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Since the chirality matrix commutes with spinor Lorentz generators, $j_5^{\mu }=\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi$ transforms as a vector under proper Lorentz transformations and as a pseudovector under parity. Contracting two copies gives the Lorentz scalar $j_{5\mu }{j}_5^{\mu }$; the extra parity signs cancel. Lorentz invariance of this interaction does not require current conservation.

A rod of rest length $L_0$ moving parallel to its length has simultaneous laboratory length $L_0/\gamma$, where $\gamma$ is the Lorentz factor. Transverse lengths are unchanged by the corresponding Lorentz transformation.

A spinor Lorentz matrix satisfies $S^{-1}{\gamma }^{\mu }S={\Lambda }^{\mu }{}_{\nu }{\gamma }^{\nu }$. Combining this identity with the transformed derivative shows that $(i{\gamma }^{\mu }{\partial }_{\mu }^{\prime }-m){\Psi }^{\prime }=S(i{\gamma }^{\mu }{\partial }_{\mu }-m)\Psi$. Thus a Lorentz transformation carries every solution of the Dirac equation to another.

A Lorentz frame at a spacetime event is an orthonormal tangent basis with metric $\mathrm{diag}(-1,1,1,1)$. Future-oriented frames are related by proper orthochronous Lorentz transformations. Any such basis can be realized at the event by normal coordinates.

A physical quantity is Lorentz invariant when its value is unchanged under Lorentz transformations. Tensor equations can be Lorentz covariant while their components transform. Fixed-vector gauge choices need not make covariance manifest, although physical gauge-invariant predictions retain the symmetry.

A Lorentz scalar is unchanged under a Lorentz transformation. Contracting every Lorentz index in a tensor expression produces a scalar when the constituent fields transform covariantly.