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.
General Relativity Theory was admired by Lev Landau as the most beautiful , many years ago. Later, a series of striking experiments showed that it is extremely successful and far reaching, too. Einstein expected that many predictions could not be tested experimentally; now there is an impressive body of evidence with practical applications in Science, and also in everyday life. This chapter is not a substitute for a full course, but will introduce the reader to the main concepts and to recent developments.
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- 1.
A more formal derivation of this result will be found shortly.
- 2.
in order to eliminate the effects of the resistance of the air.
- 3.
The strong equivalence principle states that the outcome of any local experiment (involving Gravity or not) in a free-falling laboratory is independent of velocity and location. This requires that the gravitational constant G be the same at any location in space-time.
- 4.
Again, this implies that clock go slower in a field of gravity, as they do under the action of a centrifugal force.
- 5.
I adopt the convention used by Landau-Lifschitz, Stephani and probably the majority of Authors. Moreover, the Greek indices run from 0 to 3 and the Latin indices from 1 to 3.
- 6.
g is a symmetric tensor, \(g_{\mu \nu }=g_{\nu \mu }\), while the electromagnetic tensor F is antisymmetric. In the unified field theory proposed by Einstein and Schrödinger in the 50s the field is described by a tensor with a symmetric gravitational part and antisymmetric electromagnetic contribution.
- 7.
The orbits of many body systems are chaotic over long timescales. The Solar System possesses a Lyapunov time, perhaps in the range of a hundred million years (the Lyapunov time is the characteristic timescale on which a dynamical system is chaotic) although we know that life has definitely existed for a longer time than that. However the relativistic shift has been observed through its cumulative effects on the scale of a century.
- 8.
See CWF Everitt et al., Class. Quantum Grav. 32 (2015) 224001.
- 9.
See C.W. Everitt et al., Classical and Quantum Gravity 32, 224001 (2015).
- 10.
see e.g. John D. Jackson, “Classical Electrodynamics”, Sect. 9.3.
- 11.
In some way Theory and Experiment were in competition since Georges Lemaître had predicted the redshift-distance relation in 1927.
- 12.
However,\(\varLambda \) has a competitor, known as Quintessence . This is thought to be a hypothetical scalar field; it should provide a dynamic dark energy in the sense that it generally has a density and equation of state that varies through time and space.
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Cini, M. (2018). Gravity. In: Elements of Classical and Quantum Physics. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-71330-4_8
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DOI: https://doi.org/10.1007/978-3-319-71330-4_8
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