What assumptions underly the Lorentz transformation?

[Meir Achuz]

Step 5 is the transformation law for velocities, and step 6 uses the transformation law and the invariance of c to identify beta as v/c.

 

[Sagittarius A-Star]

Step 5 is the transformation law for velocities, and step 6 uses the transformation law and the invariance of c to identify beta as v/c.

Step 5 is not yet the relativistic transformation law for velocities, although it looks so. The text in step 5 says "We can derive a transformation law for velocities ... $u' = ... = \frac{u-v}{(1-\beta u/c)} \ \ \ \ \text{(14.10)}$". Up to that point, he has for example not excluded the possibility, that $\beta$ (which he had formally introduced in the transformation formula for time, in step 3) might be zero and he gets the Galileo transformation.

First in step 6, $\beta = v/c$ is derived by demanding, that, if $u$ is the speed of light $c$, then $u'$ must be also $c$.

 

[pervect]

One point to note is that it is circular reasoning to say that the transformation of velocities formula is derived from the Lorentz transformation, because the velocity transformation formula is often a step in the derivation of the Lorentz transformation.

I'm not familiar with such a derivation. The derivation I favor requires the following three assumptions. One: if we have two observers in relative motion at the same location, light emitted by one will experience a doppler shift factor k, when received by the other, the ratio k being the ratio between the interval of emission and the interval of reception. Secondly, we need the assumption that the speed of light is constant for all inertial observers. Thirdly, to handle two observers in relative motion who are not at the same location, we add another assumption, which states that light emitted from an inertial observer and received by another inertial observer at rest with respect to the first at a different location will experience no doppler shift, that the k factor will be unity.

This is basically Bondi's argument, though Bondi did not, to my knowledge, codify the third and last assumption. I feel the third assumption is a useful approach, though, and shores up some things I don't like about Bondi's approach. I should add, I've never seen the third assuption in a published paper, but I still like it.

We don't need any arguments about velocity addition to use the above assumptions, which are simple and straightforards, and only require high school algebra to analyze. The analysis can be found in Bondi's book, "Relativity and Common Sense", and basically uses a simple radar setup to determine the relationship between the factor k and relative velocity. Looking up "k-calculus" on the internet will probably also work for those who don't want to track down Bondi's book.

 

[hartmutneff]

Everything follows from the invariance of Maxwell's wave equation. The velocity of waves is not derived from any kind of dx/dt. The velocity of a wave is more like a geometric speed. If you take a mass-spring system, with overall mass M and overall spring constant K, where the length of the system is L, the velocity of a wave is given by (KL^2/M). It is kind of a different type of velocity, as I said, not dx/dt.
Now, if you put the mass-spring system on a train, then it goes at the wrong speed, for an observer on a platform, since, classically, the velocity is KL^2/M+v, where v is the speed of the train. Therefore, the wave equation is not valid for a spring on a train.

For a mass-spring system, you can sweep this under the rug, as you can see the masses and the springs on the train. One can say, there is a preferred reference frame for a spring-mass system, the one where the springs and the masses rest.

For light, and Maxwell's wave equation, the sweeping under the rug doesn't work, as light doesn't require a medium. There is no preferred reference frame in which the medium rests, as there is no medium.

Then you have two options, you give up one the wave equation, or you fix the Galilean invariance, such that the speed of light is always the same, for every observer. If you assume a general space and time transformation, which leaves the wave equation invariant, you end up with the Lorentz transformation. This is pretty straightforward.

Once you got the Lorentz transformation, the rest, like relativistic addition of velocities, follows quite easily.

By the way, if you assumed for a minute, that light was a mass spring system, then, with the Lorentz contraction of L, the relativistic mass follows directly from the constance of KL^2/M.

 

[Sagittarius A-Star]

Can you show us this step?

I found a derivation of the LT from "relativistic velocity addition" and time dilation.

Relativistic velocity addition law derived from a machine gun analogy and time dilation only
...
1. Deriving the relativistic velocity addition law without using the Lorentz transformations
...
2. Lorentz transformations from the relativistic addition law of velocities.

Source (PDF):
https://arxiv.org/pdf/physics/0703157
via (Abstract):
https://arxiv.org/abs/physics/0703157

 

[Sagittarius A-Star]

$$u=\frac{v+w}{1+\frac{vw}{c^{2}}}$$
But which assumption exactly underlies this so that you get exactly this formula and not any other formula with approximately the same properties?

One point to note is that it is circular reasoning to say that the transformation of velocities formula is derived from the Lorentz transformation, because the velocity transformation formula is often a step in the derivation of the Lorentz transformation.

Both, the LT and the relativistic velocity composition can be derived together from the two postulates of SR and assumed linearity. I write a variant of the derivations in "Classical Electromagnetism" by Jerrold Franklin. I think, then the (former revolutionary) transformation formula for time is derived in a more intuitive way. But I like his approach to go via the velocity composition formula to derive the LT.

Definition of the constant velocity $v$:
$x' = 0 \Rightarrow x-vt=0\ \ \ \ \ \$(1)

With assumed linearity follows for the only possible transformation, that meets requirement (1):
$\require{color} x' = \color{red}A(x-vt)\color{black}\ \ \ \ \ \$(2)

With SR postulate 1 (the laws of physics are the same in all inertial reference frames) follows, that the inverse transformation must have the same form, if the sign of $v$ is reversed:
$\require{color}x = A(\color{red}x'\color{black}+vt')\ \ \ \ \ \$(3)

Eliminating $x'$, by plugging the right-hand side of equation (2) for $\require{color} \color{red}x'\color{black}$ into (3), and resolving (3) for $t'$ yields the transformation formula for time:
$t' = A(t-x\frac{1-\frac{1}{A^2}}{v})\ \ \ \ \ \$(4)

The velocity composition formula follows by calculating $dx'/dt'$ from equations (2) and (4), with $u=dx/dt$:

$u' = dx'/dt' = \frac{A(dx-vdt)}{A(dt-dx\frac{1-1/A^2}{v})} = \frac{u-v}{1-u(1-1/A^2)/v}\ \ \ \ \ \$(5)

With SR postulate 2 (the vacuum speed of light is the same in all inertial reference frames) follows:
$c = \frac{c-v}{1-c(1-1/A^2)/v}\ \ \ \ \ \$(6)
$\Rightarrow$
$A= \frac{1}{\sqrt{1-v^2/c^2}} := \gamma\ \ \ \ \ \$(7)

Plugging the the right-hand side of (7) for $A$ into (2) and (4) yields the LT.
$$x' = \frac{1}{\sqrt{1-v^2/c^2}} (x-vt)$$
$$t' = \frac{1}{\sqrt{1-v^2/c^2}} (t-vx/c^2)$$
Plugging the the right-hand side of (7) for $A$ into (5) yields the relativistic velocity composition formula.
$$u' = \frac{u-v}{1-uv/c^2}$$
All assumptions are marked in blue.

 

[vanhees71]

It's also important to remark that the Newtonian limit follows from the other possible solution, $A=1$, leading to $t=t'$.

One can go further in assuming nothing else than Postulate 1 (which is Newton's principle of inertia), homogeneity in space and time and Euclidicity of space for any inertial observer to derive the "reciprocity assumption", i.e., that if system $\Sigma'$ moves wrt. to $\Sigma$ with velocity $\vec{v}$, then $\Sigma$ moves wrt. $\Sigma'$ with velocity $-\vec{v}$ and that there are only two possibilities, i.e., Galilei-Newton or Einstein-Minkowski spacetime:

V. Berzi and V. Gorini, Reciprocity Principle and the Lorentz
Transformations, Jour. Math. Phys. 10, 1518 (1969),
https://doi.org/10.1063/1.1665000

 

[Ibix]

V. Berzi and V. Gorini, Reciprocity Principle and the Lorentz
Transformations, Jour. Math. Phys. 10, 1518 (1969),
https://doi.org/10.1063/1.1665000

Seems to be paywalled (edit: or possibly just too old). There's a "one postulate" derivation by Pal available at arxiv which I imagine is similar.

 

[vanhees71]

That's almost the same derivation, but the old paper proves the reciprocity relation from the symmetry assumptions rather than just postulating if.

 

[otennert]

My 10cent: one of the most convincing "derivations" of the Lorentz transformation is by Jean-Marc L´evy-Leblond 1976: https://www.researchgate.net/publication/252687984_One_more_derivation_of_the_Lorentz_transformation

based on the 4 assumptions:

1.) Homogeneity of spacetime and linearity of transformation
2.) Isotropy of space
3.) Group law (Lorentz transformation form a group)
4.) Causality

Identification of the velocity of light with the resulting maximum velocity is then the next, but logically independent step.

The reason why I find this derivation most convincing is that some maximum velocity (as a given parameter) is the result of very general assumptions which sound much more common-sense than putting constancy of the velocity of light as a prerequisite, which -- without prior knowledge -- is much more counter-intuitive.

 

[Sagittarius A-Star]

The reason why I find this derivation most convincing is that some maximum velocity (as a given parameter) is the result of very general assumptions

This derivation does not rule-out the Galilean case with no maximum velocity. An experiment to rule-out the Galilean case is still required.

CONCLUSION
Our four general hypotheses thus suffice to single out the Lorentz transformations and their degenerate Galilean limit as the only possible inertial transformations.

Source:
https://www.researchgate.net/publication/252687984_One_more_derivation_of_the_Lorentz_transformation

 

[Orodruin]

An experiment to rule-out the Galilean case is still required.

Experiments are required to verify all assumptions and predictions. Regardless of the path taken to the Lorentz transformations.

As I said earlier in this thread, most texts and courses follow the historical path. It is questionable if this is the best option. Lorentz transformations really aren’t anything but the Minkowski space equivalent of rotations in Euclidean space. There is no more magic to it than that. Simply assuming Minkowski space and considering transformations between orthonormal frames will do.

 

[otennert]

This derivation does not rule-out the Galilean case with no maximum velocity. An experiment to rule-out the Galilean case is still required.Source:
https://www.researchgate.net/publication/252687984_One_more_derivation_of_the_Lorentz_transformation

Yes, correct: the maximum velocity may go to infinity, which is the Galilean case. Nevertheless this derivation puts Lorentz and Galilei transformations on an equal footing, from a logical point of view. What is realized in nature is then decided by experiment.

In the end, this is not a question of physics, but of a satisfactory axiomatics. And to put the constancy of the velocity of light at the beginning, to me is not a good starting point for axiomatizing special relativity (if one may want to do so at all).

 

[Orodruin]

Yes, correct: the maximum velocity may go to infinity, which is the Galilean case.

Statements like this require an enormous grain of salt. I believe we had a discussion regarding this recently but cannot seem to find it. Someone else may recall it.

In the end, this is not a question of physics, but of a satisfactory axiomatics. And to put the constancy of the velocity of light at the beginning, to me is not a good starting point for axiomatizing special relativity (if one may want to do so at all).

If you want satisfying axiomatics, just postulate Minkowski space.

 

[otennert]

Statements like this require an enormous grain of salt. I believe we had a discussion regarding this recently but cannot seem to find it. Someone else may recall it.If you want satisfying axiomatics, just postulate Minkowski space.

Because that's so immediately obvious without prior knowledge?

And as I have hinted at: I am not propagating to axiomatize relativity at all, neither do I think that axiomatizing physics makes sense. But postulating Minkowski space and saying: these are the symmetry transformations that leave $ds^2$ invariant surely does not give any further insight into the structure of relativity at all.

With hindsight of course, we are all the wiser.

 

[Orodruin]

Because that's so immediately obvious without prior knowledge?

As I said, the historical path is somewhat cumbersome and has many pitfalls. I’d even go as far as suggesting it is largely obsolete in terms of understanding relativity. It does not need to be obvious at first look, many if not most things we teach in physics are not. What is necessary is to work out the implications and an understanding of the theory. This is simpler to do if you first learn to handle Minkowski space rather than bogging yourself down in Lorentz transformations, time dilation, length contraction, and who knows how many ”paradoxes”.

 

[Orodruin]

But postulating Minkowski space and saying: these are the symmetry transformations that leave ds2 invariant surely does not give any further insight into the structure of relativity at all.

That is simply wrong. Understanding the geometry of Minkowski space is understanding the structure of (special) relativity.

With hindsight of course, we are all the wiser.

This is the point. We have hindsight so we do not need to make students take the cumbersome historical path.

 

[strangerep]

Statements like this require an enormous grain of salt. I believe we had a discussion regarding this recently but cannot seem to find it. Someone else may recall it.

Well, there's this one, starting from my post #8.

@otennert: I've written lengthier posts over the years in threads about foundations of SR. If you do an advanced search of PF looking for "fractional linear" (and authored by me) you'll find a whole bunch.

 

[otennert]

That is simply wrong. Understanding the geometry of Minkowski space is understanding the structure of (special) relativity.

I do agree. Because the mathematics of Minkowski space, the symmetry transformations on it and its causal structure are the result of a thought process that has happened quite a while ago, and has been achieved by past generations of physicists. And I am not questioning that for teaching relativity to students in a course today it makes more sense to not follow the historical route, which I have never suggested. If you want to get a grip on relativity, you should learn the mathematics of Minkowski space.

What I am saying though is, when you go one step back for a moment and try to understand where the need, or possibility, of Lorentz transformations comes from -- and this has been the actual question at the beginning of this thread -- it gives you some more insight into the nature of relativity if you understand what minimum basic assumptions you need in order to derive these, and what changes you need to go from the non-relativistic domain to the relativistic domain. And here the little paper by Levy-Leblond (which is not following the historical route at all) gives one additional, interesting view on why Lorentz transformations emerge quite naturally from basic assumptions, that's all.

 

[otennert]

Well, there's this one, starting from my post #8.

@otennert: I've written lengthier posts over the years in threads about foundations of SR. If you do an advanced search of PF looking for "fractional linear" (and authored by me) you'll find a whole bunch.

Thank you. I will have a look at your postings.

 

[vanhees71]

Experiments are required to verify all assumptions and predictions. Regardless of the path taken to the Lorentz transformations.

As I said earlier in this thread, most texts and courses follow the historical path. It is questionable if this is the best option. Lorentz transformations really aren’t anything but the Minkowski space equivalent of rotations in Euclidean space. There is no more magic to it than that. Simply assuming Minkowski space and considering transformations between orthonormal frames will do.

Yes, I also think that this is the most elegant approach, but before I go indeed more or less through Einstein's original derivation, because it elucidates the physical way how to realize the independence of the speed of light of the velocity of the light source wrt. an inertial frame via the clock-synchronization convention. This emphasizes the necessity of the local point of view, i.e., that you need a set of (in practice of course only fictitious) standard clocks at rest in an inertial frame at each spatial point and synchronize them with light signals. That's of course a much less elegant approach than the purely mathematical one using the mathematical structure of Minkowski space as an affine pseudo-Euclidean manifold with a fundamental form of signature (1,3) (or equivalently (3,1)), but with this approach it's not clear, why this is the right space time structure!

 

[vanhees71]

This derivation does not rule-out the Galilean case with no maximum velocity. An experiment to rule-out the Galilean case is still required.Source:
https://www.researchgate.net/publication/252687984_One_more_derivation_of_the_Lorentz_transformation

That's the same derivation as the one I quoted above (with some differences in some details). I find this the most convincing derivation, because it uses only the symmetry assumptions about spacetime with the special principle of relativity and derives the reciprocity relation from them. It then follows that there are only Galilei-Newton or Einstein-Minkowski spacetime left, and of course only observation can decide between these two possibilities, and it's clear that Einstein-Minkowski is the correct description (with regard to gravitation and GR you have to make the corresponding Poincare symmetry local to get the yet most comprehensive spacetime model).

 

[Sagittarius A-Star]

Lorentz transformations really aren’t anything but the Minkowski space equivalent of rotations in Euclidean space. There is no more magic to it than that. Simply assuming Minkowski space and considering transformations between orthonormal frames will do.

Deriving the LT as hyperbolic rotation of the coordinate system is a good and "fast" approach. However, some textbooks do not simply assume Minkowski spacetime, but derive it from the two SR postulates + assuming linearity (for example Andrzej Dragan "Unusually special relativity"). Maybe, because the 2 postulates are closer to experiment.

 

[Sagittarius A-Star]

the maximum velocity may go to infinity, which is the Galilean case.

I think this would be correct, if you didn't speak about "maximum velocity", but only about the mathematical quantity "c" in the transformation equations. The value of the "maximum velocity" in SR is an artifact of the physical unit system.

GT of x to x':
https://www.wolframalpha.com/input?i=Limit[(x+-+v+t)/Sqrt[1+-+v^2/c^2],+c+->+Infinity]

GT of t to t':
https://www.wolframalpha.com/input?i=Limit[(t+-+(v+x)/c^2)/Sqrt[1+-+v^2/c^2],+c+->+Infinity]

Nevertheless this derivation puts Lorentz and Galilei transformations on an equal footing, from a logical point of view. What is realized in nature is then decided by experiment.
... And to put the constancy of the velocity of light at the beginning, to me is not a good starting point for axiomatizing special relativity (if one may want to do so at all).

Unlike many other derivations of LT, the begin of the derivation I wrote in posting #36, until including equation (5), puts also Lorentz and Galilei transformations on an equal footing.

Edit:
I must admit, that adding the group law for a transformation yields
$A= \frac{1}{\sqrt{1-v^2/c^2}} := \gamma$ and the LT, after GT and an unphysical solution were excluded, without involving SR postulate 2. That is an advantage over the derivation in my posting #36.

 

[otennert]

I think this would be correct, if you didn't speak about "maximum velocity", but only about the mathematical quantity "c" in the transformation equations. The value of the "maximum velocity" in SR is an artifact of the physical unit system.

GT of x to x':
https://www.wolframalpha.com/input?i=Limit[(x+-+v+t)/Sqrt[1+-+v^2/c^2],+c+->+Infinity]

GT of t to t':
https://www.wolframalpha.com/input?i=Limit[(t+-+(v+x)/c^2)/Sqrt[1+-+v^2/c^2],+c+->+Infinity]

What do you mean? Using the letter "c" instead of speaking of a "maximum velocity" has no more physical content -- after all "c" in the paper's derivation *is* the maximum velocity possible, whatever the value. Identification of "c" with the velocity of light is an independent step, as I mentioned. And infinity is infinity, in all sensible unit systems.

 

[Sagittarius A-Star]

after all "c" in the paper's derivation *is* the maximum velocity possible, whatever the value.

In https://www.researchgate.net/publication/252687984_One_more_derivation_of_the_Lorentz_transformation, "c" is only defined for the case $(iii) \alpha >0$ (page 275), which does not belong to the GT. The GT has no maximum velocity. But it is still remarkable, that the limit of LT is GT, as the mathematical quantity "c" approaches infinity.

And infinity is infinity, in all sensible unit systems.

No, see:

The meter is defined as the length of the path traveled by light in a vacuum during a time interval of 1/299,792,458 of a second.

Source:
https://education.nationalgeographic.org/resource/meter-defined

 

[Orodruin]

What do you mean? Using the letter "c" instead of speaking of a "maximum velocity" has no more physical content -- after all "c" in the paper's derivation *is* the maximum velocity possible, whatever the value. Identification of "c" with the velocity of light is an independent step, as I mentioned. And infinity is infinity, in all sensible unit systems.

c is technically nothing but a unit conversion factor. You have to be careful when you talk about taking limits. Technically you recover the Galilean transformations if you let c go to infinity while keeping everything else constant. However, this limit does not change Minkowski space to Galilean spacetime because regardless of the value of c, the Minkowski geometry is what it is and does not smoothly change into the geometry of Galilean spacetime in any kind of limit.

 

[robphy]

In the Lorentz transformation in standard (t,x)-variables [say, in conventional SI units] there are actually two types of speeds, “c” as the conversion constant 3e8m/s [ so t and x/c have the same units] and “c” the maximum signal speed (often seen in $v/c_{max}$). The conversion constant c doesn’t change in the looking for a Galilean limit… it’s the $c_{max}$ that tends to infinity in that limit.

 

[otennert]

c is technically nothing but a unit conversion factor. You have to be careful when you talk about taking limits. Technically you recover the Galilean transformations if you let c go to infinity while keeping everything else constant. However, this limit does not change Minkowski space to Galilean spacetime because regardless of the value of c, the Minkowski geometry is what it is and does not smoothly change into the geometry of Galilean spacetime in any kind of limit.

Correct. "c" is a parameter of the dimension of "velocity", nothing else. Which is why a priori identification with the speed of light is not justified.But it also constitutes the maximum velocity that can be reached by LTs.

Also you are correct that on the other hand, there is some subtlety in going from the Lorentz group to the Galilei group, as the group structure changes, and obviously Minkowski spacetime has a causal structure whereas Galilei spacetime hasn't. But as I see it this is beyond the scope of this paper and the initial question under consideration.

 

[otennert]

In https://www.researchgate.net/publication/252687984_One_more_derivation_of_the_Lorentz_transformation, "c" is only defined for the case $(iii) \alpha >0$ (page 275), which does not belong to the GT. The GT has no maximum velocity. But it is still remarkable, that the limit of LT is GT, as the mathematical quantity "c" approaches infinity.No, see:

Source:
https://education.nationalgeographic.org/resource/meter-defined

"c" is the maximum velocity, as in case $(iii)$. There is then no transformation that can map a velocity $v<c$ to a velocity $v\geq c$. And case $(ii)$ is when $c\to\infty$, so that $\alpha\to 0$, which is what I am saying. Case $(i)$ is subsequently discarded because it violates causality. Taking the limit at this point for formula (45) to get (43) is mathematically trivial and well-defined. It is actually not remarkable at all at this point.