Abstract
It can hardly be emphasized that one of the most important elements of Special Relativity is the Lorentz transformation. This is the reason we have spent so much space and effort to derive and study the Lorentz transformation in the early chapters of the book. One could naturally ask “After all these different derivations of the Lorentz transformation why we are not yet finished with it?” The reason is the following. Special Relativity is a geometric theory of Physics, which can be written and studied covariantly in terms of Lorentz tensors (four vectors etc.) without the need to consider a coordinate system until the very end, when one has to compute explicitly the components of the physical quantities of a problem for some observer.
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Notes
- 1.
The k − calculus is but it is very limited in its use for not introductory problems.
- 2.
Fahnline D Am J Phys 50 (1982) 50 818–821.
- 3.
See Halpern F. R. (1968) “Special Relativity and Quantum Mechanics” (Prentice Hall, Englewood Cliffs, NJ) and Goldstein Herbert “Classical Mechanics” Second Edition (1980) Addison – Wesley, Chap. 7.
- 4.
Krause J (1977) “Lorentz transformations as space-time reflections” J. Math. Physics 18,879–893.
- 5.
Basanski S L (1965) “Decomposition of the Lorentz Transformation Matrix into Skew – Symmetric Tensors” J. Math. Physics 6,1201–1202.
- 6.
Jantzen R, Capini P, Bini D: Ann Phys, 215, (1992) 1 and gr-qc/0106043.
- 7.
Urbantke H K: Found. Phys. Lett., 16, (2003) 111.
- 8.
To be precise this is a transformation in the tangent space of M 4 but due to the flatness of M 4 it can be reduced unambiguously to a transformation in M 4.
- 9.
The interested reader can find more information on this principle in Bruzzi V and Gorini V (1989) “Reciprocity Principle and the Lorentz transformation” J. Math. Physics 10, 1518 – 1524.
- 10.
The expression of the proper Lorentz transformation given in Table 15.2 coincides (after some rearrangements) with that of Krause.
- 11.
This is not necessary but it will help the reader to associate the new approach with the standard formalism.
- 12.
The set of all four types of Lorentz transformations constitutes a group. This group has four components. Of those only the subset of the proper Lorentz transformations form a group, which is a subgroup of the Lorentz group.
- 13.
See for example the book “The Physics of the Time Reversal” by Robert Sachs (1987), The University of Chicago Press.
- 14.
See Krause J (1977) “Lorentz transformations as space-time reflections” J. Math. Physics 18, 879–893.
- 15.
See also Sect. 6.3.
- 16.
Many times this transformation is written
$$\displaystyle \begin{aligned} {\mathbf{r}}^{\prime}=A \mathbf{r}-\mathbf{v}t {} \end{aligned} $$(15.60)where A is a Euclidean (orthogonal) rotation matrix and v is the velocity of Σ′ wrt Σ. This relation is not more general than (15.61), and the matrix A is not needed because relation (15.60) is a vector equation. The matrix A is needed only when (15.61) is written in a coordinate system in which case it describes the relative rotation of the 3-axes.
- 17.
The parallel transport is defined by the requirement the transported vector at the point P has the same components with the original vector at the point P ′ in the same (global) coordinate system of M 4.
- 18.
For example see F. Felice “On the velocity composition law in General Relativity” Lettere al Nuovo Cimento (1979) 25, 531–532.
- 19.
An equivalent expression is (see Ar Ben-Mehanem, Am. J. Phys. (1985) 53, p. 62–66):
$$\displaystyle \begin{aligned} \mathbf{w}=\frac{1}{(1+\mathbf{v}\cdot {\mathbf{w}}^{\prime })}\left\{ \frac{ {\mathbf{w}}^{\prime }}{\gamma _{v}}+\left( 1+\frac{\gamma _{v}-1}{\gamma _{v}} \frac{\mathbf{v}\cdot {\mathbf{w}}^{\prime }}{v^{2}}\right) \right\} \mathbf{v} . \end{aligned}$$The proof is simple. We have:
$$\displaystyle \begin{aligned} 1+\frac{\gamma _{v}}{1+\gamma _{v}}(\mathbf{v}{\mathbf{w}}^{\prime })=1+\frac{ \gamma _{v}^{2}v^{2}}{\gamma _{v}(1+\gamma _{v})}\frac{(\mathbf{v}\mathbf{w} ^{\prime })}{v^{2}}=1+\frac{\gamma _{v}^{2}-1}{\gamma _{v}(1+\gamma _{v})}{( \mathbf{v}{\mathbf{w}}^{\prime })}{v^{2}}=1+\frac{\gamma _{v}-1}{\gamma _{v}}{( \mathbf{v}{\mathbf{w}}^{\prime })}{v^{2}}. \end{aligned}$$ - 20.
Note that w i is a four-velocity hence w i w i = −1.
- 21.
Note that: v⋅ d v = vdv.
- 22.
More information on the geometry of the 3-velocity space can be found in V. Fock (1976) “The Theory of Space, Time and Gravitation” 2nd Revised Edition, Pergamon Press.
- 23.
One can also compute the connection coefficients directly from the (diagonal) metric by means of the formulae:
$$\displaystyle \begin{aligned} \Gamma_{ij}^{i} & =\frac{\partial}{\partial x^{l}}\log\sqrt{|g_{ii}|} ,\quad \Gamma_{ii}^{i}=\frac{\partial}{\partial x^{i}}\log\sqrt{|g_{ii}|} ,\quad \Gamma_{jj}^{i}=-\frac{1}{2g_{ii}}\frac{\partial g_{jj}}{\partial x^{i} }, \\ & \Gamma_{jk}^{i}=0\quad \text{ when all indices are different}. \end{aligned} $$ - 24.
In the calculations we can omit the terms containing u i because they vanish except the term u i u j which gives 1.
- 25.
See e.g. page 163 in H. Goldstein, C. Poole, J. Safko “Classical Mechanics” Third Edition (2002) Addison – Wesley Publishing Company.
- 26.
See equation (10) of the paper“Wigner’s rotation revisited” Ar Ben-Mehanem, Am. J. Phys. (1985) 53, 62–66.
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Tsamparlis, M. (2019). The Covariant Lorentz Transformation. In: Special Relativity. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-27347-7_15
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