Abstract
In this chapter we continue our program of generalization of Newtonian physical quantities to Special Relativity by considering the physical quantity angular momentum tensor. Since this quantity in Newtonian Physics it is described by an antisymmetric second order tensor it is necessary that we introduce new mathematical concepts and tools, the main one being the concept of a bivector.
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Notes
- 1.
This result is general and holds for the 1 + (n − 1) decomposition of a second order tensor.
- 2.
In a n −dimensional space the vectors E a, H a have n − 1components each hence the bivector X ab has 2(n − 1) independent components. Compare with the electromagnetic field tensor F ab which is also a bivector.
- 3.
The calculations are general and hold for n dimensions.
- 4.
E.g. a Fermi propagated frame.
- 5.
This equation is computed directly as follows:
$$\displaystyle \begin{aligned} \frac{dL^{ab}}{d\tau} & =\frac{dx^{a}}{d\tau}p^{b}+x^{a}\frac{dp^{b}}{d\tau } -\frac{dx^{b}}{d\tau}p^{a}-x^{b}\frac{dp^{a}}{d\tau} \\ & =u^{a}p^{b}+x^{a}F^{b}-u^{b}p^{a}-x^{b}F^{a} \\ & =x^{a}F^{b}-F^{a}x^{b} \\ & =M^{ab} \end{aligned} $$(14.46) - 6.
t stands for transpose. Note the method we use to compute the components by the tensor product of the corresponding matrices.
- 7.
See E.P.Wigner Rev Mod Phys. (1957) 29, 255.
- 8.
The reader may wonder why we bother to discuss the concept of spin within the limits of the non-quantum theory when we know that only a quantal description can be correct. The answer lies in the quantal theorem which states that the classical equation of motion of a dynamical variable is the quantal equation of motion of the mean value of that variable averaged over an ensemble of identical systems. Therefore the conclusions we shall draw with the classical treatment will apply to averages over many identical particles prepared in the same way, like the electrons or muons in a beam or the valence electrons in a gas of atoms in a glow tube.
- 9.
This is the 1+3 decomposition of \(\dot {S}^{a}\) wrt u a. Note that S a u a = 0 does not imply \(\dot {S}^{a}u_{a}=0.\)
- 10.
See (a) ‘Recession of the polarization of particles moving in a homogeneous electromagnetic field’ by V. Bargmann, L. Michel and V. L. Telegdi Phys Rev Letters 2 (1959), 435–436. (b) ‘Spin and orbital motions of a partticle in a homogeneous magnetic field’ by V. Henry and J. Silver Phys. Rev. 180 (1969), 1262–1263.
- 11.
Einstein A and de Haas W. J. (1915) Verhandl. Deut.Phys. Ges. 17, 152; Barnett S.J. (1935), Rev. Mod. Phys. 7, 129.
- 12.
This is the angular speed of a particle of charge q which is introduced at right angles to a uniform magnetic field of magnetic induction B ∗.
- 13.
We can always make that force centripetal by changing the direction of the speed u.
- 14.
Note that the angular velocity equals 2π∕T therefore its transformation is like T −1, that is:
$$\displaystyle \begin{aligned} \omega^{\ast}=\gamma_{u}\omega \end{aligned}$$where ω is the angular speed in Σ and ω ∗ is the angular speed in Σ∗.
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Tsamparlis, M. (2019). Relativistic Angular Momentum. In: Special Relativity. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-27347-7_14
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DOI: https://doi.org/10.1007/978-3-030-27347-7_14
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