Abstract
The notion of the coordinate time is introduced. The interrelation between constancy of c and synchronization is analyzed. Lorentz-transformations are derived and using them the relativistic effects are reconsidered. The causality paradox is discussed and the twin paradox is described from the point of view of both siblings.
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Notes
- 1.
Twin paradox vanishes only if transportation velocity is equal to zero. But then the procedure would last infinitely long.
- 2.
In the train thought experiment, for example, explosions are simultaneous in the train’s rest frame, the explosion at the rear end is the first one in the platform’s rest frame and this time order is reversed as seen from another train, moving faster in the same direction
- 3.
The velocity u is positive already for v > c 2/V According to (2.7.4), in this domain \(t_3 < t_2.\) Since v k > c 2/V it is true a fortiori in the domain v > v k .
- 4.
The velocity of Alice at θ > 0 can also be obtained directly, using (2.12.5) and (2.12.2):
$$ \frac{d\xi}{d\theta} = \frac{ dx -U\cdot dt}{ dt{\sqrt{1 - {U}^2/c^2}}} =\frac{ -V - U}{{\sqrt{1 - {U}^2/c^2}}} = +V, $$since the velocity \(\frac{dx}{dt}\) of Alice with respect to the auxiliary frame \(\mathcal{I}\) is equal to −V all the time.
- 5.
In Sect. 2.21 we will dispose of this limitation.
- 6.
In the general case the angular velocity vector is equal to \({\pmb\omega} = \frac{1}{v^2}\left({\bf v}\times{\frac{d\bf v}{dt}}\right).\) When this is substituted into (2.22.1) we obtain for the Thomas-precession angular velocity vector the expression \({\pmb\omega}_T = (1 - \gamma )\frac{1}{v^2}\left({\bf v}\times \frac{d\bf v}{dt}\right) = -\frac{\gamma}{1 + \gamma} \frac{1}{c^2}\left({\bf v}\times \frac{d\bf v}{d\tau }\right).\)
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© 2011 Péter Hraskó
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Hraskó, P. (2011). The Lorentz-Transformation. In: Basic Relativity. SpringerBriefs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17810-8_2
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DOI: https://doi.org/10.1007/978-3-642-17810-8_2
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