CHAPTER IXTHE PRINCIPLES OF RELATIVITY
MUCH of the difficulty which philosophers and laymen experience in understanding Einstein’s theory arises from a confusion between the different meanings that may be attributed to the concepts “relativity of space and motion.” For the student who already possesses some knowledge of classical science, confusions of this type, of course, are not to be feared. But in view of the general misconceptions on the subject we will mention briefly the various principles, elucidating them further as we come across them in the course of this book. We must name here:
1.The primordial mathematical relativity of space and time.
2.The kinematical or visual principle of relativity.
3.The dynamical or classical Galilean and Newtonian principle of relativity.
4.Einstein’s special principle of relativity.
5.Einstein’s general principle of relativity.
6.The radical Mach-Einstein principle of relativity.
Let us consider these various principles in their order.The mathematical type of relativityis the one we have already had occasion to mention when discussing mathematical space and time. It implies that a distance in space, even after the space has been referred to the observer’s frame, has a purely relative magnitude; so that two distances in space may be congruent or unequal according to our measuring conventions. Even after we have decided upon our measuring conventions, the magnitude of a spatial distance can only be expressed by the magnitude of our measuring rod; and if during the night all lengths were to contract in the same way, no difference could be detected when we awoke on the following day. Similar conclusions would apply to time.
Now, this mathematical aspect of the relativity of space and time would appear to be in conflict with everyday experience; for if during the night all things were to move twice as fast, there is not the slightest doubt that certain very apparent physical changes would be manifest. For instance, a rapidly rotating flywheel might burst under the tremendous strain of centrifugal stresses. However, it must be remembered that the mathematician is discussing pure amorphous mathematical space; and even though Weyl’s theory throws a new light on the relativity of magnitude by referring it to the radius of the universe as a whole, the entire question is still somewhat obscure. We need lose no more time over these purely mathematical conceptions, but shall concern ourselves with the principles of relativity whichrelate to the real universe of physics, to real space, together with its metrical field.
The kinematical or visual principle of relativity, which is at least as old as the Greeks, states that a body can be considered in motion only when referred to some other body. For instance, if we consider the particular case of the earth and sun, we would conclude, according to whether we referred motion to the earth or sun, either that the sun was rotating round the earth every twenty-four hours, or else that the earth was rotating on its axis.
The two methods of presentation would be equivalent. Again, it would be impossible for us to state whether a body was approaching us faster and faster with some accelerated motion or whether it was we who were travelling with accelerated motion towards the body, visual appearances being the same in either case. Obviously this kinematical or visual principle would connote the complete relativity of all motion and rest, hence the complete relativity of space.
But a closer study of the dynamics of material systems proves the visual principle to be untenable. It was found that the behaviour of material systems, and thus the results of mechanical experiments, were influenced by absolute acceleration and rotation through space, although they remained totally unaffected by absolute velocity.
The Newtonian or Galilean or classical or dynamical principle of relativityexpresses this elusiveness of absolute velocity or Galilean motion through space so far as mechanical experiments are concerned, while it stresses by contrast the physical significance of absolute acceleration and rotation. As is well known, it was this absoluteness or physical significance of acceleration and rotation which compelled Newton to recognise that space could not be relative. That the Newtonian principle of relativity will considerably restrict the scope of the visual principle can be gathered easily from the following example:
If a body were moving with respect to our frame of reference with a relative velocity,but with no relative acceleration, we should be justified according to the Newtonian principle in maintaining that it would be impossible to decideby mechanical experimentswhether it was we who were moving towards the body or the body that was moving towards us. To this extent the visual principle is satisfied. But, on the other hand, if a relative acceleration of rotation existed as between ourselves and the body, mechanical experimentswouldenable us to ascertain what fraction of the relative acceleration or rotation was due to our own absolute acceleration or rotation in empty space, hence also what fraction was due to the body’s motion.
We now come toEinstein’s special principle of relativity. It is an exact replica of the Newtonian principle, upholding the relativity of velocity and the absoluteness of acceleration.[36]The sole difference between the two principles is that in Einstein’s, theelusiveness of absolute velocity through space is no longer restricted to mechanical experiments and to material systems; electromagnetic and optical experiments are now placed on exactly the same footing.
While we are on the subject of the various principles of relativity, we may discuss two further types which play an important part in that extension of Einstein’s special theory known as the general theory of relativity. Here we are introduced to theGeneral Principle of Relativity. This principle states that the mathematical expressions of the laws of nature must maintain the same form regardless of our choice of a frame of reference, be it Galilean (i.e., unaccelerated through empty space), accelerated, or even squirming like an octopus, while our clocks situated throughout the frame may beat at the most capricious rates. The classical principle of relativity conceded this invariance of natural laws in the case of Galilean frames, and synchronised clocks alone, and then only in the case of the laws of mechanics. The special principle recognised that this invariance held true even for the electromagnetic laws, but, as before, only for Galilean frames. The general principle of relativity, by extending the invariance of the laws of nature to all types of motions of the frame of reference, marks the starting point for the possible relativisation of acceleration, which had heretofore stood out as aloof and as distinctly absolute.
The radicalMach-Einstein principle of relativityis the result of a natural desire to bring about the complete relativisation of all manner of motion, rotationary and accelerated, as well as uniform. This is achieved by ascribing all the dynamical effects which accompany the acceleration and rotation of material and electrodynamic systems, to motion with respect to the material universe as a whole. According to this principle, which is still highly speculative, there can exist no observable difference between the rotation of a body with respect to the universe of stars and the rotation of the stars round the body; exactly the same dynamical effects of centrifugal force would be set up in either case, so that no trace of absolute motion through empty space would be left. We see that Mach’s principle constitutes an attempt to vindicate the kinematical principle in spite of the difficulties of a dynamical nature which had been the cause of its rejection. It was in part with a view to satisfying Mach’s principle that Einstein elaborated the hypothesis of the cylindrical universe. The principle would fall if astronomical observation should prove this form for the universe to be unacceptable.