Dialog about objections against the theory of relativity

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Critic:
Many times already have the likes of mine submitted to journals a wide variety of reservations about the theory of relativity; and rarely has one of you relativists ^[1] provided an answer. We have no wish to dwell on whether this neglect was due to arrogance, or a sense of weakness, or laziness - maybe it was a particularly effective mixture of these afflictions of the soul, and then maybe it wasn't rare for the criticism to reveal that the critic simply had too little knowledge of the matter at hand. These issues will - as announced - not be discussed, but first I want to tell you this: I have come to visit you personally, so that you will not be able to back out, as you did on other occasions. For I promise you that I will stay until you have answered all of my questions.

So as not to upset you too much, and possibly even make you undertake this business (which you can't avoid anyway) with a certain pleasure, I will say this in comfort. Unlike many of my collegues, I am not so full with the status of my guild so as to make me act as a superior being with superhuman insight and certainty (like newspaper journalists about scientific literature, or playwright-critics). On the contrary, I talk as a human being, since I am aware that it is not rare for criticism to originate from lack of own thoughts. Also I have no wish to - as was lately done by one of my colleagues - jump on you like a district-attorney and accuse you of theft of intellectual property, or accuse you of equally dishounorable acts. Only the need to contribute to the clarification of several points, on which opinions still widely divert, has motivated my assault. However I must request you to grant publication of our conversation, not in the least because the shortage of paper is not the only shortage that is causing my friend, the editor of the Berolinensis, to lose sleep.

Since I notice your willingness to comply, I will immediately come to business. Ever since the special theory of relativity has been formulated, the outcome concerning the slowing influence of motion on the rate of clocks has continuously elicited opposition, and - it seems to me - with good reasons. For this outcome seems to lead inevitably to a contradiction with the foundations of the theory. To make sure we understand each other completely, let the theory now be presented with sufficient sharpness.

Let K be a Galilean coordinate system in the sense of the special theory of relativity, that is, a frame of reference, relative to which isolated, material points move in straight lines and uniformly. Also, let U1 and U2 be two identical clocks that are free from outside influences. These will run at the same pace when they are in close proximity and also at any distance from each other, if they are both at rest relative to K. However, if one of the clocks, for example U2, is relative to K in a state of uniform translational motion, then according to the special theory of relativity it should - as perceived from coordinate system K - go at a slower pace than the clock U1 that is at rest relative to K. This result is in itself highly peculiar. Grave doubts arise when one faces the following well-known thought experiment.

Let A and B be two distant points of the system K. To render the picture more precise, let A be the origin of K, and B be a point on the positive x-axis. The two clocks are initially at rest at point A. They run at the same pace, and let the positions of the hands be the same. We now impart to clock U2 a constant velocity in the positive direction of the x-axis, so that it moves towards B. At B we imagine the velocity reversed, so that clock U2 returns to A. As it arrives at A, the clock is decelerated so that it is once again at rest relative to U1. Because the change in the position of the hands of the clock, as judged from K, that might occur during the change of velocity, will not exceed a certain value, and because U2 runs slower than U1 during the motion along the length of A B (as judged from K), clock U2 must, if the length A B is sufficiently long, be running behind U1 - do you agree with that?

Relativist:
Entirely agreed. With regret I have noticed that some authors, who otherwise have a thorough understanding of the theory of relativity, wanted to avoid this inevitable result.

Critic:
Now here's the twist. According to the principle of relativity the whole affair should proceed in the same way if it is represented in a coordinate system K', that is co-moving with clock U2. Then relative to K' it is clock U1 that is moving to and fro, with clock U2 remaining at rest. It then follows that at the end U1 should run behind U2, in contradiction with the above result. Surely even the most devoted followers of the theory will not assert that in the case of two clocks that have been positioned side by side, each one is running behind the other.

Relativist:
Your last assertion is of course undisputable. However, the reason that that line of argument as a whole is untenable is that according to the special theory of relativity the coordinate systems K and K' are by no means equivalent systems. Indeed this theory asserts only the equivalence of all Galilean (unaccelerated) coordinate systems, that is, coordinate systems relative to which sufficiently isolated, material points move in straight lines and uniformly. K is such a coordinate system, but not the system K', that is accelerated from time to time. Therefore, from the result that after the motion to and fro the clock U2 is running behind U1, no contradiction can be constructed against the principles of the theory.

Critic:
I acknowledge that you have rendered my objection powerless, but I have to say that I feel convicted by your argument rather than convinced. Anyway, my objection immediately arises from its ashes when one bases oneself on the general theory of relativity. For according to this theory, coordinate systems in arbitrary states of motion are qualified, hence the proceedings described earlier can equally well be referred to the coordinate system K' that is continuously connected to U2, as to the system K.

Relativist:
It is certainly correct that from the point of view of the general theory of relativity we can just as well use coordinate system K' as coordinate system K. But it is easy to see that the systems K and K' in connection with the examined proceedings stand by no means on equal footing. While the proceedings as seen from system K can be regarded as above, a totally different picture presents itself as seen from K', as can be seen from the following comparison:

It should be kept in mind that in the left and in the right section exactly the same proceedings are described, it is just that the description on the left relates to the coordinate system K, the description on the right relates to the coordinate system K'. According to both descriptions the clock U2 is running a certain amount behind clock U1 at the end of the observed process. When relating to the coordinate system K' the behaviour explains itself as follows: During the partial processes 2 and 4 the clock U1, going at a velocity v, runs indeed at a slower pace than the resting clock U2. However, this is more than compensated by a faster pace of U1 during partial process 3. According to the general theory of relativity, a clock will go faster the higher the gravitational potential of the location where it is located, and during partial process 3 U2 happens to be located at a higher gravitational potential than U1. The calculation shows that this speeding ahead constitutes exactly twice as much as the lagging behind during the partial processes 2 and 4. This consideration completely clears up the paradox that you brought up.

Critic:
I do see that you have cleverly pulled away from the noose, but I would be lying if I would declare myself fully satisfied. The stumbling stone has not been removed; it has been relocated. You see, your consideration only shows the connection of the difficulty that was just discussed with another difficulty, that has also often been presented. You have solved the paradox, by taking the influence on the clocks into account of a gravitational field relative to K'. But isn't this gravitational field merely fictitious? Its existence is conjured up by a mere choice of coordinate system. Surely, real gravitational fields are brought forth by mass, and cannot be made to disappear by a suitable choice of coordinate system. How are we supposed to believe that a merely fictitious field could have such an influence on the pace of a clock?

Relativist:
In the first place I must point out that the distinction real - unreal is hardly helpful. In relation to K' the gravitational field "exists" in the same sense as any other physical entity that can only be defined with reference to a coordinate system, even though it is not present in relation to the system K. No special peculiarity resides here, as can easily be seen from the following example from classical mechanics. Nobody doubts the "reality" of kinetic energy, otherwise the very reality of energy would have to be denied. But it is clear that the kinetic energy of a body is dependent on the state of motion of the coordinate system, with a suitable choice of the latter one can arrange for the kinetic energy of the continuous motion of a body to assume a given positive value or the value of zero. In the special case where all the masses have a velocity in the same direction and of the same magnitude, a suitable choice of coordinate system can adjust the collective kinetic energy to zero. To me it appears that the analogy is complete.

Rather than distinguishing between "real" and "unreal" we want to more clearly distinguish between quantities that are inherent in the physical system as such (independent from the choice of coordinate system), and quantities that depend on the coordinate system. The next step would be to demand that only quantities of the first kind enter the laws of physics. However, it has been found that this objective cannot be realized in practice, as has already been demonstrated clearly by the development of classical mechanics. One could for instance consider, and this has actually been attempted, to enter into the laws of classical mechanics not the coordinates, but instead just the distances between the material points; a priori one could expect that in this way the goal of the theory of relativity would be reached most easily. The scientific development has however not confirmed this expectation. She cannot dispense with the coordinate system, and therefore has to use in the coordinates quantities that cannot be construed as results of definite measurements. According to the general theory of relativity the four coordinates of the space-time continuum are entirely arbitrary choosable parameters, devoid of any independent physical meaning. This arbitrariness partially affects also those quantities (field components) that are instrumental in describing the physical reality. Only certain, generally quite complicated expressions, that are constructed out of field components and coordinates, correspond to coordinate-independent, measurable (that is, real) quantities. For example, the component of the gravitational field in a space-time point is still not a quantity that is independent of coordinate choice; thus the gravitational field at a certain place does not correspond to something "physically real", but in connection with other data it does. Therefore one can neither say, that the gravitational field in a certain place is something "real', nor that it is "merely fictitious".

The circumstance that according to the general theory of relativity the connection between the quantities that occur in the equations and the measurable quantities is much more indirect than in terms of the usual theories, probably constitutes the main difficulty that one encounters when studying this theory. Also your last objection was based on the fact that you did not keep this circumstance constantly in mind.

You declared the fields that were called for in the clock example also as merely fictitious, only because the field lines of actual gravitational fields are necessarily brought forth by mass; in the discussed examples no mass that could bring forth those fields was present. This can be elaborated upon in two ways. Firstly, it is not an a priori necessity that the particular concept of the Newtonian theory, according to which every gravitational field is conceived as being brought forth by mass, should be retained in the general theory of relativity. This question is interconnected with the circumstance pointed out previously, that the meaning of the field components is much less directly defined as in the Newtonian theory. Secondly, it cannot be maintained that there are no masses present, that can be attributed with bringing forth the fields. To be sure, the accelerated coordinate systems cannot be called upon as real causes for the field, an opinion that a jocular critic saw fit to attribute to me on one occasion. But all the stars that are in the universe, can be conceived as taking part in bringing forth the gravitational field; because during the accelerated phases of the coordinate system K' they are accelerated relative to that and through that can induce a gravitational field, similar to how electric charges in accelerated motion can induce an electric field. Approximate integration of the gravitational equations has in fact yielded the result that induction effects must occur when masses are in accelerated motion. This consideration makes it clear that a complete clarification of the questions you have raised can only be attained if one envisions for the geometric-mechanical constitution of the Universe a representation that complies with the theory. I have attempted to do so last year, and I have reached a conception that - to my mind - is completely satisfactory; going into this would however take us too far.

Critic:
After your last statements it does seem to me that no self-contradiction of the theory of relativity can be deduced from the clock-paradox. Indeed, it now seems not unlikely to me that the theory is free from self-contradiction altogether, but it does not in itself mean that the theory should be considered in earnest. I really don't see why for the sake of some conceptual preference - namely for the concept of relativity - one ought to accept the burden of such gruesome complications and calculational difficulties. In your last answer you yourself have amply demonstrated that they are considerable. For example, would anyone get it in his head to actually use the possibility offered by the theory of relativity to relate the motions of the celestial bodies of the solar system to a geocentric coordinate system that on top of that is participating in the rotation of the Earth? Would anyone really be allowed to see this coordinate system as "at rest" and as equally valid, relative to which the fixed stars are tearing around with tremendous speed? Doesn't such an approach collide head on with common sense, and with the demand of economy of thought? I will not refrain now from repeating the drastic words that have lately been uttered by Lenard about the subject. After discussing the special theory of relativity, in which he modeled the "moving" coordinate system with a rolling train carriage, he said: "Now imagine that this train carriage makes a clearly non-uniform motion. When through the action of inertia everything inside the train is wrecked, while outside everything remains undamaged, then, in my opinion, no sane mind will draw any other conclusion than that it was the train that changed its motion with a jerk, and not the surroundings. The generalized principle of relativity demands in its simple sense, that in this case also it must be admitted that possibly it was after all the surroundings that went through the change of velocity, and that the entire accident in the train is just a consequence of this jerk of the environment, transmitted through a "gravitational effect" of the environment on the inside of the train. For the related question why the church tower adjacent to the train hasn't toppled over, while it underwent the jerk together with the surroundings - why such consequences of the jerk are so exclusively confined to the train, even though an unambiguous conclusion about the seat of the change of motion is supposed not to be had - the principle has as it would seem no satisfying answer for the simple mind.

Relativist:
There are several reasons that compel us to willingly accept the complications that the theory leads us to. In the first place, it means for a man who maintains consistency of thought a great satisfaction to see that the concept of absolute motion, to which kinematically no meaning can be attributed, does not have to enter physics; it cannot be denied that by avoiding this concept the foundation of physics has gained in consistency. Also the fact of the equality of inertia and weight of a body urgently requires an explanation. Apart from that, physics needs a method to attain an action-by-contact-theory of gravitation. Without an effective confining principle the theorists could hardly attempt the problem, because ever so many theories could be formulated, that satisfy the limited experiences in this area. Embarras de richesse is one of the most malicious opponents to make the theoretician's life miserable. The postulate of relativity reduces the possibilities in such a way that the road that the theory had to go was predetermined. Lastly the secular perihelion motion of the planet Mercury had to be clarified. This perihelion motion was certainly noticed by astronomers, and they were unsuccessful in finding an explanation on the basis of the Newtonian theory. - In asserting the equality of coordinate systems as a matter of principle it is not said that every coordinate system is equally convenient for examining a certain physical system; we see this in classical mechanics also. For example, strictly speaking one cannot say that the Earth moves in an ellipse around the Sun, because that statement presupposes a coordinate system in which the Sun is at rest, while classical mechanics also allows systems relative to which the Sun rectilinearly and uniformly moves. In examining the motion of the Earth nobody will decide to use a coordinate system of the last kind, and neither will anyone decide from considering this example that the coordinate system, whose origin is co-moving with the center of mass of the considered mechanical system, is principally privileged over other coordinate systems. It is the same in the example you mentioned. Nobody will use a coordinate system that is at rest relative to the planet Earth, because that would be impractical. However as a matter of principle such a theory of relativity is equally valid as any other. The situation, that the fixed stars are circling with tremendous velocities, when one bases an examination on such a coordinate system, does not constitute an argument against the admissibility, but merely against the efficiency of this choice of coordinates, nor does the complicated form of the relative to this coordinate system acting gravitational field, which for example would also have the components that correspond to the centrifugal force. In Mister Lenard's example the situation is similar. In terms of the theory of relativity the case may not be construed in such a way that possibly it is after all the surroundings (of the train) that experienced the change in velocity. We are not dealing here with two different, mutually exclusive hypotheses about the seat of the motion, rather with two ways of principally equal validity to represent the same factual situation. ^[2] For the decision which representation to choose only reasons of efficiency are decisive, not arguments of a principle kind. Just how little merit there is in calling upon the socalled "common sense", is shown by the following counterexample. Lenard himself says that so far no objections could be raised against the validity of the special principle of relativity (that is the principle of relativity of uniform translational motion of coordinate systems). The uniformly riding train can equally well be regarded as "at rest", the rails together with the entire surroundings can be regarded as "uniformly moving". Would the "common sense" of the train driver allow for that? He will point out that it is not the surroundings that he needs to continuously heat and lubricate, it is the locomotive, and consequently it must be in the latter that the result of his labour shows itself.

Critic:
After this conversation I have to admit that the refutation of your point of view is not as easy as it seemed to me earlier. I do have more objections up my sleeve. But before pestering you with that I want to think over our present conversation thoroughly. Before we depart, one more question, that does not concern an objection, but that I ask out of pure curiosity: how does the diseased man of theoretical physics fare, the Aether, that many of you have declared to be definitely dead?

Relativist:
Its fortunes have taken some turns, and overall one cannot say that it is dead now. Prior to Lorentz it existed as an all-pervasive fluid, as a gas-like fluid, and other than that in the most diverse forms of being, different from author to author. With Lorentz it became rigid, and embodied the resting coordinate system, respectively a privileged state of motion in the world. According to the special theory of relativity there was no longer a privileged state of motion, this meant a denial of the Aether in this sense of the preceding theories. For if there would be an Aether, then in each space-time point there would have to be a particular state of motion, that would have to play a part in optics. There is no such privileged state of motion, as has been taught to us by the special theory of relativity, and that is why there is no Aether in the old sense. The general theory of relativity also does not know a privileged state of motion in a point, that one could vaguely interpret as velocity of an Aether. However, while according to the special theory of relativity a part of space without matter and without electromagnetic field seems to be characterized as absolutely empty, e. g. not characterized by any physical quantities, empty space in this sense has according to the general theory of relativity physical qualities which are mathematically characterized by the components of the gravitational potential, that determine the metrical behavior of this part of space as well as its gravitational field. One can quite well construe this circumstance in such a way that one speaks of an Aether, whose state of being is different from point to point. Only one must take care not to attribute to this Aether properties similar to properties of matter (for example every point a certain velocity).

1. ↑ Here, "relativist" is to be understood as a supporter of the physical theory of relativity, not the philosophic relativism
2. ↑ That the tower doesn't fall over is, according to the second representation, due to the fact that it free-falls with the ground and the entire Earth in a gravitational field (which is present during the jerk), while the train is kept from free falling by external forces (braking forces). A free falling body behaves with regard to internal processes as a free floating body, isolated from all external influences.

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