Show that the light velocity c is constant under Lorentz transformations.

[peterjaybee]

Hi, this is a question from a practice paper I have. I can't think how to do this. As far as I'm aware this has to be assumed to derive the Lorents transforms, so it must be by definition true, making the question pointless. Does anyone have any thoughts or suggestions on this?

Regards,

Pete

 

[Dale]

Well, you can use the Lorentz transform to give dx' and dt' given dx and dt and v. Then simply set dx/dt = c and see what dx' and dt' equal.

 

[peterjaybee]

Its really obvious now you suggest that lol, but that's always the way with these things.

Many thanks, I would have been ripping my hair out the rest of the day without your help.

 

[utesfan100]

Actually, Lorentz assumed a preffered reference frame of the eather when originally diriving these equations. Einstein showed that the assumption of a prefferred reference frame is not needed with SR.

The process of this outline is from http://en.wikipedia.org/wiki/Lorentz_transformation

If one assumes that space is linear with time relative to velocity in an inertial frame (space is flat) then the following observations can be made:

(let [t,x,y] be in observer 1's frame and {t,x,y} be in observer 2's frame)

1) [t,Vt,0] = {t,0,0} - Invariance of the motion of observer 2
2) [t,0,0] = {t,-Vt,0} - Invariance of the motion of observer 1

The transformation will be a matrix A(V). Symmetry implies A(-V)=A^-1(v).

This is enough to find that A must take the form:

g=1/(1+k*V^2)^1/2
[t] = g*{t} + k*V*{x}
[x] = -g*{t} + g*{x}
[y] = {y}

for some k.

k=0 this becomes the Galilean transforms. MANY studies show that k = -1/c^2. This is a fundamental property of our space-time, which becomes the Lorentz equations you started with, without assuming that c is constant.

Your question is to now show that c must be constant if these transforms apply.

 

[bcrowell]

There are many different ways that this question could be interpreted. For instance, there are derivations of the Lorentz transformation that do not assume constancy of c as a postulate. A couple of examples of such presentations:

Morin, Introduction to Classical Mechanics, Cambridge, 1st ed., 2008

Rindler, Essential Relativity: Special, General, and Cosmological, 1979, p. 51

In this type of axiomatic framework, constancy of c is a theorem that has to be proved.

 

[starthaus]

Hi, this is a question from a practice paper I have. I can't think how to do this. As far as I'm aware this has to be assumed to derive the Lorents transforms, so it must be by definition true, making the question pointless. Does anyone have any thoughts or suggestions on this?

Regards,

Pete

You are right, the question is pointless indeed and your instructor should know better than that. The derivation of the Lorentz transforms assumes c=constant (see the second postulate of SR). So, the only thing that you will get, is a trivial confirmation that the derivation was consistent. Indeed, following DaleSpam's suggestion:

x'=\gamma(x-vt)
t'=\gamma(t-vx/c^2)

dx'/dt'=(dx-vdt)/(dt-v/c^2*dx)=(dx/dt-v)/(1-v/c^2*dx/dt)

If you make dx/dt=c you get dx'/dt'=c

But this is not a valid proof. You can never prove postulates. You can only disprove them and this is done by experiment only. Your instructor deserves an F. :-)

 

[starthaus]

There are many different ways that this question could be interpreted. For instance, there are derivations of the Lorentz transformation that do not assume constancy of c as a postulate. A couple of examples of such presentations:

Morin, Introduction to Classical Mechanics, Cambridge, 1st ed., 2008

Rindler, Essential Relativity: Special, General, and Cosmological, 1979, p. 51

In this type of axiomatic framework, constancy of c is a theorem that has to be proved.

Yes, there is quite a number of such papers that downgrade the second postulate to the rank of theorem.

 

[peterjaybee]

Thanks for all your help. Its interresting to see that you don't have to assume the 2nd postulate to arrive at the LT's.

 

[PhilDSP]

Actually, Lorentz assumed a preffered reference frame of the eather when originally diriving these equations. Einstein showed that the assumption of a prefferred reference frame is not needed with SR.

But isn't the "prefered" reference frame vastly more simple and superior to Einstein's conception which necessitates the need for a postulate regarding the speed of light?

Since Lorentz' and Poincare's frame for an aether is exactly the same as the object's (since an aether must be at rest for any and all objects) it disappears in effect. I haven't looked at the derivations listed where no postulate is needed for the speed of light but I suspect that they essentially utilize the "prefered" reference frame.

 

[starthaus]

But isn't the "prefered" reference frame vastly more simple and superior to Einstein's conception which necessitates the need for a postulate regarding the speed of light?

No, it isn't since all the efforts to detect it experimentally failed.

Since Lorentz' and Poincare's frame for an aether is exactly the same as the object's (since an aether must be at rest for any and all objects) it disappears in effect. I haven't looked at the derivations listed where no postulate is needed for the speed of light but I suspect that they essentially utilize the "prefered" reference frame.

None of them uses any preferred reference frame.

 

[PhilDSP]

But isn't the "prefered" reference frame vastly more simple and superior to Einstein's conception which necessitates the need for a postulate regarding the speed of light?

No, it isn't since all the efforts to detect it experimentally failed.

Efforts to detect what? A velocity of an EM medium with respect to an object absorbing radiation?

None of them uses any preferred reference frame.

I agree with you there. A so called "prefered" reference frame was not there in Lorentz' or Poincare's model. Seems to have been a misunderstanding from someone trying to understand the difference between what Einstein was saying versus vs. Poincare, Lorentz, Heavyside, Searle, Langevin, Fitzgerald, etc.,

 

[starthaus]

Efforts to detect what? A velocity of an EM medium with respect to an object absorbing radiation?

Efforts in detecting the "preferrential" frame.

I agree with you there. Something called a "prefered" reference frame was never there in Lorentz' or Poincare's model.

It has always been in Lorentz' model.

Seems to have been a misunderstanding from someone trying to understand the difference between what Einstein was saying versus vs. Poincare, Lorentz, Heavyside, Searle, Langevin, Fitzgerald, etc.,

Yes, you definitely need to read on these things. Einstein did away with the "preferrential" frame. His theory survived while all the others fell by the wayside.

 

[PhilDSP]

It has always been in Lorentz' model.

Can you explain that mathematically?

 

[starthaus]

Can you explain that mathematically?

Here is a very good reading.

 

[PhilDSP]

I think there is a misunderstanding there that involves the particular interpretation that because the medium can impart motion to a physical object it must itself undergo motion as a reaction. Granted even Poincare at one time believed that. But the mathematics doesn't contain that notion from what I have seen. And from what I've read, all of the prominent physicists I mentioned eventually discarded the notion.

 

[starthaus]

I think there is a misunderstanding there that involves the particular interpretation that because the medium can impart motion to a physical object it must itself undergo motion as a reaction. Granted even Poincare at one time believed that.

Not for long. He switched to SR after the Einstein 1905 paper.

And from what I've read, all of the prominent physicists I mentioned eventually discarded the notion.

Yes.

 

[PhilDSP]

One fundamental problem with Einstein's SR is that not only is the speed of light sometimes not equal to c, it effectively never is. The preeminence of refraction and dispersion prevent it from being so because the universe is not devoid of charge. Someone who fully knows the physics of dispersion can understand this. Einstein's SR is a quick and dirty approximation involving c that is close enough for most of the calculations we need to do in a comparative vacuum that it becomes very useful. But to base an absolute reference on an approximation is utimately begging for trouble. Lorentz theory in its later form is not plagued by that. But these considerations are probably more advanced than the OP is interested in.

 

[Chronos]

Einstein proposed spacetime twists and turns to accommodate c. This has been upheld in a variety of ways over the past century. Is 'c' a fundamental constant? That is unknowable. Is it consistent with other presumed fundamental properties of the universe - yes.

 

[Fredrik]

One fundamental problem with Einstein's SR is that not only is the speed of light sometimes not equal to c, it effectively never is. The preeminence of refraction and dispersion prevent it from being so because the universe is not devoid of charge. Someone who fully knows the physics of dispersion can understand this. Einstein's SR is a quick and dirty approximation involving c that is close enough for most of the calculations we need to do in a comparative vacuum that it becomes very useful. But to base an absolute reference on an approximation is utimately begging for trouble. Lorentz theory in its later form is not plagued by that. But these considerations are probably more advanced than the OP is interested in.

That's not a fundamental problem with SR. It's not a problem at all. And SR isn't a "quick and dirty" anything. It's a beautiful example of a classical theory, and it's also a mathematical framework in which both classical and quantum theories of matter and interactions can be defined.

SR isn't a theory about light (even though both classical and quantum theories of light can be defined in the framework of SR). It's a theory about a flat spacetime with isometries that leave a certain speed invariant. The only thing weird about all of this is that this invariant speed goes by the unfortunate name "the speed of light".

 

[PhilDSP]

Einstein proposed spacetime twists and turns to accommodate c.

Okay, that's a valid approach I think. But that means you have to rethink the definition of group velocity and phase velocity if you want to utilize SR in conjunction with wave mechanics. I don't believe anyone has tackled that.

 

[starthaus]

Einstein's SR is a quick and dirty approximation involving c that is close enough for most of the calculations we need to do in a comparative vacuum that it becomes very useful. But to base an absolute reference on an approximation is utimately begging for trouble. Lorentz theory in its later form is not plagued by that.

This claim is false, it has just been proven to you by Chronos and Fredrik.

But these considerations are probably more advanced than the OP is interested in.

I am interested in these claims, what makes you think that? Please explain.

 

[PhilDSP]

Yes, I agree with Chronos and Fredrik that SR is consistent and valid within its own framework. The problem as I see it now is that naive hybrid combinations of SR principles or mathematics with standard EM or optical methods will produce at best approximate results. And probably lead to a lot of confusion and paradoxes. That's because wave mechanics and optical methods generally make an implicit assumption that the Lorentz model applies.

One simple thing to consider is that according to Lorentzian or Maxwellian mechanics, wave propagation occurs at group velocity. Group velocity is also the rate at which a de Broglie "Matter Wave" is propagated. It's only when group velocity equals phase velocity that propagation occurs at what is known as c. Since SR on the other hand specifies that propagation occurs at c, then group velocity must take on some other meaning.

 

[starthaus]

One simple thing to consider is that according to Lorentzian or Maxwellian mechanics, wave propagation occurs at group velocity. Group velocity is also the rate at which a de Broglie "Matter Wave" is propagated. It's only when group velocity equals phase velocity that propagation occurs at what is known as c. Since SR on the other hand specifies that propagation occurs at c, then group velocity must take on some other meaning.

"Lorentzian mechanics", predicts the same exact results as SR, modulo the "presence" of the ever - elusive "preferrential" frame.
On the other hand, SR has the same exact separate concepts of group , phase and propagation velocity as the other two fields you mentioned.
So, it isn't clear what you are claiming.

 

[PhilDSP]

The difference between Einsteinian SR and that of Lorentz, as I see it, is that Lorentz states that the only bound reference is between a single particle and the medium - that the wave mechanics are such that the medium is stationary with respect to the instantaneous state of the particle. It doesn't make any (simple) claims about how another particle interacts with respect to the first particle. You have to calculate the relationships yourself. They are derivable but dependent.

In a sense that is a "prefered" reference that precludes an immediate cross-reference between particles. So at least in a superficial sense, Einstein had a point in saying so. But apparently in the Lorentz model it is not merely a "prefered" reference, it is a necessary one.

Einstein on the other hand posited that the reference must be based on the instantaneous state between the 2 particles. The state between them is not dependent on the state between each particle and the medium. That would be equivalent to what Lorentz proposed if EM propagation always occurred at c.

A Lorentzian approach means that propagation speed can take on various values while it can't in Einstein's approach. The implication with Einstein is that the concepts or measuring apparatus for time and distance vary rather than their values.

 

[Dale]

Your understanding of both Lorentz and Einstein is atrocious. What you describe here is not what either of them described. PF is for discussing mainstream science, not personal theories like this.

 

[PhilDSP]

I'm not launching a personal theory. I only want to understand the essentials of the situation. But yes, I've chosen to recast the terminology and to some extent the concepts in a way that is neutral to either viewpoint and that is a bit more specific to the issues as I perceive them. There is a background there that obviously most here have not heard or considered before. If this is too far a stretch for this forum then I'm satisfied to let it be.

 

[Dale]

When you "recast the terminology and to some extent the concepts" you are by definition launching a personal theory. Your description of both SR and LET is so far removed from the mainstream as to be completely unrecognizable.