Index Notation and Invariants

Building Lorentz scalars from four-vectors and tensors is the practical core of covariance: every measurable that all observers agree on is a fully contracted object. This page collects the standard invariants and the index discipline that produces them. Convention: $\eta =\mathrm{diag}(+,-,-,-)$, $c=1$. See four-vectors & tensors for the transformation rules.

Building scalars

A Lorentz scalar has no free indices. The basic recipe: contract every upper index with a lower one using the metric. Common invariants:

Common four-vectors

• Wave four-vector $k^{\mu }=(\omega /c,\mathbf{k})$, $k^2=0$ for light; $k\cdot x$ invariant powers the Doppler/aberration formulas.
• Current $j^{\mu }=(c\rho ,\mathbf{j})$, conservation ${\partial }_{\mu }{j}^{\mu }=0$.
• Potential $A^{\mu }=(\varphi /c,\mathbf{A})$, giving $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$.

Discipline

• Contract one upper with one lower index; never two of a kind.
• A free index must appear once per term, same position, both sides.
• Scalars are frame-independent; vectors/tensors transform — match ranks.

This machinery underlies covariant electromagnetism and the dispersion relation.