What assumptions underly the Lorentz transformation?

[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.

 

[vanhees71]

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.

That's a subtle issue. On the one hand you are right: The particular value of $c$ is just a convention defining the unit of lengths in terms of the unit of time in any given system of units. In the SI they make unit of time (the second, s) the most fundamental unit, because time measurements are among the most precise measurements possible. It's still the hyperfine transition of Cs-133 used to define the second, but that may change in not too far future since there are more accurate realizations possible (either an atomic clock in the visible-light range or the nuclear Th clock). Then the unit of length (the meter, m) is defined by setting the limit speed of relativity to a certain value. Since with very high accuracy the photon is massless the realization of $c$ in measurements is simply the speed of electromagnetic waves in a vacuum.

On the other hand all this of course hinges in the existence of the limiting speed and the validity of the relativistic spacetime model. If the world were Galilean, there'd be no fundamental natural constant with the dimensions of a velocity and time and lengths units would have to be defined independently of each other with some "normals" (as it was before 1983, when the second was defined as today and the meter independently by some wavelength of a certain Kr-86 line).

 

[Orodruin]

Another interesting thing that is worth pointing out regarding the Lorentz transformation:

It is a symmetry of the wave equation and can be derived from just looking for symmetries of the wave Lagrangian. Not just the wave equation for light but for any wave equation with $c$ being the wave speed. Of course, this in itself does not say anything about the Lorentz transformation being fundamentally related to the spacetime structure.

 

[Meir Achuz]

That is conservation of (angular) momentum.
Tell that to gravity.

 

[robphy]

Then the unit of length (the meter, m) is defined by setting the limit speed of relativity to a certain value.
…

On the other hand all this of course hinges in the existence of the limiting speed and the validity of the relativistic spacetime model. If the world were Galilean, there'd be no fundamental natural constant with the dimensions of a velocity and time and lengths units would have to be defined independently of each other with some "normals" (as it was before 1983, when the second was defined as today and the meter independently by some wavelength of a certain Kr-86 line).

From a geometric viewpoint, the Galilean structures lacking a fundamental natural constant of velocity
Is similar to
Euclidean space not having a fundamental length constant or length-scale (think “radius”… associated with curvature) [among the classic Riemannian-signature nonEuclidean geometries] as one has for an elliptic/spherical space or a hyperbolic space.
Similarly, the Minkowski and Galilean structures also lack a fundamental length-scale, unlike their curved analogies.

 

[Sagittarius A-Star]

"c" is the maximum velocity, as in case $(iii)$.
...
And case $(ii)$ is when $c\to\infty$, so that $\alpha\to 0$, which is what I am saying.

In case $(iii)$, $\alpha$, and therefore also the maximum velocity $c$ are constants of nature, as described in https://www.researchgate.net/publication/252687984_One_more_derivation_of_the_Lorentz_transformation when discussing case $(iii)$ (see page 276):

Clearly, as in case (i), the numerical value of α depends on the initial choice of units for space and time coordinates, so that, physically, there is but one situation here.

As example, take the SI unit system, according to which $c=3 \cdot 10^8 m/s$. It makes no sense to write:
$\require{color}\lim_{3 \cdot 10^8 \frac{m}{s} \rightarrow \infty} {\frac{1}{\sqrt{1-v^2/c^2}} (t-\color{red}\frac{vx}{c^2}\color{black})}$

If you would instead use a physical unit system that yields a double value for $c$ by changing the definition of "1 meter", then the denominator of the red fraction would quadruple, but also the numerator would quadruple. The transformation would stay the same.

 

[otennert]

In case $(iii)$, $\alpha$, and therefore also the maximum velocity $c$ are constants of nature, as described in https://www.researchgate.net/publication/252687984_One_more_derivation_of_the_Lorentz_transformation when discussing case $(iii)$ (see page 276):

As example, take the SI unit system, according to which $c=3 \cdot 10^8 m/s$. It makes no sense to write:
$\require{color}\lim_{3 \cdot 10^8 \frac{m}{s} \rightarrow \infty} {\frac{1}{\sqrt{1-v^2/c^2}} (t-\color{red}\frac{vx}{c^2}\color{black})}$

This is of course does not make any sense at all, as it is not the way you consider limits in mathematics. The point of the paper on this is that it does not matter of all what value $c$ has from a conceptual point of view. It might be anything. It is a natural constant, obviously, but again it does not need to be the speed of light.

The fact that it is the speed of light is an independent, next step, but outside the paper -- and as it turns out, experimental evidence supports this.

 

[Sagittarius A-Star]

The fact that it is the speed of light is an independent, next step, but outside the paper -- and as it turns out, experimental evidence supports this.

Experimental evidence supports, that the defined constant "speed of light" applies also to the maximum speed in physics, as described in SR. As argued in #65, a re-definition of that constant would not change the transformation at all.

The speed of light in vacuum cannot be measured, because it is only a conversion factor in the unit system.

 

[otennert]

For the logic of the paper, and for the original question "what are the basic assumptions underlying LTs", it is completely irrelevant what the actual value of $c$ is, which unit system is in use, how the metre is defined, or whether $c$ is actually the speed of light or not. Switching from an old definition of the metre to a new one, or from $m/s$ to $miles/hour$ or to so-called "natural units" where simply $c=1$, does not make any difference at all to that logical argument.

The basic 2 options are: is the velocity parameter $c$, which is a natural constant, finite or infinite (in the paper's nomenclature: is $\alpha>0$ or is $\alpha=0$)? That's cases $(ii,iii)$ in the paper. And it is a mathematical triviality to get from (45) to (43) by taking $\lim_{c\to \infty}$.

Everything else is to be separated from that or has to be decided otherwise, and of course has already been done so.

Don't get me wrong: I don't disagree with your physical statements, I am just not fine with your logical reasoning. But maybe there is a misunderstanding here?

 

[PeroK]

The basic 2 options are: is the velocity parameter $c$, which is a natural constant, finite or infinite (in the paper's nomenclature: is $\alpha>0$ or is $\alpha=0$)? That's cases $(ii,iii)$ in the paper. And it is a mathematical triviality to get from (45) to (43) by taking $\lim_{c\to \infty}$.

There definitely was a thread about this recently and, IMO, Newtonian space and time is not geometrically the limit of Minkowski spacetime as $c \to \infty$. In the sense that the geometries do not converge. As any good maths student will tell you, plugging in $c = \infty$ is not taking a limit.

If we say that there is no invariant speed, then we have Newtonian physics. That is both simple and unimpeachable mathematically.

 

[otennert]

There definitely was a thread about this recently and, IMO, Newtonian space and time is not geometrically the limit of Minkowski spacetime as $c \to \infty$. In the sense that the geometries do not converge. As any good maths student will tell you, plugging in $c = \infty$ is not taking a limit.

I remember that thread, and I responded there that this is correct, as Newton/Galilei spacetime has no causal structure, but also is outside the scope both of the paper and the initial question under consideration.

In the paper under discussion, to get from eq. (45) to (43), is by letting $c\to\infty$, that's basic math and well-defined. That's all.

Of course: when the line of reasoning of the paper is finished, the next logical steps are to investigate:
- Is a maximum velocity (which is then a natural constant) given in nature? (--> answer: yes)
- Could it be identified with the speed of light? (--> answer: yes)
- What implications does the fact that there is a finite maximum speed have? (e.g. causal structure/timelike/spacelike/null curves etc.)

Also correct, but again outside the scope of both the paper and the initial question: in order to go from the Lorentz group to the Galilei group, simply taking $c\to\infty$ is not sufficiently well-defined at all, as the structure of both the group and its generators change, as is well-known. The mathematical procedure to do so is called a group contraction and has in this specific case been demonstrated by Inönu and Wigner 1953.