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

[PeterDonis]

I'm curious what the OP thinks of two other derivations given in the same Wikipedia article:

Derivation using spherical wavefronts of light:

https://en.wikipedia.org/wiki/Deriv...transformations#Spherical_wavefronts_of_light

Einstein's popular derivation:

https://en.wikipedia.org/wiki/Deriv...transformations#Einstein's_popular_derivation

 

[Dale]

The derivation in the Wikipedia article assumes hyperbolic rotation, as has already been remarked. It's not just the name of the subsection, it's also the next assumption introduced (because the form of the equations "suggest" it) after linearity.

Sure, but there is no point in complaining to the OP about it. It isn't his wording. He is complaining about this derivation too.

 

[PeterDonis]

there is no point in complaining to the OP about it. It isn't his wording.

Yes, but he said he wasn't complaining about the derivation because of that term. I think he should.

 

[PeterDonis]

when choosing the ST interval in its unanimously accepted form (no matter if you consider 1+1D or 1+3D), you are already assuming that the change of perspective between the two frames consists of a hyperbolic rotation

The interval doesn't need to be assumed. The fact that $(ct)^2 - x^2 - y^2 - z^2 = 0$ for a spherical wave front of light in any inertial frame can be derived from the fact that the speed of light is $c$ in any inertial frame. The derivation you pointed at doesn't make that explicit (which is another issue with that derivation). But the "spherical wavefronts of light" derivation that I referenced, which is later in the same Wikipedia article, does.

 

[Dale]

Yes, but he said he wasn't complaining about the derivation because of that term. I think he should.

I think then that all three of us agree that this specific derivation is not good for a variety of reasons.

 

[PeterDonis]

I think then that all three of us agree that this specific derivation is not good for a variety of reasons.

Yes, agreed.

 

[Sagittarius A-Star]

The interval doesn't need to be assumed.

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: $\lambda \mu = 1$.

 

[Saw]

Many comments since my last one, I will reply to them collectively:

- The derivation I pointed to takes as introduction at least the first part of the other one called "Spherical wavefront", so any criticism on the former applies to the latter.
- But please note that my criticism exclusively refers to the part *up to* the formulation of the ST interval. *After that* the derivation introduces linearity and, following consequences of linearity, it hints that the form of the equations suggests hyperbolic rotation, but these are new assumptions that were not deemed necessary for deriving the ST interval.
- I have by now abandoned the argument about the teachings to be drawn from a 1+1D derivation, as the same point can be made relying on the 1 +3D one.
- My criticism (as noted, circumscribed to the part until derivation of ST interval) is only this:
* based on invariance of c postulate, you get c\Delta t = \sqrt {\Delta {x^2} + \Delta {y^2} + \Delta {x^2}}
* based on relativity postulate, you can presume that the same interval that is valid for O will be valid for O'
* but, in order to jump from the former expression to {(c\Delta t)^2} = \Delta {x^2} + \Delta {y^2} + \Delta {z^2} and hence to the ST interval {(c\Delta t)^2} - (\Delta {x^2} + \Delta {y^2} + \Delta {z^2}) = 0 you cannot rely on the algebraic trick "squaring both sides" because that is valid only for solving equations and finding unknowns, not for guessing which form take the dimensions in an invariant interval, which in turn depends on the nature of the change of perspective that applies in the case at hand.
- Although I initially intended to restrict the thread to what the derivation misses, I have already revealed what I had in mind as to what it misses: it should admit that the fact that the difference of perspective between frames in spacetime consists of a hyperbolic rotation is a "prius" to the ST interval. In other words, the third assumption of the derivation should come first, as a sort of third postulate.
- Ok, the null vector itself would not rotate because it is the eigenvector of the rotation matrix, but the ST interval is one that assumes that the values of ct and x, y, z transform as appropriate for a change of perspective corresponding to a hyperbolic rotation.
- Now if you said that any of the last two statements is wrong, I would be disconcerted and need your guidance as to why.
PS: BTW, I noted once that the eigenvalues are the Doppler factor (so the null vector would dilate by such factor when the transformation matrix is applied to it?); but if you explain to me how and why, that'd be great.

 

[Dale]

* based on invariance of c postulate, you get c\Delta t = \sqrt {\Delta {x^2} + \Delta {y^2} + \Delta {x^2}}

This is incorrect. $\Delta t$ can be negative, but this expression cannot have a negative $\Delta t$. The correct form for the invariance of c postulate is $c^2\Delta t^2=\Delta x^2+\Delta y^2+\Delta z^2$ from the beginning.

you cannot rely on the algebraic trick "squaring both sides" because that is valid only for solving equations and finding unknowns, not for guessing which form take the dimensions in an invariant interval

On the contrary. When you go from the correct starting expression to yours you need to include both the positive and the negative square roots.

This is a perfectly valid algebraic operation and your restriction doesn’t make sense. If an equation is true then it remains true under correct algebraic manipulations regardless of why you are doing them. That is indeed the point of algebra.

 

[Sagittarius A-Star]

PS: BTW, I noted once that the eigenvalues are the Doppler factor (so the null vector would dilate by such factor when the transformation matrix is applied to it?); but if you explain to me how and why, that'd be great.

You can calculate the longitudinal relativistic Doppler factor by Lorentz-transforming the four-frequency of a photon, that moves in x direction.

The four-frequency of a massless particle, such as a photon, is a four-vector defined by
$N^{a}=(\nu ,\nu {\hat {\mathbf {n} }})$
where $\nu$ is the photon's frequency and $\hat {\mathbf {n} }$ is a unit vector in the direction of the photon's motion. The four-frequency of a photon is always a future-pointing and null vector.

Source:
https://en.wikipedia.org/wiki/Four-frequency

 

[PeterDonis]

it should admit that the fact that the difference of perspective between frames in spacetime consists of a hyperbolic rotation is a "prius" to the ST interval. In other words, the third assumption of the derivation should come first, as a sort of third postulate.

No, it can't, because, as has already been pointed out and as you acknowledge, the transformation does not rotate null vectors, it dilates them.

- Ok, the null vector itself would not rotate because it is the eigenvector of the rotation matrix, but the ST interval is one that assumes that the values of ct and x, y, z transform as appropriate for a change of perspective corresponding to a hyperbolic rotation.

This is wrong. The SR (I assume you meant this instead of "ST") interval makes no such assumption. If it did, it would be wrong since the interval has to apply to null vectors.

- Now if you said that any of the last two statements is wrong, I would be disconcerted and need your guidance as to why.

Isn't it obvious? Once more: a Lorentz boost does not rotate null vectors. So you cannot base any derivation of the LT equations on hyperbolic rotation if you want it to apply to null vectors. And of course if you're starting off your derivation by looking at spherical wave fronts of light, you want your derivation to apply to null vectors.

PS: BTW, I noted once that the eigenvalues are the Doppler factor (so the null vector would dilate by such factor when the transformation matrix is applied to it?)

Yes, of course, that's what eigenvalue and eigenvector mean.

but if you explain to me how and why, that'd be great.

I'm not sure what needs explaining. Yes, the eigenvalues of the LT are the Doppler factors, and the eigenvectors are the null vectors that get dilated by the Doppler factors. This is just another way of saying that Lorentz boosts dilate null vectors and don't rotate them.

If you're asking why there are two Doppler factors, with one the reciprocal of the other, that should be obvious: they are for the two possible directions light can move relative to the direction of the Lorentz boost. Note that transformations of lightlike vectors that aren't in the same direction as the boost are more complicated; we haven't discussed those at all in this thread.

 

[PeterDonis]

you cannot rely on the algebraic trick "squaring both sides"

If this argument were valid, it would prove too much: it would prove that the Pythagorean theorem is not correct in Euclidean space.

I would suggest thinking carefully about what justifies the Pythagorean theorem (i.e., summing squares of coordinate deltas) in Euclidean space. The justification for using squares of coordinate deltas in Minkowski spacetime will be the same.

 

[Saw]

I think that we can narrow down to and organize the discussion around three issues:

1) if in order to jump:
- from what is obvious, i.e. that the spatial path traversed by light in a given frame, as composed by the Pythagorean combination of the three spatial dimensions, is equal to the time elapsed in such frame times the speed of light
c\Delta t = \sqrt {\Delta {x^2} + \Delta {y^2} + \Delta {x^2}}
- to what is already the ST interval in the form of
{(c\Delta t)^2} = \Delta {x^2} + \Delta {y^2} + \Delta {z^2}
the argument "I have squared both sides" is sufficient justification or you need "something else", some "added justification"
2) what this "else" would be, which in my opinion is (being very generic) hypothesizing that the new dimension, cT, is related to the old ones, X-Y-Z, in a manner that precisely justifies squaring both sides and which also has to do with the Pythagorean Theorem, but in a different manner, in order to account for the different sign
3) if in particular this "else" can be simply saying that the transformation is going to be a hyperbolic rotation, so I identify the ST interval with the equation of the hyperbola, the unit hyperbola

To me 1) is very clear, but I would not like to look stubborn. At a given time, we could agree that the matter has been discussed enough and leave the thread (without need to close it!) until someone comes with a new comment. But as of now I don't lose the hope of convincing you. Some arguments:

This is incorrect. $\Delta t$ can be negative, but this expression cannot have a negative $\Delta t$.

As already stated, if you want to get rid of negative time you can always stipulate that what goes into the ST interval is the absolute value of cT or either square both sides but immediately squareroot them, which leads to this other (wrong) interval:
\sqrt {{{(c\Delta t)}^2}} - \sqrt {(\Delta {x^2} + \Delta {y^2} + \Delta {z^2})} = \sqrt {{{(c\Delta t')}^2}} - \sqrt {(\Delta x{'^2} + \Delta y{'^2} + \Delta z{'^2})}
Of course, to avoid this ugly expression you must square both sides, but the subtlety that I favor is just admitting that you do it for a reason that goes beyond the pure algebraic trick.

This is a perfectly valid algebraic operation and your restriction doesn’t make sense. If an equation is true then it remains true under correct algebraic manipulations regardless of why you are doing them. That is indeed the point of algebra.

Sure. The equation remains true under the algebraic manipulation of squaring both sides. But the equation would also remain true under the algebraic manipulation of raising both sides to the 11th power, which would lead to a ST that is false. That is because what is valid for the algebraic purpose (solving for unknowns) may not be valid to guess what remains invariant under transformations. We have a set of operations that would be valid for the first purpose (squaring both sides, squaring and squarerooting to get absolute values, raising both sides to any other power...), but only the first is apt for the second purpose and that is due to "something else".

As to 2),

If this argument were valid, it would prove too much: it would prove that the Pythagorean theorem is not correct in Euclidean space.

I would suggest thinking carefully about what justifies the Pythagorean theorem (i.e., summing squares of coordinate deltas) in Euclidean space. The justification for using squares of coordinate deltas in Minkowski spacetime will be the same.

Well, it is clear to me that we can add up vectors through the Pythagorean theorem ("PT") when they are orthogonal to each other. That is what you do for X-Y-Z with the expression:
c\Delta t = \sqrt {\Delta {x^2} + \Delta {y^2} + \Delta {x^2}}
The question is how and why the new dimension cT can join the team of dimensions "somehow" related by the PT. I have thought of two routes:
a) The resultant of combining X-Y-Z through the PT in Euclidean manner combines with cT through the hyperbola equation. In this case, cT is orthogonal to X-Y-Z in the Minkowski way (dot product with negative sign is zero).
b) As reflected in the Loedel diagram, what is (in another sense) perpendicular is cT with (simplifying) X' and cT' with X, so that the ST interval is the height of the right triangle.
How the two things match is, though, a subtle question which I find beyond my reach as of now. Comments are welcome, but really the OP was not meant to discuss this.

In any case, the fact that the orthogonality between cT and X-Y-Z is implicit or embedded in the ST interval itself seems clear to me. As a proof: look at how the Wikipedia derivation continues in the second section "Linearity". At a given point they find a term that is remininscent of the last term of the Law of Cosines and they equate it to 0 after "comparing coefficients" and noting that the term of comparison does not contain a counterpart. To be expected, because the ST interval is built by assuming orthogonality.

As to 3), what was puzzling me was that in the ST diagram the null vector remains intact, how could it be dilated by the eigenvalue = the Doppler factor?

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).

Said this, does this mean that you are unfavoring now any derivation of the ST interval that takes as illustration a null vector? 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 or you can repeat it with lightlike (what we have done so far) and timelike (which is the other derivation of the light clock though experiment, which Iike, although it seems not be in fashion nowadays), I would not know how to express it with spacelike, since we would present an impossible scenario of something traveling FTL... 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.

Ultimately, this is a question of semantics: I think that you can perfectly say that the ST interval is what remains invariant under the "hyperbolic rotation" of a vector, no matter what kind of vector we are talking about and regardless whether the effect of the rotation is change of direction or dilation. Take the example of a right triangle where the height is overlapping with the Y axis and you reflect it precisely over the Y axis. You would say that the figure as a whole has been "reflected", even if a particular side (the height) has remained untouched (not even dilated in this case) in the course of the reflection.

 

[Sagittarius A-Star]

As to 3), what was puzzling me was that in the ST diagram the null vector remains intact, how could it be dilated by the eigenvalue = the Doppler factor?

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, that is correct. I think a Minkowski diagram is incomplete, if the scales on each axis are omitted, as it is unfortunately often done, for example in the SR book of Leonard Susskind. It is otherwise a great book.

You will find the Doppler factor in equations (1) and (2) of the following LT derivation of Macdonald, because the $T+X$ and $T-X$ are light-cone coordinates. This LT derivation has similarities to the discussed one of Einstein.

Source ("World’s Fastest Derivation of the Lorentz Transformation"):
http://www.faculty.luther.edu/~macdonal/LorentzT/LorentzT.html

The reason, why that is the Doppler factor is, that the four frequency $\mathbf N$ of a photon is a null-vector and therefore transforms in the same way. Reason, why the four frequency of a photon is a null-vector: The four-momentum of a photon is also a null-vector and, for movement in x direction, can be written as:
$\mathbf P = \frac{E}{c^2}(c, c, 0, 0) = \frac{h \nu}{c^2}(c, c, 0, 0) = \frac{h}{c}(\nu, \nu, 0, 0) = \frac{h}{c}\mathbf N$.

 

[Sagittarius A-Star]

As already stated, if you want to get rid of negative time you can always stipulate that what goes into the ST interval is the absolute value of cT or either square both sides but immediately squareroot them, which leads to this other (wrong) interval:
\sqrt {{{(c\Delta t)}^2}} - \sqrt {(\Delta {x^2} + \Delta {y^2} + \Delta {z^2})} = \sqrt {{{(c\Delta t')}^2}} - \sqrt {(\Delta x{'^2} + \Delta y{'^2} + \Delta z{'^2})}

The question is, if you want to write an equation, that is only valid for event pairs with a spacetime interval of zero. For arbitrary event pairs, you need the quadratic form.

I think that you can perfectly say that the ST interval is what remains invariant under the "hyperbolic rotation" of a vector, no matter what kind of vector we are talking about and regardless whether the effect of the rotation is change of direction or dilation.

In the following video, a derivation of the LT via "hyperbolic rotation" is shown, including the analogy to Euclidean rotation.
https://www.physicsforums.com/threads/videos-andrzej-dragan-course-on-relativity.1011307/

 

[Dale]

the Pythagorean combination

The Pythagorean theorem is $a^2+b^2=c^2$ to begin with. So the starting point for this proof is $\Delta x^2+\Delta y^2+\Delta z^2=c^2\Delta t^2$. Your argument makes 0 sense. You are making a huge deal of your starting point and your starting point is incorrect. If you weren’t making a big deal of the starting point that wouldn’t matter. But it is the crux of your argument and it is wrong. The Pythagorean theorem is the squared statement, so if you insist that the starting point is important, then that is the one to use.

to avoid this ugly expression you must square both sides, but the subtlety that I favor is just admitting that you do it for a reason that goes beyond the pure algebraic trick

The reason doesn’t matter. We are doing all of the algebraic steps in any proof because we already know that they get us where we want to go in the end. We are allowed to do the algebraic "tricks" as long as we perform them correctly, regardless of our reasons for doing them. This restriction you are asserting is irrelevant.

But the equation would also remain true under the algebraic manipulation of raising both sides to the 11th power, which would lead to a ST that is false.

No, it wouldn’t be false. If you do valid algebra then you will have the same set of events as a solution. The algebraic operations are designed expressly to leave those unchanged. It won’t make it false, just ugly.

what is valid for the algebraic purpose (solving for unknowns) may not be valid to guess what remains invariant under transformations.

This is simply not true. The thing that is invariant is the set of events that forms the light cone. The algebra doesn’t change that. I don’t understand how you can have passed algebra class and believe this point you are making.

In short, I disagree with both your starting point and your reasoning from the starting point

 

[Dale]

The equation remains true under the algebraic manipulation of squaring both sides. But the equation would also remain true under the algebraic manipulation of raising both sides to the 11th power, which would lead to a ST that is false.

To show that this is wrong is rather easy. As I said above, the whole point of algebra is to leave the set of solutions unchanged.

Here is a set of events that satisfies the invariance of c (in units where c=1) based on the Pythagorean theorem. It is a standard light cone with the apex at the origin.

If I take the square root of both sides to get the corrected version of your expression then I get the exact same light cone.

I wasn't sure if you meant the 11th power of your square root formula or the 11th power of the original interval but it doesn't really matter. Here is the 11th power of the square root formula. It produces the exact same light cone.

The algebraic operations are valid. The motivation is irrelevant. Taking the 11th power does not produce a false equation, just an ugly one.

Your argument is flat out wrong from start to finish.

 

[Saw]

@Dale, thanks for your development, although I find it hard to follow, I am not sure if we are talking about the same thing.

What I meant is the following: instead of squaring both sides, another valid algebraic operation is raising both sides to the 11th power and, unless I messed up somewhere, it seems that this way we get to a ST interval that does not work:

 

[Saw]

The question is, if you want to write an equation, that is only valid for event pairs with a spacetime interval of zero. For arbitrary event pairs, you need the quadratic form.

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. I am thinking that this is hard to do on the basis of a null vector, because this is after all the "simple situation": no matter if you have 3 spatial dimensions, it is still a simple situation because here all you need to know is whether light can reach the second event in time and it does simply because $c\Delta t = \Delta x.$ It is with a timelike interval that you need some additional information, the number of ticks that lapse between two events in a clock present at both, so this is a more complete and hence a better scenario to get the "full" ST interval, which can then be generalized to the simpler case (null vector) or the remaining case (spacelike vector), which is harder to take as reference for the derivation.

 

[Dale]

unless I messed up somewhere

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.

 

[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.