In the following, the notation of Mansouri-Sexl is used. They chose the coefficients a, b, d, e of the following transformation between reference frames:
where T, X, Y, Z are the Cartesian coordinates measured in a postulated preferred frame (in which the speed of light c is isotropic), and t, x, y, z are the coordinates measured in a frame moving in the +X direction (with the same origin and parallel axes) at speed v relative to the preferred frame. And therefore is the factor by which the interval between ticks of a clock increases when it moves (time dilation) and is factor by which the length of a measuring rod is shortened when it moves (length contraction). If and and then the Lorentz transformation follows. The purpose of the test theory is to allow a(v) and b(v) to be measured by experiment, and to see how close the experimental values come to the values predicted by special relativity. (Notice that Newtonian physics, which has been conclusively excluded by experiment, results from )
The value of e(v) depends only on the choice of clock synchronization and cannot be determined by experiment. Mansouri-Sexl discussed the following synchronization schemes:
- Internal clock synchronization like the Poincaré-Einstein synchronization by using light signals, or synchronization by slow clock transport. Those synchronization schemes are in general not equivalent, except the case when a(v) and b(v) have their exact relativistic value.
- External clock synchronization by choosing a "preferred" reference frame (like the CMB) and using the clocks of this frame to synchronize the clocks in all other frames ("absolute" synchronization).
By giving the effects of time dilation and length contraction the exact relativistic value, this test theory is experimentally equivalent to special relativity, independent of the chosen synchronization. So Mansouri and Sexl spoke about the "remarkable result that a theory maintaining absolute simultaneity is equivalent to special relativity." They also noticed the similarity between this test theory and Lorentz ether theory of Hendrik Lorentz, Joseph Larmor and Henri Poincaré. Though Mansouri, Sexl, and the overwhelming majority of physicists prefer special relativity over such an aether theory, because the latter "destroys the internal symmetry of a physical theory".
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