Transformation of Lengths and Relativistic Invariants

Let us examine what effect this tfansformation has on the measurement of distances in one frame from the other. As before we assume a one-dimensional case where an inertial frame S is moving with velocity v relative to a stationary frame S. The distance between two points jcJ and x 2 in S is 

The same two points are moving with speed v relative to S, where we measure their distance as 

If lengths are not conserved between frames, we might want to look for other quantities that are invariant under the transformations in (2.11). We have seen that these transformations mix time and position, and so we would expect that any invariant quantity should involve both variables. We can take a clue from the constancy of the speed of light. From the expression for the distance traveled by a light signal in (2.4), we may write 

By postulate 2 this quantity is conserved in a transformation between frames for a light signal, or equivalently a photon. What about a particle traveling at a slower speed  

At this point we may easily generalize to the full three-dimensional spatial coordinate space. The complete set of transformation equations, known as the Lorentz 

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