Abstract
In this chapter we introduce the basic concepts of special relativity—i.e., spacetime, 4-vectors, the relativistic form of Newton’s 2nd law, and the relativistic Lagrangian formulation, which could serve as the starting point for a course devoted to relativity or relativistic dynamics.
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Notes
- 1.
These equations for the electric and magnetic fields \(\mathbf {E}\) and \(\mathbf {B}\) are written in MKS units. The quantities \(\rho \) and \(\mathbf {J}\), which appear on the right-hand side, are the electric charge density and electric current density, respectively, which act as sources for the fields. \(\varepsilon _0\) is a physical constant called the permitivity of free space (\(\varepsilon _0 = 8.85\times 10^{-12}~\mathrm{C}^2/\mathrm{N}\cdot \mathrm{m}^2\)), and \(\mu _0\) is the permeability of free space (\(\mu _0 = 4\pi \times 10^{-7}~\mathrm{N/A^2}\)). See e.g., Griffiths (1999) for more details.
- 2.
There is also an additional minus sign that comes from the fact that \(\cosh ^2\chi -\sinh ^2\chi = 1\), while \(\cos ^2\theta + \sin ^2\theta =1\).
- 3.
With respect to any other inertial reference frame \(x^{\alpha '}\equiv (ct',x',y', z')\), we use the transformation law for vector components to get \(p^{\alpha '}\).
- 4.
Recall that spacetime vectors with up indices \(A^\alpha \) are not identical to spacetime vectors with down indices \(A_\alpha \); they are related by application of \(\eta _{\alpha \beta }\) or \(\eta ^{\alpha \beta }\). See Appendix A.2 for more details.
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Benacquista, M.J., Romano, J.D. (2018). Special Relativity. In: Classical Mechanics. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-68780-3_11
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DOI: https://doi.org/10.1007/978-3-319-68780-3_11
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