A What assumptions underly the Lorentz transformation?

[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.

 

[PeroK]

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.

That mathematics is flaky at best. For every value of $c$ we have the same Minkowski geometry. The limit cannot be, therefore, what you get by plugging in $c = \infty$.

 

[otennert]

That mathematics is flaky at best. For every value of $c$ we have the same Minkowski geometry. The limit cannot be, therefore, what you get by plugging in $c = \infty$.

But this is not what I am saying, and I do agree with your statement on geometry.

 

[PeroK]

What you and the paper are saying:

For all $v$, the transformation $L(v) \to G(v)$ as $c \to \infty$.

But, what you need is:

As $c \to \infty$: for all $v < c$ we have $L(v) \to G(v)$.

In other words, it's nonsensical to increase $c$, as the maximum speed, but bound $v$ (all allowable speeds) by some initial $c_0$.

To make your argument valid you would need a fixed bound on $v$ while $c$ increases without limit.

 

[otennert]

What you and the paper are saying:

For all $v$, the transformation $L(v) \to G(v)$ as $c \to \infty$.

Yes. It is basic math to go from (45) to (43) by taking the limit $c\to\infty$. No physical context needed for this at all, everything is well-defined.

But, what you need is:

As $c \to \infty$: for all $v < c$ we have $L(v) \to G(v)$.

In other words, it's nonsensical to increase $c$, as the maximum speed, but bound $v$ (all allowable speeds) by some initial $c_0$.

To make your argument valid you would need a fixed bound on $v$ while $c$ increases without limit.

This I do not understand. What would the fixed bound $c_0$ be necessary for? In Galilean spacetime, there is no such fixed bound. Any object can have any relative velocity to any other object, and also simultaneity is absolute, as can be seen in (43).

Of course, we all know, that this is not realized in nature. Nonetheless, the same basic and very general assumptions that lead to LT also lead to GT as an option. This is the whole point of the argument.

I might miss to get your point on this, but if you are referring to the fact that of course the limit $c\to\infty$ is somehow too simple to convert a Lorentzian spacetime with all its causal structure into a Galilean one without, I do totally agree! As I stated before, the mathematical procedure is called a group contraction which implies i.a. a rescaling of the group generators before a limiting process can then be made.

The different geometries of Newton/Galilei vs Lorentz/Minkowski are rooted exactly where the different algebraic properties of the Galilei group and algebra vs Lorentz group and algebra are. But from a logical point of view, this is a consequence of the fact that there is a limiting velocity $c$ in existence instead of none as in Galilei, and not another a priori assumption.

 

[Sagittarius A-Star]

But maybe there is a misunderstanding here?

That's likely. In the paper they avoid misunderstanding by mentioning in case $(ii)$ only $\alpha = 0$ and not $c$.

 

[vanhees71]

The point is that you discuss two different (related) aspects: (a) the Galilei group as a "deformation" of the Poincare group and (b) the spacetime geometries. The former in a well-defined mathematical sense can be seen as taking "$c \rightarrow \infty$", while that's not the case for the latter.

 

[PeroK]

@otennert as in the previous thread your conclusion that as $c \to \infty$ all gamma factors tend to $1$ is false. The flaw in your basic maths is misplacing the universal quantifier for $v$ independent of $c$.

For example, try taking the limit with $v = \frac c 2$. Which is perfectly valid.

 

[vanhees71]

Of course the deformation from the Poincare to the Galilei group has to be taken at fixed group parameter, $v$, not at fixed $\beta=v/c$.

 

[otennert]

@otennert as in the previous thread your conclusion that as $c \to \infty$ all gamma factors tend to $1$ is false. The flaw in your basic maths is misplacing the universal quantifier for $v$ independent of $c$.

For example, try taking the limit with $v = \frac c 2$. Which is perfectly valid.

You have lost me completely now. Are we both looking at the same formula?

$\lim_{c\to\infty}\frac{x-vt}{(1-v^2/c^2)^{1/2}}=x-vt$

Can you please point me to my basic math error here?

 

[PeroK]

You have lost me completely now. Are we both looking at the same basic formula?

$\lim_{c\to\infty}\frac{x-vt}{(1-v^2/c^2)^{1/2}}=x-vt$

Can you please point me to my basic math error here?

You're missing the existential and universal quantifiers. That's a sort of pointwise convergence that tells you nothing about the overall geometry.

It's similar to the difference between pointwise and uniform continuity.

 

[otennert]

I keep saying I am not talking about overall geometry at all, neither does the paper. The whole discussion of this thread at the beginning and the paper I have cited revolves around the very simple question: what are the basic assumptions for Lorentz transformations? And the paper gives these, both LT and GT are the result from these, and by letting $c\to\infty$, you end up from LT to GT, as in (45) to (43).

No statement at all is given on the implications on geometry, on group structure, on any non-trivial properties that a Lorentz spacetime has over a Galilean spacetime.

And I do agree with you that there are such implications, and the transition from Lorentz to Galilei algebra is non-trivial, spacetime geometry is different etc.etc. But this is a completely different discussion! And I am happy to have this, but still it is not what has been asked for in the first place, and is something that is not addressed by the paper at all!

 

[PeroK]

By omitting the quantifier $\forall v <c$ you hide the flaw and hide the invalidity of your limit.

 

[vanhees71]

That may well be true, but the "non-relativistic limit" of some formula/theory usually involves a formal expansion in powers of $1/c$.

I think the question, in which sense Galilei-Newton spacetime is a limit of Einstein-Minkowski spacetime is more complicated. I've not yet seen any formal discussion of that. Maybe one can work it out by looking at the corresponding "limit" which deforms a Minkowski diagram (for one-dimensional motion of course) in its analogue for Newtonian physics, which is pretty strange though, so that nobody ever discusses it in textbooks. I once thought about it, but then I found it pretty useless for presenting it to students in the introductory lecture of special relativity ;-)).

 

[Sagittarius A-Star]

$\lim_{c\to\infty}\frac{x-vt}{(1-v^2/c^2)^{1/2}}=x-vt$

Wolfram Alpha agrees without explicit quanifier.

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]

 

[otennert]

By omitting the quantifier $\forall v <c$ you hide the flaw and hide the invalidity of your limit.

I am beginning to see what you mean. What makes you think that $v$ is not be held fixed while letting $c\to\infty$? That was my assumption all the time. You seem to require that somehow the ratio $v/c$ needs to be kept constant, but I am not seeing the reason why...?

 

[vanhees71]

The Galilean limit often can be derived as the limit $\beta=v/c \rightarrow 0$, but also a bit depends on the theory you are looking at. E.g., for mechanics the limit $\beta \rightarrow 0$ is indeed usually getting you to the non-relativistic approximation.

For the Maxwell equations it's another business, and you have to distinguish between different "Galilean limits" ("electric" and "magnetic" ones):

M. LeBellac, J. M. Levy-Leblond, Galilean electromagnetism, Nuovo Cim. 14B, 217 (1973)

Generally, I also do not understand what @PeroK is after.

 

[otennert]

The Galilean limit often can be derived as the limit $\beta=v/c \rightarrow 0$, but also a bit depends on the theory you are looking at. E.g., for mechanics the limit $\beta \rightarrow 0$ is indeed usually getting you to the non-relativistic approximation.

For the Maxwell equations it's another business, and you have to distinguish between different "Galilean limits" ("electric" and "magnetic" ones):

M. LeBellac, J. M. Levy-Leblond, Galilean electromagnetism, Nuovo Cim. 14B, 217 (1973)

Generally, I also do not understand what @PeroK is after.

Intriguing! I will give it a read. Actually I would never have imagined someone would investigate the Galilei limit of a Gauge theory at the core of which are massless particles.

 

[PeroK]

I am beginning to see what you mean. What makes you think that $v$ is not be held fixed while letting $c\to\infty$? That was my assumption all the time. You seem to require that somehow the ratio $v/c$ needs to be kept constant, but I am not seeing the reason why...?

It's a question of what we choose first. If I choose any $v$, then you let $c \to \infty$, then $\gamma(v) \to 1$. Fair enough.

But, if you choose $c$, then I can choose any $v < c$ and we have the full range of gamma factors and the same geometry. Still Minkowski and no nearer to Galilean. So, there is no convergence from Minkowski to Gallilean

The question is which of these is correct?

 

[robphy]

"$c\rightarrow\infty$" is for physical intuition (for a layperson or a physics student).

A more mathematical sound approach (which avoids "limiting processes" as much as possible)
is to use a dimensionless parameter that I call
$$E=0,$$ which is one option from $\{-1,0,1\}$
(in my drafts and posters, I call it $\epsilon^2$)
which is essentially the sign of the dimensionful quantities with units of a squared-inverse-speed
used by

• Berzi (who calls it $K$ in #49 of the Why is Minkowski Spacetime Noneuclidean, in the article linked by @vanhees71, see also #37 in this current thread )
• Levy-Leblond (who calls it $\alpha$ in the paper in #40 linked by @otennert )
• Ehlers calls this the "causality constant" (#35 from Why is Minkowski Spacetime Noneuclidean)
• Synge would call it an "indicator" (since he used it to handle timelike and spacelike quantities in a unified way).

(An inverse quantity is sometimes used to avoid issues of infinity,
and may be more physical than the historically-defined quantity.
Example: the inverse temperature $\beta$ thermodynamic beta to handle issues of negative temperature.)From #42 in Why is Minkowski Spacetime Noneuclidean
one can write
$\left( \begin{array}{c} t' \\ \frac{x'}{c_{light}} \end{array} \right) = \left( \begin{array}{cc} \frac{1}{\sqrt{1-E\beta^2}} & \frac{E\beta}{\sqrt{1-E\beta^2}} \\ \frac{\beta}{\sqrt{1-E\beta^2}} & \frac{1}{\sqrt{1-E\beta^2}} & \end{array} \right)\\ \left( \begin{array}{c} t \\ \frac{x}{c_{light}} \end{array} \right)$
where $\beta=v/c_{light}$ where $c_{light}=3\times10^8\ \mbox{m/s}$ [a fixed quantity, playing the role of a convenient conversion constant].
For $E=0$ (galilean) or $E=+1$ (minkowskian) , one could think of $E$ as if it were $$\left(\frac{c_{light}}{c_{max}}\right)^2,$$
as implemented in code, for example, https://www.desmos.com/calculator/kv8szi3ic8 .
(As I said in #58 , this "accounting" approach disentangles

• "c" as a space-time unit conversion constant [which is an issue of history].
• "c" as maximum-signal-speed [which is an issue of physics]

)

So, by primarily using this parameter $E$ or its equivalent [as used above],
we can avoid (or at least minimize) issues of taking limits to infinity
and move on to the other likely-more-interesting mathematical structures of the physics problem.

 

[otennert]

It's a question of what we choose first. If I choose any $v$, then you let $c \to \infty$, then $\gamma(v) \to 1$. Fair enough.

But, if you choose $c$, then I can choose any $v < c$ and we have the full range of gamma factors and the same geometry. Still Minkowski and no nearer to Galilean. So, there is no convergence from Minkowski to Gallilean

The question is which of these is correct?

By "choose" you mean "hold fixed", correct? If so, then by your first statement you agree with my reasoning, at least from a mathematical perspective. I am happy to see that. I am saying: this is the correct way of looking at the problem.

This means in your second statement "if you choose $c$..." you mean hold $c$ fixed and let $v$ go to some limit, but which one? To $c$, as this is the natural constant representing the maximum velocity? But that limit "$v=c$" does not exist, as the LT only exist for $v<c$ so I don't see what limit can be taken here. $v\to\infty$ does not exist either, for the same reason.

The other option you seem to have implied in your posting #77 is to look at some kind of limit where both $v$ and $c$ go to $\infty$, while the ratio $v/c$ is held constant. But this is actually no limiting procedure at all, because it effectively only rescales velocities, while keeping the overall structure of the theory invariant.