Derivation of space-time interval WITHOUT Lorentz transform?

[maverick_starstrider]

Hi,

I'm reading a book on SR/Field theories that simply posits the space-time interval and from that defines a Lorentz transform as any transformation which leaves the interval invariant. My question is how do we posit the space-time interval in this manner using only the postulates of special relativity. What does it represent (if we cannot invoke the concept of a Lorentz transform). What is its derivation? One normally goes the other way around by deriving the Lorentz transform and showing that the spacetime interval is an extension of a euclidian norm that is invariant under it.

 

[DrGreg]

Try www.physicsforums.com/showthread.php?t=213184


P.S. The link given in post #3 of that thread is broken, it has since moved to http://autotheist.synthasite.com/bondi1.php

 

[maverick_starstrider]

Try www.physicsforums.com/showthread.php?t=213184


P.S. The link given in post #3 of that thread is broken, it has since moved to http://autotheist.synthasite.com/bondi1.php

Can it be done without making reference to some sort of thought or gedanken experiment?

 

[DrGreg]

Can it be done without making reference to some sort of thought or gedanken experiment?

What sort of argument isn't a thought experiment? Give an example of the sort of argument you are looking for.

 

[maverick_starstrider]

What sort of argument isn't a thought experiment? Give an example of the sort of argument you are looking for.

Something akin to Landau's derivation of classical mechanics in his first chapter of "Mechanics". By postulating something like the principle of extremal action and then by postulating certain symmetries of our system obtaining all the relevant equations.

Can I go from something like:

1) The speed of light is constant in all reference frames
2) Euler-Lagrange Mechanics
3) The laws of physics are the same in all reference frames
4) Some other postulates of a similar nature

to special relativity without invoking a thought experiment or Maxwell's equations?

 

[Pengwuino]

I believe there is a problem in Jackson's E/M text where you derive the Lorentz transformations using simply isotropy and homogeneity and a maximum speed limit.

 

[maverick_starstrider]

I believe there is a problem in Jackson's E/M text where you derive the Lorentz transformations using simply isotropy and homogeneity and a maximum speed limit.

That sounds very promising, I'm going to look into it, thanks.

 

[maverick_starstrider]

That sounds very promising, I'm going to look into it, thanks.

Yes, 11.1 and 11.2. Thanks very much, I'm trying to solve it right now. So from this one can obtain the requirement of Lorentz Invariance and then I suppose one can determine what quantities are invariant under it (with certain other requirements) to define the spacetime interval.

 

[Matterwave]

You can get the ds^2 is invariant by merely assuming homogeneous and isotropic space and homogeneous time. It's done in Landau and Lifgarbagez Classical Theory of Fields section 2.

 

[omg!]

landau's argument goes like this:

for two infinitely close events, he defines the infinitesimal interval to be

ds^2=c^2 dt^2 - dx^2

we know that ds=0 iff the events describe the propagation of signals at the speed of light.
in this case, by the constancy of the speed of light, ds'=0 also in any another inertial system.

Then he argues that "ds and ds' are infinitesimal on the same order", and that therefore
ds^2 = a(V)ds'^2
where V is the relative speed of the inertial systems, from which he derives straightforwardly that ds^2=ds'^2.
Based on equality for infinitesimal intervals, he then concludes that s=s' in general.

homogeneity space, time and isotropy of space are only used to argue that a cannot be a function of coordinates and time.

There are two points that do not convince me:
1. what exactly does he mean with "ds and ds' infinitesimal on the same order" and why is it the case?
2. how do you formally show that ds^2=ds'^2 \Rightarrow s=s'?

 

[strangerep]

Maverick,

If you're really keen to investigate this sort of thing deeply, here's what I regard as the current state-of-the-art treatment.

What matters is the group transformations between inertial frames. Knowing the group, one can deduce a line element from the Casimir(s) of the group. So let's start with:

1) The relativity postulate (RP): the laws of physics are identical in all inertial (i.e., unaccelerated) frames.

but

2) Don't assume the light postulate (LP) that light signals in vacuum are propagated rectilinearly, with the same speed $c$ at all times, in all directions, in all inertial frames. It turns this LP is unnecessary.

Then ask: "what is the most general transformation compatible with the RP?". I.e., consider a (differentiable) transformation $(x,t)\to(x',t')$ such that (defining velocities $u := dx/dt$, $~u':=dx'/dt'$),
$$
\def\Ddrv#1#2{\frac{d #1}{d #2}}
a := \Ddrv{u}{t} = 0 ~~\implies~~ a' := \Ddrv{u'}{t'} = 0 ~.
$$
It turns out the the most general such transformation is of fractional linear form:
$$
x' ~=~ \frac{Ax + Bt + C}{ax + bt + 1} ~,~~~~~
t' ~=~ \frac{Dx + Et + F}{ax + bt + 1}
$$
with 8 parameters $A,B,C,D,E,F,a,b$ which satisfy a certain determinantal condition (which I won't bother writing here). A somewhat-simplified derivation of this transformation can be found in Appendix 6 of this paper:

S. S. Stepanov,
Fundamental Physical Constants & the Principle of Parametric Incompleteness,
Available as arXiv:physics/9909009.

(Don't worry about the main body of that paper -- only appendix 6 is relevant here.)

Now consider three inertial frames, denoted k, k', and k'' such that the origin at rest in k has constant 3-velocity $u$ in k', and the origin at rest in k' has constant velocity $-u$ in k. Also assume that at $t=0$ in k, the spatial origins coincide, and also at $t'=0$.

If we also require that velocity boost transformations in a given direction form a group, with single parameter $u$, and that $u=0$ corresponds to the identity transformation, then by specializing the fractional-linear transformations above to this case, one can derive the following results:

a) There exists a frame-independent constant $c$ with dimensions of speed (but now regarded only as a parameter whose value must be determined by experiment).

b) The velocity addition law arising from composing 2 boosts has the form:
$$
u'' ~=~ \frac{u + u'}{1 + \frac{u'u}{c^2}}
$$
A sketch of the calculation can be found in Section 2 of this paper:

S. N. Manida,
Fock-Lorentz transformations and time-varying speed of light,
Available as: arXiv:gr-qc/9905046 .

(Don't be put off by the "time-varying speed of light" phrase in the title. It's not what you might think. In any case, only Section 2 is relevant here.)

[As a "bonus", Manida also shows that the fractional-linear transformations for boosts also admit another invariant parameter R with dimensions of length -- taken to be another empircally-determined parameter, but necessarily cosmologically "large". In the limit $R\to\infty$, one recovers the usual Lorentz transformations of SR.]

Armed with this knowledge of the group, one can find an invariant line element. For $R=\infty$, it's the Minkowski line element. For $R<\infty$ it's slightly different.

When I first saw these derivations it seemed to me utterly remarkable that existence of an invariant limiting speed $c$ can be derived from nothing more than the RP, differentiability of coordinate transformations, and the assumption that parallel velocity boosts form a group.

(Oh, and of course one still needs to assume local spatial isotropy to extend the above to 3 space dimensions.)

 

[samalkhaiat]

Yes, 11.1 and 11.2. Thanks very much, I'm trying to solve it right now. So from this one can obtain the requirement of Lorentz Invariance and then I suppose one can determine what quantities are invariant under it (with certain other requirements) to define the spacetime interval.

See post #9 in
www.physicsforums.com/showthread.php?t=420204

Sam

 

[lugita15]

If we also require that velocity boost transformations in a given direction form a group, with single parameter $u$

What is the physical motivation for assuming that parallel boosts are closed under composition? After all, in SR boosts in general are not closed under composition, so can't a slightly more general theory make the composition of two parallel boosts some transformation other than a boost in the same direction? Or do other symmetries like isotropy of space preclude that possibility? Also, rather than parallel boosts forming a one-parameter group, what physical reason do we have for rejecting the possibility that parallel boosts just form a semigroup, defined by two parameters like the time evolution operators in a dissipative system?

There exists a frame-independent constant $c$ with dimensions of speed (but now regarded only as a parameter whose value must be determined by experiment).

Does the argument eliminate the possibility that this constant is infinity, i.e. that the world is classical? It would be very interesting if it did, because that would have given Newton a purely theoretical reason to reject his theory of mechanics.

 

[strangerep]

What is the physical motivation for assuming that parallel boosts are closed under composition? After all, in SR boosts in general are not closed under composition, so can't a slightly more general theory make the composition of two parallel boosts some transformation other than a boost in the same direction?

The setup of the inertial frames is very similar to the usual setup in SR -- if we restrict to very small neighborhoods of the origins. I.e., the usual SR derivations of Lorentz transformations set things up such that the origins coincide at some time in frame $k$, and the clocks in both frames $k, k'$ are reset to 0 accordingly. Similarly, we assume that a spatial rotation is performed so that (at least locally) the orthogonal directions in both frames coincide at that time. This is essentially the standard textbook setup in SR, except that to derive Fock-Lorentz-Manida transformations we only require this in small neighborhoods of each origin.

That such parallel boosts should be closed is based on physical intuition that for any pair of inertial frames whose origins coincide at a time t=0 (and clock-synchronized in the other frame so that at the event t'=0 also), there should exist a relative velocity between them. After all, we have only synchronized the frames when their origins coincide, but we have not constrained the speed of one origin relative to the other -- this was the whole point of trying to derive transformations between observers in relative motion.

Also, rather than parallel boosts forming a one-parameter group, what physical reason do we have for rejecting the possibility that parallel boosts just form a semigroup,

If the $k'$ frame has velocity $u$ relative to the $k$ frame, then $k$ should have velocity $-u$ relative to the $k'$ frame. Such considerations imply that each boost transformation should have an inverse. Hence they form a group rather than a semigroup.

Does the argument eliminate the possibility that this constant is infinity, [...]

No.

 

[lugita15]

That such parallel boosts should be closed is based on physical intuition that for any pair of inertial frames whose origins coincide at a time t=0 (and clock-synchronized in the other frame so that at the event t'=0 also), there should exist a relative velocity between them.

If the $k'$ frame has velocity $u$ relative to the $k$ frame, then $k$ should have velocity $-u$ relative to the $k'$ frame. Such considerations imply that each boost transformation should have an inverse.

I agree that these two observations are intuitively obvious, but I fear they arise from the same intuition that gives rise to Galilean velocity addition and the like.

But put yourself in the shoes of someone who only knows Newtonian mechanics, and has heard that there is a new-fangled theory called relativity which changes the compositions and relations of velocities. Would you have any reason to suspect that boosts in general may not be closed, but boosts in a straight line are? Would you have any reason to suspect that the Galilean velocity addition formula may be wrong but the notion that the u and -u symmetry is still correct?

 

[strangerep]

I agree that these two observations are intuitively obvious, but I fear they arise from the same intuition that gives rise to Galilean velocity addition and the like.

That would be a distortion, imho. The main difference in the foundations of Newtonian and Einsteinian relativity is that time is absolute in the former. In both versions of the "inertial frame" concept, acceleration is zero, but in Einsteinian relativity we admit a larger set of transformations which are allowed to muck around with time.

But put yourself in the shoes of someone who only knows Newtonian mechanics, and has heard that there is a new-fangled theory called relativity which changes the compositions and relations of velocities. [...]

The velocity addition formulas are derived, not postulated. Your "someone" should study the theory in more precise detail, rather than making such guesses. :-)

[Maverick, feel free to object if this is wandering too far off-topic...]

 

[Samshorn]

Don't assume the light postulate (LP) that light signals in vacuum are propagated rectilinearly, with the same speed $c$ at all times, in all directions, in all inertial frames. It turns this LP is unnecessary.

That's a common misconception, that often arises when people first realize that there are only three logically consistent forms of continuous relativistic transformations (I say continuous to exclude the linear fractional varities), namely Euclidean, Galilean, and Minkowskian. Ever since about 1906 people have been re-discovering this, and they get so excited that they rush into print with the claim that special relativity can be derived based only on the relativity principle. Of course, that's not true, since one can only derive a general form that encompases the three possible cases. Remember that the function of the light principle (or something equivalent to it) was never anything other than to (first) distinguish between those cases, and (second) quantify the parameter and link it with measurable physically meaningful observables. Both of these are indispensable for the physically meaningful theory of special relativity.

When I first saw these derivations it seemed to me utterly remarkable that existence of an invariant limiting speed $c$ can be derived from nothing more than the RP, differentiability of coordinate transformations, and the assumption that parallel velocity boosts form a group.

Right, that's the enthusiasm that grips people for awhile when they are first exposed to this group theoretic approach, but note that it is based on conflating the cases when speeds are limitless and when they are limited. This linguistic abuse comes about by treating "infinity" as a number, encouraged by the cute little symbol. But that symbol just signifies that the quantity is unlimited, so it makes no sense to refer to "c" as a limiting speed in the case when "c" is infinite, because "infinite" means unlimited. Of course, even after the huge conceptual step of asserting that c is finite (which is warranted only by something like the supposedly "unnecessary" LP), it also makes no sense to disregard the significance of the numerical value of this finite constant in physically meaningful terms, which is the full content of the LP.

Oddly enough, almost everyone who gets excited and starts to write a paper on this ("hey, special relativity can be derived from only the relativity principle!") at some point realizes and acknowledges in the text the necessity and function of the light principle (or something equivalent), tacitly admitting that the relativity principle alone is obviously NOT sufficient, but somehow they can't bear to change their headline. That's why there have been so many papers over the past 100 years with titles and abstracts claiming the LP is unnecessary, even though the papers themselves admit (with varying degrees of candor) that it is indispensible.

 

[strangerep]

Right, that's the enthusiasm that grips people for awhile when they are first exposed to this group theoretic approach, but note that it is based on conflating the cases when speeds are limitless and when they are limited. This linguistic abuse comes about by treating "infinity" as a number, encouraged by the cute little symbol. But that symbol just signifies that the quantity is unlimited, so it makes no sense to refer to "c" as a limiting speed in the case when "c" is infinite, because "infinite" means unlimited.

In the derivations, I understood "c" as a parameter which was not fixed by the original postulate (RP). I.e., "c" was to be regarded as a parameter to be experimentally determined (e.g., via tests of the velocity addition law).

Of course, even after the huge conceptual step of asserting that c is finite [...]

Again, I understood that in Manida's derivation, no such arbitrary assertion is being made. Rather, the parameter "c" is to be determined by analyzing consequences of the transformation formulae, and conducting experiments to test them, such as tests of the velocity addition law -- which then place experimental bounds on the value of "c". One finds of course that the value coincides (within experimental error bounds) with the speed of light.

[...] there have been so many papers over the past 100 years with titles and abstracts claiming the LP is unnecessary, even though the papers themselves admit (with varying degrees of candor) that it is indispensible.

References please?

 

[Samshorn]

In the derivations, I understood "c" as a parameter which was not fixed by the original postulate (RP).

Right, it doesn't fix "c" even to the extent of determining whether it is finite or infinite, and hence it doesn't even distinguish between theories that possesses the unique features of special relativity (relativity of simultaneity, time dilation, length contraction, partial temporal ordering of events, the light cone structure) from those that don't, let alone provide any physically meaningful quantitative content.

"c" was to be regarded as a parameter to be experimentally determined.

Of course, both the relativity principle and the light principle are based on experimental knowledge. This does nothing to justify the claim that we can dispense with either of those principles.

I understood that in Manida's derivation, no such arbitrary assertion is being made.

First, it isn't "Manida's derivation". As I said, newbies have been rushing into print with that "derivation" for over a century. Second, the principles on which special relativity is founded are not "arbitrary" at all, they are distillations of our empirical knowledge. Third, in the absence of any empirically-based principle fixing the speed of light (or some equivalent physical attribute of phenomena), the group theoretic derivation cannot deliver special relativity, as is well known, and as you yourself have admitted (albeit indirectly).

References please?

I guess it's not surprising that you're unacquainted with the vast literature on this subject, since you can call it "Manida's derivation" without laughing. Well, let's see, just to get you started, Poincare already gave the group-based derivation of the Lorentz transformation in his 1905 Palermo paper (published in 1906), but people usually regard Ignatowski's 1910 paper (Einige allgemeine Bermerkungen zum Relativitatsprinzip", Verh. Deutch. Phys. Ges., 12, 1910, pp788-96) as the beginning of the fad of claiming to derive special relativity from the relativity principle alone. Then there was Frank and Roth (1911), Pars (1921), Lalan (1937), Arzelies (1966, with a good bibliography of many more references), Berzi and Gorini (1969), Sussmann (1969), Lee and Kalotas (1975), Levy-Leblond (1976), Toretti (1983), Lucas and Hodgeson (1990, Spacetime and Electromagnetism, a whole book devoted to various derivations of the Lorentz transformation), etc., etc... As Toretti said, "Ignatowski's work (1910) has been repeated and refined by numerous authors, some of whom independently lighted on the same ideas, unaware that they had long been available in well known journals".

An amusing example of the fallacy in an easily accessible secondary source is in Rindler's "Essential Relativity" (1969, 1977), which contains a section entitled "Special Relativity without the Second Postulate". After giving the group theoretic derivation, the section concludes with

"The relativity principle itself necessarily implies that either all inertial frames are related by Galilean transformations or by Lorentz Transformations with positive c^2. The role of a "second postulate" in relativity is now clear: it has to isolate one or the other of these transformation groups... However, in order to determine the universal constant c^2 the postulate must be quantitative. For example, a statement like "simultaneity is not absolute", while implying A Lorentz group fails to determine c^2... We shall see later that relativistic mass increase, or the famous formula E = mc^2, and others, could all equally well serve as second postulates."

The important thing to notice is that Rindler has just explained why we NEED a second postulate, either the light speed postulate or something equivalent to it ... even though the heading of the section is "special relativity without the second postulate". So it goes.

 

[Histspec]

References please?

See also http://arxiv.org/abs/1112.1466, accepted for publication in "Journal of High Energy Physics". It cites 25 relevant papers (ref. 1-25) with derivations of the Lorentz transformation, apparently without (?) a priori appeal to the light speed postulate, starting with Ignatowski 1910.

 

[strangerep]

First, it isn't "Manida's derivation".

I didn't say that Manida was the first. I only mentioned Manida's exposition of it because that's the one I've studied closely most recently.

I guess it's not surprising that you're unacquainted with the vast literature on this subject,

My "references please?" request was in reference to your statement that ``almost everyone who gets excited and starts to write a paper on this ("hey, special relativity can be derived from only the relativity principle!") at some point realizes and acknowledges in the text the necessity and function of the light principle (or something equivalent),...''
I.e., I hoped you would give a specific vector to where someone is guilty of that.

I am not "unacquainted" with the literature, but apparently less acquainted than yourself. I am indeed aware of some of the references you mentioned (and a number of others that you didn't). I thank you for the ones I was not aware of.

But since you're well-acquainted with the relevant literature, can you tell me which derivations of this are performed in the full generality of fractional linear transformations, and don't invoke some principle to force linearity of the transformations at an early stage? (That was the context I was interested in when I decided to study Manida's paper.)

since you can call it "Manida's derivation"

You seem to have extrapolated one small phrase into a much wider meaning that was not my intent.

without laughing.

Since you're relatively new here, I'll remind you that the PF rules do not permit rudeness, and the tone of your posts is bordering on that. This, and your quick tendency to put words in my mouth, makes it difficult to have a constructive discussion.

An amusing example of the fallacy in an easily accessible secondary source is in Rindler's "Essential Relativity" (1969, 1977), which contains a section entitled "Special Relativity without the Second Postulate". [...]

I've studied some of Rindler's other works, but not that one. I'll do so before replying to this point.

 

[strangerep]

See also http://arxiv.org/abs/1112.1466, accepted for publication in "Journal of High Energy Physics". It cites 25 relevant papers (ref. 1-25) with derivations of the Lorentz transformation, apparently without (?) a priori appeal to the light speed postulate, starting with Ignatowski 1910.

Thanks very much. I'll take a look.

 

[Fredrik]

An amusing example of the fallacy in an easily accessible secondary source is in Rindler's "Essential Relativity" (1969, 1977), which contains a section entitled "Special Relativity without the Second Postulate". After giving the group theoretic derivation, the section concludes with

"The relativity principle itself necessarily implies that either all inertial frames are related by Galilean transformations or by Lorentz Transformations with positive c^2. The role of a "second postulate" in relativity is now clear: it has to isolate one or the other of these transformation groups... However, in order to determine the universal constant c^2 the postulate must be quantitative. For example, a statement like "simultaneity is not absolute", while implying A Lorentz group fails to determine c^2... We shall see later that relativistic mass increase, or the famous formula E = mc^2, and others, could all equally well serve as second postulates."

The important thing to notice is that Rindler has just explained why we NEED a second postulate, either the light speed postulate or something equivalent to it ... even though the heading of the section is "special relativity without the second postulate". So it goes.

That title makes it weird, but e.g. "Relativity without the second postulate" would have been OK. (Symmetry assumptions lead to the conclusion that we're dealing with either "Galilean relativity" or "Einsteinian relativity", and then it's just a matter of choosing between the two).

 

[m4r35n357]

Hmm, I thought I understood this following the thread here: https://www.physicsforums.com/showthread.php?t=605094, but now I'm not sure. I took away the impression that it is perfectly valid to derive the three possible "versions" of relativity, and allow experiment (eg. muon lifetimes) to differentiate between the one with the speed limit and those without. Then one can make further experimental links between that top speed and the speed of light, add the LP, then call it "Special Relativity".
If I'm mistaken could someone explain what is wrong with this perspective?

 

[Samshorn]

My "references please?" request was in reference to your statement that ``almost everyone who gets excited and starts to write a paper on this ("hey, special relativity can be derived from only the relativity principle!") at some point realizes and acknowledges in the text the necessity and function of the light principle (or something equivalent),...'' I.e., I hoped you would give a specific vector to where someone is guilty of that.

One specific vector points to this very thread, where it was argued that special relativity can be derived based purely on the principle of relativity, even while acknowledging that the purported derivation doesn't establish whether "c" is finite or infinite, and hence doesn't even establish whether simultaneity is relative or absolute, whether there is such a thing as time dilation, length contraction, or whether E=mc^2, or whether the events of spacetime are partially ordered with a null-cone structure, etc., etc. Hopefully it goes without saying that something which doesn't determine any of this (even qualitatively, let alone quantitatively) can hardly be called a derivation of special relativity. Rindler is another vector. But we don't have to search hard for these examples, because it should go without saying that every purported derivation of special relativity that fails to determine whether c is finite or infinite (let alone the numerical value of c in physically meaningful units) is "guilty of that". What's lacking here is not references.

Can you tell me which derivations of this are performed in the full generality of fractional linear transformations, and don't invoke some principle to force linearity of the transformations at an early stage? (That was the context I was interested in when I decided to study Manida's paper.)

Einstein himself discussed the justification for limiting the transformations to the purely linear ones, especially in some review articles around 1910-1912 as I recall, but only in somewhat vague terms. Pauli's 1921 encyclopedia article on relativity was more explicit, giving the usual and (by then) well known justifications for excluding them, e.g., we require finite coordinates map to finite coordinates. Most other good derivations in the literature, if you read them carefully, include some justification sufficient to exclude linear fractional transformations (rather than just assuming it), such as requring that continuous worldlines map to continuous worldlines. Rindler discusses linear fractional transformations too, and the various reasons why they may be considered unphysical. You can also find web articles on this subject - try googling "lorentz linear fractional". The point is that people have not failed to consider this, i.e., it's not a new subject.

 

[Samshorn]

Hmm, I thought I understood this following the thread here: https://www.physicsforums.com/showthread.php?t=605094, but now I'm not sure. I took away the impression that it is perfectly valid to derive the three possible "versions" of relativity, and allow experiment (eg. muon lifetimes) to differentiate between the one with the speed limit and those without. Then one can make further experimental links between that top speed and the speed of light, add the LP, then call it "Special Relativity".
If I'm mistaken could someone explain what is wrong with this perspective?

Already explained. See the previous posts in this thread. By saying that the light principle is based on empirical observation you do not justify the claim that we can dispense with it. The relativity principle is based on experimental evidence as well, so by your reasoning we can dispense with it, and simply "derive" special relativity as follows: "Experiment differentiates between special relativity and every other possible theory". That misses the whole point of theoretical physics, which is not just to compile a cataloge of experimental results, but to glean from experiment a small number of succinct principles from which all the results can be logically deduced, and around which we can organize our knowledge. This includes principles such as conservation of energy and momentum, etc. Among these empirically based principles is relativity, but that by itself is obviously not sufficient to logically deduce special relativity. We need some other physical principle (i.e., some other general fact gleaned from experience), such as the light principle. We cannot dispense with some other principle to arrive at the physically meaningful theory of special relativity.

Again, simply pointing out that our principles can be gleaned from experience (of course they can!) does not imply that we can dispense with them.

 

[TrickyDicky]

which derivations of this are performed in the full generality of fractional linear transformations, and don't invoke some principle to force linearity of the transformations at an early stage?

A good explanation can be read here: http://www.mathpages.com/home/kmath659/kmath659.htm

 

[TrickyDicky]

Curiously (SL 2,C), a.k.a the Mobius group, a.k.a. fractional linear transformations group is considered the gauge group of General relativity.

 

[strangerep]

[...] Most other good derivations in the literature, if you read them carefully, include some justification sufficient to exclude linear fractional transformations (rather than just assuming it), [..]

That wasn't my question -- I already knew all that. I was asking whether you knew of a references that doesn't restrict early to linear transformations, but rather works through the LFTs. Since you did not address the question I actually asked, I guess you don't know, which is disappointing since you had led me to believe you were well-acquainted with the literature.

Unfortunately, since apparently you do not read my posts carefully and thoughtfully before replying, I see no point in continuing this discussion with you unless it returns to constructive content.

 

[strangerep]

which derivations of this are performed in the full generality of fractional linear transformations, and don't invoke some principle to force linearity of the transformations at an early stage?

A good explanation can be read here: http://www.mathpages.com/home/kmath659/kmath659.htm

Thanks. I've only skimmed it so far, but it seems a bit closer to the sort of thing I was looking for, -- i.e., a more balanced, open-minded perspective. I'll read it more carefully tomorrow.

BTW, who is the author of that document? (It wasn't crystal clear from the home page.)
I also failed to find a date on it.

 

[strangerep]

[...] Then one can make further experimental links between that top speed and the speed of light, add the LP, then call it "Special Relativity".
If I'm mistaken could someone explain what is wrong with this perspective?

The only clarification I'd suggest is as follows:

Working with a finite, but undetermined value of "c" in the derivation, one can derive the LP by analyzing what happens in the limit as $v/c \to 1$. One can also carry out all the usual Wignerian analysis of the unitary irreducible representations of this group to determine other properties of particles in the same limit. Part of the miscommunication in this thread may be due to the unfortunate coincidence that "LP" may stand for "Light Postulate", (meaning something we assume at the beginning of a derivation), or it may stand for "Light Principle" which can be interpreted more flexibly.

 

[lugita15]

BTW, who is the author of that document? (It wasn't crystal clear from the home page.)
I also failed to find a date on it.

His name is Kevin Brown. His website mathpages.com is excellent, containing countless interesting articles on topics in math and physics. He's even published these articles in book form.

I especially enjoyed his book Reflections on Relativity:
http://mathpages.com/rr/rrtoc.htm

 

[Samshorn]

That wasn't my question -- I already knew all that. I was asking whether you knew of a references that doesn't restrict early to linear transformations, but rather works through the LFTs.

The references I mentioned do "work through the LFTs" in order to examine the behavior of such transformations and determine that they are not suitable to represent the relationship between standard inertial coordinate systems. Torretti, for example, is especially detailed and explicit in his treatment. But obviously when people like Einstein and Pauli say succinctly that LFTs don't map finite coordinates to finite coordinates, and don't map continuous worldlines to continuous loci of coordinates, and don't maintain homogeneity and isotropy, etc., they have "worked through the LFTs". So it's hard to see what more you could want. Are you asking for references that conclude LFTs actually ARE useful representations of the relationship between standard inertial coordinate systems? Or are you just looking for more slow-witted references that take longer to figure out that LFTs are not suitable? Or are you seeking references that hypothesize a universe in which space and time are not homogeneous, such that spacetime LFTs are physically meaningful?

Part of the miscommunication in this thread may be due to the unfortunate coincidence that "LP" may stand for "Light Postulate", (meaning something we assume at the beginning of a derivation), or it may stand for "Light Principle" which can be interpreted more flexibly.

It's more than just a miscommunication, it's a fundamental misunderstanding about the inductive-deductive method in physics. Read Newton on this subject: "In this philosophy [physics] particular propositions are inferred from the phenomena, and afterwards rendered general by induction... from thence to deduce other phenomena..." This is absolutely basic to how physics is done. From observations of phenomena, such as that momentum always seems to be conserved, we apply induction (necessarily incomplete) to generalize this into a principle (momentum conservation), and THEN we ASSUME this, i.e., we take this principle as a formal postulate in the deductive construction of a theory "from thence to deduce other phenomena". That's how the inductive-deductive process of physics works. In this way, both the relativity principle and the light principle are inferred from observation, and then applied as postulates in the formal development of a theory.

Unfortunately, newbies often misunderstand this process, thinking that "postulates are something we assume at the beginning of a derivation", overlooking the fact that, in science, the beginning of the deductive derivation is not the beginning of the inductive-deductive process. Unlike pure mathematics, which is wholly deductive (as usually conceived) from arbitrary postulates, the postulates that represent physical principles underlying physical theories represent the distillation of our most secure empirical knowledge.

The misconception that the physical principles (and hence the postulates) on which special relativity is founded are simply baseless hypotheses is what leads newbies to get excited when they discover that those principles are actually well-founded in observation. They think "Hey! We can replace a baseless hypothesis with experimental results!" That's a complete misunderstanding, because both the relativity principle and the light principle are already distillations of experimental results. That's where those propositions came from. They are not arbitrary hypotheses that are merely assumed without foundation. Indeed they are the aspects of our empirical knowledge that are the most secure. (See Poincare and Einstein on the advantages of "principle theories".) They merely play the role of formal postulates in the deductive phase of theory building.

 

[m4r35n357]

Already explained. See the previous posts in this thread. By saying that the light principle is based on empirical observation you do not justify the claim that we can dispense with it. The relativity principle is based on experimental evidence as well, so by your reasoning we can dispense with it, and simply "derive" special relativity as follows: "Experiment differentiates between special relativity and every other possible theory". That misses the whole point of theoretical physics, which is not just to compile a cataloge of experimental results, but to glean from experiment a small number of succinct principles from which all the results can be logically deduced, and around which we can organize our knowledge. This includes principles such as conservation of energy and momentum, etc. Among these empirically based principles is relativity, but that by itself is obviously not sufficient to logically deduce special relativity. We need some other physical principle (i.e., some other general fact gleaned from experience), such as the light principle. We cannot dispense with some other principle to arrive at the physically meaningful theory of special relativity.

Again, simply pointing out that our principles can be gleaned from experience (of course they can!) does not imply that we can dispense with them.

Well my understanding as it is (!), the relativity principle gives us a top speed, and Maxwell's equations give us invariant light speed, and I don't think anyone is seriously trying to belittle or dispense with the light postulate because without it those other two are completely independent. That's where I'm coming from; as you can tell I had some trouble understanding your answer . . . .

 

[Samshorn]

Well my understanding as it is (!), the relativity principle gives us a top speed

No, the relativity principle does not give us a "top speed", because infinity is not a top speed, it is a word meaning there is no top speed. Galilean relativity does not have a top speed, and therefore it doesn't exhibit relativity of simultaneity or time dilation or length contraction or a null cone structure or any of the other unique features of special relativity that arise when there is a top speed, and yet it is perfectly consistent with the relativity principle.

and Maxwell's equations give us invariant light speed...

No, Maxwell's equations do not give us invariant light speed, because they do not, in themselves, contain any information as to how relatively moving systems of coordinates in which Maxwell's equations hold good are related to each other. (Also, we know that Maxwell's equations are not correct, see QED.)

I don't think anyone is seriously trying to belittle or dispense with the light postulate...

Not even the people (including you) who have explicitly claimed that we can dispense with the light postulate?

...as you can tell I had some trouble understanding your answer . . . .

Yes.

 

[m4r35n357]

No, the relativity principle does not give us a "top speed", because infinity is not a top speed, it is a word meaning there is no top speed. Galilean relativity does not have a top speed, and therefore it doesn't exhibit relativity of simultaneity or time dilation or length contraction or a null cone structure or any of the other unique features of special relativity that arise when there is a top speed, and yet it is perfectly consistent with the relativity principle.

No, Maxwell's equations do not give us invariant light speed, because they do not, in themselves, contain any information as to how relatively moving systems of coordinates in which Maxwell's equations hold good are related to each other. (Also, we know that Maxwell's equations are not correct, see QED.)



Not even the people (including you) who have explicitly claimed that we can dispense with the light postulate?



Yes.

I most certainly did not specify Galilean relativity, and I don't know why you think I did. In fact I was referring to the Lorentzian solution (I was picked up earlier because I "left it to be determined by experiment") which does specify a top speed. I also explicitly stated that I am not attempting to dispense with the light postulate.
It would appear you have misunderstood me as much as I have you, so I will save you the effort of arguing against stuff you think I said & just leave it. Sorry it didn't work out.

 

[TrickyDicky]

Going back to the fractional linear transformations, I think it is important to stress that their unphysicality is not sufficiently granted just by the homogeneity of the space but also by its flat geometry (euclidean in the galilean case or minkowkian in the SR case) as it is explicit in Pauli's quote from the mathpages link :"All writers start with the requirement that the transformation formulae should be linear. This can be justified by the statement that a uniform rectilinear motion in K must also be uniform and rectilinear in K’. Furthermore it is to be taken for granted that finite coordinates in K remain finite in K’. This also implies the validity of Euclidean geometry and the homogeneous nature of space and time."

 

[strangerep]

Going back to the fractional linear transformations,
[...]
Pauli's quote from the mathpages link :"All writers start with the requirement that the transformation formulae should be linear. This can be justified by the statement that a uniform rectilinear motion in K must also be uniform and rectilinear in K’. Furthermore it is to be taken for granted that finite coordinates in K remain finite in K’. This also implies the validity of Euclidean geometry and the homogeneous nature of space and time."

I'd be interested in a discussion about the LFTs and the extent to which one may be reasonably justified in relaxing some of the criteria mentioned in that quote. But such a discussion would certainly diverge too far from the original topic of this thread. Also, it is not a mainstream subject, hence probably belongs over in the BTSM forum. :-)