Derivation of space-time interval WITHOUT Lorentz transform?

[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. :-)