The Principle of Relativity — Story, Setting & Ideas

1888 the magnetic effect due to a rotating dielectric (the condenser remaining stationary) to be proportional to K - 1, and not to K. Finally Eichenwald in 1903 found that when both condenser and dielectric are rotated together, the effect observed was quite independent of K, a result quite consistent with the two previous experiments. The Rowland effect proportional to K, together with the opposite Röntgen effect proportional to 1 - K, makes the Eichenwald effect independent of K.

All these experiments together with those of Blondlot and Wilson made it clear that the electromagnetic effect due to a moving dielectric was proportional to K - 1, and not to K as required by Hertz’s theory. Thus the above group of experiments with moving dielectrics directly contradicted the Hertz-Heaviside theory. The internal discrepancies inherent in the classic ether theory had now become too prominent. It was clear that the ether concept had finally outgrown its usefulness. The observed facts had become too contradictory and too heterogeneous to be reduced to an organised whole with the help of the ether concept alone. Radical departures from the classical theory had become absolutely necessary.

There were several outstanding difficulties in connection with anomalous dispersion, selective reflection and selective absorption which could not be satisfactory explained in the classic electromagnetic theory. It was evident that the assumption of some kind of discreteness in the optical medium had become inevitable. Such an assumption naturally gave rise to an atomic theory of electricity, namely, the modern electron theory. Lorentz had postulated the existence of electrons so early as 1878, but it was not until some years later that the electron theory became firmly established on a satisfactory basis.

Lorentz assumed that a moving dielectric merely carried away its own “polarisation doublets,” which on his theory gave rise to the induced field proportional to K - 1. The field near a moving dielectric is naturally proportional to K - 1 and not to K. Lorentz’s theory thus gave a satisfactory explanation of all those experiments with moving dielectrics which required effects proportional to K - 1. Lorentz further succeeded in obtaining a value for the Fresnelian convection coefficient equal to 1 - 1/μ^2, the exact value required by all optical experiments of the moving type.

We must now go back to Michelson and Morley’s experiment. We have seen that both parts of the beam are situated in free ether; no material medium is involved in any portion of the paths actually traversed by the beam. Consequently no compensation due to Fresnelian convection of ether by moving medium is possible. Thus Fresnelian convection compensation can have no possible application in this case. Yet some marvellous compensation has evidently taken place which has completely masked the “absolute” velocity of the earth.

In Michelson and Morley’s experiment, the distance travelled by the beam along OA (that is, in a direction parallel to the motion of the platform) is 2_l_β², while the distance travelled by the beam along OB, perpendicular to the direction of motion of the platform, is 2_l_β. Yet the most careful experiments showed, as Eddington says, “that both parts of the beam took the same time as tested by the interference bands produced. It would seem that OA and OB could not really have been of the same length; and if OB was of length _l_, OA must have been of length _l_/β. The apparatus was now rotated through 90°, so that OB became the up-stream. The time for the two journeys was again the same, so that 0B must now be the shorter length. The plain meaning of the experiment is that both arms have a length _l_ when placed along O_y_ (perpendicular to the direction of motion), and automatically contract to a length _l_/β, when placed along O_x_ (parallel to the direction of motion). This explanation was first given by Fitz-Gerald.”

This Fitz-Gerald contraction, startling enough in itself, does not suffice. Assuming this contraction to be a real one, the distance travelled with respect to the ether is 2_l_β and the time taken for this journey is 2_l_β/_c_. But the distance travelled with respect to the platform is always 2_l_. Hence the velocity of light with respect to the platform is

$$ \frac {2l}{\frac {2l\beta}{c}} = \frac {c}{\beta} $$

a variable quantity depending on the “absolute” velocity of the platform. But no trace of such an effect has ever been found. The velocity of light is always found to be quite independent of the velocity of the platform. The present difficulty cannot be solved by any further alteration in the measure of space. The only recourse left open is to alter the measure of time as well, that is, to adopt the concept of “local time.” If a moving clock goes slower so that one ‘real’ second becomes 1/β second as measured in the moving system, the velocity of light relative to the platform will always remain _c_. We must adopt two very startling hypotheses, namely, the Fitz-Gerald contraction and the concept of “local time,” in order to give a satisfactory explanation of the Michelson-Morley experiment.

These results were already reached by Lorentz in the course of further developments of his electron theory. Lorentz used a special set of transformation equations[2] for time which implicitly introduced the concept of local time. But he himself failed to attach any special significance to it, and looked upon it rather as a mere mathematical artifice like imaginary quantities in analysis or the circle at infinity in projective geometry. The originality of Einstein at this stage consists in his successful physical interpretation of these results, and viewing them as the coherent organised consequences of a single general principle. Lorentz established the Relativity Theorem[3] (consisting merely of a set of transformation equations) while Einstein generalised it into a Universal Principle. In addition Einstein introduced fundamentally new concepts of space and time, which served to destroy old fetishes and demanded a wholesale revision of scientific concepts and thus opened up new possibilities in the synthetic unification of natural processes.

Newton had framed his laws of motion in such a way as to make them quite independent of the absolute velocity of the earth. Uniform relative motion of ether and matter could not be detected with the help of dynamical laws. According to Einstein neither could it be detected with the help of optical or electromagnetic experiments. Thus the Einsteinian Principle of Relativity asserts that all physical laws are independent of the ‘absolute’ velocity of an observer.

For different systems, the _form_ of all physical laws is conserved. If we chose the velocity of light[4] to be the fundamental unit of measurement for all observers (that is, assume the constancy of the velocity of light in all systems) we can establish a _metric_ “one-one” correspondence between any two observed systems, such correspondence depending only the _relative_ velocity of the two systems. Einstein’s Relativity is thus merely the consistent logical application of the well known physical principle that we can know nothing but _relative_ motion. In this sense it is a further extension of Newtonian Relativity.

On this interpretation, the Lorentz-Fitzgerald contraction and “local time” lose their arbitrary character. Space and time as measured by two different observers are naturally diverse, and the difference depends only on their relative motion. Both are equally valid; they are merely different descriptions of the same physical reality. This is essentially the point of view adopted by Minkowski. He considers time itself to be one of the co-ordinate axes, and in his four-dimensional world, that is in the space-time reality, relative motion is reduced to a rotation of the axes of reference. Thus, the diversity in the measurement of lengths and temporal rates is merely due to the static difference in the “frame-work” of the different observers.

The above theory of Relativity absorbed practically the whole of the electromagnetic theory based on the Maxwell-Lorentz system of field equations. It combined all the advantages of classic Maxwellian theory together with an electronic hypothesis. The Lorentz assumption of polarisation doublets had furnished a satisfactory explanation of the Fresnelian convection of ether, but in the new theory this is deduced merely as a consequence of the altered concept of relative velocity. In addition, the theory of Relativity accepted the results of Michelson and Morley’s experiments as a definite principle, namely, the principle of the constancy of the velocity of light, so that there was nothing left for explanation in the Michelson-Morley experiment. But even more than all this, it established a single general principle which served to connect together in a simple coherent and fruitful manner the known facts of Physics.

The theory of Relativity received direct experimental confirmation in several directions. Repeated attempts were made to detect the Lorentz-Fitzgerald contraction. Any ordinary physical contraction will usually have observable physical results; for example, the total electrical resistance of a conductor will diminish. Trouton and Noble, Trouton and Rankine, Rayleigh and Brace, and others employed a variety of different methods to detect the Lorentz-Fitzgerald contraction, but invariably with the same negative results. _Whether there is an ether or not, uniform velocity with respect to it can never be detected._ This does not prove that there is no such thing as an ether but certainly does render the ether entirely superfluous. Universal compensation is due to a change in local units of length and time, or rather, being merely different descriptions of the same reality, there is no compensation at all.

There was another group of observed phenomena which could scarcely be fitted into a Newtonian scheme of dynamics without doing violence to it. The experimental work of Kaufmann, in 1901, made it abundantly clear that the “mass” of an electron depended on its velocity. So early as 1881, J. J. Thomson had shown that the inertia of a charged particle increased with its velocity. Abraham now deduced a formula for the variation of mass with velocity, on the hypothesis that an electron always remained a _rigid_ sphere. Lorentz proceeded on the assumption that the electron shared in the Lorentz-Fitzgerald contraction and obtained a totally different formula. A very careful series of measurements carried out independently by Bücherer, Wolz, Hupka and finally Neumann in 1913, decided conclusively in favour of the Lorentz formula. This “contractile” formula follows immediately as a direct consequence of the new Theory of Relativity, without any assumption as regards the electrical origin of inertia. Thus the complete agreement of experimental facts with the predictions of the new theory must be considered as confirming it as a principle which goes even beyond the electron itself. The greatest triumph of this new theory consists, indeed, in the fact that a large number of results, which had formerly required all kinds of special hypotheses for their explanation, are now deduced very simply as inevitable consequences of one single general principle.

We have now traced the history of the development of the restricted or special theory of Relativity, which is mainly concerned with optical and electrical phenomena. It was first offered by Einstein in 1905. Ten years later, Einstein formulated his second theory, the Generalised Principle of Relativity. This new theory is mainly a theory of gravitation and has very little connection with optics and electricity. In one sense, the second theory is indeed a further generalisation of the restricted principle, but the former does not really contain the latter as a special case.

Einstein’s first theory is restricted in the sense that it only refers to uniform rectilinear motion and has no application to any kind of accelerated movements. Einstein in his second theory extends the Relativity Principle to cases of accelerated motion. If Relativity is to be universally true, then even accelerated motion must be merely _relative motion between matter and matter_. Hence the Generalised Principle of Relativity asserts that “absolute” motion cannot be detected even with the help of gravitational laws.

All movements must be referred to definite sets of co-ordinate axes. If there is any change of axes, the numerical magnitude of the movements will also change. But according to Newtonian dynamics, such alteration in physical movements can only be due to the effect of certain forces in the field.[5] Thus any change of axes will introduce new “geometrical” forces in the field which are quite independent of the nature of the body acted on. Gravitational forces also have this same remarkable property, and gravitation itself may be of essentially the same nature as these “geometrical” forces introduced by a change of axes. This leads to Einstein’s famous Principle of Equivalence. _A gravitational field of force is strictly equivalent to one introduced by a transformation of co-ordinates and no possible experiment can distinguish between the two._

Thus it may become possible to “transform away” gravitational effects, at least for sufficiently small regions of space, by referring all movements to a new set of axes. This new “framework” may of course have all kinds of very complicated movements when referred to the old Galilean or “rectangular unaccelerated system of co-ordinates.”

But there is no reason why we should look upon the Galilean system as more fundamental than any other. If it is found simpler to refer all motion in a gravitational field to a special set of co-ordinates, we may certainly look upon this special “framework” (at least for the particular region concerned), to be more fundamental and more natural. We may, still more simply, identify this particular framework with the special local properties of space in that region. That is, we can look upon the effects of a gravitational field as simply due to the local properties of space and time itself. The very presence of matter implies a modification of the characteristics of space and time in its neighbourhood. As Eddington says “matter does not cause the curvature of space-time. It is the curvature. Just as light does not cause electromagnetic oscillations; it is the oscillations.”

We may look upon this from a slightly different point of view. The General Principle of Relativity asserts that all motion is merely relative motion between matter and matter, and as all movements must be referred to definite sets of co-ordinates, the ground of any possible framework must ultimately be material in character. It _is_ convenient to take the matter actually present in a field as the fundamental ground of our framework. If this is done, the special characteristics of our framework would naturally depend on the actual distribution of matter in the field. But physical space and time is completely defined by the “framework.” In other words the “framework” itself _is_ space and time. Hence we see how _physical_ space and time is actually defined by the local distribution of matter.

There are certain magnitudes which remain constant by any change of axes. In ordinary geometry distance between two points is one such magnitude; so that δ_x²_ + δ_y²_ + δ_z²_ is an invariant. In the restricted theory of light, the principle of constancy of light velocity demands that δ_x²_ + δ_y²_ + δ_z²_ - _c²_δ_t²_ should remain constant.

The _separation ds_ of adjacent events is defined by _ds²_ = -_dx²_ - _dy²_ - _dz²_ + _c²dt²_. It is an extension of the notion of distance and this is the new invariant. Now if _x_, _y_, _z_, _t_ are transformed to any set of new variables _x₁_, _x₂_, _x₃_, _x₄_, we shall get a quadratic expression for

$$ ds^2 = g_{1\;1}x_{1}^2 + 2g_{1\;2}x_{1}x_{2} + ... = \sum g_{i\;j}x_{i}x_{j} $$

where the _g_’s are functions of _x₁_, _x₂_, _x₃_, _x₄_ depending on the transformation.

The special properties of space and time in any region are defined by these _g_’s which are themselves determined by the actual distribution of matter in the locality. Thus from the Newtonian point of view, these _g_’s represent the gravitational effect of matter while from the Relativity stand-point, these merely define the non-Newtonian (and incidentally non-Euclidean) space in the neighbourhood of matter.

We have seen that Einstein’s theory requires local curvature of space-time in the neighbourhood of matter. Such altered characteristics of space and time give a satisfactory explanation of an outstanding discrepancy in the observed advance of perihelion of Mercury. The large discordance is almost completely removed by Einstein’s theory.

Again, in an intense gravitational field, a beam of light will be affected by the local curvature of space, so that to an observer who is referring all phenomena to a Newtonian system, the beam of light will appear to deviate from its path along an Euclidean straight line.

This famous prediction of Einstein about the deflection of a beam of light by the sun’s gravitational field was tested during the total solar eclipse of May, 1919. The observed deflection is decisively in favour of the Generalised Theory of Relativity.

It should be noted however that the velocity of light itself would decrease in a gravitational field. This may appear at first sight to be a violation of the principle of constancy of light-velocity. But when we remember that the Special Theory is explicitly _restricted_ to the case of unaccelerated motion, the difficulty vanishes. In the absence of a gravitational field, that is in any unaccelerated system, the velocity of light will always remain constant. Thus the validity of the Special Theory is completely preserved within its own _restricted_ field.

Einstein has proposed a third crucial test. He has predicted a shift of spectral lines towards the red, due to an intense gravitational potential. Experimental difficulties are very considerable here, as the shift of spectral lines is a complex phenomenon. Evidence is conflicting and nothing conclusive can yet be asserted. Einstein thought that a gravitational displacement of the Fraunhofer lines is a necessary and fundamental condition for the acceptance of his theory. But Eddington has pointed out that even if this test fails, the logical conclusion would seem to be that while Einstein’s law of gravitation is true for matter in bulk, it is not true for such small material systems as atomic oscillator.

From the conceptual stand-point there are several important consequences of the Generalised or Gravitational Theory of Relativity. Physical space-time is perceived to be intimately connected with the actual local distribution of matter. Euclid-Newtonian space-time is _not_ the actual space-time of Physics, simply because the former completely neglects the actual presence of matter. Euclid-Newtonian continuum is merely an abstraction, while physical space-*time is the actual framework which has some definite curvature due to the presence of matter. Gravitational Theory of Relativity thus brings out clearly the fundamental distinction between actual physical space-time (which is non-isotropic and non-Euclid-Newtonian) on one hand and the abstract Euclid-Newtonian continuum (which is homogeneous, isotropic and a purely intellectual construction) on the other.

The measurements of the rotation of the earth reveals a fundamental framework which may be called the “inertial framework.” This constitutes the actual physical universe. This universe approaches Galilean space-time at a great distance from matter.

The properties of this physical universe may be referred to some world-distribution of matter or the “inertial framework” may be constructed by a suitable modification of the law of gravitation itself. In Einstein’s theory the actual curvature of the “inertial framework” is referred to vast quantities of undetected world-matter. It has interesting consequences. The dimensions of Einsteinian universe would depend on the quantity of matter in it; it would vanish to a point in the total absence of matter. Then again curvature depends on the quantity of matter, and hence in the presence of a sufficient quantity of matter space-time may curve round and close up. Einsteinian universe will then reduce to a finite system without boundaries, like the surface of a sphere. In this “closed up” system, light rays will come to a focus after travelling round the universe and we should see an “anti-sun” (corresponding to the back surface of the sun) at a point in the sky opposite to the real sun. This anti-sun would of course be equally large and equally bright if there is no absorption of light in free space.

In de Sitter’s theory, the existence of vast quantities of world-matter is not required. But beyond a definite distance from an observer, time itself stands still, so that to the observer nothing can ever “happen” there. All these theories are still highly speculative in character, but they have certainly extended the scope of theoretical physics to the central problem of the ultimate nature of the universe itself.

One outstanding peculiarity still attaches to the concept of electric force—it is not amenable to any process of being “transformed away” by a suitable change of framework. H. Weyl, it seems, has developed a geometrical theory (in hyper-space) in which no fundamental distinction is made between gravitational and electrical forces.