Elements of Classical and Quantum Physics pp 135-163 | Cite as

# Gravity

## Abstract

Hidden in the Special Relativity, there are important hints that led Einstein to its generalization. The paradox named after Paul Eherenfest dates back to 1909. He considered a rigid cylinder rotating around its axis. By symmetry, the section of the cylinder must remain circular, and the radius R should not be affected by the motion, since it is always orthogonal to the velocity. But the circumference can be visualized as a polygon with many sides, and they move parallel to the velocity \(v=\omega R\). So, Eherenfest concluded that the length of the circumference in the laboratory frame *K* should be \(2\pi R \sqrt{1-\frac{v^{2}}{c^{2}}}\), and this was a striking paradox. The problem was somewhat obscured by complications concerning the elastic response of the material constituting the cylinder, and by the practical impossibility of performing this experiment in the laboratory. However, Einstein pointed out the weak point of the above argument: it is not clear how Eherenfest would determine the length of the moving circle. The thought experiment must be done correctly. For example, the cylinder could be measured when it is fixed and afterwards it could be set in motion; but in this case, Special Relativity cannot tell what the effects of the acceleration are. The safe procedure requires adopting the reference K\({^\prime }\) which is rotating with the cylinder. For reasons of symmetry, a circumference in K is also a circumference in K\({^\prime }\), but in K\({^\prime }\) a length along the circumference is a proper length, and one can measure it using small rods. An observer in the inertial system K could count them, but would find that they are Lorentz contracted. Therefore, the solution of the paradox is that in K\({^\prime }\) more rods are needed, and so the length is *increased* to \(\frac{2\pi R}{\sqrt{1-\frac{v^{2}}{c^{2}}}},\) while in K the length is \(2\pi R\), as it should be, according to the Euclidean geometry. The physical difference between K and K\({^\prime }\) is that K is inertial, while in K\({^\prime }\), there are inertial forces. The Euclidean rules do not apply in a curved space. A somewhat similar situation occurs in a straight route from the North Pole to Rome then along the 41.9th parallel; it would result that the parallel is shorter than \( 2 \pi \) times the Rome-pole distance, because the Earth is almost spherical, and so plane Geometry does not apply. We met this argument already - recall Eqs. ( 7.17) and ( 7.29). This analogy suggests that the anomalous length of the circumference is the result of a curvature of space-time, and also of three-dimensional space, which is related to the accelerated path. Thus, Einstein’s crucial point is that the Euclidean Geometry holds in inertial systems but not in accelerated ones. The observer in K\({^\prime }\) should feel inertial forces and note that the clocks that are further from the origin run slower than those that are nearer. We have already seen that Classical Mechanics allows us to choose any reference system and the inertial forces are automatically generated by the Lagrangian formalism in a simple way; therefore, the extension of the theory to include accelerated systems is a logical necessity in the first place. The above example reveals that the inertial forces are related to a more general geometry of space-time, and this is in line with the well-known fact that they produce accelerations (e.g., centrifugal and Coriolis) that are *mass-independent*. An elephant and a mosquito receive the same acceleration from a rotating platform. But this mass-independence of the acceleration has another time-honored, celebrated occurrence, namely, Gravity.