# Frame of Reference Transformation

Using the Lorentz transformations and the index notation,

Where

This can be expressed in matrix form as

Letting the indices run from 0 to 3, we can write

Where is the Lorentz transformation matrix for motion in the z direction.

Using the Einstein convention, this can be written as

If we take the 4-vector quantities to be on an infinitesimally small scale, then there exists a linear relationship between the transformation. Following the rules of partial differentiation,

Expressing this in matrix form

Again, using a summation over the indicies

Using the Einstein convention

The Lorentz transformations are also invariant in that they are just a rotation, i.e. Det . The inner product is preserved,

Where