Einstein's special relativity theory

There is no known application of Einstein's theory of special relativity to seismic imaging. But some of the mathematical methods are related, and now is the appropriate time to take a peek at this famous theory.

In 1887 the Michelson-Morley interferometer experiment established with high accuracy that light travels in all directions at the same speed, day and night, winter and summer. We have seen that the dispersion relation of the scalar wave equation is a circle centered at the origin, meaning that waves go the same speed in all directions. But if the coordinate system is moving with respect to the medium, then the dispersion relation loses directional symmetry. For light propagating in the vacuum of outer space, there seems to be no natural reference coordinate system. If the earth is presumed to be at rest in the summer, then by winter, the earth is moving around the sun in the opposite direction. The summer coordinates relate to the winter coordinates by something like $x' = x\, - \,2 \,v_{\rm earth} t$.While analysis of the Michelson-Morley experiment shows that such motion should have a measurable asymmetry, measurements show that the predicted asymmetry is absent. Why? One theory is the ``ether'' theory. Ether is a presumed substance that explains the paradox of the Michelson-Morley experiment. It is presumed to be of minuscule density and viscosity, allowing us to imagine that it is somehow dragged around the earth in such a way that earthbound experimenters are always moving at the same speed as it is. Other measurements, however, also contradict the presumption of ether. Just as wind refracts atmospheric sound waves, ether should cause a measurable refraction of starlight, but this is not observed.

Einstein's explanation of the experiments is based on a mathematical fact that you can easily verify. Let a coordinate frame be defined by

$$\begin{eqnarray} z' \ \ \ &=& \ \ \ z\ -\ {v\,t \over \sqrt { 1\,-\,v^2 / c^2 } ... ... \ { t \ -\ {v \over c^2 } \ z \over \sqrt { 1\,-\,v^2 / c^2 } }\end{eqnarray}$$ (45) (46) (47)

The amazing thing about this transformation, which you can easily prove, is that it converts the equation $P_{xx} + \, P_{zz} = c^{-2} P_{tt}$ to the equation P_{x' x'} + P_{z' z'} = c^-2 P_{t' t'} . The transformed wave equation is independent of velocity v which is what led Einstein to his surprising conclusions.