What assumptions underly the Lorentz transformation?

[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.

 

[otennert]

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

I don't share the view that ##c\to\infty## is for the laypeople only. After all, in mathematics as well as in physics there are a couple of situations where some quantity ##C## goes to infinity, which normally is the case when some other quantity ##D## is suppressed by the inverse of that factor and by taking that limit is essentially eliminated (as usually is ##C## at the same time, which is what happens when going from the Lorentz transformation formula to the Galilei transformation formula).

You give an example yourself: the inverse temperature ##\beta## goes to ##\infty## when ##T\to 0##. Depending on what you want to derive, it may make it much more transparent to see the impossibility to reach that value.

Regarding your differentiation between ##c_{light}## and ##c_{max}##: if ##c_{light}\neq c_{max}## then ##c_{light}## is no natural constant, it is actually not constant at all any more, because the natural constant is ##c_{max}##. The velocity of light would be dependent on the reference frame as e.g. is the windspeed on Earth. Subsequently, in your nomenclature, the 2 cases ##E=0## and ##E=+1##, which you define as ##E=\left(\frac{c_{light}}{c_{max}}\right)^2##, would be:
- a photon at rest (##c_{light}=0##)
- a photon at the speed ##c_{max}##, which however is impossible as there is no LT that maps some ##v<c_{max}## to ##v=c_{max}##. Of course, you may correct your value range by saying ##E\in[0,1)## so you have a half-open interval, which would at least be mathematically consistent.

Either way, your interpretation of ##E=0## representing Galilei, as you write, and ##E=1## representing Lorentz, is not correct.

Amemdment: actually, if ##c_{max}## has a finite value, you have Lorentz. The value of ##c_{light}## is completely irrelevant. It might be ##c_{max}##, which is the world we live in, it may be not, still we would live in a Lorentz spacetime, but with some very exotic properties of light.

 

[otennert]

I'm not sure, what you mean by "absolute velocity", but you can indeed ask, what are the symmetry transformations for a spacetime model, in which the special principle of relativity holds in addition to the other usual symmetries, i.e., homogeneity of time and space as well as Euclidicity of space for any inertial observer. With these assumptions you indeed get only two possible spacetime models: Galilei-Newton spacetime without any additional fundamental parameter or Einstein-Minkowski spacetime, which introduces a "limiting speed" as a fundamtental parameter, which empirically is given by the speed of light in vacuo. A nice paper deriving this is

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

I've just seen this posting. I've never heard of this paper but will study it and see how it differs from Levy-Leblond. Thanks for mentioning this!

 

[robphy]

Regarding your differentiation between ##c_{light}## and ##c_{max}##: if ##c_{light}\neq c_{max}## then ##c_{light}## is no natural constant, it is actually not constant at all any more, because the natural constant is ##c_{max}##. The velocity of light would be dependent on the reference frame as e.g. is the windspeed on Earth. Subsequently, in your nomenclature, the 2 cases ##E=0## and ##E=+1##, which you define as ##E=\left(\frac{c_{light}}{c_{max}}\right)^2##, would be:
- a photon at rest (##c_{light}=0##)
- a photon at the speed ##c_{max}##, which however is impossible as there is no LT that maps some ##v<c_{max}## to ##v=c_{max}##. Of course, you may correct your value range by saying ##E\in[0,1)## so you have a half-open interval, which would at least be mathematically consistent.

You are misunderstanding the approach.

##c_{light}## is a constant (like the speed of sound is).. akin to a convenient conversion unit between meters and feet. It does not vary.
##c_{max}## is the parameter that varies between theories.

Either way, your interpretation of ##E=0## representing Galilei, as you write, and ##E=1## representing Lorentz, is not correct.

Amemdment: actually, if ##c_{max}## has a finite value, you have Lorentz. The value of ##c_{light}## is completely irrelevant. It might be ##c_{max}##, which is the world we live in, it may be not, still we would live in a Lorentz spacetime, but with some very exotic properties of light.

Plug in the specific values of E, regardless of the relation of E to other quantities.
Do you get Euclidean, Galilean, and Minkowski accordingly?

The approach I use is based on Yaglom, Levy-Leblond (the “Additivity, rapidity, relativity” paper and the “Galilean Electromagnetism” paper), Ehlers frame theory, etc.

An old poster that needs to be written up into a paper https://www.aapt.org/doorway/Posters/SalgadoPoster/Salgado-GRposter.pdf
goes beyond just formulating the idea (not just talking about the idea)… it is implemented to discuss physical situations to show how it can be explicitly used to unify the approach to encompass Galilean and Minkowskian relativity.
See the references at the end.
( As mentioned earlier, here is
a visualization to support my approach https://www.desmos.com/calculator/kv8szi3ic8 .)

 

[Dragrath]

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.

Hmm so one issue I see with this assumption is that while it works purely within the idealized special relativity is this no longer applies within the extension of special relativity to general relativity as spacetime curvature induces gravitational or cosmological doppler effects.

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).

Hmm interesting historical note on the meter and yeah how units are defined is nontrivial even if the choice is somewhat arbitary as not all methods have the same accuracy.

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.

So one relevant caveat related to these assumptions here which sticks out is that Homogeneity and isotropy of spacetime are assumed to be valid which while it is natural in special relativity may actually be much more problematic in the context of general relativity for some subtle meta-mathematical reasons which may become relevant in terms of solving some current unsolved problems in physics and thus consequently requiring more complicated work on velocity vector addition at cosmological scales.

The problems relate to the defining properties of all systems of partial differential equations specifically how for every system of partial differential equations there exists an unique solution for all valid initial conditions, which in turn means that information on the initial conditions must be conserved at cosmological scales.

Starting from the proof of the "No big crunch theorem" proved by Matthew Kleban & Leonardo Senatore
in Inhomogenous and anisotropic cosmology(https://iopscience.iop.org/article/10.1088/1475-7516/2016/10/022/meta), you can show that at least in the limiting case of any arbitrarily nontrivial flat or open universe with 3 dimensions of space and one of time which is initially expanding that their proof falsifies the assumption that at some sufficiently large scale spacetime can be treated as relatively uniform and isotropic. In the context of systems of differential equations it isn't hard to mathematically prove such an assumption explicitly requires that information conservation be violated i.e. information must be destroyed for the assumption to hold. Given that the conservation of information is a foundational postulate of quantum mechanics this indicates homogeneity and isotropy are not a valid postulate. Oh and you also automatically get the laws of thermodynamics and the arrow of time as universal constraints on the variation of the metric within such a universe. (Though the 2nd law of thermodynamics and arrow of time in such a case appear to directionally depend on whether the universe is initially overall expanding of contracting)

In particular since the recent experimental test for the pure kinematic dipole assumption by Nathan J. Secrest et al(https://iopscience.iop.org/article/10.3847/2041-8213/abdd40) which is needed to prevent the CMB dipole from falsifying the cosmological principal being valid anywhere within the observable universe, there is now strong experimental evidence that we can't treat space as either isotropic or homogeneous at any scale within the observable universe, i.e. a significant component of the CMB dipole must arises from some cosmological effects related to the overall large scale structure of the Universe. An implication of this is that there is that the general relativistic extension of special relativistic velocity transforms will need mathematical modification based on large scale anisotropies.

 

[otennert]

You are misunderstanding the approach.

##c_{light}## is a constant (like the speed of sound is).. akin to a conversion unit between meters and feet. It does not vary.
##c_{max}## is the parameter that varies between theories.
Plug in the specific values of E, regardless of the relation of E to other quantities.
Do you get Euclidean, Galilean, and Minkowski accordingly?

The approach I use is based on Yaglom, Levy-Leblond (the “Additivity, rapidity, relativity” paper and the “Galilean Electromagnetism” paper), Ehlers frame theory, etc.

An old poster that needs to be written up into a paper https://www.aapt.org/doorway/Posters/SalgadoPoster/Salgado-GRposter.pdf
goes beyond just formulating the idea… it is implemented to discuss physical situations.
See the references at the end.

OK, so ##c_{light}## is just another numerical constant, but ##c_{max}## is a natural constant, not necessarily the speed of light, correct? It is somewhat misleading then to call it ##c_{light}## which suggests it is the speed of light -- which actually only makes sense if there is the speed of light, i.e. constant. Otherwise I can't see the physical significance. There is neither the speed of sound, although in this case of course an observer's rest frame is implied with regards to the medium (=air) -- which however is impossible to define for light unless there is a medium at rest.

But anyway, so you are saying that ##E=0## is equivalent to ##\alpha=0## in Levy-Leblond, and ##E=1=1/c_{max}## is equivalent to ##\alpha>0## in some suitable units, nothing else, correct? Understood.

I will have a look at your poster.

 

[robphy]

But anyway, so you are saying that ##E=0## is equivalent to ##\alpha=0## in Levy-Leblond, and ##E=1=1/c_{max}## is equivalent to ##\alpha>0## in some suitable units, nothing else, correct? Understood.

I will have a look at your poster.

At the level of the “E”, yes.
The idea is not new and has been around for awhile, often rediscovered without being aware of earlier approaches.

I have been using “E” here at PF ever since I implemented the idea in Desmos to demonstrate the unified geometrical relationships explicitly in a familiar context, beyond merely classification or abstract formula.

The ##\epsilon^2## approach (inspired by Yaglom) is a gateway to developing a computational technique akin to using complex numbers as vectors in Euclidean geometry.
(That opens up another can of worms that I decided to bottle up for now… but it’s in the poster.
I claim it is all consistent but I need to learn more projective geometry to make it more understandable and palatable to a more general audience, as well as fortify proofs and calculations.)

While foundational “interpretations” may be argued about, my interpretation leads to a concrete realization that yields the appropriate formulas and geometric constructions to describe (when appropriately reduced) standard physics in standard physics notation.

In short, “E” appears in unified formulas for physics. Choosing “E=0” gets you the PHY 101 formulas, “E=1” gets you special relativistic formulas, “E=-1” gets you the Euclidean analogue…. Ideally developed with a common unified approach (common geometric construction). You don’t have unlearn everything… but loosen up rigid viewpoints to allow the transition between signatures.

 

[Orodruin]

Ok, so here's the thing:

If there is an invariant speed ##V##, then spacetime is Minkowski regardless of what that invariant speed is. We all agree on this as far as I can tell. The question is what happens afterwards in the line of thought. On the one hand, yes, taking the limit ##V\to \infty## formally reduces the Lorentz transformations to the Galilei transformations as long as everything else is kept constant. However, the objection here is that calling this limit ##V \to \infty## is somewhat misleading from the Minkowski geometry point of view. The reason for this is that the natural thing to do would be to use the same units for time and space directions such that the metric diagonal becomes ##\pm (1,-1,-1,-1)## depending on convention. This is in complete analogy with using the same units for the x- and y-axes in Euclidean space. In this regard, the invariant speed ##V## is simply a scaling choice between time and space units and such a scaling choice can never affect the geometry. Therefore, letting ##V\to \infty## while keeping ##v## fixed geometrically corresponds to only allowing small rapidities, which would be more accurately referred to as ##v/V \to 0##.

Compare to the Euclidean case in two dimensions. Geometrically it would be natural to use the same units for both directions, but for some applications it may be of more relevance to use nautical miles in one direction and fathoms in another. The conversion factor between those units is roughly ##K = 1012## fathoms/NM. A rotation now takes the form
$$
x' = \Gamma ( x - ky/K^2), \qquad y' = \Gamma (y - kx),
$$
where ##\Gamma = 1/\sqrt{1+(k/K)^2}## and ##k## is a rotation parameter (##k = K \tan\theta## where ##\theta## is the rotation angle). Formally, for ##K\to \infty## with everything else fixed, we would find
$$
x' = x, \qquad y' = y - kx.
$$
Geometrically, nothing changed, but our rotations of fixed ##k## became smaller and smaller rotations as ##K## increased and the formal limit no longer displays the same group structure as that of the full geometry. I think this is why many, including myself quite often, do not really like talking about the limit as ##c \to \infty##, with ##c## being just an arbitrary unit conversion factor unrelated to the actual geometry.

You give an example yourself: the inverse temperature β goes to ∞ when T→0. Depending on what you want to derive, it may make it much more transparent to see the impossibility to reach that value.

This is a bit of a misdirected analogy as it involves a physical quantity going into a limit. The more relevant analogy would be letting ##k_B##, which is effectively a conversion factor between units of temperature and units of energy, go to zero. Neither is really relevant of course, as the big question is typically how ##k_B T## relates to a typical difference in energies.

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##.

... which is the same as in the Euclidean case saying that what you want to fix is for some reason the parameter ##k## above rather than the geometrically somewhat cleaner rotation angle ##\theta##.

 

[Vanadium 50]

To me, the OP looks to be asked and answered, yet this thread keeps going in circles. Does that make it an example of Thomas Precession?

 

[otennert]

To me, the OP looks to be asked and answered, yet this thread keeps going in circles. Does that make it an example of Thomas Precession?

No, it's a closed timelike curve. ;)

 

[Orodruin]

No, it's a closed timelike curve. ;)

There are no closed timelike curves in Minkowski space!

 

[otennert]

There are no closed timelike curves in Minkowski space!

Does flat spacetime with periodic boundary conditions count as somewhat Minkowski? If not, I give up.

 

[Orodruin]

Does flat spacetime with periodic boundary conditions count as somewhat Minkowski? If not, I give up.

No, that’s a different geometry. As evidenced by the existence of closed timelike curves.

 

[vanhees71]

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?

I'm still puzzled. The Lorentz boost with fixed ##\vec{v}## goes to a Galilei boost when taking ##c \rightarrow \infty##. So on this very elementary level, there's not such a complicated issue in "deforming" the Poincare group to the Galilei group.

It gets more subtle in the context of quantum theory, where you deal with ray representations, and doing the "deformation" in the correct way there you end up not with the Galilei group but with a central extension of it (despite the fact that anyway you end up with the covering group too, enabling half-integer representations of the rotation group in both Poincare and Galilei groups). Proper unitary representations of the Galilei group don't lead to useful dynamics of the quantum theory (Enönü and Wigner).

 

[vanhees71]

Ok, so here's the thing:

If there is an invariant speed ##V##, then spacetime is Minkowski regardless of what that invariant speed is. We all agree on this as far as I can tell. The question is what happens afterwards in the line of thought. On the one hand, yes, taking the limit ##V\to \infty## formally reduces the Lorentz transformations to the Galilei transformations as long as everything else is kept constant. However, the objection here is that calling this limit ##V \to \infty## is somewhat misleading from the Minkowski geometry point of view. The reason for this is that the natural thing to do would be to use the same units for time and space directions such that the metric diagonal becomes ##\pm (1,-1,-1,-1)## depending on convention. This is in complete analogy with using the same units for the x- and y-axes in Euclidean space. In this regard, the invariant speed ##V## is simply a scaling choice between time and space units and such a scaling choice can never affect the geometry. Therefore, letting ##V\to \infty## while keeping ##v## fixed geometrically corresponds to only allowing small rapidities, which would be more accurately referred to as ##v/V \to 0##.

Of course you can NOT use "natural units" when you want to take the Newtonian limit, because if there is no "limiting speed", ##V##, there simply is no "natural unit" for velocities/speeds and also no natural way to measure time intervals and distances in the same unit. So before you take the limit ##V \rightarrow \infty## you have to fix space and time units.

In Newtonian mechanics there is no such connection between space and time. The geometry is completely different: In Newtonian mechanics you have time and at each point in time a Euclidean space. If I remember the formalism right that's a kind of fiber bundle, while in special relativity you have a pseudo-Euclidean affine 4D manifold. I don't know, how to formally describe the limit ##V \rightarrow \infty## to deform the Minkowski spacetime to the Newtonian spacetime. I'm sure, there should be some literature about this somewhere.

Compare to the Euclidean case in two dimensions. Geometrically it would be natural to use the same units for both directions, but for some applications it may be of more relevance to use nautical miles in one direction and fathoms in another. The conversion factor between those units is roughly ##K = 1012## fathoms/NM. A rotation now takes the form
$$
x' = \Gamma ( x - ky/K^2), \qquad y' = \Gamma (y - kx),
$$
where ##\Gamma = 1/\sqrt{1+(k/K)^2}## and ##k## is a rotation parameter (##k = K \tan\theta## where ##\theta## is the rotation angle). Formally, for ##K\to \infty## with everything else fixed, we would find
$$
x' = x, \qquad y' = y - kx.
$$
Geometrically, nothing changed, but our rotations of fixed ##k## became smaller and smaller rotations as ##K## increased and the formal limit no longer displays the same group structure as that of the full geometry. I think this is why many, including myself quite often, do not really like talking about the limit as ##c \to \infty##, with ##c## being just an arbitrary unit conversion factor unrelated to the actual geometry.

I don't think that this is a good analogy, because here you stay within a fixed Euclidean geometry of the plane. There it indeed doesn't make sense to arbitrarily choose different units along different directions and then take a limit of the conversion factor.

This is a bit of a misdirected analogy as it involves a physical quantity going into a limit. The more relevant analogy would be letting ##k_B##, which is effectively a conversion factor between units of temperature and units of energy, go to zero. Neither is really relevant of course, as the big question is typically how ##k_B T## relates to a typical difference in energies.

That's of course true. So it's a difference to have a concrete physical situation, of which you make some approximation in certain limits.

In our context an example is the above quoted work by LeBellac and Levy-Leblond on the different "Galilean limits" of Maxwell's equations. After reading this paper I guess even the most anti-relativity sceptic should get convinced that relativity is the right foundation of electromagnetic theory ;-)).

 

[PeroK]

I'm still puzzled. The Lorentz boost with fixed ##\vec{v}## goes to a Galilei boost when taking ##c \rightarrow \infty##. So on this very elementary level, there's not such a complicated issue in "deforming" the Poincare group to the Galilei group.

It gets more subtle in the context of quantum theory, where you deal with ray representations, and doing the "deformation" in the correct way there you end up not with the Galilei group but with a central extension of it (despite the fact that anyway you end up with the covering group too, enabling half-integer representations of the rotation group in both Poincare and Galilei groups). Proper unitary representations of the Galilei group don't lead to useful dynamics of the quantum theory (Enönü and Wigner).

Here's an example. Let$$S_n = \{\frac 1 n, \frac 2 n, \dots \frac {n-1} n\}$$For any ##k## we have:$$\lim_{n \to \infty} \frac k n = 0$$But, it would be wrong to conclude that$$\lim_{n \to \infty} S_n = \{0\}$$The pointwise argument fails to capture the limiting behaviour of the set.