times), but the valid algebraic operation that I carried out to accommodate such concern does not lead to the good ST interval, as shown here:

Do you want me to just square both sides, without square rooting? Sure, I can do that, it is also a valid algebraic operation. But please give me a reason for doing that with all its letters, other than "reflect that time cannot be negative" because that is ensured by my operation and a reason also other than "this way you get by chance the correct answer as proved by experiments", because that is only an ad hoc or trial and error approach. But don't pretend that you are "deriving" it based on two single postulates like relativity and invariance of c because you are not.

Others may have other comments, if you still have yours, that is of course perfect, but I kindly suggest that we leave this particular matter here. I really appreciate and thank you for the time you have taken to object my view and have actually learnt a lot from your comments, but we are not going to persuade each other on this point, I myself surrender!

 

[Saw]

check an equation, that you intend to work only for a null-interval, with a test interval of ##2s \cdot c##. That cannot work.

Uhh? No, I don't intend that at all! You misunderstand me. What I have done is deriving the ST interval, as people do, based on a null interval and then declared, as people do, that such form of the ST interval is of general application, i.e. it is valid for all intervals, also to a timelike one where proper time is 2s.

It is just that, to derive that *generally valid* ST interval, I have applied the same logic as people use: I have raised both sides of the equation to 11th power or squared both sides and then square rooted them (in the latter case to avoid negative time). Incredibly, in spite of the fact that both are valid algebraic operations and that the second accommodates our only declared concern... the thus derived ST interval does not work...

 

[PeterDonis]

my starting point does reflect that time cannot be negative

Which is wrong. It is perfectly possible for the difference in coordinate times between two events to be negative.

the valid algebraic operation that I carried out to accommodate such concern does not lead to the good ST interval

What do you mean by "the good interval"? For an interval that describes a spherical wave front of light (in the forward light cone, which is what you have been implicitly assuming, and what the Wikipedia article also implicitly assumes), ##ct## to any power is equal to ##\sqrt{x^2 + y^2 + z^2}## to any power. That is what @Dale has been telling you. So if your purpose is to describe a spherical wave front of light, the algebraic operations "raise to the 2nd power" and "raise to the 11th power" do both give you correct equations, just as @Dale said. If you are just looking at null intervals, there is no reason to prefer the "square root" form to any other; all of them pick out the same spherical wave front of light, so all of them are equally valid as descriptions of that spherical wave front of light.

If, on the other hand, you are trying to derive Lorentz boost equations that will apply to any interval, not just a null interval that describes a spherical wave front of light, then you have to look at other constraints besides the ones imposed by a spherical wave front of light. In other words, you have to look at other intervals, where it is not the case that ##ct = \sqrt{x^2 + y^2 + z^2}##, because the interval is not null. And you have to restrict yourself to operations that are valid for those cases as well. None of your arguments have done that, and that is why you keep getting pushback.

 

[PeterDonis]

What I have done is deriving the ST interval, as people do, based on a null interval and then declared, as people do, that such form of the ST interval is of general application

No, that is not what you've done. "People" don't derive the interval that way.

to derive that *generally valid* ST interval, I have applied the same logic as people use: I have raised both sides of the equation to 11th power or squared both sides and then square rooted them (in the latter case to avoid negative time). Incredibly, in spite of the fact that both are valid algebraic operations and that the second accommodates our only declared concern... the thus derived ST interval does not work...

And the reason, as I pointed out in post #74 just now, is that these operations are not valid for intervals that are not null intervals. So it's no surprise that they give you wrong answers. But if you just look at null intervals only, they are valid operations and they do give you equations that correctly describe null intervals--but only null intervals. Which is all you can expect when you use operations that are only valid for null intervals.

 

[Dale]

Now my starting point does reflect that time cannot be negative (this has also been pointed out many times),

That is wrong. ##\Delta t## can be negative. This is why you should start with the correct version which is ##\Delta t^2 = \Delta x^2 + \Delta y^2 + \Delta z^2##. This formula works correctly for negative ##\Delta t##. Your wrong formula does not.

 

[PeterDonis]

I have now realized that the Doppler factor is also the scale at which the second frame is drawn in the Minkowski diagram, so the null vector "is" dilated in this diagram even if you dont see it on the page, right? Thanks for guiding me to this is insight (if it is correct at all).

Yes, it's correct.

Said this, does this mean that you are unfavoring now any derivation of the ST interval that takes as illustration a null vector?

No. What I am saying is that your method of using the null interval is not the one that is used in the derivations you referenced. The derivations you referenced do not obtain the general form of the interval by arguing from the equation for a null interval and arbitrarily restricting to the squared form of that equation.

Well, the problem with deriving the ST interval is that there are three possible displays, depending on whether you choose a timelike, lightlike or spacelike vector... You have to choose one reference and then generalize the result to the others

No, you don't. All you have to do is restrict yourself to operations that are valid regardless of whether the vector is timelike, null, or spacelike.

or you can repeat it with lightlike (what we have done so far)

No, that's not what the "spherical wave front" arguments do.

and timelike (which is the other derivation of the light clock though experiment, which Iike, although it seems not be in fashion nowadays)

I'm not sure what you're referring to here. Can you give a reference?

I would not know how to express it with spacelike, since we would present an impossible scenario of something traveling FTL...

Not at all. Considering a spacelike vector in no way requires you to assume that some observer is traveling on a spacelike worldline.

In any case, I don't see a problem in saying that we are in face of a hyperbolic rotation even if the reference is precisely the vector that acts as eigenvector of the rotation and therefore gets only dilated, without changing direction.

The issue is not the term "hyperbolic rotation"--yes, boosts are often called that even though they dilate null vectors instead of rotating them.

The issue is whether a derivation that makes use of hyperbolic rotation equations is implicitly making some assumption that is only valid if the rotation is an actual rotation, not a dilation.

 

[Dale]

Now my starting point does reflect that time cannot be negative (this has also been pointed out many times)

Again, this is wrong. Your formula misses half of the light cone. Half of the events that have a null interval from the origin are missing with your incorrect formula: ##\Delta t=\sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2}##

You need to start with the right formula: ##\Delta t^2=\Delta x^2 + \Delta y^2 + \Delta z^2##

I kindly suggest that we leave this particular matter here. I really appreciate and thank you for the time you have taken to object my view and have actually learnt a lot from your comments, but we are not going to persuade each other on this point

Your point is demonstrably wrong. Your starting equation does not cover the full set of events with a null spacetime interval from the apex event.

You may stop making your incorrect point any time you like whether I have persuaded you or not, but I will continue to state that it is wrong each time you continue to make it.

instead of squaring both sides, another valid algebraic operation is raising both sides to the 11th

And I already demonstrated that doing so leads to the same set of events for the null interval.

 

[Sagittarius A-Star]

Uhh? No, I don't intend that at all! You misunderstand me.

Sorry for that! Then I still do not understand, why you put so much effort in showing, that other powers than square do not work for arbitrary intervals (for example for ##2s \cdot c##), as this is anyway clear to all.

I understand, that the main question from you is now, how to derive the Minkowski spacetime geometry for 1 time dimension and 1 spatial dimension (= without y and z), what was not done in the discussed Wikipedia article. They should have made this clearer by putting a "##=0##" behind the second equation, as they did it behind the first equation.

Certainly. The question is only how to get to the quadratic form, if that can happen a little magically, through an algebraic operation, or rather it must happen, but based on a hypothesis on how ct, on the one hand, and x-y-z, on the other hand, combine together.

My favorite approach is to use the LT derivation of Macdonald and multiply equations (1) and (2):
##T+X = \gamma (1+v)(T' + X') \ \ \ \ \ (1)##
##T-X = \gamma (1-v)(T' - X') \ \ \ \ \ (2)##
Multiplication (and using the formula for ##\gamma##, derived at the end of the paper):

##T^2-X^2 = {T'}^2 - {X'}^2##.

 

[robphy]

My favorite approach is to use the LT derivation of Macdonald and multiply equations (1) and (2):
##T+X = \gamma (1+v)(T' + X') \ \ \ \ \ (1)##
##T-X = \gamma (1-v)(T' - X') \ \ \ \ \ (2)##
Multiplication (and using the formula for ##\gamma##, derived at the end of the paper):

##T^2-X^2 = {T'}^2 - {X'}^2##.

See Bondi’s Relativity and Common Sense for a physical motivation (using radar measurements) for that algebraic maneuver. (Secretly, it’s because of the eigenbasis of the boost.)

 

[PeterDonis]

deriving the ST interval

One of the issues in this thread is that you keep vacillating about what you want to derive. Do you want to derive the Minkowski interval? Or do you want to derive the Lorentz boost equations? They're not the same thing. Logically, if you have one, you can derive the other from the fact that Lorentz boosts leave the Minkowski interval invariant. But that still doesn't make them the same.

The derivation you referenced in the OP, and the others from the same Wikipedia article, are derivations of the Lorentz boost equations. They are not derivations of the Minkowski interval. The fact that a spherical wave front of light obeys an equation that can be made to look like the Minkowski interval (which is a consequence of the fact that that interval is zero for light) does not mean that any derivation that involves a spherical wave front of light must be a derivation of the Minkowski interval.

So far nobody in this thread has referenced a derivation of the Minkowski interval, by some means that does not involve deriving the Lorentz boost equations (and then, implicitly, observing that those transformations leave the Minkowski interval invariant).

 

[Sagittarius A-Star]

So far nobody in this thread has referenced a derivation of the Minkowski interval, by some means that does not involve deriving the Lorentz boost equations (and then, implicitly, observing that those transformations leave the Minkowski interval invariant).

Currently, I don't completely understand, if my following posting contained such thing.

Yes. According to Wikipedia, Einstein started his derivation of the LT by adding and subtracting the equations
##\begin{cases}
x' - ct' = \lambda (x - ct) \\
x' + ct' = \mu (x + ct)
\end{cases}##
From this one can also derive the invariance of the spacetime interval, by multiplying the equations. The result is:
##{x'}^2 - c^2{t'}^2 = \lambda \mu (x^2 - c^2t^2)##.
From reciprocity between both frames can be concluded: ##\lambda \mu = 1##.

Reason: Einstein did not describe in detail, if his equations (3) and (4) are valid for all events and if yes, why. I don't understand, if those equations are only a "good guess", which resulted by chance in the correct LT.

Source:
https://en.wikisource.org/wiki/Rela...mple_Derivation_of_the_Lorentz_Transformation

 

[Dale]

So far nobody in this thread has referenced a derivation of the Minkowski interval, by some means that does not involve deriving the Lorentz boost equations (and then, implicitly, observing that those transformations leave the Minkowski interval invariant).

My personal preference is to assume the invariance of the Minkowski metric and then derive the LT as a transform that preserves the form of the interval. Before deriving the LT I would demonstrate the experimental results that come directly from the interval (specifically the invariance of c and time dilation) as motivation for the interval.

 

[PeterDonis]

Einstein did not describe in detail, if his equations (3) and (4) are valid for all events

It must, because he makes use of the equation for the worldline of the origin of the primed frame, ##x' = 0##, in the unprimed frame. In other words, his derivation makes use of both the behavior of light rays (in both opposite directions along the ##x## axis) and the behavior of a timelike inertial worldline (which could be any arbitrary timelike inertial worldline, since we can pick any such as the worldline of the origin of the primed frame). So it must be valid for all events. (Showing how the above is sufficient to include spacelike separated events as well is left as an exercise for the reader.)

 

[Saw]

One of the issues in this thread is that you keep vacillating about what you want to derive. Do you want to derive the Minkowski interval? Or do you want to derive the Lorentz boost equations? They're not the same thing. Logically, if you have one, you can derive the other from the fact that Lorentz boosts leave the Minkowski interval invariant. But that still doesn't make them the same.

The derivation you referenced in the OP, and the others from the same Wikipedia article, are derivations of the Lorentz boost equations. They are not derivations of the Minkowski interval. The fact that a spherical wave front of light obeys an equation that can be made to look like the Minkowski interval (which is a consequence of the fact that that interval is zero for light) does not mean that any derivation that involves a spherical wave front of light must be a derivation of the Minkowski interval.

So far nobody in this thread has referenced a derivation of the Minkowski interval, by some means that does not involve deriving the Lorentz boost equations (and then, implicitly, observing that those transformations leave the Minkowski interval invariant).

Good point, for the sake of clarification.

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*. I have also underlined it several times.

The derivation you referenced in the OP, and the others from the same Wikipedia article, are derivations of the Lorentz boost equations. They are not derivations of the Minkowski interval. The fact that a spherical wave front of light obeys an equation that can be made to look like the Minkowski interval (which is a consequence of the fact that that interval is zero for light) does not mean that any derivation that involves a spherical wave front of light must be a derivation of the Minkowski interval.

Well, the particular derivation that I linked to clearly seems to conceive its first section as a first step of the LT derivation consisting precisely in a "ST derivation". At least that is how I interpreted its wording, copied below for convenience:

Anyhow, the positions about the specific issue that I mentioned are clear. You are telling me that I should start with the formula where c*time interval is already squared and that otherwise I would miss "half of the light cone". I have to assimilate that, but say I stand corrected.

But then I am very interested in knowing how you would derive yourselves the ST interval. There are other ways, but it seems to me that in fact taking this as a first step is a very reasonable way of facing the derivation of the LTs: first, in agreement with relativity principle, you stipulate that the two frames will solve the problem at hand in the same way, thanks to combining their respective values in a given way, according to a certain formula, and then you proceed to guess the transformation rule that converts values from one basis to another, preserving the invariance of such ST interval.

My personal preference is to assume the invariance of the Minkowski interval and then derive the LT as a transform that preserves the form of the interval. Before deriving the LT I would demonstrate the experimental results that come directly from the interval (specifically the invariance of c and time dilation) as motivation for the interval.

You have anticipated my question. So you would also start with the ST interval as first step and for this purpose, as motivation for the interval, rely on invariance of c and time dilation. But would you start with the equation of the spherical wave front (null vector) or another? Since you mention time dilation, would you rely by chance on the light clock thought experiment (timelike vector)? What about the others?

 

[PeterDonis]

I think that the OP is clear in that my concern is only about the way to derive the *ST interval*.

Then every single reference given in this thread so far is irrelevant (as well as most of the discussion), because none of them are about deriving the interval. All of them are about deriving the Lorentz boost equations.

Do you have any reference for a derivation of the interval?

 

[PeterDonis]

Well, the particular derivation that I linked to clearly seems to conceive its first section as a first step of the LT derivation consisting precisely in a "ST derivation". At least that is how I interpreted its wording

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.

 

[PeterDonis]

You are telling me that I should start with the formula where c*time interval is already squared and that otherwise I would miss "half of the light cone".

If your goal is to describe the entire light cone, yes, that's what you have to do. But that has nothing whatsoever with "deriving the interval".

 

[PeterDonis]

then you proceed to guess the transformation rule that converts values from one basis to another, preserving the invariance of such ST interval.

Once again: the equation for a spherical wave front of light, in itself, is not "the interval". It's just an equation for a spherical wave front of light.

If you want to derive "the interval", one obvious way to do it is to first derive the Lorentz boost, and then derive the interval by looking at what the boost leaves invariant. That's basically how you would do it with any of the derivations you have referenced, since all of them are derivations of the Lorentz boost and none of them start with the interval (they start with the equation for a spherical wave front of light, which, as above, is not the same thing).

If you want to derive the interval first, and then derive the Lorentz boost by asking what group of transformations will leave that interval invariant, you would have to start with some other derivation entirely, not any of the ones that have been discussed in this thread.

 

[Dale]

But would you start with the equation of the spherical wave front (null vector) or another?

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.

Since you mention time dilation, would you rely by chance on the light clock thought experiment (timelike vector)?

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.

 

[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.