Abstract
In the previous chapters we developed the basic concepts of relativistic motion, that is, the time and spatial distance. The methodology we followed was the definition of these concepts, in agreement with the corresponding concepts of Newtonian theory, which is already a “relativistic” theory of motion.
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Notes
- 1.
Affine parametrization means that the length of the tangent vector \( \frac {dx^{i}}{d\tau }\) has constant length along the world line. Proper time is the affine parameter (modulo a linear transformation) for which this constant equals − c 2.
- 2.
This is consistent with the so called Reciprocity Principle which is frequently stated in Special Relativity and expresses the relativity of relativistic motion.
- 3.
The solution of the problem is a particular case of the sequence \(a_{r}= \frac {\kappa a_{r}+\lambda }{\mu a_{r}+\rho },\) whose terms a r converge to a common value x. In this case, we set \(x=\frac {\kappa x+\lambda }{\mu x+\rho }\) and compute the roots ρ 1, ρ 2 of the quadratic equation. Then the following recursion formula holds
$$\displaystyle \begin{aligned} \frac{a_{r+1}-\rho_{1}}{a_{r+1}-\rho_{2}}=A\frac{a_{r}-\rho_{1}}{ a_{r}-\rho_{2}} \end{aligned}$$where A is a constant. [Study the case ρ 1 = ρ 2]. In our example a r = v r and κ = 1, λ = a, μ = 1, ρ = a. The equation \(x=\frac {x+a}{ax+1}\Rightarrow ax^{2}+x=x+a\Rightarrow x=\pm 1\) hence the reduction relation is:
$$\displaystyle \begin{aligned} \frac{v_{r+1}-1}{v_{r+1}+1}=A\frac{v_{r-1}}{v_{r+1}}\text{ or }\zeta _{r+1}=A\zeta_{r}. \end{aligned}$$In order to calculate A, we consider the terms ζ 2 = Aζ 1 and replacing we find \(A=\frac {1-a}{1+a}\) etc.
- 4.
The concept of relative four-vector cannot be extended to theories of Physics formulated over a non-linear space i.e. curved spaces (e.g. General Relativity). The reason is that the relative vector involves the difference of vectors defined at different points in space, therefore one has to “transport” one of the vectors at the point of application of the other, an operation which involves necessarily the curvature of the space.
- 5.
There is a significant difference, which must be pointed out. The relative (relativistic) velocity v 21 refers to the velocity of the particle 2 in the proper frame Σ1 of particle 1, therefore involves the relativistic measurement of velocity (not photon!) and the inequality \(\left \vert {\mathbf {v}}_{12}\right \vert <c\) is expected to hold. Indeed, if we consider two particles, one moving along the positive x −axis with speed c∕2 and the other moving along the negative x −axis with speed c∕2, then using (6.35) we compute that the (relativistic) relative speed of the particles equals 4c∕5 < c, whereas the Newtonian relative speed is c.
- 6.
The interested reader can find more on the Wigner angle in e.g. A. Ben-Menahem (1983) “Wigner’s rotation revisited” Am. J. Phys. 53, pp 62–66.
- 7.
In this section we follow the notation that two indices in a velocity indicate the first quantity with reference to the second. For example the velocity of Σ1 wrt Σ2 will be denoted as v 12. Concerning the angles we follow the notation that the angle between the velocities v 10, v 20 of Σ1 and Σ2 in Σ0 will be denoted by A 12.
- 8.
As we shall show in Sect. 15.4.3 , when we study the covariant form of the Lorentz transformation, the 3-velocity space is a three dimensional Riemannian manifold of constant negative curvature whose metric is Lorentz covariant. Such spaces are known as Lobachevsky spaces.
- 9.
See for example B.P. Peirce, A short Table of integrals Ginn, Boston 1929 formulae 631 and 632 and for a theoretical treatmentA. Ungar Foundations of Physics (1998) 28, 1283–1321.
- 10.
L.H. Thomas (1927), Philosophical Magazine 3, 1.
- 11.
More on the nature of spin we shall discuss later in the chapter on the angular momentum in Special Relativity
- 12.
These vectors are 1-forms but this is not crucial for our considerations here.
- 13.
Apply the identity A × (B ×C) = (A ⋅C)B − (B ⋅A)C.
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Tsamparlis, M. (2019). Relativistic Kinematics. In: Special Relativity. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-27347-7_6
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