Abstract
This chapter deals with the integration of tensor fields over various domains of spacetime. It starts by introducing the notion of 4-volume and the integration of a differential 4-form over a four-dimensional domain. Then submanifolds of dimension 1, 2 or 3 are defined, as well as submanifolds with boundary. The integration over a submanifold is discussed for differential forms and more general tensor fields. Flux integrals are also considered. Finally, Stokes’ theorem is introduced, and various applications of it are presented.
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Notes
- 1.
Or a Riemann–Darboux integral: since we shall consider only piecewise continuous functions, we shall not make any distinction.
- 2.
Let us recall that the concept of manifold has been defined in Sect. 7.2.1.
- 3.
In practice, the explicit mention of \(\boldsymbol{\rho }\) is often skipped, and one speaks about the “oriented submanifold \(\mathcal{V}\)”.
- 4.
This definition assumes that \(\mathcal{V}\) is entirely covered by a single adapted coordinate system. Now, for certain submanifolds, various coordinate systems can be required. The integral must then be decomposed in a sum via a process called partition of unity. We shall not enter in these technical considerations here; cf. e.g. Berger and Gostiaux (1988).
- 5.
Let us recall that, in the present context, x 0 stands for the first coordinate of a system adapted to \(\mathcal{V}\); it is thus not necessarily a time coordinate.
- 6.
We have already encountered this type of hypersurface in Sect. 9.3.5.
- 7.
Wolfgang Pauli (1900–1958): Austrian theoretical physicist, who authored fundamental works in quantum mechanics and received the Nobel Prize in Physics in 1945 for the discovery of the exclusion principle. His contribution to relativity is mostly the large encyclopedia article (Pauli 1921), which he wrote at the age of 21, at the request of his thesis advisor Arnold Sommerfeld (cf. p. 27). He also got interested in the relativistic treatment of gravitation (cf. Sect. 22.2.4).
- 8.
George G. Stokes (1819–1903): British physicist and mathematician, of Irish origin, known for his works in fluid dynamics (Navier–Stokes equation) and optics.
References
Berger M. & Gostiaux B., 1988, Differential Geometry: Manifolds, Curves, and Surfaces, Springer-Verlag (New York).
Laue M., 1907, Die Mitführung des Lichtes durch bewegte Körper nach dem Relativitätsprinzip, Annalen der Physik 23, 989; http://gallica.bnf.fr/ark:/12148/bpt6k153304.image.f993 Eng. tr.: http://en.wikisource.org/wiki/The_Entrainment_of_Light_by_Moving_Bodies_According_to_the_Principle_of_Relativity
Pauli W., 1921, Relativitätstheorie, in Encyklopädie der mathematischen Wissenschaften, vol. V19, Teubner (Leipzig); Eng. tr. (with supplementary notes by the author) in Pauli (1958).
Penrose R., 2007, The Road to Reality, Vintage Books (New York).
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Gourgoulhon, É. (2013). Integration in Spacetime. In: Special Relativity in General Frames. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37276-6_16
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DOI: https://doi.org/10.1007/978-3-642-37276-6_16
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