Abstract
This chapter introduces the basics of particle description in special relativity, starting by the concept of worldline in Minkowski spacetime. The proper time is then defined as the metric length along a timelike worldline. The concepts of 4-velocity and 4-acceleration of a massive particle are introduced next. The case of massless particles is then contemplated, with the associated notions of null geodesic and light cone. The twin paradox is discussed in depth, via Langevin’s traveller, thereby producing a nice illustration of all the concepts introduced in the chapter. Some experimental realizations of the twin paradox are also presented. The chapter ends with the study of geometrical properties of timelike worldlines.
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Notes
- 1.
Minkowski spacetime is however the arena for relativistic quantum field theory.
- 2.
It is however still not a norm in the mathematical meaning, for it does not satisfy the triangle inequality \(\|\overrightarrow{\boldsymbol{v}} + \overrightarrow{\boldsymbol{w}}\| \leq \|\overrightarrow{\boldsymbol{v}}\| +\| \overrightarrow{\boldsymbol{w}}\|\).
- 3.
Let us recall that the notation \(\overrightarrow{\boldsymbol{u}} \cdot \overrightarrow{\boldsymbol{u}}\) stands for \(\boldsymbol{g}(\overrightarrow{\boldsymbol{u}},\overrightarrow{\boldsymbol{u}})\).
- 4.
As for the 4-velocity, which has not the dimension of a velocity, cf. Remark 2.9.
- 5.
Paul Langevin (1872–1946): French physicist, known for his work on the magnetic properties of materials and Brownian motion. As a friend of Einstein since 1911, he contributed a lot in the diffusion of relativity in France (Paty 1999a). He was also president of the French League of Human Rights from 1944 to 1946.
- 6.
We shall define more precisely the concept of observer in Chap. 3, the present version being sufficient for the purpose of this section.
- 7.
Let us recall that \(\mathrm{d}(\sinh u) =\cosh u\,\mathrm{d}u\) and \(\sqrt{1 {+\sinh }^{2 } u} =\cosh u\).
- 8.
- 9.
An example of such spaces is a torus or, more generally, any compact domain with periodic boundary conditions.
- 10.
We shall make precise the notion of velocity in Chap. 4.
- 11.
If the timelike constraint was relaxed, then \(\mathcal{L}\) could move “backward in time” and t would not be a good parameter along it.
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Gourgoulhon, É. (2013). Worldlines and Proper Time. In: Special Relativity in General Frames. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37276-6_2
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