The Problem of Reciprocity and Non-Reciprocity
in Special Relativity Theory

(Francisco J. Müller)


Abstract: The implications of the two relativistic postulates are analyzed in terms of their reciprocity and symmetry aspects. Whereas the first postulate entails a perfect equivalence of all frames, the second postulate, as reflected by the Einstein-Lorentz equations, implies an asymmetry of values (not of forms) that clashes not only with the first postulate equivalence but with the possibility of experimental verifications, all of which are "asymmetric" in each concrete case. A double index notation is proposed at the end, in order to show more clearly these asymmetries, and to interpret in a realistic way, the referential (physical) meanings of the four space/time variables. Lorentzian versus Einsteinian relativities are contrasted. The former seems to cope better with the asymmetric experimental realities. The latter only does so by subreptitiously changing the meaning of the physical variables.

    In the introduction of his 1905 relativity paper (1) Einstein gives a hint that the 2nd relativistic postulate about the universal constancy of the speed of light might be "apparently irreconcilable with the former" postulate, that is, with the first Postulate about the formal validity of physical laws for all frames of reference. Few physicists have pondered why Einstein posited this dialectical contradiction at the very beginning of his paper. It is the purpose of this article to dig into this question. A superficial interpretation of the possible contradiction views it as the clash between the classical, or Galilean composition of velocities, and the counter-intuitive "scandal" that light seems to travel at the same speed for all relatively moving observers. But there are more hidden aspects to this problem. The prototype of this light "scandal" was provided by the null Michelson-Morley experiment. This experiment, however, have always been performed in a "proper frame", without even the possibility of observing it from the viewpoint point of a moving frame (for example, from the Sun)(2). Yet, the theory elaborated by Einstein, introduces a perfectly "reciprocal" treatment of the experiment, (in its equivalent gedanken version) both in a proper and in a non-proper frame.

    After all the theoretical work was done by Einstein, arriving at the Lorentz equations, the reciprocity and symmetry of such equations have been greatly studied and formalized. Group theory assimilated the equations as part of its program. This theory is perhaps the best mathematical tool to investigate such symmetry properties of physical phenomena as rotations, nuclear particles, relativity, Quantum symmetries, hypersymmetries, anti-symmetries, etc. etc.

Relativistic formalization versus physical realities

    An essential aspect of these formalizations is to consider variables like x,y,z,t with primed and unprimed characters. The operators, matrices, generators, etc. are said to "transform" one set of un-primed variables into the primed ones, and viceversa. Reciprocity and symmetry are, here, practically synonyms. Reciprocity, however, usually refers only to the relationship (also termed "inverse") between one set of variables and another "corresponding" set. In contrast, symmetry can be discovered within one such single set. For example the elements Aij and Aji are equal in a "symmetric" matrix or tensor. In contrast the word "anti-symmetric" has been used to described the case when Aij = -Aji . When the reciprocity simply consists in changing (+) and (-) signs some authors still consider it "symmetric", others "anti-symmetric" and some even "asymmetric".

    Leaving the heavens of pure mathematics, the physical use of such primed and unprimed quantities have been mostly to denote variables pertaining to a "moving" versus a "non-moving"(resting) frame of reference. Yet, on account of the relativity principle and the equivalence of all frames (First Postulate), nobody takes seriously the labels "moving" versus "non-moving" any more. Being perfectly "interchangeable" or "reciprocal", the primes and unprimes become, again, a mere mathematical formality. This leads to the familiar interpretation of the Lorentz equations, for example, as a "rotation" in 4-dimensional space. To the performance of succesive Lorentz steps as mere multiplications in group theory. And to the "raising" of the time variable to an "equivalent" status as the space variables through the artifact of an X4 = ict terminology. This enables to write the "fundamental relativistic interval dS" in a very beautiful way.

    But the danger of all these mathematical myths is that, by taking them too seriously, (and isolating them from their referential meanings) the physical reality that they "signify" becomes more and more obscured , ignored and even confusing. This confusion applies especially to the concept of "event".

The elusive "event" (physical versus mathematical)

    A first problem that arises in this respect is the nature, the uniqueness, or even the existence, of the "event" (the space/time point) that is being "observed", "seen", "transformed", by a certain number of observers, (minimally by two) in relative "motion". Mathematically speaking this event can be common to any set of reference frames, regardless of their "motions", because the world of Mathematics is intrinsically static and motionless. If we mark a point XY in the plane of a Cartesian system K and then refer that "same point" to another frame K' that has been "rotated" respect the first, there is absolutely no difficulty in expressing X' = RX and Y' = RY where R is the "rotation matrix". The original point does not "belong" any more to the K system as to the K' one. In fact, nothing has to move and, indeed, nothing has moved.

    But when we return to physical reality a given "event" is not only a set of coordinates, but a physical entity, a "happening", a phenomenon. This could be a Coulomb force between two (real) charges; or a photon being emitted or received, or a ticking clock, etc. Then it is dramatically important to distinguish in which frame the said entity or event is AT REST and in which it is seen as a "moving" reality. This physical event is certainly not interchangeable and non-reciprocal. In theory yes, we could set the physical entities in ANY of the two frames, but in each concrete experiment one and only one of the frames is the proper one.

    It is in this sense that the Lorentz equations yield "asymmetric values", although being formally symmetric (or anti-symmetric). Panofsky and Phillips, very well established authors of the 1960's, explicitly recognized this when they described the Lorentz contraction, X<X', saying (3):

"This relation is asymmetrical in X and X', since it gives the relation between measurement of a proper length X' (at rest) in E ' and an improper length X (not at rest) in E .

    In another place, however, they say that Lorentz equations show that "except for the sign of v, [the frames] E and E ' are equivalent, in agreement with the second relativity postulate".(4) (Both emphases are mine).

    So for these authors the (+/-) anti-symmetry of the velocity v, is in one place judged as an "asymmetry", yet in another as an "equivalence". Be this as it may, the important point is not to lose "track" of where the event is proper and where it is not. The labels of "moving" and "resting" are absolutely interchangeable and in that sense they are "trivialized" Yet, the primed and unprimed notation of formalized group and transformation theory only pays attention to such irrelevant tags, and is oblivious of the most important point: the residence of the unique physical reality that confers one frame the character of being "proper" and the other "non-proper". Even worse, by ignoring the residence of that unique physical event the primed/unprimed notation opens the way to the removal, altogether of the unique physical event. Then the scenario of the twin paradox is wide opened and all its concomitants debates start to proliferate. Let me expand on this.

Methodological origin of the clock or twin paradox

    In the twin paradox the time of each twin plays the role of a proper time. Each twin's clock is resting in each twin's frame. Why should the "moving" twin return younger, when in its own proper clock nothing has absolutely happened according to the first relativity Postulate? From where comes the "asymmetry" of their times or "ages"?

    We know the standard relativistic answer: "from the fact that only the 'traveling' twin accelerated". And we know also the standard non-relativistic objection: "accelerations are as relative and as interchangeable as the velocities. If distances are relative and their first derivatives are so, why not their second derivatives?" "Because a force, indeed" answer the relativists, "must have been used by the 'moving' twin to produce the acceleration". But again the non-relativists accuse the relativists of introducing a "non-inertial" element in their theory (a force!), thus violating the logical presuppositions under which the whole Lorentzian deduction was made. This debate can keep going on and on. I do not pretend to end it here. But what I want to point out here is the origin of the debate, which is this: a mis-application of the Lorentz transformation.

    The Lorentz equations, at least as conceived by its original author in 1904 (5), were only geared to study how that unique physical reality of which I have spoken before, (an electron, a ray of light, etc), is "seen", or "measured" by TWO observers in relative motion. We can call this a ternary structure: two observers and ONE physical event. (2O1E for short). In contrast, the twin paradox scenario uses a binary structure: two clocks or "twins" in relative motion and no common event independent of their own clocks or "ages" to be studied or observed. Such a binary scenario leaves the theory totally empty and devoid of objective purpose. It is futile, in search of some objectivity, to speak of the twins as "observers" and the clocks as "events". That language is but a picturesque antropomorphism that has creeped only too much into relativistic textbooks. (Einstein himself is partially guilty of this antropomorphism when using expressions like: the observers "judge", or "ascertain", or "declare", etc, etc).

Critical comment

    What, then, should we conclude here about the debates on the clock paradox? That they have been a waste of time. Relativists will keep insisting in their "asymmetrizing" acceleration factor, while non-relativists will insist in the intrinsic equivalence of all truly inertial frames. In reality, the struggle here is between the equivalence, reciprocity and formal symmetry that springs from the First Postulate on the one hand, and on the other, the (numerical) asymmetry that results from the Lorentz equations (2nd Postulate) when applied to a concrete situation. Indeed, we have returned here to that "apparent contradiction" between the two Postulates that Einstein envisaged at the beginning of his paper. The fact that the Lorentz equations are in harmony with c = c for both (or all) frames does not remove the clash between the asymmetry of the numerical values it predicts for the rest of the physical realities, in comparison with the perfect equality and equivalence predicted for those same realities by the First Postulate.

    In reality the reason why physicists do not perceive this methodological (even logical) "mis-match" between the two relativistic postulates is that they have gone too far into mathematics and have abandoned too much the real physical qualities. They imply, following Minkowski, that those qualities are mere "shadows", and only their mathematical connections in a four-dimensional (mythical) language remain formally invariant.

    In this respect it is frequently said that relativity theory ended the "mechanistic" philosophy of the etherists. But in reality relativity has "ended" all physical qualities, (electromagnetism included), in the name of pure games of kinematically related frames. This is not only mechanistic, but "geometristic". The world of relativity is, thus, a time-less, motionless world. And if we still remember Aristotle we will agree with him when he said that "if we do not understand motion we cannot understand Nature". So relativity does not "understand" Nature at all, since "nothing moves" in the relativistic eternal and timeless heavens.

    This critical comment, however, does not entail, in the least, that the effects experimentally tested like time dilation, mass increase, etc, are not real. But only that they cannot be predicted from a symmetristic, equivalential, reciprocating, version of relativity theory as Einstein's version. It seems that only the Lorentzian (non-symmetristic) version of relativity makes sense. To consider this I will summarize the previous verbal analysis in a more concise and symbolic way, using a "double index" notation, a notation that I think has been long time overdue in relativistic textbooks.

Analysis using a "double index" notation

    Let A and B be two frames of reference in relative motion. We shall not stipulate which is moving and which is not. Let any property T (like time) be labelled with two subindexes: the first indicating: where the event or object is; and the second by whom or in which frame the observation is made. Thus:

1- TAA means: the "time" of an event in A as measured by A (a proper value)

2- TAB means: the "time" of an event in A as measured by B (a non-proper value )

3- TBB means: the time of an event in B as measured by B (a proper value)

4- TBAmeans: the time of an event in B as measured by A (a non-proper or a "coordinate" value)

Whenever the two subindexes coincide, we have a proper value. Whenever they are mixed, we have a non-proper value (or "coordinate" value) The previous T variables should be understood as dT's in the case of time intervals. For simplicity I will use the single letter T . With this notation, and the additional symbol L(+/-v) for a Lorentz transformation we can express the First Postulate writing that:

(I) TAA = T BB (valid also for all TCC , T DD , T EE , etc)

    And the 2nd Postulate by means of the reciprocal Lorentz equations:

(IIa) TAA =L(v)T AB

(IIb) TBB =L(-v)TBA

    Notice that in the Lorentz equations the first subindexes have to be equal, because both frames, A and B are refering to a unique common object either in A, (Eq.IIa), or in B (Eq.IIb). Thus, an equation like T AA = L(v)T BA would be totally meaningless.

    Are all equations I and II applicable simultaneously in each concrete situation? Are all four "referential" parameters defined above simultaneously meaningful? Let us see the possible answers both in an Einsteinian (reciprocal) relativity and in Lorentz's (non-reciprocal) kind.

    For both versions of relativity, the First Postulate should be always applicable. Unfortunately, due to the limitations of the primed/unprimed notation, there is not even a way to express Eq-I as done above. To write T = T' with this purpose would be to invite total confusion when the Lorentz equations later yield  T different from T' . Especially in this aspect we see the greater advantage of the double index notation.

    Now in the Lorentzian context equations II can only be applied one at a time in each case. Suppose we are dealing with (IIa), so that the event is proper in A and not proper in B, (regardless of which "moves" or not). Then TBB and TBA are "empty". No part of this experiment refers to them and, hence, no paradox arises. We could later repeat an "identical" experiment in B and apply (IIb), in which case TAA and TAB will represent nothing of "that" experiment. The conclusion of the experiment is simply, in the first case, that A has a proper time or "age" T AA and B sees it non-properly as TAB and in the second, reciprocal case, that B has a proper time or "age" TBB and A sees it non-properly as TBA . The two reciprocal cases do not "mix" or "apply" together in one single experiment.

    In Einsteinian relativity, however, both equations II are taken as perfectly reciprocal (which they are) and as applicable simultaneously but in a "mixed" way. (Typically this is the clock paradox bynary scenario) What happens here is that relativists speak of the "age" of each twin or clock, in their own proper frames. These "ages" should be denoted by TAA and TBB in the double index notation. That being the case, the only conclusion should be what the First Postulate establishes, namely that TAA = TBB . No possible asymmetric aging can arise. But relativists insist that one frame, for example A, is "really moving" and asymmetrically accelerating. Then they apply (IIa) to conclude that TAA = L(v)TAB , a result which they express verbally by saying that the "age" of twin A is smaller (younger) than the "age" of twin B. But the problem here is that TAB is not the proper "age" of twin B, but only the non-proper "view" or measurement that B has of A's age. The proper age of B is still TBB, a fact that relativists overlook simply because they lack the adequate double index notation that I am proposing here.

    Relativists "transform" conceptually the non-proper measurement TAB made by B, into the proper age TBB of B. This happens, as explained above, because in the bynary scenario there is no fixed, unique, common object to be observed. Then the subjectivity of each observer takes the place of the object to be measured. This is what produces the "mixing-up" of equations in Einstein's "symmetristic" relativity which I mentioned above.

    This unwarranted "transformation" of the physical meaning of the mathematical variables is, perhaps, the most persistent and irritating error both in Einstein's original paper and its descendants. Parameters like x,y,z, and ct that were used to describe the motion of a "light ray", are suddenly, without justification, used for representing the "positions and times" of a moving clock or rod. Again this all can happen because the primed/unprimed notation used exclusively by relativists is totally "blind" to the referential physical meaning of the variables. The prime/unprime notation simply indicates "moving" vs "non-moving" systems, a tag that is perfectly equivalent, reciprocal and interchangeable. But the proper vs non-proper elements of the experiments are, in contrast, non-reciprocal and physically asymmetric. This physical asymmetry is the only thing that experiments can show and verify, a fact that means that in practice, only Lorentzian relativity, with its preferred ether frame, is the one in harmony with experiments. The only way that Einsteinian symmetristic relativity can cope with experimental facts is by converting itself to the Lorentzian scenario as each situation demands. A few examples of this suffices.

The experimental asymmetries oppose Einstein's symmetristic relativity

1 - The GPS community, for example, must use a reference frame fixed to the earth and not rotating with it, in order to account for the Sagnac effect that takes place between the GPS satellites and the terrestrial stations. This is an example of a preferred frame of reference, breaking all reciprocities and symmetries. Likewise, the gyrolasers, nowadays used in commercial aircraft, are based on the "asymmetric" times of flight of two opposing beams of light, S/(c+v) and S/(c-v), observed by proper observers corotating with the gyrolaser. Hence they can deduce the speed v from their observations. Again, Lorentz "wins" and Einstein "looses".

2 - In the Hafele-Keating experiment the results did not show the symmetry they were supposed to show. Any school boy knows that the term v2/c2 has even parity (+v and v yield the same result). So clocks flying to the East must have behaved exactly like those flying to the West. But they did not. The clocks that flew westward were accelerated (!) respect the resting Washington clocks, while the eastbound clocks did show the expected time dilation. To explain these asymmetric results the authors had to adopt the "viewpoint" of observers fixed to the North pole, (ie, not rotating with the Earth, just as the GPS community does). Another triumph for non-reciprocal relativity. In adopting this "polar viewpoint" Hafele-Keating were implying that the observers at rest in Washington (themselves) could see what the North pole observer "should" have seen. If that is the case, then why use any relativistic transformation at all? Regardless of the sloppiness of Hafele-Keating's experiment it showed a remarkably "neat" violation of the symmetristic and transformational relativistic scenario.

Final conclusion

    In reality what happens in all this is that when we have two really inertial, (non-communicable) frames of reference, only one side of the story can be verified experimentally in each case. As mentioned above, we cannot perform the MM experiment from a non-proper frame, neither can we observe fast mesons, or for that matter relativistic electrons, simultaneously in the non-proper and proper frame. So physical reality, by selecting in each case a proper frame and not another, breaks the theoretical symmetry of Einstein's relativity. From all this two major conclusions can be derived:

1 - Einstein's symmetristic theory can never be proven right because of methodological impossibilities. But then it cannot be disproved either, for the same reason. Relativists will always supply, in their favor, what the "imaginary" observer "must" have seen. Hence, strictly speaking, applying Popper's falsification methodology, relativity theory is not a scientific theory at all. It springs from ideal, impossibly symmetristic, thought experiments.

2 - When only primed and unprimed variables are used everything works perfectly in the mathematical heavens of group theory. But when the more "messy" but physically more revealing double index notation is used we see how symmetristic relativity becomes asymmetric by subreptitiously changing the physical meaning of the variables.



1) Einstein, A, "The Electrodynamics of Moving Bodies", (1905) Annalen der Physik, vol.17, 891.

2) Attempts at performing MM's experiment with a star light as source are vitiated by the fact that once the light enters the lens of the apparatus it is re-radiated internally, hence, becoming again an inmediate "proper" source, just like a terrestrial one.

3) Panofsky, W and Phillips, M "Classical Electricity and Magnetism", Addison-Wesley, Reading, Mass., 1962, pg. 291.

4) Panofsky and Phillips, Ibid. pg. 292.

5) Lorentz, H A, "Electromagnetic Phenomena in a System Moving with any velocity less than that of Light," Proc. Acad. Sc. Amsterdam, vol. 6, 809 (1904).

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Francisco J. Müller is the actual President of the Natural Philosophy Alliance (see the section "Alternative Physics On Line" in this same volume of Episteme), and is known for a criticism of relativity grounded even on original experimental results (see point 16 in the second list of the quoted section).

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