Test theories of special relativity give a mathematical framework for analyzing results of experiments to verify special relativity.
An experiment to test the theory of relativity cannot assume the theory is true, and therefore needs some other framework of assumptions that are wider than those of relativity. For example, a test theory may have a different postulate about light concerning one-way speed of light vs. two-way speed of light, it may have a preferred frame of reference, and may violate Lorentz invariance in many different ways. Test theories predicting different experimental results from Einstein's special relativity, are Robertson's test theory (1949), and the Mansouri–Sexl theory (1977) which is equivalent to Robertson's theory. Another, more extensive model is the Standard-Model Extension, which also includes the standard model and general relativity.
Robertson–Mansouri–Sexl framework
Basic principles
Howard Percy Robertson (1949) extended the Lorentz transformation by adding additional parameters. He assumed a preferred frame of reference, in which the two-way speed of light, i.e. the average speed from source to observer and back, is isotropic, while it is anisotropic in relatively moving frames due to the parameters employed. In addition, Robertson used the Poincaré–Einstein synchronization in all frames, making the one-way speed of light isotropic in all of them.
A similar model was introduced by Reza Mansouri and Roman Ulrich Sexl (1977). Contrary to Robertson, Mansouri–Sexl not only added additional parameters to the Lorentz transformation, but also discussed different synchronization schemes. The Poincaré–Einstein synchronization is only used in the preferred frame, while in relatively moving frames they used "external synchronization", i.e., the clock indications of the preferred frame are employed in those frames. Therefore, not only the two-way speed of light but also the one-way speed is anisotropic in moving frames.
Since the two-way speed of light in moving frames is anisotropic in both models, and only this speed is measurable without synchronization scheme in experimental tests, the models are experimentally equivalent and summarized as the "Robertson–Mansouri–Sexl test theory" (RMS). On the other hand, in special relativity the two-way speed of light is isotropic, therefore RMS gives different experimental predictions than special relativity. By evaluating the RMS parameters, this theory serves as a framework for assessing possible violations of Lorentz invariance.
Theory
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).