Abstract
An elementary geometric fact, stated as the intercept theorem, makes an observed clock run visibly slower, if it moves away in the line of sight and to run visibly faster by the inverse factor, if it approaches the observer with the same velocity. This Doppler effect of light in the vacuum is particularly simple, because, different from the Doppler effect of sound, it depends only on the relative velocity of the light source and its observer. We employ a referee to determine whether moving clocks are equal and how the times between pairs of events compare. This time endows spacetime with a geometric structure, the distance, which is similar to but also different from Euclidean distance. From the Doppler effect we determine the addition of velocities, time dilation and length contraction and clarify the related paradoxes.
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Notes
- 1.
\(\kappa \) and \(\nu \) are the Greek letters kappa and nu.
- 2.
Figure 2.13 depicts the worldlines of the observer \(\fancyscript{O}\) and the clock \(\fancyscript{C}\) in a plane. However, we consider the general case in which the worldline of the observer is parallel to the plane and does not intersect the worldline of the clock. Note that in spacetime diagrams the frequency of light is not a property of a light ray but pertains to the distance of two events on parallel light rays.
- 3.
\(\theta \) and \(\varphi \) are the Greek letters theta and phi.
- 4.
Without mentioning it explicitly we shall consider different copies of \(\mathbb{ R} ^4\), e.g. spacetime or the set of four-velocities, four-momenta or four-accelerations. Vectors from different spaces cannot be added, because they differ in units. e.g. a velocity \(\mathbf{ {v}}\) cannot be added to a position \(\mathbf{ {x}}\). What can be added is the image \(\mathbf{ {v}}t\) of a velocity \(\mathbf{ {v}}\) under the linear map \(t\), which maps it to the space of positions. Though vectors from different four-spaces cannot be added, their directions can be compared, because, as we shall see, the Lorentz group acts on each of these spaces and the \(x\)-direction, for example, is the set of vectors which is invariant under rotations around the \(x\)-axis and under boosts in \(y\)- and \(z\)-directions (also compare page 89).
- 5.
The reader has to deduce from the context whether the length squared or the \(y\)-component of a vector is meant.
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Dragon, N. (2012). Time and Distance. In: The Geometry of Special Relativity - a Concise Course. SpringerBriefs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28329-1_2
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DOI: https://doi.org/10.1007/978-3-642-28329-1_2
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