Relativity Discussion Forum

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Comment by Louis Vlemincq
I have discovered your website with great interest: I had been looking a very long time for such a cold, rational view of the current accepted science theories by an expert in the field. Until now, I had only been able two find two equally unappealing categories, the science integrists that elevate the truths of 'orthodox science' to the rank of unquestionable dogmas, and the crackpots trying to challenge these established theories with such blatantly wrong arguments that they are ridiculed or ignored by the mainstream scientists.

Though not a scientist myself, I'm what you could call a scientific literate: I've studied electronics and I hold what you'd call in the UK a Bsc in electronics. During these studies, I was taught the basics of physics, including for instance the theory of Relativity. At the time, I was slightly disconcerted, as is normal, I suppose, but didn't really question the theory: the maths seemed right and I had little time to concentrate on it, since I had many other chapters of the syllabus to absorb.
Later however, my discomfort grew; and I felt something had to be wrong somewhere, but I was unable to find any flaw. Unlike the afore-mentioned crackpots, I was aware that challenging the theory was not that easy: if you accepted the premises and the reasoning, you had to come to similar conclusions, and any alternative theory using the same bases would merely paraphrase the original theory. Sporadic reflections on the subject spanning over many year progressively led me to believe that some 'illegal operation' had implicitly been made during the calculations. It had to be connected to speed, as speed is so central to the theory. The problem, I thought, was to put on an equal footing the propagation of a wave in the vacuum and the displacement of a material object across the same vacuum; these are quite different phenomenons, each obeying its own set of rules. It just happens that they are both expressed in m/s, but even that may be an artefact of the choosen system of units or insufficient knowledge from our part: after all we could very well choose a system in which weight and mass merge in a single unit and be happy with it 99% of the time, but it would make life extremely difficult for physicists. Therefore, making them coexist happily in the same equations seemed rather 'risqué'.
Your own explanation that the transit time of the light is dependant only on the distance between the source and the observer at the instant of the emission adresses in a nice and elegant manner my objection to the conventionnal view.
I've the feeling that a whole sector of science has been brought to a standstill for almost a century due to this misconception.
If we look at it in another way we can see that this theory has been absolutely sterile: I know of no application based on Relativity. You find numerous experiments that take into account relativistic effects, but that's clearly not comparable; after all, these effects can be explained by alternative theories, often more simply. On the other hand, the quantum theory, as ugly, muddled and anti intuitive as it may be, is the source of almost everything that mades up our modern world. I certainly do not say quantum theory is the ultimate answer, some theory of everything, but it does have a real operationnal power, and a real value. Even the good old Maxwell theory is still extremely useful for engineering purposes today, although it has well out-lived its sell-by date. And by the way, the fact that the light 'knows' its point of arrival at the instant it is emitted sounds strangely quantic to me.
The science needs more people like you, courageous enough to look at existing beliefs in a sober, detached and objective manner.

Reply
I appreciate your comments on my website. Although I have studied Relativity like any other physics student, I had very much the same feeling as you had, i.e. that something was wrong but I did not know exactly what. I did not bother about it too much because my subsequent field of work did not require any relativistic theory. Only in more recent years did I have some time (due to unemployment) to deal with this matter more closely, and I found that the error is in fact the vectorial addition of velocities that Einstein applied when deriving the Lorentz transformation, i.e. he maintained the usual concept of velocities (although it was well known from experiments that this does not apply for light), and in order to compensate for this error he then applied a second one by re-scaling the time and space co-ordinates accordingly. With a more objective approach he should have noticed that the only way the travel time can be truly independent of any velocity involved is when it depends just on the distance (your note that this reminds you somewhat of quantum theory is quite relevant I think, because light is in fact produced by quantum mechanical processes in atoms, so this connection is definitely there; but as mentioned towards the botton of my page regarding the Speed of Light, the actual reason for the invariance of c is the circumstance that, unlike conventional waves, light waves must be able to travel in vacuum i.e. without a carrier medium, and I can't see any other logically consistent way of achieving this than by making the travel time of a light signal only dependent on distance but not velocity).
I had actually recently a couple more positive emails regarding this point, so I am mildly optimistic that this view might catch on on a wider scale. The problem is that, as you mentioned yourself, many people argue against Relativity with arguments that are quite obviously themselves flawed, hence even involuntarily strengthening the position of Relativity (if only for the reason that the latter is well established) and making it difficult for any justified criticism and suggestions. In this sense, I am so to speak caught between two fronts (also with regard to some other issues addressed on my website), but I am confident that it will eventually be more of an advantage than a handicap.

Comment by Jan Zwarts
Reaction on your Mathematical Inconsistencies in Einstein's Derivation of the Lorentz Transformation:
Your words: "From these equations ((1),(2) and (1a),(2a)), he concludes then (3) and (4)".
Now read what Einstein writes: "Obviously this, ((1)<=>(2)), will be the case when relation (3) is fulfilled in general". In other words, (3) implies the postulate that the speed of light is the same for each observer. ((1) <=> (2)) does not imply (3) and that is not an error!! Nobody (except you ?) requires that.
(3) is an assumption, actually a brilliant idea.

Reply
In his derivation of the Lorentz Transformation, Einstein does not at all indicate that his equations (3) and (4) are assumptions (which obviously would need a specific justification). Above his Eq.(4), he merely speaks of a 'condition' and the nature of his derivation suggests that he apparently thinks this condition is both sufficient and necessary for the invariance of the speed of light in different reference frames. It surely is the former (as x'-ct'=λ.(x-ct) implies that x'=ct' if x=ct), but, as mentioned on my page, the assumption it would also be necessary is not only unjustified and physically implausible, but it leads in fact to algebraic inconsistencies if one applies the resultant Lorentz transformation to light signals travelling in opposite directions (as shown by Eqs.(8)-(14) on my page) .

Jan Zwarts (2)
To derive the Lorentz Transform one has to assume the speed of light postulate, the relativity postulate (no preferred observer) and that the relation is linear in x and t. This last assumption is what Einstein does in (3) and (4). The words he uses are clear to me that it is an assumption, a generalisation. And valid for all possible x, t, x' and t'. Do you have a better idea?
The algebraic inconsistence you mention in Eqs.(8)-(14) is a violation of the Lorentz Transform by yourself. In relations (8) and (9) the t and t' can not both be the same !! I give you an exercise: Write in (9) : x2' = -c.t2'. Now prove that t'/t2' = (a-b)/(a+b) = (c-v)/(c+v).
Deriving the Lorentz Transform is one thing, applying it correctly is another. And that is not easy !

Reply (2)
t' must be the same in (8) and (9), otherwise it would violate the relativity postulate: if you send two light signals into opposite directions in the primed frame, then the signal travelling into the positive direction must reach the point x at the same time as the signal travelling into the negative direction reaches point -x; so if you have x2'=-x1', then with your suggestion it would follow that t2'=t'.
t and t' are simply the independent variables here, as defined for instance by synchronized clocks distributed throughout each frame; they do by definition not depend on location; suggesting that they do not only violates the relativity postulate, but is also mathematically inconsistent: as an even simpler illustration for what Einstein has actually done here, consider the example in the box on my page regarding the Lorentz Transformation.

Jan Zwarts (3)
You mix up two things: 1) -c.t' is the coordinate of the negative light signal in the primed system. 2) x2' is the coordinate x2 of the unprimed system (=-c.t) as observed from the primed system. These coordinates are not the same !!
The Lorentz Transform has the relativity property (just reverse the relation to see that). By assuming in (8) and (9) the t in both to be the same and correctly applying the Lorentz Transform, one finds: t'/t2' = (a-b)/(a+b) = (c-v)/(c+v). Assuming t' to be the same in (8) and (9) one finds a similar relation for t/t.. This shows the relativity postulate.

Reply (3)
According to the definition, x2'=-ct', so I don't know how you can say that the two sides are not the same.

Einstein's own definition (see the first paragraph up to Eq.(2) at http://www.bartleby.com/173/a1.html ), clearly states that x' is the coordinate with which the light signal transmits in the primed frame. It is not, as you are asserting, the coordinate of the light signal in the unprimed frame (x) as seen from the primed frame. Because of the invariance of c, the latter has nothing to do with with how far the light travels for observers stationary in the primed frame.
Consider two detectors located (stationary) at x' and -x' in the primed frame. Then the invariance of c requires that these detectors must register two light signal sent into opposite directions from the origin at the same time t', whatever the velocity of the light source at the moment of emission was. Detectors located stationary at x and -x in the unprimed frame will also detect the light pulses at the same time (t) and there is thus no connection between the two frames. Whatever happens at the detectors in the unprimed frame does not concern those in the primed frame and vice versa.

Being able to reverse the Lorentz transformation does not prove that the latter is correct. Where the Lorentz transformation fails here is if you change x to -x in the Lorentz transformation, x' does not also change to -x' as is required by the invariance of c according to the above argument.
This shows that any kind of velocity dependent transformation between the two frames (i.e. Einstein's 'assumption' you mentioned earlier) does in fact violate the postulate of the invariance of c.

Jan Zwarts (4)
The Lorentz Transform is a coordination transformation. I will write it down for you:
Transformation of space-time coordinate (x1=ct,t) (from your (8)) gives x1'=act-bct; ct'=act-bct; resulting in: x1'=ct' and t'=(a-b)t.
Transformation of space-time coordinate (x2=-ct,t) (from your (9)) gives: x2'=-act-bct; ct2'=act+bct; resulting in: x2'=-ct2' and t2'=(a+b)t.
You see that t' and t2' are different which you cannot ignore.

As result of the Lorentz transformation above, x2' is the coordinate x2(=-ct) as observed from the primed system. Now you define x2'=-ct', which is different and not the result of the transformation of (x2,t) as shown above. But in your "proof of inconsistencies in the Lorentz Transform" you do as if this is the case in the transformation (11)! Here you create a conflict, a violation of the Lorentz Transform. You want to prove that the Lorentz Transform is invalid, but then you must apply the transform correctly until you (hope to) come to a contradiction. That is not what you do and therefore your proof is worthless.

If you write your final (8) and (9) as:
(8) x2=-x1
(9a) x2'=-c(a+b)t=-ct'(a+b)/(a-b)=-x1'(a+b)/(a-b),
then everything is fine and consistent because:
(13a) x1'(a+b)=(ax1+bct)(a-b)
Now substitute (10):
(14a) (ax1-bct)(a+b)=(ax1+bct)(a-b) This results in 0=0.

Some additional remarks on your critics of the Einstein derivation:
(1) and (1a) are the equations for two different light signals, one in positive direction and one in negative direction, both starting at t=0 in the origin of the ref. system. (For instance at t=2 the x-coordinates are 2c and -2c respectively). Why do you say that they should hold at the same time (thus holding only for the intersection (x,t)=(0,0))? Einstein does not require that. Only you do. Einstein did not make an error, you are wrong yourself. You don't mention what Einstein is looking for: a relation which implies that (1) and (2) as well as (1a) and (2a) are equivalent. This means that the constraints are (1)<=>(2) and (1a)<=>(2a), which is the speed of light postulate. With the relations (3) and (4) these constraints are satisfied where the λ and μ appear to depend only on v and c. A great result.

You write also: "Some people actually interprete Eqs.(3) and (4) as general linear relationships between the variables x',t' and x,t not necessarily related to Eqs.(1)-(2a) anymore." Correct, except for the last part, the relation is given above. You say that this is an invalid generalisation. I don't understand why you say so. It is valid because the constraints are satisfied.

The Lorentz Transform has the constant speed of light and relativity properties. Actually, the transform has at least these properties. By the Einstein generalisation to a linear relation, additional properties could have been introduced. I know one, the transform is bijective which was not a requirement at forehand.

Reply (4)
The fact that (in general) t and t2' are different is exactly the inconsistency that I mentioned:
you have x1'=ct' and x2'=-ct2'=-ct'.(a+b)/(a-b). Now the invariance of c requires that for any time t', x2'=-x1' (i.e. the signal spreads isotropically). Now this is obviously only possible if (a+b)/(a-b)=1 i.e. if b=0 (note that I have not made any assumptions here, but strictly applied algebraic identities and the initial constraint).

With your (or rather Einstein's) interpretation, the Lorentz transformation practically becomes its own constraint, with the original constraint not only being lost but actually being violated in the process. This is clearly a flawed mathematical logic (see also the example in the box which I have added now to my page regarding the Derivation of the Lorentz Transformation).

You should ask yourself why Einstein actually wanted to have a velocity dependent transformation between the coordinates at all. The answer obviously is that essentially he wanted to apply the usual concept of 'speed' (i.e. the Galilei transformation) to light, although the invariance of c implies that there is in fact no velocity dependence. It is this obvious conceptual contradiction which is responsible for the mathematical inconsistencies associated with his 'assumption' (I am quoting the word here because he doesn't point out at all in his derivation that there is an assumption being made, which on its own pretty much proves that either he was simply mathematically incompetent or intended his theory as a hoax).

Jan Zwarts (5)
Mr. X does not believe the Pythagorean theorem a2+b2=c2. He gives a proof by contradiction. He thinks that a+b=c. From this follows that a2+b2+2ab=c2. And according to Pythagoras this becomes 2ab=0. So a=0 or b=0. This, concludes Mr. X, reduces the theorem to nothing.

This is exactly the same as you do Thomas. You think that x2=-ct' which contradicts the Lorentz Transform in the same way as a+b=c contradicts the Pythagorean. And by the way, that the invariance of c requires that for any time t', x2'=-x1', is not true. I will try to show you that later.

Reply (5)
If the constraint for the sides of a triangle is a+b=c, then this does indeed imply a=0 or b=0 (the triangle reduces to a straight line).

What Einstein effectively did was to change the constraint such that a2+b2=c2 with a,b and c all different from zero. He would have probably proceeded like this here: he replaces the constraint a+b=c with a+b=c' where c'=k*c (with k being some scaling factor). This new constraint now satisfies the Pythagorean theorem without a or b being zero if k=√[1+2ab/(a2+b2)] (as is easily found by inserting it into the theorem). However, the new constraint has not only nothing to do with the original one anymore but is in fact algebraically inconsistent with it unless indeed a=0 or b=0.

Jan Zwarts (6)
I cannot follow you anymore. But one thing is sure. Your "proof" of invalidity of the Lorentz Transform fails.
I will now prove compliance with the invariance of c and the relativity postulate of the Lorentz Transform (LT) with your example. I give the transformations again:
LT( ct, t) gives x1'=ct' (1) , t'=t(a-b)
LT(-ct2,t2) gives x2'=-ct2' (2) , t2'=t2(a+b)
Note that I use t2, but it is no problem to take t2=t (the same t as in (1)), so
LT(-ct, t) gives x2'=-ct2' (2a) , with now t2'=t(a+b).
Note also that t2' is not equal to t' (the t' of (1)). This is due to the fact that the situation is not symmetric. Change v in -v i.e. b in -b and see what happens: t' and t2' switch their values as well as x1' and x2' (except the sign). This shows the relativity postulate.
From (1) and (2) it is evident that the invariance of c postulate is satisfied in both. And also in (2a), because (2a) is not essentially different from (2). You are wrong if you say it is not. One needs not to require x2'=-x1' (absolutely not).
This actually finishes the proof.

What happens if you make x2'=-x1'. Then we have:
x2'=-x1'= -ct' = -ct(a-b) = -ct(a+b)(a-b)/(a+b) = -ct2'(a-b)/(a+b).
To satisfy invariance of c, x2' has to be equal to -ct2' as in (2a). This requires that b=0. But that is only necessary because you want that x2'=-x1'. With your own wrong ideas you conclude that b=0.
That x2' is not equal to -x1' may seem strange, but that is typically SRT. Not so easy to understand. But it has probably to do with the asymmetry as I showed above.

Reply (6)
It is one thing to understand a logically consistent theory and another to believe you understand a logically inconsistent theory because you don't see the logical flaws in it:
as I have repeatedly explained and also clarified through further examples, the Lorentz transformation does actually not satisfy the original 'Relativity Postulate' but something else that you are making up in the process of 'deriving' it. This is invalid mathematical reasoning. The relativity postulate is uniquely algebraically defined as "if x=ct then x'=ct' (and vice versa)" and "if x=-ct then x'=-ct' (and vice versa)". There is nothing more and nothing less to it and any assumption that the x' and t' would mean something different in both conditions is algebraically flawed. The fact that you have to change b to -b in the Lorentz transformation in order to preserve the relativity postulate just proves this.

I really don't see how I could add any more arguments in order to make you realize this. I know it takes time to break out of long established ways of thinking, but I am sure if you try to critically reflect on this issue without assuming already a priori that Einstein must be right, you will eventually understand the point I am making.

Jan Zwarts (7)
You mention only the invariant speed of light postulate, also a relativity aspect indeed: both observers experience the same speed of light. But SRT IS more. The observers experience actually all aspects the same. Most famous are time dilation and length contraction. Both observers see the other clock run slower and the other one shorter in length at ANY relative speed between them.
To derive the Lorentz Transform one has also to require that the inverse transform is the same transform with v changed sign. (Only the speed of light postulate is not enough). I call this the relativity postulate but that gives obviously a misunderstanding with you. The reason to change b into -b is to show you this last aspect of relativity. I am still thinking about the asymmetry aspect, I am not quite sure if that is the reason that t' and t2' are different.

Reply (7)
The fact that v (or 'b' in the above formulae) changes sign for the inverse transform is not a separate requirement for the Lorentz transformation. It is a trivial consequence of assuming a linear transformation and holds also for the Galilei transformation (x'=x-vt obviously implies x=x'+vt). But as I have explained above, this is not consistent with the invariant speed of light postulate if one changes the signs of x and x' unless v=0.
The only true postulate here is actually the invariance of the speed of light as it is based on experimental evidence. The Lorentz transformation, 'time dilation' and 'length contraction' are merely consequences of Einstein's mistaken attempt to apply the usual concept of 'speed' (i.e. essentially a Galilei transformation) to light signals whilst maintaining the invariance of c in different reference frames. Not only has this led to the mathematical inconsistencies we have been discussing here, but also to logical paradoxes like the Twin Paradox. Note also for instance that the Lorentz transformation, if correctly applied, does actually not result in a Stellar Aberration, so in this case (and may be others as well) the apparent agreement with observations and experiments is only due to making further mistakes when practically applying the Lorentz transformation. Other phenomena like the 'relativistic' dynamics of charged particles for instance could for instance well be explained by velocity dependent forces (see my page regarding the Newtonian Relativistic Electrodynamics).
So the apparent observational and experimental support for the theory of Special Relativity should not really be an argument for making a logically flawed theory acceptable (I just wanted to mention this as a concluding remark as some people always come up with the argument that Relativity would be experimentally verified; regarding the conceptual and mathematical consistency of the theory, I don't think there is much more to add to our discussion above, so I suggest we let the reader make up his/her own mind about it on this basis).

Jan Zwarts (8)
One last remark. I promised to think about the asymmetry matter:
The difference between x1' and x2' as well as t' and t2' is indeed due to the asymmetry of the situation.
Call unprimed ref. system O, the primed O'.
1) One light signal travels in positive direction relative to O and O' while O' travels in positive direction relative to O.
2) The other light signal travels in negative direction relative to O and O' while O' travels in positive direction relative to O.
By changing v in -v the asymmetry is reversed. That this would show the relativity postulate in this context, as I said, is not correct.

Lets look at an example : take c=1, v=0.6, t=10.
=> Lorentz factor = 1/√(1-v2) = 1.25. x1=ct => x1=10, x1' =(10-0.6.10).1.25=5, t'=(10-0.6.10/c2).1.25=5 ; (note that x1'=ct').
x2=-ct => x2=-10, x2' =(-10-0.6.10).1.25=-20, t2'=(10+0.6.10/c2).1.25=20 ; (and here x2'=-ct2').
In the Galilean world where x'=x-vt and t'=t the result would be (with c relative to O):
x1' = 10-0.6.10=4, t'=10
x2' =-10-0.6.10=-16, t2'=10
Note that the ratio x2'/x1' is the same in both worlds (here -4).
There can be a substantial difference between x1' and -x2'. And even greater in the Lorentz world. The reason why x1' and x2' are different in the Lorentz world is the same as in the Galilean world.

Reply (8)
The circumstance that x2'/x1' is the same for the Galilei and Lorentz transformation just illustrates that there is in fact no difference between the two as far as the inapplicability of the transformation for the propagation of light signals is concerned:
the invariance of c is defined by the conditions x1-ct =0 <=> x1'-ct' =0 and x2+ct =0 <=> x2'+ct' =0 where t and t' are unique variables (see even Einsteins own definitions in this respect). The symmetry of these conditions is clear from the situation I mentioned already earlier above: if one has a detector mounted at x1' and another at x2'=-x1', then the above conditions require that these detectors are reached within the same time from the origin (regardless of any velocity of the light source), i.e. t' must be the same in x1'-ct' =0 and x2'+ct' =0. A value t2'≠t' in the latter equation is therefore inconsistent with the invariance of c, and any conclusions based on this assumption are thus mathematically and physically flawed.
The simple fact is that the condition for the invariance of c is symmetric, whereas the Galilei transformation is (in general) asymmetric, and they can not be reconciled with each other without introducing algebraic inconsistencies.

Comment by William Campbell
I came across your website while looking for data about current views on lightspeed.
I believe the error is the consideration that lightspeed is constant. It can be slowed, so in one sense, it is variable, an indisputable logic. That we have not observed (or cannot due to physical limitations) velocities higher than the agreed-upon finite value does not "prove" that the higher velocities do not exist. As with our past of scientific discovery, when so many times "what is seen is true, what is not is false," when false is better regarded as "yet to be determined."

I have pondered this for some time, and recent events add to it. 1. the error in the Hubble telescope. Is a miniscule increase in lightspeed off-planet possibly responsible? Or is it without question, a manufacturing defect? 2. the Pioneer probes. Perhaps they are precisely where they should be, and only appear closer because the electromagnetic radiation, over the greatest distance we have ever thrown it, is gaining speed, and thus, "seems" closer.

This returns to experiments that can be conducted on Earth, regarding the change in velocity, and the radiation's recovery to previous speed. I assume you are familiar with researchers "parking" a beam of light within a super-cold atom cloud. And by manipulation of the coupling beam, the subject beam could be slowed, stopped, even absorbed by the cloud, then released at will, at which point it resumes as before. This may all seem old news, but an important (and perhaps overlooked) observation should be applied to this feat. Does light slow down? And when released, accelerate? Or, does it go from speed A, to speed B, then stop, and when released, instantaneously resume speed A? This is an important clue. If there is no acceleration or deceleration, isn't that an important observation that may change opinions of electromagnetic radiation? For example, that it is not associated with matter, i.e., not any form of particle.

For light received from distant locations, perhaps the radiation is travelling at far greater velocity, and is only slowed during its journey through the universe. If experiments on Earth suggest otherwise, one might consider that the scale on which the test is conducted is not sufficient to reveal unknown properties of electromagnetic radiation. Does it accrue velocity when unhindered, over great distance? How could we possibly know, limited to one location in the universe, with tests of such small scale examining a phenomenon so incredibly swift? And, are forces within our Solar system holding it back during our experiments? We are on a planet with a magnetic field. Does this affect lightspeed? Send light through a strong magnetic force, does it slow down? Or have these experiments already been conducted, and determined? Another consideration: as inhabitants of Earth, our access to nearby natural light is limited to one star, our sun. Perhaps meaningless, though perhaps not. All factors must be considered if one is hoping to arrive at undeniable constants for any characteristic of a universe so vast, which we desire apply to all portions of that vast universe, not only our local portion. Until the experiment spans the proper scale, one cannot be certain. At least, as certain as modern physics appear to be, that lightspeed, from one supposedly expanding end of the universe to the other, is invariable at all points in-between. More evidence is required to support that supposition, which we well know, so much more is based upon.

Reply
The situation regarding the speed of light is actually rather confusing: first of all, as you indicated already yourself, if light propagates in a medium it is indeed not constant but, apart from being slower than in a vacuum, it also depends to a certain degree on the frame of reference (depending on the refractive index). Secondly, as mentioned already by me on my page regarding the Speed of Light (see towards the bottom), magnetic or electric fields could also 'break' the invariance of c as the light might 'attach' itself to these fields.
Thus, for discussing the constancy of the speed of light, one has to neglect the effect of a physical medium and/or electromagnetic fields i.e. one has to assume that light propagates in a perfect field-free vacuum. However, it is exactly in this case that things really start to get confusing: as shown on my page regarding the Lorentz Transformation, a constant speed of light is actually theoretically impossible if the usual concept of 'speed' (which implies a vectorial velocity addition) is used. The point is that the theory of Special Relativity nevertheless attempts to apply the usual concept of 'speed' and then tries to correct this error by re-defining space and time units. If corresponding experiments indicate a constant speed of light on the basis of this flawed theoretical approach, one has actually to conclude that, using the traditional definition for 'speed', the speed of light was actually not constant in these experiments (thus confirming the theoretical impossibility I mentioned). The question is thus whether using a different definition of 'speed', the speed of light can actually be constant. As indicated on my page regarding the Speed of Light, the only way to achieve this is to assume that the time it takes a light signal to travel from the source to the observer is completely independent of any relative motion of the two i.e. it depends only on their distance at the moment the signal is emitted. It is in fact obvious that this must be so as otherwise the speed of light could not be defined at all considering the circumstance that electromagmetic fields can travel in vacuum i.e. without any medium. To what extent this is confirmed by observations I can't be sure at the moment as it would require a corresponding re-evaluation of the data, but discrepancies like the apparent Anomalous Pioneer Acceleration or Anomalies with the GPS system might be associated with this.
Regarding the spatial variation of the speed of light: the speed could only be different if there are physical reasons for this, e.g. if there is change in the physical medium or electromagnetic fields. It could explain certain phenomena but surely not the error in the Hubble space telescope as a change in the speed of light would not affect the focus of a reflecting telescope.

In any case, the constancy of the speed of light just applies to, well, light, but does not affect the circumstance that other physical entities can have arbitrary velocities relative to each other.

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