Abstract
After all the preparations in the last six chapters, we can finally deal with the first of the various very surprising effects of the special theory of relativity: the relativity of simultaneity.
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Notes
- 1.
Sometimes, one talks about “\(3+1\) dimensions” (or “\(1+1\) dimensions”) to make clear that 3 (or 1) space dimensions plus the time dimension are considered.
- 2.
- 3.
This works, but only if the transport of the clocks is very slow. We will see the reason for this in Sect. 9.10.
- 4.
Note that, to define when two events at different locations are simultaneous, we need to know when two events at the same location are simultaneous.
- 5.
Actually, this synchronization method should be called Einstein-Poincaré synchronization, because Poincaré had already used it five years before Einstein.
- 6.
If, in exceptional cases, they represent accelerated reference frames, this circumstance will be explicitly noted.
- 7.
This concept is taken from W. Rindler, “Relativity” (Oxford University Press, 2006), the book mentioned in the Preface.
- 8.
O tempora, o mores!: In Galileo’s and Newton’s time, there were, of course, neither trains nor rockets. That’s why Alice was standing on the pier at that time and Bob passed her in a ship. In Einstein’s times, Alice already had replaced the ship with a train and the pier with a train station. These days, Alice sits in a rocket and travels through outer space.
- 9.
If the term “space” occurs in the expression “Euclidean space” or something similar then “space” is meant in the mathematical sense. Otherwise, “space” refers to the three-dimensional physical space.
- 10.
Therefore, before introducing the principle of the absolute speed of light in Sect. 6.1, from a logical point of view, we should have presented Einstein’s synchronization method. The synchronization method without the principle of the absolute speed of light is useful, while the principle without the synchronization method is not. On the other hand, the principle suggests a synchronization method: At \(t_A=0\), send a light pulse to B, and when it arrives, set the time of clock B to \(t_B = \ell /c\). Indeed, Einstein in his epoch-making paper on special relativity, first defined simultaneity [Einstein05a]: “Die letztere Zeit [gemeinsame „Zeit“] kann nun definiert werden, indem man durch Definition festsetzt, dass die „Zeit“, welche das Licht braucht, um von A nach B zu gelangen, gleich ist der „Zeit“, welche es braucht, um von B nach A zu gelangen.” and in English: “The latter time [common “time”] can now be defined by stating by definition that the “time” which the light needs to get from A to B is equal to the “time” it takes to get from B to A.”.
- 11.
Although some physicists are still arguing, and propositions to measure one-way velocities sometimes appear, this seems to be the consensus in the community. But if, indeed, we could measure one-way velocities, this would imply that only a particular one of the synchronization methods would be correct. However, for the effects predicted by special relativity, this would not change anything.
- 12.
In Reichenbach synchronization, each inertial frame could have its own \(\epsilon \).
- 13.
Note that, for \(\epsilon =0\) or \(\epsilon =1\), one of the two velocities \(c_\pm \) would become infinity.
- 14.
Remember that the isotropy of the speed of light and the Einstein synchronization method are intertwined.
- 15.
Most of this wisdom comes from the three papers by Mansouri and Sexl [MansouriSexl77a,MansouriSexl77b,MansouriSexl77c], who performed a similar derivation of the Lorentz transformation from experimental findings as Robertson [Robertson49]. Robertson, however, took Einstein’s synchronization method for granted, and Mansouri and Sexl also considered other synchronization schemes. The conventionality of simultaneity was pointed out earlier by Reichenbach [Reichenbach58]. Anderson et al. [Anderson+98] extended Mansouri and Sexl’s analysis considerably, illuminated the effects on physics and contributed strongly to a convergence of different opinions. Selleri [Selleri94] then formulated the derivation that we followed here. Rizzi et al. [Rizzi+08] disputed and proved wrong several of the physical implications stated by Selleri (but not his derivation). A nice and readable summary was written by de Abreu and Guerra
- 16.
Whatever that may mean ...
- 17.
The dotted red half-line L would be a message with light speed and the dashed black line S a “line of simultaneity” for Bob. Each half-line that starts at \(E_3\) and passes between L and S in the negative x-direction is, for Bob, faster than light.
- 18.
The relative velocity between two inertial observers must be smaller than the speed of light. It can be arbitrarily close to the speed of light, but cannot reach it. See also Sect. 13.4.
- 19.
We assume that the influence of gravitation can be neglected.
- 20.
As we already mentioned, for the standard synchronization procedure, we assume that the light pulse travels with the speed of light. This is not true in fiberglass, for which the speed of light in an inertial frame typically is about 2/3 of the speed of light in vacuum. For our discussion, this is not a problem, as we could use mirrors to keep the light pulse going around the Earth.
- 21.
To determine the phase shift, one can measure it first in an inertial frame and then in the rotating reference frame. For the problem at hand, it is not possible to stop the Earth’s rotation. But we can “reverse” the Earth’s rotation by just reversing the Sagnac interferometer.
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Strohm, T. (2023). Relativity of Simultaneity. In: Special Relativity for the Enthusiast. Springer, Cham. https://doi.org/10.1007/978-3-031-21924-5_7
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DOI: https://doi.org/10.1007/978-3-031-21924-5_7
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