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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.

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.

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.