Special Relativity and Linear Algebra

Before Einstein’s publication in 1905 of his theory of special relativity, the mathematical manipulations that were a product of his theory were in fact already known. The so called Lorentz transformations were tricks that had been found that allowed the speed of light to propagate in all directions at the same speed, which accounted for its strange behavior when traveling through the now infamous “ether.” Though Lorentz found the transformations well before him, it was not until Einstein decided to discard the ether and postulate the constancy of the speed of light as a physical law did the theory of relativity truly blossom. For a complete treatment of relativistic space and time, Einstein’s General Theory of Relativity is needed. His special theory makes one assumption that is shown to be false in the more general case: space is linear. In fact, many fascinating effects arise from the non-linearity of space, such as black holes and the curvature of the Universe. But, particularly in the absence of enormous masses, space can be closely approximated by linear means. This motivates a linear algebraic approach to special relativity. To proceed, I will put forward the postulates of special relativity, define the mathematical terminology necessary, motivated certain mathematical assumptions with physical arguments, and use the power of linear algebra to deduce the Lorentz transformations. This paper is aimed at an audience familiar with linear algebra.