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For $d{s}_5^2={\eta }_{ab}d{x}^{a}d{x}^b+(dy+A_{a}d{x}^a{)}^2$, with $A$ independent of $y$, an affine parameter $\lambda$ makes $\xi =dy/d\lambda +A_{a}d{x}^a/d\lambda$ and the tangent norm $\mathcal{N}=g_{AB}{\dot{x}}^A{\dot{x}}^B$ constant. The four-dimensional interval satisfiesConsequently the formula $d{\lambda }^2=d{u}^2/(1-{\xi }^2)$ assumes the additional normalization $\mathcal{N}=1$; affine parametrization alone does not set this value. With base metric signature $+---$ and $u=c\tau$, the projected curve is timelike only for $\mathcal{N}>{\xi }^2$. The normalized case gives $d\tau /d\lambda =\sqrt{1-{\xi }^2}/c$. The positive fibre term and the $+---$ base together have two positive directions; the same sign convention must be retained when interpreting the reduction.

The simply connected Bertotti-Robinson geometry is the product of the universal cover of two-dimensional anti-de Sitter spacetime and a two-sphere with equal radius $m>0$. In metric signature $(-,+,+,+)$,Its radial Penrose diagram is an infinite conformal strip of two-dimensional anti-de Sitter spacetime. The metric tensor $-{\lambda }^{2}d{t}^2+m^2{\lambda }^{-2}d{\lambda }^2+m^{2}d{\Omega }_2^2$ is a patch of it, with $\lambda =m(\sin T-\sin X)/\cos X$. Its Killing vector field ${\partial }_t$ extends across alternating degenerate Killing horizons $\lambda =0$. Both factors are geodesically complete, giving a maximal analytic extension of this patch without periodic time identification.

A nonzero causal vector is timelike or null: $g(v,v)\le 0$ with metric signature $(-,+,+,+)$. The zero vector is commonly included when stating the dominant energy condition.

For future-directed particles with total four-momentum $P$ and metric signature $(-+++)$, the positive square root of $-P_{\mu }{P}^{\mu }$ is their energy in the frame where total spatial momentum vanishes. Its invariant definition remains valid at a collision in spacetime.

Let $X$ be a Conformal Killing vector field with conformal Killing factor $\lambda$, and let $U$ be the tangent vector of a geodesic with affine parameter $s$. The product rule, metric compatibility and geodesic equation giveEvery null geodesic therefore has the conserved quantity $g(X,U)$. On a timelike geodesic with proper time as parameter, the derivative is $-\lambda$ in metric signature $(-,+,+,+)$. With a nonaffine parameter satisfying ${\nabla }_{U}U=fU$, an extra term $fg(X,U)$ appears; the conserved expression for a null geodesic is tied to its affine parameter normalization.

In units $c=1$ and metric signature $(-,+,+,+)$, take a fixed unit timelike vector field $n$ and $\alpha >0$. The displayed equation for a scalar field is tensorial; in arbitrary coordinates use the flat Levi-Civita connection in place of ordinary second derivatives. In the rest inertial frame of $n$ it becomes $-(1+\alpha ){\varphi }_{tt}+{\nabla }^2\varphi =0$, with characteristic speed $1/\sqrt{1+\alpha }$. Its fixed $n$ selects a preferred frame. A boost that does not preserve $n$ is not a symmetry with this background held fixed, although transforming the components of every quantity gives an equivalent coordinate description. This separates general covariance from a substantive relativity principle.

On a scalar, the covariant wave operator is $\square \varphi =|g{|}^{-1/2}{\partial }_a(|g{|}^{1/2}{g}^{ab}{\partial }_b\varphi )$. For the metric signature $(-+++)$ it has the flat-space form $-{\partial }_t^2+{\nabla }^2$. This divergence expression is convenient for separation of variables and for integrating the Klein-Gordon equation by parts.

Definitions of the Riemann curvature tensor can differ by an overall sign, independent of the metric signature. Reversing that definition reverses the Ricci tensor, Ricci scalar and Einstein tensor, as well as the curvature side of the geodesic deviation identity and the Palatini identity. A derivation must choose the contraction and derivative ordering explicitly before combining formulas from different conventions. Changing a metric signature alone is a separate operation.

In units $c=1$, a slowly moving isolated source has trace-free mass quadrupole moment $Q^{ij}=\int T^{00}(x^i{x}^j-{\delta }^{ij}|\mathbf{x}{|}^2/3)d^{3}x$. The retarded linearized gravity solution and the stress-energy quadrupole identity give the displayed physical spatial strain, with $\mathbf{n}=\mathbf{x}/r$ and the transverse-traceless projector $\Lambda$. If the spatial metric is written $g_{ij}=-{\delta }_{ij}-2E_{ij}$ for metric signature $(+---)$, its trace-free Lorenz-gauge coefficient before the transverse projection is $E_{ij}=G{\ddot{Q}}_{ij}(t-r)/r$. Thus $H^{\mathrm{TT}}=2E^{\mathrm{TT}}$. Far distance alone does not permit dropping source-dependent retardation: the quadrupole approximation also requires the source size to be small compared with the radiation wavelength.

With a mostly-plus metric signature, $e^{iS}$, and gauge-fixing term $-(\partial \cdot A{)}^2/2$, the free adjoint gauge-field kernel is $-p^2{\eta }_{\mu \nu }{\delta }_{ab}$. Inverting it gives the displayed Feynman propagator. This convention uses the same denominator and sign as a massless real scalar propagator.

The fixed-target production threshold concerns the incident accelerator beam, whereas antiproton mass from momentum and velocity concerns an outgoing secondary beam. To see the distinction, consider a free proton hitting a stationary free proton, producingThe final electric charge is $+2e$ and final baryon number is $3-1=2$, as initially. Approximate all four final rest masses by the same proton rest mass $m$. With metric signature $(+---)$, define the squared centre-of-mass energy by $W^2=c^2(P_1+P_2{)}^2$. The incident proton has total energy $E=m{c}^2+K$, and the target four-momentum is $(mc,0)$. Four-momentum conservation and the mass shell relation giveAt threshold the four final particles have zero relative momentum in their centre-of-momentum frame, so $W_{\mathrm{thr}}=4m{c}^2$. HenceOnly $2m{c}^2$ of new rest energy is created. The larger incident kinetic energy is required because the final system must also carry the initial laboratory momentum. The incident total energy at threshold is $7m{c}^2$; neither this nor $K_{\mathrm{thr}}$ is the selected secondary momentum.

The 1955 authors explicitly qualified the stationary-nucleon threshold: target nucleon motion inside an atomic nucleus can lower the required incident energy.^[ref]

The kinematic reason is visible for two on-shell incoming nucleons:A target momentum directed against the beam increases $W^2$. Applying this illustration to a bound nucleon also requires binding energy and residual-nucleus recoil; the free stationary-target $6m{c}^2$ result is therefore not a universal sharp threshold for a copper target. A low observed secondary momentum is entirely compatible with a much larger accelerator kinetic energy.

The four-acceleration is the covariant derivative of four-velocity along a massive observer's worldline: $A^{\mu }=d{U}^{\mu }/d\tau +{\Gamma }_{\nu \rho }^{\mu }{U}^{\nu }{U}^{\rho }$. In Cartesian Minkowski spacetime coordinates the Christoffel symbols vanish, leaving the ordinary proper time derivative. Differentiating the constant velocity norm gives $U\cdot A=0$; in the instantaneous rest frame, $A=(0,\mathbf{a})$, so $A\cdot A=-a^2$ with $+---$ metric signature. The magnitude $a$ is the proper acceleration.

For uniform circular motion in an inertial laboratory, the Lorentz factor is constant. With metric signature $+---$, four-acceleration is $A^{\mu }=(0,{\gamma }^2\mathbf{a})$ and proper acceleration has magnitude ${\gamma }^2{v}^2/r$. The instantaneous spatial acceleration is perpendicular to velocity, even though the particle has nonzero proper acceleration.

For coordinates $x^{\mu }=(ct,x,y,z)$, the four-gradient of a scalar field is its covector of coordinate derivatives. Under a constant Lorentz transformation, ${\partial }_{\mu }^{\prime }=({\Lambda }^{-1}{)}^{\nu }{}_{\mu }{\partial }_{\nu }$. In metric signature $(+,-,-,-)$, raising its index gives ${\partial }^{\mu }=(c^{-1}{\partial }_t,-\nabla )$. Contracting it with a four-vector field produces the Lorentz scalar four-divergence. In particular ${\partial }_{\mu }{J}^{\mu }=0$ is charge continuity in every inertial frame.

A massive observer's four-velocity is the tangent $U^{\mu }=d{X}^{\mu }/d\tau$ to its worldline, parametrized by proper time. Its invariant squared Minkowski norm is $c^2$ in the $+---$ metric signature, and $-c^2$ in the opposite convention. Unlike the coordinate values $X^{\mu }$, the tangent transforms as a vector under a general coordinate change.

In metric signature $(+---)$, a future-pointing timelike four-vector has positive time component larger than its spatial magnitude. Two such vectors have strictly positive Minkowski metric scalar product, since $P\cdot Q=P^0{Q}^0-\mathbf{P}\cdot \mathbf{Q}\ge P^0{Q}^0-|\mathbf{P}||\mathbf{Q}|>0$. In the opposite signature the scalar-product sign reverses, while the causal condition is the same. The future timelike cone is closed under addition, which rules out a null photon momentum equalling a sum of two massive future four-momenta.

Each Killing vector field supplies a conserved quantity for freely moving test particles: it turns a geometric symmetry into a first integral of the geodesic equation. Let $\gamma (\lambda )$ be a geodesic with affine parameter $\lambda$, tangent vector $U^{\mu }=d{x}^{\mu }/d\lambda$, and ${\nabla }_{U}U=0$. Wherever a Killing vector field $K$ is defined along the geodesic,The statement applies to Riemannian manifolds and to timelike geodesics, null geodesics and spacelike geodesics in a Lorentzian manifold. It requires an affine parameter; for a massive test particle, proper time is one choice. See Tong, section 4.3.2, equation (4.32).

For a worked example in the Euclidean plane, take $\gamma (t)=(1+t,2-t)$ with affine parameter $t$ and velocity $U=(1,-1)$. The Euclidean Killing vector fields for translation and rotation giveThe rotational field along the geodesic is $K_{\mathrm{rot}}(\gamma (t))=(t-2,1+t)$: it changes from $(-2,1)$ at $t=0$ to $(-1,2)$ at $t=1$, yet its pairing with $U$ stays $-3$. With toy mass $m=1$ and $t$ interpreted as time, the first constant is the $x$ component of linear momentum, and the second is the signed $z$ component of angular momentum about the origin. Their numerical values use toy units. The geodesic is straight; the rotation flow orbits are circles.

This conservation law follows from metric compatibility and the Killing equation:The geodesic equation kills the acceleration term, and the symmetric product $U^{\mu }{U}^{\nu }$ selects the vanishing symmetric derivative. This tensor contraction is the whole proof outline; see Carroll, equation (5.43).

In a stationary spacetime, a chosen time-translation Killing vector field $K$ gives Killing energy. With metric signature $(-+++)$, $c=1$, and a massive test particle's four-momentum $p^{\mu }=m{U}^{\mu }$, the physical normalization isHere $\Phi$ generates the axial rotational symmetry when one exists. In coordinates with $K={\partial }_t$ and $\Phi ={\partial }_{\varphi }$, these are $E=-p_t$ and $L_z=p_{\varphi }$. Both are conserved quantities along the geodesic, even if the metric has a time-angle cross term. A future timelike $K$ makes $E$ positive for a future massive particle; the scale of $E$ depends on the normalization of $K$. Not every spacetime has a global timelike Killing vector field, and a stationary field can become spacelike in an ergoregion of a stationary spacetime. Conservation on the field's domain does not imply a universal global positive energy; see Tong, sections 4.3.1-4.3.2 on the timelike-domain qualification and conserved charges.

For metric signature $(+---)$, a stationary perfect fluid follows a timelike Killing vector field $K$ with four-velocity $U=K/N$. Its Killing-flow four-acceleration is $U^b{\nabla }_b{U}_a=-{\nabla }_a\log N$. The relativistic Euler equation and stationarity of the pressure give the displayed hydrostatic equilibrium equation. This uses stationarity along $K$ and does not require hypersurface orthogonality.

With metric signature $(+---)$ and a future timelike stationary Killing vector $K$, positive energy is $E=g(P,K)$ for future physical momentum $P$. A common $(-+++)$ convention instead uses $-g(P,K)$. The normalization of $K$ fixes the observer or asymptotic time scale.

With metric signature $(+---)$, normalize a timelike Killing vector field by $U^a=K^a/N$, $N^2=K^a{K}_a$. The Killing equation gives $K^b{\nabla }_b{K}_a=-\frac{1}{2}{\nabla }_a{N}^2$, while $K^b{\nabla }_{b}N=0$. Thus the four-acceleration of its normalized flow is the displayed gradient. With metric signature $(-+++)$ the corresponding lower-index formula has the opposite sign.