Based on the wave equation, some authors say that the distinction is based on invariance as opposed to conformal invariance Yet, if one looks at the line elements, (and I haven't seen that done before) Lorentz's ds^2=c^2dt^2-dr^2 is invariant while Voigt's is not ds^2=c^2dt^2/gamma^2-dr^2(1-v^2/c^2 sin^2A), where gamma is the Lorentz factor and A is the angle between the two velocity planes.. That is, ds^2=0, Lorentz predicts that the speed of light is c while Voigt's gives 1/gamma times the inverse transform (c^2-v^2)/(1-v^2/c^2 sin^2A), of the square of the velocity which is the relativistic difference of two velocities. Note that the relativistic difference of two velocities cannot be expressed by the relativistic composition law when the velocities lie in different planes. The correct relativistic composition law is given by (16.11) in Fock's book The Theory of Space, Time and Gravitation, and it will only reduce to linear relations for parallel (16.12) and perpendicular velocities (16.13). [This places in doubt his relations (16.07) and (16.08).]
The point being that Voigt gives an inverse relation to that given by the relativistic composition of velocities (up to a multiplicative constant, 1/gamma). So if one is greater than the speed of light the other must be less than c. Moreover, the square of the velocity composition law, is the hyperbolic metric of velocity space. This is mentioned in Fock's book, and developed further in A New Perspective on Relativity. But Voigt says that its inverse is also a velocity, up to a multiplicative factor of gamma!
Since we are always dealing with out-and-back propagation, the arithmetic-geometric-harmonic mean equality holds. If the arithmetic mean is
1/2(c+v+c-v)=c, the other two will be less. On the contrary if c+v and c-v are inverse velocities, then the harmonic mean is c, while the other two are greater than c. In other words, the one way speed of light is independent, and all that must be satisfied is that the out and return trips average to c, which is an experimentally determined fact. Therefore, in the extreme case, one speed can be infinite while the other c/2 to satisfy the harmonic mean of zero, 2/(2/c+1/infinity)=c. Obviously, this implies a space synchronization instead of the velocity synchronization used by Einstein.
This applies to the Doppler effect, which is what Voigt's original intention. According to Einstein, v is the velocity relative to an observer. In Essen's (The Special Theory of Relativity: A Critical Analysis) words: "Instead of obtaining values of c+v or c-v for the velocity of light, an observer obtains the values [sic] c. Thus it appears that there should be no Doppler change in frequency, and yet this effect is known to exist."
This has also been emphasized by Beckmann in Einstein Plus Two: "The velocity of light is constant with respect to the local gravitational field through which it propagates."
Voigt's transform was the same as that now known as the Lorentz Transform but with an additional scaling factor of gamma applied to all coordinates:
https://en.wikipedia.org/wiki/Woldemar_Voigt#The_Voigt_transformation
That means most aspects are the same when applied to EM but it fails when we apply it to matter. The gamma factor in the y' and z' terms means a moving object would expand or shrink in the transverse direction resulting in the "Rod and Tube Paradox" shown in the attached illustration.
Question: Do both transforms predict that the speed of light is c regardless of the fact that the observer is traveling a constant speed v?
BL: Do both transforms predict that the speed of light is c regardless of the fact that the observer is traveling a constant speed v?
Yes, the t' and x' coordinates are both altered by the same factor which cancels when you calculate dx/dt.
The change of scale for y' and z' doesn't matter for an EM plane wave of course but it obviously fails when you try to extend it to matter.
Just set the line element for the Voigt transform equal to zero: you don't get c as the speed of light like you do in the Lorentz transform. The Lorentz factors don't cancel out in the line element in the Voigt transform.
How can x'/t'=x/t=c, where c is a universal constant? That would imply that x and t, or x' and t' are both not independent.
Yet, if x'=x-vt then dividing through by c gives t'=t-vx/c^2, when the above relations are used, which are two of the Voigt equations. They are also Lorentz's because, as you say, the Lorentz factor cancels out when you take their ratio. It is no wonder that the velocity addition law indicates that c is the ultimate speed. It is built in.
But, when you solve ds=0 for Voigt you get the inverse velocity to within a Lorentz factor, G, than you would if you were to solve the velocity addition law:
G_3^=G_1G_2(1-v_1v_2/c^2), so if one is less than c, the other must be greater than c.
BL: But, when you solve ds=0 for Voigt you get the inverse velocity to within a Lorentz factor ...
Start with the Lorentz version:
ds2 = dt2 - dx2
For light, dt = dx hence ds =0.
For Voigt, multiply both terms by ɣ:
ds2 = (ɣ dt)2 - (ɣ dx)2 = ɣ2 (dt2 - dx)2 = 0
You've made a mistake in you maths somewhere. Check the Wikipedia reference I gave you before, you'll see:
Wiki: "If the right-hand sides of his equations are multiplied by ɣ, they become the modern Lorentz transformation. ... Since Voigt's transformation preserves the speed of light in all frames, the Michelson–Morley experiment and the Kennedy–Thorndike experiment can not distinguish between the two transformations."
Yes, you are right. I dropped the time-slippage term, vx/c^2. The same is done when showing that the Fresnel coefficient emerges from the relativistic addition law by dropping powers of v/c greater than first. Von Laue used this in support of relativity. But, if this is the case, the Fresnel coefficient is not exact, and must contain higher order terms in v/c. None have been reported, at least to the best of my knowledge.
From the addition law, Einstein claimed that c is a universal constant, the same in all directions. So you need terms of higher power in v/c to establish that. So the latter conclusion is wrong if Fresnel is exact.
The problem lies with the frame transformation equations. x and t are no longer independent because their ratio is always c in any frame. So x'=G(x-ut) and t'=G(t-ux/c^2), G the Lorentz factor. Introducing this into the two equations give x'=Kct and t'=Kt, where K=(1-v/c)^(1/2)/(1+v/c)^(1/2), the relativistic Doppler term. Eliminating t in the first gives x'/t'=c. This is for Lorentz. For Voigt, y''=y/G and z'=z/G. Their time derivatives, dy'/dt'(1-u/c)=dy/dt/G or dy'/dt=K.dy/dt, which show that in the y- and z-directions, the velocities are Doppler shifted. The speed of light is not invariant in the y- and z-directions.
Fresnel's is a first-order optic effect whereas Einstein's is a second-order effect in which the sum of back-and-forth speeds of light is always c.
The Wiki article you cite says that Voigt and Lorentz can be distinguished by the magnitude of time dilation. Both transforms are exacerbated by the fact that x and t are not independent variables. So to derive the time dilation effect you cannot consider x as fixed and consider two different times [cf. French, "Special Relativity", p. 100]. If x is fixed so is t because their ratio, c, is constant. In Lorentz's case t'=K t, while in Voigt's case t'=t(1-u/c), the distinction between relativistic and classical Doppler shifts. However by holding x constant, you get t'=Gt, which is what the Wiki article claims has been confirmed by the Ives Stilwell experiment. Ives had something much different to say.
Under the condition that Fresnel holds exactly (for no exceptions have been found), the two expressions for t'/t are equivalent. No higher powers than first are considered in K, and the square root is expanded to first order in v/c.
In fact, you can derive the Fresnel coefficient the invariance of the time difference in Hoek's experiment under velocity v inversion [Beckmann, "Einstein Plus Two", p. 35]:
[c/n+(e-1)v]^(-1)-[c-v]^(-1)=[c/n-(e-1)v]^(-1)-(c+v)^(-1)
where n is the index of refraction and e is Fresnel's coefficient. Neglecting higher powers of v/c gives e=1-1/n^2. However, it is usually derived in expanding the relativistic law of addition [French, p. 132] and neglecting higher powers in v/c.
That the two-way speed of light is c follows from the harmonic mean, H=c::
[c/n+v/n^2]^(-1)+(c-v)^(-1)=(n+1)/H,
again when terms of higher powers in v/c are neglected.
The arithmetic mean is A=[c-v(1-1/n)]/n, and the geometric mean G={c[c-v(1-1/n)]/n}^(1/2), again provided higher powers in v/c are neglected. Thus G^2=A.H, when higher powers in v/c are neglected. But, now there is no limit on the one-way speed of light. In fact, from the harmonic-geometric-arithmetic mean equality (two independent variables) if the harmonic mean is c, the other two means can be greater than c.
The velocity addition law for v_1, v_2 is (G is again the Lorentz factor)
G'=G_1G_2(1-v_1.v_2/c^2)
and if this is to be the same the inverse of the speed obtained from ds=0 (to within a factor G), the time slippage term ux/c^2 has to be dropped. But this is not because it is of higher power than v/c, rather because it gives the Lorentz time dilation factor. As I mentioned above the elimination of the time-slippage term is effectuated by considering two different times as the same position in space [French, p. 100]. But, x and t are not independent because their ratio must be c. Fixing x means fixing t also.
First-order and second-order optical effects have been confused, which is further aggravated by the fact that c is a universal constant so that x and t, (or x' and t'), are not independent variables. Note the first-order optical effects place no condition on the speed of light only the second-order do.
Dear Berard,
BL: The speed of light is not invariant in the y- and z-directions.
The same gamma factor applies to y and z as to x so the speed should also be invariant in those directions.
BL: The Wiki article you cite says that Voigt and Lorentz can be distinguished by the magnitude of time dilation. ... In Lorentz's case t'=K t, while in Voigt's case t'=t(1-u/c), the distinction between relativistic and classical Doppler shifts. However by holding x constant, you get t'=Gt,
The Ives-Stilwell experiment was designed to measure t'/t directly. It is usually used to distinguish between the Galilean Transform where t'=t and the Lorentz Transform which includes gamma, but if it can do that, it must also be able to distinguish between Lorentz and Voigt which has a gamma2 factor. I think you need to calculate the Doppler shift as a function of the particle velocity for each of the three hypotheses. I don't see that you need Fresnel coefficients to do that, I think you're making the task more complex than it needs to be.
Dear George:
GD: The same gamma factor applies to y and z as to x so the speed should also be invariant in these directions.
In Lorentz's case, the Gs are in x and t while in Voigt's case they are in y and z.
Consider first Voigt:
x'=x-ut
t'=t-ux/c^2
Introduce the former into the later, divide by t' and use x'/t'=c. There then results:
t'=(1-u/c)t. (1)
Introduce this into the first equation reduces it to an identity c-u=c-u when x/t=c is used. For y or z, you have y'=y/G. Dividing by t, and using (1)
I get (y'/t')(1-u/c)=(y/t)/G. Thus, y'/t'=(y/t)K^(-1). The velocities in the y- and z- directions are Doppler shifted. Unlike the velocities x/t=c and x'/t'=c, the velocities y'/t' and y/t are not invariant.
Now do the same for Lorentz:
x'=G(x-ut)
t'=G(t-ux/c^2).
It is only when the slippage term is neglected that I get t'=Gt. This is not (1) above. Books on relativity use two times and one x to subtract the x-dependence out. But this is incorrect since x and t are not independent.
Alternatively eliminating x with the aid of the first equation gives
t'=G(1-u^2/c^2)t - ux'/c^2.
Now neglect of the slippage term gives the inverse relation:
t'/t=1/G
which, again, is not (1) above!
The point I made before is that if Fresnel is exact, the velocity addition law (parallel velocities) can't be because the Fresnel coefficient would be e=1-1/n^2-v/nc) to second order in v/c, i.e. a function of the velocity. One must therefore distinguish between first- and second-order optical effects.
BL: In Lorentz's case, the Gs are in x and t while in Voigt's case they are in y and z.
Using loose wording, the coefficients for Lorentz are:
x -> x' : gamma
t -> t' : gamma
y -> y' : 1
z -> z' : 1
while for Voigt they are:
x -> x' : gamma2
t -> t' : gamma2
y -> y' : gamma
z -> z' : gamma
In other words Voigt = Lorentz followed by a re-scaling of all 4 by an additional gamma term. Rescaling both distance (in any direction) and time by the same factor cannot change velocity, twice the distance in twice the time is the same speed.
BL: The point I made before is that if Fresnel is exact, the velocity addition law (parallel velocities) can't be ...
The law of additions is exact (the clearest version IMHO is the simple addition of rapidities). I haven't studied Fresnel coefficients so I can't say if it is common to use an approximation but that might be the case.
Out and back are both straight line paths in SR so there is no relevance for precession. The question is about the difference between the Lorentz and Voigt transforms.
I've seen your contributions on many threads and in very few was it relevant. I just get tired of "mission posters" butting in everywhere to promote their pet theories and disrupting what might otherwise have been a sensible and professional discussion. RG has too many of those, I hope you won't join their ranks.
Best regards
George
My apologies to Bernard as this is somewhat off-topic.
Dear Preston,
PG: An electromagnetic wave has structure. It has an orientation with respect to the direction of propagation.
Certainly, the field vectors are perpendicular to the direction of propagation.
PG: Considering a single wave, one can label the component closest to the destination as A, and the furthest from the destination and label it B.
As the fields are transverse, both are equally distant. Of more importance is the polarisation which can be clockwise or anti-clockwise but there is no part that reaches the reflector before the other, unless you are talking about a train of waves.
PG: Experiments being discussed by Dr. Lavenda involve the two way speed of light.
He does mention that but the original question was about the Lorentz and Voigt transforms which operate solely on frame coordinates, they don't even involve light (and if they did it would be one way, event to event propagation). That was the point of my reply, the difference between the transforms is a scaling factor applied to all four coordinates.
Thomas precession applies to curvilinear paths so I can certainly see how it would be of interest in say the effect on light polarisation during gravitational lensing, but not to the Lorentz Transforms which don't necessarily even need to be applied to light.
PG: Your perspective about "mission posters" and "pet theories" is incredibly judgmental and condescending.
As I said, I hope that is not going to apply to you.
PG: Being completely comfortable in the tremendously inadequate current mainstream physics, and without a new theory means that you will not have solutions to questions, such as Dr. Lavela's about differences between gravitational waves and electromagnetic waves or ether and space time.
On the contrary, I'm working on outreach explanations of gravitational waves myself, so if he has questions, while I am no expert, I could probably help with that. Since the observations by the LSC show no deviations from GR, there is no justification for any new theory. That's a major problem, without some observed discrepancy from GR, there is no way to test any new theory so an explanation based on GR is as good as any other.
Please let me recap.
1. There seems to be confusion between a first-order and second-order Doppler effect. Instead of looking at the hyperbolic line element, consider the relation between the phases. In Voigt's case, the primed and unprimed variables will be related by the classical Doppler shift (1+v/c), while in Lorentz's case they will be given by the relativistic factor of the square root of the ratio (1+v/c)/(1-v/c). So they are not "equivalent". Essen asks if there is no speed greater than c what is c+v, which we know to exist from the Doppler effect.
2. GD claims that I am looking into technicalities when I compare Fresnel drag and the velocity addition law. First let me say that the addition law is planar, there are no angles between velocity planes. Fresnel's drag law was used in support of the velocity addition law by von Laue. This means that the quadratic term in the velocity must be neglected. Why? Although it is known that there is also an additional contribution due to dispersion, as derived by Lorentz, but the dispersive term is velocity independent. Taking the quadratic term into account, would make the drag force a linear function of the velocity. Speed increases and so does the force, bringing into play accelerations. We are outside the realm of special relativity. So the quadratic term must be neglected. Does that mean that the planar addition law is approximate? No, but it does not belong to Euclidean geometry.
3. The linear addition law was important to Einstein for from it he deduced that there can be no speed greater than c, and it is isotropic in all directions. But, this pertains to the two-way speed of light (c.f. Ives' paper below).
4. The one-way speed is a different beast. What is definitely correct is the Lobachevsky (velocity) line element because it corresponds to the Beltrami metric. It can be derived in many different ways (viz. "A New Perspective on Relativity") least of all from the velocity addition law even allowing the two velocities to lie in different planes. (This brings in an angle dependency between the two planes.) However, I have not been able to derive it from the Lorentz transform. If you consult Fock, "Space Time and Gravitation" you will see that the first (vector) equation in (16.01) is not what you would consider a Lorentz transformation of the vector r. I would have expected to see something like r'=r-vr/c. So I don't know how to get from the Lorentz transform to the hyperbolic line element. In my original question I can get it from Voigt at the expense of having to neglect the time slippage term. This quadratic term is analogous to the quadratic term that is neglected in the Fresnel drag derivation of the planar velocity addition law.
5. The hyperbolic line element can, however, be derived exactly from the Kennedy-Thorndike line element eq (1) of Phys. Rev. 42 (1932) 400. Actually, what you get is the inverse velocity. This is because they started out with a line element involving time contraction. Rather if you consider length contraction you come out with the inverse velocity as I show in "Seeing Gravity" eqn (5.61) where there is a misprint: "b" should be "v". The linear term in v/c cancels out when the back-and-forth times are summed. You then get the hyperbolic line element reported by Fock.
6. Ives considers the one-way speed of light in Proc. Am. Phil. Soc. 95 (1951) 125 entitled "Revisions of the Lorentz Transformations." In eqn (5) he considers different speeds for the out-and-back speeds of light. His eqn (7) is a statement of the arithmetic-harmonic-geometric equality (for two independent variables). He sets the geometric mean equal to the speed of light. Since the geometric mean is the smallest of the three, the others must be greater than c. He doesn't recognize this though. He relinquishes the definition of clock time in the Lorentz transform, x/c, and adds a factor involving "q", the "self-measured velocity of the moved clock." (I have no idea of what that is or where it came from.) So the Lorentz transform is not applicable, also according to Ives, for the one-way speed of light, and neither are any of the transformations with arbitrary k in them, where k is the Lorentz coefficient. (This includes the Voigt transformation.)
7. PG: Angles enter when the two velocity planes are not parallel. Thomas precession is derived from time derivative of the angle dependent term in the hyperbolic metric eqn (10.1.21) in "A New Perspective on Relativity." However, for rotational phenomena, the equivalence principle will not work. The pertinent line element happens to be also the Beltrami metric where observers on the rotating disc cannot distinguish their positions from the readings of their clocks and rulers because they shrink or magnify along with the inhabitants. (The Beltrami metric is conformal to the space component of the Schwarzschild inner solution line element eqn (9.6.3) in "A New Perspective..."
I missed this earlier, so let me recap too:
BL: 2. GD claims that I am looking into technicalities when I compare Fresnel drag and the velocity addition law.
You asked "What is the difference between the Voigt and Lorentz transforms?" and the answer I gave is that the Voigt transforms are the same as those named after Lorentz but with an additional scaling factor of gamma in all dimensions. I still believe that to be an accurate and complete response.
Since that extra factor creates a contradiction when applied to a rod passing through a tube (see the attached graphic), it passes through in one frame but is too big in the other, the Voigt transforms can be trivially dismissed as not describing reality, although they may be of academic interest if you want to investigate for example what effect they would have on Doppler shift or other effects like Fresnel drag or the velocity addition law.
Although both transformations are obtained using the idea of invariance, they are actually very different. Voigt derived his transformation by requiring that the wave equation governing the propagation of a signal in the elastic solid medium has the same form in the stationary and in the moving coordinate system. The Lorentz transform follows from the assumption that the shape of a spherical wavefront is preserved (“The Classical Theory of Fields” by Landau & Lifshitz, par. 4; the text just below eq. 4.2).
The most serious problem with the derivation of the Voigt transform is evident after reading of the original paper on the Doppler effect, where Voigt assumes that
x’=x-v t, t’=t-x v /c^2
in order to obtain the desired form of a wave equation in the ‘primed system’. The justification for this is: “da ja sein muss”, “as it must be”
A succinct summary of the relation between the Voigt and the Lorentz transforms is given by Pauli (1921):
“As long ago as 1887, in a paper still written from the point of view of the elastic solid theory of light, Voigt mentioned that it is mathematically convenient to introduce a local time t′ into a moving reference frames.... In this way the wave equation △φ − (1/c^2)∂2φ/∂t2 = 0 could be made to remain valid in the moving reference frame, too. These remarks, however, remained completely unnoticed, and a similar transformation was not again suggested until 1892 and 1895, when H. A. Lorentz published his fundamental papers on the subject.”
(From: Relativitätstheorie in Encyklopädie der Mathematischen wissenschaften, Teubner, Leipzig. English translation: Theory of Relativity 1958, Pergamon, New York)
Main difference between both transforms are:
-Voigt Transform does not generate a symmetry group
-The kinetic energy of the moving body and the Doppler Effect are not the same under the Voigt and Lorentz transforms
Surprisingly the mass-energy conservation law appears to be the same in both transforms (See Gluckman (1976): Voigt kinematics and electrodynamic consequences,Foundations of Physics, June 1976, Volume 6, issue 3, pp 305–316)
The topic of the Voigt transformation remains interesting from the point of view of the theory of conformal transformations.
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