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A passive Lorentz transformation changes the components used to describe the same events and fields while preserving the standard Minkowski metric matrix: ${\Lambda }^T\eta \Lambda =\eta$. This expresses the relation between standard clock-normalized inertial frames. An arbitrary coordinate relabeling remains possible, but generally changes that metric matrix. Active boost invariance of a physical system additionally requires the relevant dynamical symmetry; a preferred physical background can obstruct it even when every equation is written in tensor notation.

An antisymmetric tensor combines the electric polarization and magnetization of a medium. For Minkowski metric $(-,+,+,+)$ and units with light speed one, choose $M^{i0}=P_i$ and $M^{ij}={ϵ}_{ijk}{M}_k$, together with $F_{i0}=E_i$ and $F_{ij}={ϵ}_{ijk}{B}_k$. Then the Lorentz scalar interaction is $M^{\mu \nu }{F}_{\mu \nu }/2=\mathbf{P}\cdot \mathbf{E}+\mathbf{M}\cdot \mathbf{B}$. The bound charge and bound current have four-current $j_b^{\mu }={\partial }_{\nu }{M}^{\mu \nu }$, giving ${\rho }_b=-\nabla \cdot \mathbf{P}$ and ${\mathbf{J}}_b={\partial }_t\mathbf{P}+\nabla \times \mathbf{M}$. This current is identically conserved because partial derivatives commute and $M$ is antisymmetric.

For a nonzero null momentum and a nontrivial BRST charge on the gauge/ghost one-particle sector, closed vector states obey $k\cdot \epsilon =0$ and exact vector states have polarization proportional to $k$. The Faddeev-Popov ghost field state is exact and the antighost field state is not closed. With mostly-plus Minkowski metric, choose $k=(\omega ,0,0,\omega )$; closure sets ${\epsilon }^3={\epsilon }^0$, so the norm is $|{\epsilon }^1{|}^2+|{\epsilon }^2{|}^2$. Quotienting by the null longitudinal polarization leaves two positive-norm transverse polarizations per group index. Self-adjointness and nilpotence alone do not establish positivity on an arbitrary state space: the nonzero momentum, nontrivial charge and stated indefinite inner products are essential here.

With Minkowski metric signature $(+---)$ and source coupling $-{\mu }_0{J}_{\mu }{A}^{\mu }$ added to the Proca action, a massive vector field coupled to a conserved current obeysIn vacuum this gives $(\square +m^2)A^{\nu }=0$ after imposing the dynamically derived Lorenz constraint in Proca theory. The explicit mass term breaks the gauge invariance of the massless Maxwell theory.

For a free massive Proca field in the mostly-plus Minkowski metric, the covariant inverse of the quadratic action and the canonical time-ordered product of vector potentials differ locally. With Fourier transform $e^{iq\cdot (x-y)}$, the covariant Proca propagator is $-i({\eta }_{ab}+q_a{q}_b/m^2)/(q^2+m^2-i0)$, while the literal canonical two-point correlation function subtracts $i{\delta }_a^0{\delta }_b^0/m^2$ in momentum space. The time component is nondynamical; its physical-mode numerator is $|\mathbf{q}{|}^2/m^2$, rather than $-1+(q^0{)}^2/m^2$. The difference is an instantaneous Dirac delta function, not another propagating state. Both expressions agree away from coincidence. This distinction matters when identifying a covariant Green function with a canonical correlator in a constrained field theory.

A Lorentz transformation with determinant $+1$ that preserves future-directed timelike four-vectors. Equivalently, it belongs to the identity component of the Lorentz group. For the Minkowski metric $(-,+,+,+)$, its matrix satisfies ${\Lambda }^T\eta \Lambda =\eta$ and ${\Lambda }^0{}_0\ge 1$. Such transformations preserve the future light cone, so the retarded Green function of the d'Alembert operator transforms as a Lorentz scalar.

Expand the Einstein tensor about the Minkowski metric. The quadratic term in the metric perturbation defines $t_{\mu \nu }=-G_{\mu \nu }^{(2)}[h,h]/(8\pi )$, a symmetric tensor relative to the background. For a first-order vacuum solution, the expanded contracted Bianchi identity gives ${\partial }^{\mu }{t}_{\mu \nu }=0$. Equivalently, a second-order correction $j$ obeys $G^{(1)}[j]=8\pi t$, whose flat divergence vanishes identically. This conservation is an on-shell statement. The local $t$ depends on perturbative gauge and the background split; conservation alone does not make it a gauge-independent local gravitational stress-energy tensor.

The worldline einbein imposes the relativistic mass-shell condition. With the Minkowski metric of signature $(-,+,\dots ,+)$, its first-class constraint is $\phi =(p^2+M^2)/2$.

In the $(t,z)$ plane, $u=t-z$ is retarded time and $v=t+z$ is advanced time. The radial part of the Minkowski metric is $-d{t}^2+d{z}^2=-dudv$.

For the Lorentz algebra convention $[M_{\mu \nu },M_{\rho \sigma }]=i(M_{\mu \sigma }{\eta }_{\nu \rho }+M_{\nu \rho }{\eta }_{\mu \sigma }-M_{\mu \rho }{\eta }_{\nu \sigma }-M_{\nu \sigma }{\eta }_{\mu \rho })$, choose $\eta =\mathrm{diag}(s,-s,-s,-s)$, $s=\pm 1$, ${ϵ}_{123}=1$, $J_i={ϵ}_{ijk}{M}_{jk}/2$, and $K_i=M_{0i}$. The resulting rotation bracket is $[J_i,J_j]=si{ϵ}_{ijk}{J}_k$ as well as the two displayed brackets. Changing the Minkowski metric signature while holding this generator convention fixed reverses all coefficients. A simultaneous redefinition of the generators can instead preserve the familiar form. Conventions must be specified before comparing algebra signs.

An automorphism preserving the physically specified background structure of spacetime. With only a Minkowski metric and fixed orientations, the appropriate external symmetries are the corresponding Poincaré group elements. An added preferred vector field reduces this group. This is distinct from merely changing coordinates.

For positive lapse $N$, this time reparametrization gives a local Minkowski metric. If the new time coordinate ranges over the entire real line, it is a global isometry to Minkowski space. In particular $N=\cosh t$ gives $T=\sinh t$ and complete nonperiodic timelike geodesics. A lapse depending on the spatial coordinate instead can describe a different curved geometry; exchanging the arguments is not a harmless relabeling of the same metric.

A timelike basis polarization in covariant photon quantization has positive Minkowski metric squared length, conventionally ${ϵ}^{0\mu }=(1,0)$ with signature $(+,-,-,-)$. It is distinct from the spatial longitudinal polarization ${ϵ}^{3\mu }=(0,\hat{\mathbf{p}})$. The extra minus sign in the covariant oscillator commutator makes its one-photon state a negative-norm photon state. Neither this temporal polarization nor an isolated longitudinal photon survives the Gupta-Bleuler null-state quotient.

For $J=\mathrm{diag}(1,-1)$, the matrices satisfying $A^{T}JA=J$ and $A_{11}>0$ form the time-orientation-preserving Lorentz group in one spatial dimension. They are exactly $A=H(t)D_{\epsilon }$, where $H(t)=\left(\begin{array}{cc}\cosh t& \sinh t\\ \sinh t& \cosh t\end{array}\right)$, $D_{\epsilon }=\mathrm{diag}(1,\epsilon )$ and $\epsilon =\pm 1$. Preservation of the Minkowski metric makes the first column a future-pointing unit timelike vector; Minkowski inner product orthogonality determines the second up to sign. The identity $D_{\epsilon }H(s)=H(\epsilon s)D_{\epsilon }$ gives the multiplication law $(t,\epsilon )(s,\eta )=(t+\epsilon s,\epsilon \eta )$ and inverse $(-\epsilon t,\epsilon )$.

At a nonzero Euclidean momentum $k$, the transverse projector of a vector field is $P_T^{ab}={\delta }^{ab}-k^a{k}^b/k^2$. In a Lorentzian Minkowski metric, its mixed-index form is $(P_T{)}^a{}_b={\delta }_b^a-p^a{p}_b/p^2$, defined for $p^2\ne 0$. It obeys $P_{T}p=0$ and $P_T^2=P_T$. A nonzero null four-momentum is insufficient for this particular decomposition because the denominator vanishes.

A world line is an object's history as a curve of spacetime events. In an inertial frame, a particle at rest has constant spatial coordinates, so its world line is vertical in a spacetime diagram with $ct$ upward. A particle at constant velocity $v$ along one axis has $x=x_0+vt$, a straight sloping world line. The coordinate representation changes with the inertial frame, while the physical history is the same.

A massive observer follows a future-directed timelike curve. Along a differentiable world line with velocity $\mathbf{v}(t)$, its clock accumulatesThe Minkowski metric therefore measures clock time along the curve, not just the difference between two coordinate readings. In the straight $O$-to-$A$ example, $v=0.75c$ for $4\mathrm{s}$, giving $4\sqrt{1-0.75^2}\mathrm{s}=\sqrt{7}\mathrm{s}$. If the observer changes velocity along the path, use the integral rather than assigning the straight-segment formula to the whole path.