Lorentz Invariance and Relativistic Symmetry Principles

Abstract

We consider the four-dimensional space with coordinates x _μ (μ = 0, 1, 2, 3), which — assuming the most general case — may be complex numbers. The absolute value of the position vector is given by (summation convention)

$$s = \sqrt {{g_{\mu v}}{x^\mu }{x^v}} = \sqrt {{x_\mu }{x^\mu }} ,$$

(16.1)

and the length s can also take complex values. Examining an orthogonal transformation a _μν , which relates each point with coordinates x _μ to new ones x _′μ

$${x'^v} = {a^v}_\mu {x^\mu }$$

(16.2)

the absolute value should remain unchanged by this transformation (this is the fundamental, defining condition for orthogonal transformations), i.e.

$$s' = \sqrt {{{x'}_\mu }{{x'}^\mu }} = \sqrt {{x_\mu }{x^\mu }} = s$$

(16.3)

.