Alternate derivation of Lorentz Trans.

[timetraveldude]

Has anybody come up with a way to derive the LT not based on the constantcy of the speed of light in all inertial reference frames?

 

[HallsofIvy]

WHY would one want to? The constancy of the speed of light was the experimental data that led to the Lorenz transformation. If the speed of light was not constant why would one want or need the Lorenz transformation?

 

[Fredrik]

I'm not sure what you (timetraveldude) have in mind here. The Lorentz tranformation is equivalent to the statement that the speed of light is the same to all inertial observers. I don't think the question really makes sense. Any postulate that you can use as a starting point for a derivation of the Lorentz tranformation will include that stuff about the speed or light, whether it's apparent or not.

 

[arildno]

I think Lorentz himself had another way...
If I remember correctly, he made some ad-hoc assumption that objects contracted in the direction of motion.
This is of course only of historical interest; I think Lorentz attempted to account for the Michelson-Morley experiment while simultaneously saving the ether hypothesis.

Einstein, of course, presented a far more elegant, and convincing, chain of argument for the necessity of the Lorentz transformation.

 

[timetraveldude]

WHY would one want to? The constancy of the speed of light was the experimental data that led to the Lorenz transformation. If the speed of light was not constant why would one want or need the Lorenz transformation?

I am not sure if you are correct. As I understand it, Maxwell's equations were not invariant under the Galilean transformations. Einstein felt that the laws are physics should be the same in all inertial reference frames. So either the Galilean transformations were wrong or Maxwell's equations.

 

[timetraveldude]

Simultaneity post

Maybe you should see my other post in this forum. The relativity of simultaneity argument is based on light but if you use sound instead of light all inertial observers will agree whether two events were simultaneous. On the other hand if you can derive the Lorentz transformations without using the 2nd postulate (i.e. through a different means) then the relativity of simultaneity is preserved.

 

[timetraveldude]

I'm not sure what you (timetraveldude) have in mind here. The Lorentz tranformation is equivalent to the statement that the speed of light is the same to all inertial observers. I don't think the question really makes sense. Any postulate that you can use as a starting point for a derivation of the Lorentz tranformation will include that stuff about the speed or light, whether it's apparent or not.

This is not true. The Lorentz transformations only describe distance and time transformations between coordinate systems if there is a universal speed limit.

The impression I get from the people here is that they have memorized the details but do not know how to think critically. I have proved that you can derive the Lorentz transformations without utilizing the 2nd postulate of SR. Am I the only one? There was a paper in 1972 that did this exact thing but using a different method from mine. The idea is basically that if you are teaching a mechanics course and want to incorporate SR without reference to electro-dynamics you need a different way of introducing the Lorentz transformations.

 

[jcsd]

The problem is for you argument to have any validity you need something that travels on a null worldline, light does whereas sound does not.

 

[Fredrik]

This is not true. The Lorentz transformations only describe distance and time transformations between coordinate systems if there is a universal speed limit.

OK. When I think of "the speed of light" I don't even think of light. To me "the speed of light" is just a name that represents the universal speed. That's why I thought your suggestion sounded so strange. But OK, you don't want to do a derivation that doesn't involve a universal speed, you want to do a derivation that doesn't involve light (or anything else from the classical or quantum theory of electrodynamics). That's a different story.

I've seen a derivation like that once, or at least a part of it. Unfortunately I don't remember who wrote it. Their idea was to assume nothing at all about the properties of space, except rotational and translational invariance, and try to determine the most general rule for addition of velocities. The result they got was of course not the non-relativistic "u+v" but a relativistic-looking formula (u+v)/(1+Kuv), where K was a non-negative real number that couldn't be determined from the postulates they had started with.

The constant K can of course be identified with 1/c², but there's no need to do that just yet. Instead, we can use the velocity addition formula as the starting point of a derivation of the rest of special relativity, including the Lorentz transformation.

 

[timetraveldude]

The problem is for you argument to have any validity you need something that travels on a null worldline, light does whereas sound does not.

My argument is perfectly valid. Again you are using as evidence what I am questioning. If you want to remain in the realm of logical thinkers you need to understand this is not acceptable.

 

[jcsd]

My argument is perfectly valid. Again you are using as evidence what I am questioning. If you want to remain in the realm of logical thinkers you need to understand this is not acceptable.

The problem is that sound is totally irrelevant to considerations of simultaneity in relativity. Special relativity is inertanally self-consistent so any questioning of well-know results such the fialure of simulatenity at distance cannot come from poniticating it must come from experimental evidmnece, yet you offer none.

 

[cragwolf]

To answer the OP, yes, there are many alternative derivations of the Lorentz Transformations, some of which do not assume the constancy of the speed of light. I'll just list a few that don't assume the constancy of the speed of light:

Y.P.Terletskii, "Paradoxes in the Theory of Relativity", Plenum Press, New York, 1968, P17
R.Weinstock, "New Approach to Special Relativity", Am. J. Phys. 33 640-645 (1965)
A.R.Lee and T.M.Kalotas, "Lorentz Transformation from the First Postulate", Am. J. Phys. 43 434-437 (1975)
J.M.Levy-Leblond, "One more Derivation of the Lorentz Transformation", Am. J. Phys. 44 271-277 (1976)
A.Sen, "How Galileo could have derived the Special Theory of Relativity", Am. J. Phys. 62 157-162 (1994)
J.H.Field, "Space-Time Exchange Invariance: Special Relativity as a Symmetry Principle", [http://arxiv.org/physics/0012011 ]

 

[timetraveldude]

To answer the OP, yes, there are many alternative derivations of the Lorentz Transformations, some of which do not assume the constancy of the speed of light. I'll just list a few that don't assume the constancy of the speed of light:

Y.P.Terletskii, "Paradoxes in the Theory of Relativity", Plenum Press, New York, 1968, P17
R.Weinstock, "New Approach to Special Relativity", Am. J. Phys. 33 640-645 (1965)
A.R.Lee and T.M.Kalotas, "Lorentz Transformation from the First Postulate", Am. J. Phys. 43 434-437 (1975)
J.M.Levy-Leblond, "One more Derivation of the Lorentz Transformation", Am. J. Phys. 44 271-277 (1976)
A.Sen, "How Galileo could have derived the Special Theory of Relativity", Am. J. Phys. 62 157-162 (1994)
J.H.Field, "Space-Time Exchange Invariance: Special Relativity as a Symmetry Principle", [http://arxiv.org/physics/0012011 ]

Thank you. You are the first person I have met in this thread who actually thinks.

 

[Garth]

Have you tried K calculus? Developed by Milne in his Kinematic cosmology in the 1930's and used by d'Inverno in "Introducing Einstein's Relativity".
Garth

 

[timetraveldude]

Have you tried K calculus? Developed by Milne in his Kinematic cosmology in the 1930's and used by d'Inverno in "Introducing Einstein's Relativity".
Garth

Thanks. WOW! Two useful posts in a row. This is a violation of statistics.

It is amazing that the people who make the most useless posts are the so called mentors.

 

[Hurkyl]

Just to make sure you realize, these approaches will derive the constancy of the speed of light.

 

[quantumdude]

Fredrik: The Lorentz tranformation is equivalent to the statement that the speed of light is the same to all inertial observers.

timetraveldude: This is not true.

Yes, it is true.

Einstein started with the constant speed of light postulate and Maxwell's equations. Requiring the latter to be covariant, he derived the Lorentz transformation. But you could just as easily start from the Lorentz transformation and derive from that the speed of light postulate, and of course the covariance of Maxwell's equations.

It is amazing that the people who make the most useless posts are the so called mentors.

Lose the attitude.

 

[quantumdude]

Never mind folks. Timetraveldude is just another alias for our beloved protonman and tenzin.

He won't be joining us in this thread anymore.

 

[fixizrox]

Yes, it is true.

Einstein started with the constant speed of light postulate and Maxwell's equations. Requiring the latter to be covariant, he derived the Lorentz transformation. But you could just as easily start from the Lorentz transformation and derive from that the speed of light postulate, and of course the covariance of Maxwell's equations.

The lorentz transformations say nothing about the speed of light being the same in all inertial reference frames. I derived the LT without any reference at all to light.

 

[quantumdude]

The lorentz transformations say nothing about the speed of light being the same in all inertial reference frames.

Of course they do. As has been said repeatedly, you can derive the speed of light postulate from the LT.

I derived the LT without any reference at all to light.

Good for you. Now use it to derive the relativistic velocity addition law, and then then Einstein's speed of light postulate.

 

[quantumdude]

Has anybody come up with a way to derive the LT not based on the constantcy of the speed of light in all inertial reference frames?

I derived the LT without any reference at all to light.

Looks like you had the answer to your own question all along.

It makes one wonder why you bothered to ask it here.

 

[Fredrik]

Of course they do. As has been said repeatedly, you can derive the speed of light postulate from the LT.

I think timetraveldude knows that. I think he understands that the Lorentz transformations imply the existence of a velocity that's the same to all inertial observers. What he's trying to make a big deal of here, is that there's nothing in the Lorentz transformations that explicitly mentions light. Sure, they mention the speed of light (the universal velocity), but they don't say that this is the same thing as the speed of light (photons/electromagnetic waves). That's why he won't accept that the Lorentz transformation is equivalent to the speed of light postulate.

 

[Fredrik]

Timetraveldude, if you have derived the Lorentz transformations without using light, that's not really a big deal. As I mentioned before, the most general velocity addition law that's consistent with rotational and translational invariance has been shown to be (u+v)/(1+Kuv), where K is just a constant to be determined later.

If K is not 0, we can define a constant c that has dimensions of velocity: c²=1/K. The velocity addition formula can be used to derive the Lorentz tranformations, and this will lead us to the idea of Minkowski space. Now, if we try to construct a theory of light that's consistent with the idea that Minkowski space is an accurate representation of space and time, we will eventually end up with QED, Maxwell's equations, and the identification c = the speed of light.

 

[Haelfix]

Umm, lorentz invariance is *required* for Maxwells equations to be self consistent. Now, the identification is that the universal speed is precisely C, the somewhat arbitrary constant that appears in those equations.

You are free of course to pick a higher value.. call it vmax, but then you contradict experiment. So experiment ultimately fixes the consistency of the theory.

 

[Fredrik]

I agree. I just think it's interesting that even if you've never heard of Maxwell's equations, and have no idea what the speed of light is, it's still possible to realize that SR (with some universal velocity) is at least a possibility.

 

[Stingray]

I agree. I just think it's interesting that even if you've never heard of Maxwell's equations, and have no idea what the speed of light is, it's still possible to realize that SR (with some universal velocity) is at least a possibility.

Interestingly, you can go even further in that "what if" scenario. If you write down Newtonian physics (including Newtonian gravity, but not electromagnetism) in a spacetime-like geometric form, you find that it is very natural to add a 1-parameter extension to the theory. If that parameter is set equal to 1/c^2, you have general relativity!

 

[fizixrox]

Timetraveldude, if you have derived the Lorentz transformations without using light, that's not really a big deal. As I mentioned before, the most general velocity addition law that's consistent with rotational and translational invariance has been shown to be (u+v)/(1+Kuv), where K is just a constant to be determined later.

If K is not 0, we can define a constant c that has dimensions of velocity: c²=1/K. The velocity addition formula can be used to derive the Lorentz tranformations, and this will lead us to the idea of Minkowski space. Now, if we try to construct a theory of light that's consistent with the idea that Minkowski space is an accurate representation of space and time, we will eventually end up with QED, Maxwell's equations, and the identification c = the speed of light.

Fredrik,

I assumed that the idea that you can derive an equation for a universal speed limit independent of the constantcy of the speed of light was not original. But I teach SR and have come to know it a bit. It often occurred to me to think of the 'c' in the Lorentz transformations as a universal speed limit and no t the speed of light. This was the result of contemplating some of the pedagogical derivations of time dilation and Lorentz contraction. They were not rigorous enough to be conclusive.

I then did a literature review and found that this idea has been taken up and written about in academic journals.

I would be curious to know if you have any references to "the most general velocity addition law that's consistent with rotational and translational invariance has been shown to be (u+v)/(1+Kuv)" as you mentioned above.

Congrads on an excellent post.

 

[Project]

Has anybody come up with a way to derive the LT not based on the constantcy of the speed of light in all inertial reference frames?

There are many derivations of the Lorentz transformation that do not use Einstein's second postulate. See http://www.everythingimportant.org/relativity/special.pdf for example.

 

[Project]

There are many derivations of the Lorentz transformation that do not use Einstein's second postulate. See http://www.everythingimportant.org/relativity/special.pdf for example.

Also see http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000043000005000434000001

 

[zwicky]

Just to make sure you realize, these approaches will derive the constancy of the speed of light.

I think these approaches will derive the existence of a fundamental velocity scale instead. Experimentally equal to the speed of light in vacuo for not to short wave length.