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Henri Poincaré, Space and time (2) – Physical and psychological relativity

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The psychological time, the Bergsonian duration, from which the scientist’s time has come out, serves to classify the phenomena that occur in the same consciousness; it is powerless to classify two psychological phenomena which have for their theater two different consciences or, a fortiori, two physical phenomena. One event is happening on Earth, another on Sirius; how will we know if the first is anterior to the second, or simultaneous, or posterior? it can only be by convention.

But we can consider the relativity of time and space from an entirely different point of view. Consider the laws to which the world obeys; they can be expressed by differential equations; we find that these equations are not altered, if we change the rectangular axes of coordinates, these axes remaining fixed; neither if one changes the origin of time, nor if one replaces the fixed rectangular axes by moving rectangular axes, but whose movement is a rectilinear and uniform translation. Let me call the relativity as psychological if it is considered from the first point of view and physical if it is to the second. You see right away that physical relativity is much more restricted than psychological relativity. We have said, for example, that nothing would be changed, if we multiply all the lengths by the same constant, provided that the multiplication carried at once on all the objects and all the instruments; now, if we multiply all the coordinates by the same constant, it is possible that our differential equations are altered. They would be so if the system were brought back to rotating rotary axes since it would be necessary to introduce the ordinary centrifugal force and the compound centrifugal force; this is how Foucault’s experiment could highlight the rotation of the Earth. There is something here that shocks our ideas about the relativity of space, ideas based on psychological relativity, and this disagreement has seemed embarrassing to many philosophers.

Let’s take a closer look at the question. All parts of the world are in solidarity and no matter where Sirius is, it is probably not absolutely without action on what is going on here. If, then, we want to write the differential equations governing the world, either these equations will be inaccurate, or they will have to depend on the state of the whole world. There will not be a system of equations for the terrestrial world, and another for the world of Sirius, there will be one that will apply to the whole universe. Now, we do not observe the differential equations directly; what we observe is the finite equations which are the immediate translation of the observable phenomena and from which the differential equations are deduced by differentiation. The differential equations are not altered when we make one of the changes of axes of which we have spoken, but it is not the same of the finite equations; the change of axes would force us to change the integration constants. The principle of relativity does not apply to finite equations directly observed, but to differential equations.

Now, how can one go from finite equations to differential equations of which they are integral? it is necessary to know several particular integrals differing from each other by the values ​​attributed to the constants of integration, then to eliminate these constants by differentiation; only one of these solutions is realized in nature, although there is an infinity of possibilities; to form the differential equations, one would have to know not only the one that is realized, but all those that are possible.

Now, if we have only one system of laws applicable to the whole universe, observation will give us only one solution, that which is realized; for the universe is drawn only from one copy; and this is a first difficulty.

Moreover, by virtue of the psychological relativity of space, we can only observe what our instruments can measure; they will give us, for example, the distances of the stars, or the various bodies which we have to consider; they will not give us their coordinates in relation to fixed or moving axes which have only a purely conventional existence. If our equations contain these coordinates, it is by a fiction which can be convenient, but which is only a fiction; if we want our equations to directly translate what we observe, distances will have to be included among our independent variables, and then other variables will disappear by themselves. This will be our principle of relativity, but it no longer makes sense; it only means that we have introduced in our equations auxiliary variables, parasitic, which represent nothing tangible and that it is possible to eliminate them.

These difficulties will vanish if we do not hold to absolute rigor. The various parts of the world are united, but if the distance is great, the action is so weak that it is right to neglect it; and then our equations will be divided into separate systems, one applying to the terrestrial world alone, the other to the solar world, the other to the world of Sirius, or even to much smaller worlds such as the table of a laboratory.

And then it is no longer true to say that the universe is drawn only to a single copy; there can be many tables in a laboratory; it will be possible to repeat an experiment by varying the conditions; we will not know either a single solution, the only one realized, but a large number of possible solutions and it will become easy to pass from finite equations to differential equations.

On the other hand, we will know not only the mutual distances of the various bodies of one of these small worlds, but their distances from the bodies of the neighboring little worlds. We can arrange for only the seconds to vary, the former remaining constant. It will then be as if we had changed the axes to which the first small world was reported. The stars are too far to act substantially on our terrestrial world, but we see them, and thanks to them we can bring this terrestrial world to axes related to these stars; we have the means of measuring at once the mutual distances of the terrestrial bodies and the coordinates of these bodies with respect to this system of axes which is foreign to the terrestrial world. The principle of relativity thus takes on meaning: it becomes verifiable.

Let us observe, however, that we have obtained this result only by neglecting certain actions, and yet we do not consider our principle merely approximate; we attribute to it an absolute value; seeing that it remains true, no matter how far apart our little worlds are from each other, we agree that it is true for the exact equations of the universe; and this convention will never be in want, since, applied to the whole universe, the principle is unverifiable.

Now, let’s go back to the case we talked about earlier; a system is sometimes referred to fixed axes, sometimes to rotating axes; will the equations governing it change? Yes, answers the ordinary Mechanics; is it correct ? What we observe is not the coordinates of the bodies, but their mutual distances; we could therefore seek to form the equations to which these distances obey, by eliminating the other quantities, which are only parasitic variables and inaccessible to observation. This elimination is always possible; only if we had kept the coordinates, we would have arrived at differential equations of the second order; those which we will obtain after having eliminated all that is not observable, will be on the contrary of the 3rd order, so that they will give place to a greater number of possibilities. On this account the principle of relativity will still apply to this case; when we move from fixed axes to revolving axes, these equations of the 3rd order will not vary. What will vary are the 2d order equations that define the coordinates; now, these are, so to speak, integrals of the first, and as in all the integrals of the differential equations, there is an integration constant, it is this constant which does not remain the same when we move from fixed axes to rotating axes. But, as we suppose our system completely isolated in space, that we regard it as the whole universe, we have no way of knowing whether it is turning; it is therefore the equations of the third order which express what we observe.

Instead of considering the whole universe, let us now consider our little separate worlds without mechanical action on each other, but visible to each other; if one of these worlds turns, we will see then that it turns; we will recognize that the value to be attributed to the constant of which we have just spoken depends on the speed of rotation, and this is how the convention habitually adopted by the mechanics will be justified.

We thus see what is the meaning of the principle of physical relativity; it is no longer a simple convention; it is verifiable and therefore it might not be verified; it is an experimental truth, and what is the meaning of this truth? It is easy to deduce from the foregoing considerations; it signifies that the mutual action of two bodies tends to zero when these two bodies move indefinitely away from each other; it means that two distant worlds behave as if they were independent; and it is better understood why the principle of physical relativity has less extension than the principle of psychological relativity; it is no longer a necessity due to the very nature of our mind; it is an experimental truth to which experience imposes limits.

This principle of physical relativity can be used to define space; it provides us, so to speak, with a new measuring instrument. Let me explain: how could the solid body be used to measure, or rather to construct space? By conveying a solid body from one position to another, we recognized that it could be applied first to one figure and then to another, and we agreed to consider these two figures as equal. From this convention came geometry. Each possible movement of the solid body thus corresponded to a transformation of space in itself, not altering the shapes and magnitudes of the figures; and the geometry is only the knowledge of the mutual relations of these transformations, or to speak the mathematical language, the study of the structure of the group formed by these transformations, that is to say of the group of the movements of the solid bodies .

That being said, here is another group, that of transformations which do not alter our differential equations; here is another way of defining the equality of two figures; we will no longer say: two figures are equal when the same solid body can be applied to one and to the other; we will say: two figures are equal when the same mechanical system, rather distant from neighboring systems to be regarded as isolated, placed first so that its different material points reproduce the first figure, and then so that they reproduce the second, then behave in the same way.

Do the two conceptions differ essentially one from the other? No; a solid body takes its form under the influence of the mutual attractions and repulsions of its different molecules; and this system of forces must be in equilibrium. To define the space so that a solid body retains its shape when it is moved, is to define it in such a way that the equilibrium equations of this body are not altered by a change of axes; now, these equilibrium equations are only a particular case of the general equations of Dynamics, which, according to the principle of physical relativity, must not be modified by this change of axes.

A solid body is a mechanical system like any other; the only difference between our old definition of space and the new is that it is wider, in that it allows the solid body to be replaced by any other mechanical system. Moreover, the new convention does not only define space, it defines time. It teaches us what two simultaneous instants are, what is two equal times or one double time of another.

One last remark: the principle of physical relativity, as we have said, is an experimental fact, just as the properties of natural solids; as such, it is susceptible of incessant revision; and geometry must escape this revision; for that it must become again a convention, that the principle of relativity be regarded as a convention; we have said what is its experimental meaning, it signifies that the mutual action of two very distant systems tends to zero when their distance increases indefinitely; experience tells us that this is almost true; it can not teach us that this is entirely true, since the distance of the two systems will always remain finite. But nothing prevents us from supposing that this is entirely true; nothing would prevent us even if experience gave the principle an apparent contradiction; suppose that the mutual action, after having diminished when we increase the distance, then begins to grow; nothing would prevent us from admitting that for an even greater distance it would decrease again and finally reach zero. Only then does the principle appear to us as a convention, which removes it from the experience. It is a convention which is suggested to us by experience, but which we adopt freely.

What is the revolution that is due to the recent progress of Physics? The principle of relativity, in its old form, had to be abandoned, it is replaced by the principle of relativity of Lorentz. It is the transformations of the “Lorentz group” that do not alter the differential equations of Dynamics. If we suppose that the system is no longer related to fixed axes, but to axes animated by a translation movement, we must admit that all bodies deform, that a sphere, for example, transforms into an ellipsoid whose minor axis is parallel to the translation of the axes; time itself must be profoundly modified; here are two observers, the first linked to the fixed axes, the second to the moving axes, but believing each other at rest. Not only this figure, which the first looks like a sphere, will appear to the second like an ellipsoid; but two events which the first will regard as simultaneous will no longer be so for the second.

Everything happens as if time was a fourth dimension of space; and as if the four-dimensional space resulting from the combination of ordinary space and time could rotate not only around an axis of ordinary space, so that time is not altered, but around any axis. For the comparison to be mathematically correct, it would be necessary to attribute purely imaginary values ​​to this fourth coordinate of space; the four coordinates of a point of our new space would not be x, y, z and t, but x, y, z, and t√-1. But I do not insist on this point; the essential thing is to notice that in the new conception space and time are no longer two entirely distinct entities and that we can consider separately, but two parts of the same whole and two parts which are as closely entwined with each other so that we can no longer separate them easily.

Another remark: I once sought to define the relation of two events occurring in two different theaters by saying that one will be regarded as anterior to the other if it can be considered as the cause of the other. This definition becomes insufficient; in this New Mechanics, there is no effect that is transmitted instantly; the maximum transmission speed is that of the Light; under these conditions it may happen that the event A cannot be (by virtue of the sole consideration of space and time) nor the effect nor the cause of the event B, if the distance from the places where they occur is such that the Light can not be transported in time either from the place of B instead of A, nor from the place of A instead of B.

What will be our position in front of these new conceptions? Are we going to be forced to change our conclusions? Certainly not: we had adopted a convention because it seemed convenient to us, and we said that nothing could force us to abandon it. Today some physicists want to adopt a new convention. It is not that they are constrained; they think this new convention more convenient, that’s all; and those who are not of this opinion may legitimately preserve the old, in order not to disturb their old habits. I believe between us that this is what they will do for a long time.

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