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)

$$ s = \sqrt {g\mu \nu x^\mu x^\nu } \, = \,\sqrt {x\mu x^\mu }, $$

(16.1)

and the length s can also take complex values. Examining an orthogonal transformation αμν, which relates each point with coordinates x _ß to new ones x^1μ:

$$ x'^{\nu \,\,} = \,\alpha ^\nu \,\mu ^{x^\mu } $$

(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)