Abstract
The physical measurement of the 4-rotation of a given observer is first discussed. Then uniformly rotating observers are introduced, as well as the associated corotating observers, forming the so-called rotating disk. The problem of clock synchronization among a set of corotating observers is investigated, with the application to the definition of a timescale at the surface of the Earth. The Ehrenfest paradox regarding the transition of a rotating disk from rest to motion is discussed in detail. Finally the Sagnac effect is investigated and various applications of it are presented.
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Notes
- 1.
Let us recall that t, the proper time of \(\mathcal{O}\), is also the proper time of \(\mathcal{O}_{{\ast}}\).
- 2.
Lev D. Landau (1908–1968): Soviet theoretical physicist, 1962 Nobel Prize in Physics for the explanation of superfluidity; Landau contributed to many fields of physics, among which relativistic hydrodynamics. He wrote with Evgeny Lifshitz a course covering all theoretical physics of the twentieth century (Landau and Lifshitz 1975).
- 3.
Evgeny M. Lifshitz (1915–1985): Soviet theoretical physicist, former student of Landau, specialist of solid state physics and general relativity.
- 4.
Theodor Kaluza (1885–1954): German mathematician, mostly known for his work in theoretical physics, especially for the so-called Kaluza–Klein theory (1921)—an attempt to unify gravitation and electromagnetism (the only known interactions at that time) in a five-dimensional space. Polyglot, Kaluza was speaking not less than 17 languages!
- 5.
t ground and t plane were noted, respectively, T and T ′ in Sect. 2.6.6.
- 6.
In Sect. 2.6.6, the considered quantity was T ′ − T = t plane − t ground, rather than t ground − t plane.
- 7.
Additionally, one must correct from the general relativistic redshift mentioned above, because \(\mathcal{O}^{\prime}\) is higher in the Earth gravitational potential than \(\mathcal{O}_{{\ast}}\), who is located at the Earth centre.
- 8.
Paul Ehrenfest (1880–1933): Austrian physicist (naturalized Dutch in 1922), known for his work in quantum mechanics and famous for the clarity of his physics lecture at the University of Leiden. He got depressed and committed suicide, as his thesis advisor, Ludwig Boltzmann, did 27 years before.
- 9.
Actually, Ehrenfest considered a cylinder, rather than a disk; but this changes nothing to the present discussion, since the height of the cylinder plays no role.
- 10.
\(\varphi _{{\ast}}\) stands for the azimuthal coordinates related to the inertial coordinates (x ∗ , y ∗ ) by \(x_{{\ast}} = r\cos \varphi _{{\ast}}\), \(y_{{\ast}} = r\sin \varphi _{{\ast}}\) and \(r:= \sqrt{x_{{\ast} }^{2 } + y_{{\ast} }^{2}}\).
- 11.
Carlton W. Berenda (1911–1980): American physicist and philosopher of sciences.
- 12.
Nathan Rosen (1909–1995): American–Israeli physicist, assistant of Einstein at Princeton; he is the “R” of the famous EPR paradox in quantum mechanics; he is also known for the Einstein–Rosen bridge in general relativity.
- 13.
Georges Sagnac (1869–1928): French physicist, one of the pioneers of X-ray studies in France (he notably discovered X-ray fluorescence); he got interested in the optics of moving bodies, in the framework of the aether theory, of which he was a proponent. He was a friend of Paul Langevin (cf. p.40), Émile Borel (cf. p.215) and Pierre and Marie Curie.
- 14.
Let us recall that the unit vectors \(\overrightarrow{\boldsymbol{n}}\) and \(\overrightarrow{\boldsymbol{e}}\,^{\prime}_{1}\) have been defined in Sect. 13.3.3.
- 15.
Henry G. Gale (1874–1942): American astrophysicist, editor of the Astrophysical Journal from 1912 to 1940.
- 16.
Oliver J. Lodge (1851–1940): British physicist and writer, who performed important studies in electromagnetism, notably on wireless telegraphy; he also invented a type of spark ignition for internal combustion engines (the so-called Lodge Igniter).
- 17.
Paul Harzer (1857–1932): German astronomer working at Kiel Observatory.
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Gourgoulhon, É. (2013). Rotating Observers. In: Special Relativity in General Frames. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37276-6_13
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