Rjbuenker
RE: "Derivation" section of Length contraction: Why not just use Einstein's two postulates directly?
editThe current discussion of Fitzgerald-Lorentz length contraction in Wikipedia is based entirely on the Lorentz transformation (LT) of the special theory of relativity (SR). I object in particular to the section entitled "Derivation" which begins with "Length contraction can simply be derived from the Lorentz transformation as it was shown, among ..."
I would like to point out that it is possible to use Einstein’s two postulates of relativity directly to predict changes in the lengths of objects upon acceleration without involving the LT.
The method of choice for measuring distances is the Global Positioning System (GPS). Atomic clocks are used to measure the elapsed time for light to travel between two points and this value is multiplied with the speed of light (c) to obtain the corresponding distance. Although this technique is designed to measure distances on the earth’s surface, it can also be adapted to make a definitive test of the relativistic length contraction hypothesis, as shown below.
The length of a metal bar is measured by using two identical (proper) clocks on the ground. The elapsed time for light to traverse the metal bar is found to be L/c s for both clocks, indicating that the length of the bar is L m at this point in the experiment. The bar and one of the clocks are then put into orbit and the measurement is repeated. Consistent with the relativity principle (Einstein’s first postulate 1), no change in the length measurement is found on the satellite. However, because of the effects of time dilation, it is known that the onboard clock has slowed by a factor of Q>1 (the effects of the gravitational red shift are neglected at this stage of the argument). The corresponding elapsed time on the clock left behind on the earth’s surface is therefore deduced to be Q times L/c s, a larger value than before. The speed of light on the satellite is still equal to c for the observer on the ground (Einstein’s second postulate 1). The conclusion is therefore that the bar has expanded as a result of being accelerated into orbit; its length has increased from L m to QL m. Moreover, the increase in length is the same in all directions because the local time measurement on the satellite is completely independent of the orientation of the metal bar to the observer on the ground. The observer on the satellite is unaware of this change, which simply means that the lengths of all objects on the satellite have increased by the same factor (uniform scaling of distance and time).
Confirmatory evidence is found in the results of transverse Doppler measurements. They show that a decrease in light frequency (caused by time dilation at the light source) is always accompanied by a corresponding increase in the wavelength of the radiation. Again, the effect is the same in all directions. This result is clearly a consequence of the constancy of the speed of light, just as in the GPS example considered first. The slowing down of clocks can be looked upon as an increase in the unit of time in the moving rest frame. In order for the speed of light to be unaffected by this change, it is obviously necessary for the unit of velocity to remain constant, which means that the unit of distance, i.e. the length of the meter stick, must increase in the same proportion as the unit of time.
The above examples show that isotropic length expansion accompanies time dilation, not the anisotropic length contraction predicted by Einstein in his 1905 paper.1 Deduction of length expansion is based solely on the two relativistic postulates (relativity principle and the constancy of the speed of light). Since the latter have received extensive experimental verification, not the least through the experience with GPS distance measurements, there is no reason to doubt the correctness of this conclusion.
What it shows is that the theory of special relativity is not internally consistent. If one uses the above postulates directly to predict changes in length upon acceleration, the answer is opposite to what is deduced on the basis of the Lorentz transformation (LT). There is a simple explanation for this discrepancy. Einstein made an assumption in deriving the LT which was not declared as such. He claimed [see the four equations at the bottom of p. 900 in his original paper1] that a function φ defined there only depends on v, the relative speed of the two rest frames under discussion (S and S’). One obtains a qualitatively different result if one chooses φ instead so as to satisfy the basic assumption of the GPS methodology: the rates of clocks in relative motion are strictly proportional to one another, i.e. t=Qt’ in the notation used in the above example. The resulting alternative LT still satisfies Einstein’s two postulates, but predicts that the lengths of objects expand rather than contract when clock rates slow as a result of acceleration. More details may be found elsewhere.2
1 A. Einstein, Ann. Physik 17, 891 (1905).
2 R. J. Buenker, Apeiron 15, 254 (2008).
Rjbuenker (talk) 17:21, 25 April 2011 (UTC)
The following exchange has taken place between Dr. Buenker and two colleagues, one (Dr. A) a distinguished author of a book on relativity and the other (Dr. C), an associate editor of a respected physics journal. The remarks from Dr. C below were sent directly to Dr. A.
Dr. C to Dr. A: To the observer in S, the ends of the bar are in motion. It’s not simple to deduce a length from this measurement, and that’s why you need the LT, and that gives contraction. There are two parts of the path, going in the direction of v and opposite v. And since the bar’s angle to v matters, the length will depend on the angle.
General Reply I: Dr. A simply assumes the LT is correct. He ignores the fact that Einstein used an undeclared assumption to arrive at the LT, and therefore that it is by no means proven that the LT is correct. The point of the present discussion is to show that Einstein’s two postulates in conjunction with the definition of time dilation obtain a different result, namely isotropic length expansion accompanying time dilation, without involving the LT. Dr. C does not deny this in the above response, which indicates that he has no counter-argument to the procedure that avoids the LT entirely. The following reply was sent directly to Dr. C:
Dr. Buenker to Dr. C (April 26, 2011): The point where we diverge in our views is when you say “To S, the ends of the bar are in motion…” The observer in S does not even have to see the bar to measure its length. He can deduce the value with arbitrary accuracy if he knows a) the elapsed time for light to traverse the bar that is measured locally in S’ and b) how much faster his proper clock in S runs than the proper clock in S’. He can wait for days before receiving (e.g. by e-mail) the value in a) to make his calculation. He also has to know what the “conversion factor” for elapsed times was at the time of the local measurement in S’.
The other point of disagreement is your position that the LT must give the correct answer. I invite you to look at Einstein’s 1905 paper on p. 900. He claims that the function φ defined there only depends on the relative speed v of S and S’. That is an undeclared assumption. It needs to be verified experimentally. It is possible to derive an alternative LT that also satisfies Einstein’s two postulates but is, at the same time, consistent with the assumption of strict proportionality of the rates of proper clocks in different inertial systems (t=Qt’). That version of the LT also leads directly to the conclusion that lengths expand when clock rates slow.
There is one other experimental fact that should be considered in the present context. Measurements of the transverse Doppler effect have shown that the frequency of light waves emitted from moving sources decrease by the same fraction as the corresponding wavelength increases. The observer moving with the light source does not notice any change in either quantity, so how does one explain this? Various authors have pointed out that the decrease in frequency can be understood as the result of the slowing down of all clocks in the rest frame of the light source. But if that’s true, and I believe it is, then by the same argument one has to conclude that the diffraction gratings in S’ have all increased in dimension by the same fractional amount in all directions as the frequencies have decreased. In closing, let me break down the argument in four easy steps: 1)The elapsed time ΔT measured locally for light to traverse the metal bar in S is L/c s (2nd Postulate); 2)The corresponding elapsed time measured locally in S’ after the bar is stationary there is also ΔT = L/c s (1st Postulate); 3)If the proper clocks in S’ have slowed by a factor of Q relative to those in S, the observer in S can safely deduce that the corresponding elapsed time on his proper clock is ΔT’ = QL/c s (Q>1, definition of the amount of time dilation); 4)The current length of the bar measured in S based on the elapsed time measurement in S’ is therefore cΔT’ = QL m, showing that isotropic length expansion has accompanied the time dilation in S’ (2nd Postulate again).
Dr. C to Dr. Buenker (April 28, 2011): Thanks for your quick and thoughtful response. We may have different and irreconcilable philosophies about the exercise of gedanken experiments in spacetime physics. To me, the only true results in frame S are from experiments done in frame S, not results inferred from an experiment in S' via rules that may or may not be in accord with the basic postulates. Inference from one frame's result to another is verboten---each frame's experience of a set of events must make sense in that frame. The only connection between frames is the identification of the same events (spacetime points) that have different (x,y,z,t) depending on the frame. For that reason, I am not inclined as you are to deduce the length of the bar in S' from surmises in frame S. Also, relative to your modified Lorentz transformation, I do not understand the operational meaning of u_x. Could you please explain what that quantity is?
Dr. Buenker to Dr. C (April 30, 2011): The practice that you want to exclude is used continuously in GPS technology. We wouldn’t have GPS devices in our cars today if local clock readings for events on satellites (S’) could not be reliably converted to readings on clocks located on the earth’s surface (S). We have to be consistent with our assumptions in making logical arguments. Once it is known that the conversion factor for clock rates on the satellite is Q> 1 in the GPS technology, it is imperative that we use the same value for this conversion factor in all other applications. To exclude this assumption in a justifiable manner, it is necessary to show why a different conversion factor needs to be used in the present case. You have failed to do this is your comments to date. I submit that the reason for this is quite simply that there is no justification for excluding this assumption. The quantity ux =x/t is the component of the velocity of the object being measured in S that is parallel to v (ux’ is the analogous quantity in S’). It appears in Einstein’s velocity transformation in all three equations:
ux’ = (ux – v )/(1 – v ux /c2), uy’ = uy/γ(1 – v ux /c2), uz’ = uz/γ(1 – v ux /c2).
The alternative LT is obtained by combining the VT with the GPS proportionality relation, t’=t/Q, which serves as the temporal equation in the latter transformation (no space-time mixing). The VT is used in textbooks to measure distances. It ensures that any two observers agree on the speed u of some object (including a light pulse, of course) that moves between two fixed points. The observer in S’ with the slower clock measures a shorter value for the latter distance as Δr’=u Δt’. The other one in S knows that the corresponding time on his clock is Δt=Q Δt’ (Q>1), so he concludes that the distance between the same two points is Δr = u Δt = u Q Δt’ = Q Δr’. Thus, Δr> Δr’: isotropic expansion, not anisotropic length contraction. Actually, the observer in S doesn’t have to assume anything about the relative rates of clocks in S and S’ in this example. He just has to use his own proper clock and length standard to measure the speed of the object and the elapsed time for it to travel between the two fixed points. When he does this, he will automatically obtain u and Δt. The two results for the respective distance measurements can be compared at some later date. If it does not turn out that Δr = Q Δr’, that would be proof that either the VT is not valid or that the conversion factor is something other than Q.
General Reply II (May 7, 2011): Dr. C has not replied to the main point in the discussion as of the present date, namely how to justify Einstein’s assumption that the function φ defined on p. 900 in his 1905 paper (ref. 1 above) can only depend on v, and therefore must have the constant value of unity. The LT rests squarely on this assumption, so it is not correct to say that only the two postulates are required in his original derivation. The present approach relies instead on experimental data acquired over the past 50 years to determine φ. They show that elapsed times in S and S’ are strictly proportional to one another: t=Qt’. The alternative LT is derived by combining the experimental proportionality equation with the VT. Isotropic length expansion results from the alternative LT, so the inconsistency with the GPS experiment that is present when the LT is used is completely eliminated. So also is the supposed symmetry in the timing results of two observers in relative motion as well as the necessity of claiming that remote non-simultaneity of events is the unavoidable consequence of Einstein’s two postulates of the special theory of relativity.
RE: The Lorentz transformation for frames in standard configuration
editMy objection is that the wording of the article on Lorentz transformation, in particular the section on Einstein's derivation of the LT, is not sufficiently general. Einstein’s original derivation1 contains the following equations [γ=(1-v2/c2)-0.5]:
x’ = γφ(x – vt) y’ = φy z’ = φz t’ = γφ(t – vxc-2).
The function φ appears on the right-hand sides of all four equations. He states without explanation that φ only depends on the relative speed v of S and S’. It should be clearly understood that this is an additional assumption in his derivation of the Lorentz transformation (LT) besides the two postulates of relativity. He then goes on to prove that the only value of φ under this assumption is unity. Note that the relativistic velocity transformation (VT) obtained later in his paper can be obtained without making this assumption since φ cancels upon division of the equations for x’, y’ and z’ by that for t’. This cancellation is also the reason for including φ in the above general equations. The VT has the following form (ux’=x’/t’ etc.):
ux’ = (1 – vux/c2)-1(ux - v) = η (ux - v)
uy’ = γ-1 (1 – vux/c2)-1 uy = η γ-1 uy
uz’ = γ-1 (1 – vux/c2)-1 uz = η γ-1 uz.
Instead of assuming that φ=1, one can obtain a GPS-compatible version of the general LT by combining the VT equations with the proportionality relation: t’= Q-1t = Q’t:
x’ = η Q-1(x - vt)
y’ = η (γQ)-1y
z’ = η (γQ)-1z
t’= Q-1t
This result is equivalent to setting φ equal to η (γQ)-1 in the general LT. Note that the GPS-compatible value of φ is not only a function of v but also of ux (the parallel component of the speed of the object of the measurement), contrary to Einstein’s original assumption.
The alternative LT (GPSLT) is clearly consistent with Einstein’s two postulates of relativity and also with the VT. It can be inverted in the usual way by interchanging the primed and unprimed symbols and changing the sign of v (note that ηη’= γ2). Unlike Einstein’s LT, the GPSLT is consistent with the ancient principle of the objectivity/rationality of measurement (PRM) because it does not assume that two clocks in motion must each be running slower than one another at the same time. It predicts instead that the respective proportionality factors for the two observers in S and S’ are the reciprocal of one another (QQ’=1). It also is consistent with the principle of remote simultaneity (PRS) because if Δt = 0, Δt’ must vanish as well. No violations of either the PRM or the PRS have ever been observed.
It needs to be emphasized that the GPSLT derivation is specific to the space/time variables. It does not affect the four-vector relation for energy/momentum. An analogous “normalization” function which does have a value of unity occurs in the derivation (based on Hamilton’s dE=vdp relation) of the latter transformation. This is necessary to ensure that the relativistic kinetic energy approaches the non-relativistic value at low relative speeds. More details may be found elsewhere.2
1A. Einstein, Ann. Physik 17, 891 (1905).
2 R. J. Buenker, Apeiron 15, 382 (2008).
Rjbuenker (talk) 18:27, 26 April 2011 (UTC)