Abstract
In the last chapter we introduced the idea that, using the same approach followed in Chap. 3, one can define four-dimensional objects which are the counterparts of the 3D Euclidean scalars, vectors, and tensors in the Minkowski spacetime.
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- 1.
For example, from what has been previously shown, a relativistic compatible dynamics in Euclidean geometry is difficult to conceive and to understand.
- 2.
Which is exactly why we can tell that the particle is moving at the speed of light, because \(v=\left| \mathrm {d}\mathbf {x}/\mathrm {d}t\right| =c\).
- 3.
It is evident that in this case Eq. (6.1.1) does not hold even if \(\mathrm {d}t\) can still be defined.
- 4.
Or, which is the same, the velocity \(\mathbf {v}\) as measured in the rest frame \(\bar{S}\) of \(\varvec{w}\).
- 5.
We remind the reader that these equations had been derived in the case of \(\mathbf {v}=0\).
- 6.
The second equation can be obtained by the relation \(\left( \mathbf {a}\times \mathbf {b}\right) ^{2}=a^{2}b^{2}-\left( \mathbf {a}\cdot \mathbf {b}\right) ^{2}\), which allows us to rewrite Eq. (6.1.11) as
$$ \mathbf {a}\cdot \mathbf {a}=\gamma ^{6}\frac{\bar{v}^{2}\bar{a}^{2}-\left( \bar{\mathbf {v}}\times \bar{\mathbf {a}}\right) ^{2}}{c^{2}}+\gamma ^{4}\bar{a}^{2}. $$In fact, for \(\mathbf {a}_{\parallel }\) this becomes
$$ a_{\parallel }^{2}=\gamma ^{6}\frac{\bar{v}^{2}\bar{a}_{\parallel }^{2}}{c^{2}}+\gamma ^{4}\bar{a}_{\parallel }^{2}=\gamma ^{6}\bar{a}_{\parallel }^{2}\left( \frac{1}{\gamma ^{2}}+\frac{\bar{v}^{2}}{c^{2}}\right) =\gamma ^{6}\bar{a}_{\parallel }^{2}\left( 1-\frac{\bar{v}^{2}}{c^{2}}+\frac{\bar{v}^{2}}{c^{2}}\right) =\gamma ^{6}\bar{a}_{\parallel }^{2}.$$ - 7.
And therefore of an absolute concept of simultaneity among events.
- 8.
Or at least no one in agreement with the experimental data.
- 9.
For example, the definition of kinetic energy satisfies the basic requirement of coinciding with its classical definition when \(v\ll c\).
- 10.
To convince ourselves of this it is sufficient to notice that we can assign an energy \(E=h\nu \) to a photon with frequency \(\nu \) and to remember that the fact that light can exert a pressure is shown in many cases and at different levels, from small didactic experiments to objects the size of solar sails.
- 11.
The latter both because \(m=0\) and because for massless particles \(\mathrm {d}s^{2}=0\).
- 12.
Not to be confused with the Lagrangian of the variational approach.
- 13.
It is thus clear that the introduction of the einbein has introduced one more “coordinate” in the Lagrangian, which, however, is not a true dynamical coordinate because \(\partial L/\partial \dot{e}=0\). From this point of view, therefore, the function of the einbein is to show the local gauge freedom of the problem by writing a singular Lagrangian and making it possible to fix a specific gauge.
- 14.
We recall that L has to be a Lorentz scalar to guarantee the covariance of the resulting equations with respect to any transformation of the Poincaré group.
- 15.
One could equivalently say that \(-c^{2}\mathrm {d}\tau ^{2}=\mathrm {d}s^{2}=\eta _{\alpha \beta }\mathrm {d}x^{\alpha }\mathrm {d}x^{\beta }\) and therefore \(\mathrm {d}\tau =\sqrt{-\eta _{\alpha \beta }\dot{x}^{\alpha }\dot{x}^{\beta }/c^{2}}\,\mathrm {d}\tau \) so that \(\mathrm {d}\tau \) is actually a quantity related to the four-velocity of the particle.
- 16.
The aberration of a light was detected in the seventeenth century in the form of the so-called stellar aberration. The first explanation was given by Bradley in 1727, who derived the non relativistic equation of the light aberration.
- 17.
In considering the tangent, however, we are are introducing a sign ambiguity.
- 18.
As a further note, if the system is not only singular, but also \(\partial ^{2}L/\partial \dot{q}_{i}\partial \dot{q}_{j}=0\) for all i, j then it transforms to
$$ \frac{\partial ^{2}L}{\partial \dot{q}_{i}\partial q_{j}}\dot{q}_{j}=\frac{\partial L}{\partial q_{i}}-\frac{\partial ^{2}L}{\partial \dot{q}_{i}\partial \lambda } $$that in its turn can be solved only if
$$ \mathrm {det}\left( \frac{\partial ^{2}L}{\partial \dot{q}_{i}\partial q_{j}}\right) \ne 0. $$If \(\partial L/\partial \dot{q}_{j}=0\) for a specific j, then the coordinate \(q_{j}\) is said to be non propagating and the Lagrangian is non dynamic for that coordinate. A Lagrangian is simply non dynamic as a whole if the condition holds for all the \(q_{j}\).
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Vecchiato, A. (2017). Special Relativity in Minkowskian Spacetime. In: Variational Approach to Gravity Field Theories. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-51211-2_6
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DOI: https://doi.org/10.1007/978-3-319-51211-2_6
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