## 3

## Mathematical Background

##### 3.1 LINEAR ORTHOGONAL TRANSFORMATION AND ITS INVARIANTS

The central problem in the theory of relativity is to formulate the laws of physics and, in particular, the equations of the electromagnetic field, in a form which is invariant under transformations to another reference system that is moving uniformly with respect to the first. In this chapter, we will confine ourselves to systems of rectangular Cartesian coordinates.

The position of a particle *P* in a given Cartesian coordinate system *OXYZ* is given by the vector

**r** = *x*_{1}**i**_{1} + *x*_{2}**i**_{2} + *x*_{3}**i**_{3} (3.1)

where **i**_{1}, **i**_{2}, **i**_{3} are respectively unit vectors along the *X*-, *Y*-, and *Z*- axes. These unit vectors are called *base vectors*. They define the coordinate system or basis ...