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Geometric or field data held fixed when a theory's dynamical degrees of freedom vary. The Minkowski metric is background structure in ordinary special relativity; the spacetime metric tensor is dynamical in general relativity. A fixed timelike vector field can introduce a preferred direction even when an equation is written using tensor notation.

With Minkowski metric signature $+---$ and $\mathrm{tr}({\gamma }^{\mu }{\gamma }^{\nu }{\gamma }^{\rho }{\gamma }^{\sigma }{\gamma }_5)=4i{ϵ}^{\mu \nu \rho \sigma }$, define $H^{\mu \nu }(a,b)=a^{\mu }{b}^{\nu }+a^{\nu }{b}^{\mu }-g^{\mu \nu }a\cdot b+i{ϵ}^{\mu \nu \rho \sigma }{a}_{\rho }{b}_{\sigma }$. Then $\mathrm{tr}[/a{\gamma }^{\mu }(1-{\gamma }_5)/b{\gamma }^{\nu }(1-{\gamma }_5)]=8H^{\mu \nu }(a,b)$. The symmetric-tensor contraction gives twice the sum of the two possible dot-product pairings; the Levi-Civita symbol contraction, including the two factors of $i$, gives twice their difference. Their sum is the displayed identity, explaining the single pairing in left-chiral weak rates.

The prescription replaces partial derivatives in flat-spacetime matter equations by the Levi-Civita connection and the Minkowski metric by the curved metric tensor. It implements minimal gravitational coupling and reproduces flat-spacetime physics at a point in a local inertial frame. It is not a unique rule for rearranged higher-derivative equations: commuting partial derivatives before replacement can conceal a Riemann curvature tensor term afterwards. The derivative-order ambiguity in covariant Maxwell potential equations is an explicit example. The equivalence principle removes uniform gravitational acceleration locally but does not erase tidal curvature over a finite experiment.

For Minkowski metric $(+---)$ and $\mathcal{L}=-G_{\mu \nu }^{\ast }{G}^{\mu \nu }/2+M^2{B}_{\mu }^{\ast }{B}^{\mu }$, use coordinates $B^{\mu },B^{\ast \mu }$. The spatial canonical momenta are ${\pi }_i=G^{\ast 0i}$, ${\pi }_i^{\ast }=G^{0i}$; the temporal momenta vanish. Constraint preservation gives ${\partial }_i{\pi }_i=M^2{B}_0^{\ast }$. Eliminating $B_0$ and its conjugate gives the positive reduced Hamiltonian density displayed above, up to a spatial surface term. In particular the massive Gauss law has a mass-dependent right-hand side; setting it to zero removes a physical longitudinal particle polarization.

For a source-free plane electromagnetic wave in vacuum, let $n^{\mu }=(1,\mathbf{s})$ be its null propagation Lorentz four-vector. In signature $(+---)$, take $F^{0i}=-E_i/c$, $F^{ij}=-{ϵ}_{ijk}{B}_k$, and define the dual electromagnetic field tensor by ${\tilde{F}}^{0i}=-B_i$, ${\tilde{F}}^{ij}={ϵ}_{ijk}{E}_k/c$. Lower both indices with the Minkowski metric. Direct contraction givesBoth contractions vanish exactly when the transverse plane electromagnetic wave Maxwell equations hold. Being zero covectors, they remain zero in every inertial frame. Thus the transformed spatial wavevector is perpendicular to both transformed fields, even though the separate three-dimensional dot products are not themselves Lorentz scalars for arbitrary fields.

One contraction alone is insufficient without the other Maxwell equations. In particular, adding a magnetic field parallel to $\mathbf{s}$ leaves $n^{\mu }{F}_{\mu \nu }$ unchanged but violates $\mathbf{s}\cdot \mathbf{B}=0$. Conversely the two scalar perpendicularity conditions alone do not impose the required relation $\mathbf{B}=\mathbf{s}\times \mathbf{E}/c$ between field strengths and directions.

For Minkowski metric $\mathrm{diag}(1,-1)$, the displayed gamma matrices satisfy the Clifford algebra and ${\gamma }^5={\gamma }^0{\gamma }^1$ has chirality eigenvalues $+1,-1$. Ordering the spinor as $({\psi }_R,{\psi }_L{)}^{\mathsf{T}}$ gives $i({\partial }_t+{\partial }_x){\psi }_R=m{\psi }_L$ and $i({\partial }_t-{\partial }_x){\psi }_L=m{\psi }_R$. A plane wave has the Dirac Hamiltonian $p{\sigma }_3+m{\sigma }_1$, whose square is $(p^2+m^2)1$, proving $E^2=p^2+m^2$.

For gamma matrices ${\gamma }^0={\sigma }_3$, ${\gamma }^1=i{\sigma }_1$, ${\gamma }^2=i{\sigma }_2$ and Minkowski metric $\mathrm{diag}(1,-1,-1)$, a one-axis spatial reflection acts with ${\gamma }^1$ on the two-component Dirac field. It preserves the kinetic term and reverses the Dirac mass term. Time reversal acts with ${\gamma }^2$ and includes conjugation of the operator coefficients; it has the same mass reversal. Their composition preserves a real mass. Charge conjugation can use ${\psi }_C={\gamma }^1{\psi }^{\ast }$, because ${\gamma }^{\mu }{\gamma }^1=-{\gamma }^1{\gamma }^{\mu \ast }$. The mass bilinear is charge-conjugation even when Grassmann field exchange signs are included, and the kinetic action is invariant up to a boundary term. These statements concern the classical free action.

The equivalence principle states that gravity couples universally to energy-momentum and that a freely falling local frame removes uniform gravitational acceleration at a point. Tidal curvature remains measurable over a finite region.

Imagine releasing a ball inside a small falling elevator. Relative to the cabin it initially floats, because cabin and ball share the same free fall. The weak equivalence principle says this trajectory is independent of the test particle's composition for identical initial conditions. The Einstein equivalence principle further identifies local nongravitational experiments with those of special relativity. It is a statement about experiments in a local inertial frame, not just equality of two masses.

The word local is essential. Coordinates can make the metric tensor equal to the Minkowski metric and its first derivatives vanish at one event. They generally cannot erase the Riemann curvature tensor, which involves second derivatives. Two separated balls can therefore approach or recede even inside the falling cabin. A sufficiently extended experiment detects these tidal forces; a long observation can accumulate them even when the cabin is small. The equivalence principle does not identify an entire planetary gravitational field with acceleration in flat spacetime. See Tong's discussion of normal coordinates and local equivalence.

This covariant scalar-force model uses the Minkowski metric and the displayed momentum law, with $\square \Phi =-4\pi G{T}^i{}_i$. For a massive test particle, write $p^i=m{u}^i$, $u^i{u}_i=1$. Contracting the force law with $p_i$ gives constant rest mass; the acceleration is $d{u}^i/d\tau ={\partial }^i\Phi -u^i(u^j{\partial }_j\Phi )$, independent of the mass. A static weak field obeys the Poisson equation and yields Newtonian acceleration $-\nabla \Phi$. For a photon, $p\cdot dx=0$ and $dx$ is parallel to $p$, so $d{p}^i=-p^{i}d\Phi$: photon energies change but their unparametrized paths remain straight. The motion law has the same unparametrized curves as the conformal metric $e^{2\Phi }\eta$, but the field equation must not be silently identified with a different scalar-gravity theory.

The generators of spacetime translations in a quantum field theory. With the mostly-minus Minkowski metric, $P\cdot a=H{a}^0-\mathbf{P}\cdot \mathbf{a}$. A momentum eigenstate satisfies $P^{\mu }|p⟩=p^{\mu }|p⟩$, where $p^{\mu }$ is its four-momentum. After subtracting the vacuum energy, a translation-invariant vacuum state has eigenvalue zero.

A four-vector is a vector whose four vector components transform under a Lorentz transformation as $U^{\prime \mu }={\Lambda }^{\mu }{}_{\nu }{U}^{\nu }$. Its individual vector components generally change between inertial frames; the Minkowski metric combines them into invariant scalar products. Listing four quantities is not enough to make a four-vector: their transformation law matters.

The most direct example is the displacement between two spacetime events, written $(c\Delta t,\Delta x,\Delta y,\Delta z)$. Multiplying the time difference by the speed of light gives all four vector components the same length units. A position four-vector describes an event relative to a chosen origin, while a displacement does not depend on a common translation of the origin. Four-momentum and four-velocity use the same transformation rule for different physical quantities.

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.

For the mostly-minus Minkowski metric, this product anticommutes with every gamma matrix and is anti-Hermitian in the standard representation. The conventional Hermitian chirality matrix is $iG$. Nevertheless $\bar{\psi }G\psi$ is Hermitian, because ${\gamma }^{0}G{\gamma }^0=-G$. Therefore a real interaction coefficient in $-g\varphi \bar{\psi }G\psi$ is compatible with Hermitian dynamics and gives the Feynman vertex $-igG$.

Set the inertial cylindrical angle to $\Phi =\varphi +\omega t$. The Minkowski metric becomes $d{s}^2=(c^2-{\omega }^2{r}^2)d{t}^2-2\omega r^{2}dtd\varphi -d{r}^2-r^{2}d{\varphi }^2-d{z}^2$. For a massive geodesic parametrized by proper time, its cyclic coordinates give conserved covariant four-momentum components $p_t=m[(c^2-{\omega }^2{r}^2)\dot{t}-\omega r^2\dot{\varphi }]$, $p_{\varphi }=-m{r}^2(\dot{\varphi }+\omega \dot{t})$ and $p_z=-m\dot{z}$. Their combination gives $\dot{t}=(p_t-\omega p_{\varphi })/(m{c}^2)$. The radial Euler-Lagrange equation gives the displayed coordinate acceleration, although the free worldline is straight in inertial coordinates. The conserved inertial energy is $p_t-\omega p_{\varphi }$, and the inertial axial angular momentum is $-p_{\varphi }$.

A homogeneous quadratic polynomial in the transverse coordinates $x,y$ is a harmonic function exactly when the sum of its $x^2$ and $y^2$ coefficients is zero. Thus it has the displayed form. For the Minkowski metric $dudv+d{x}^2+d{y}^2$, a metric perturbation $h=\psi d{u}^2$ has zero trace, zero divergence and $\square h=0$, since ${\partial }_v\psi =0$ and ${\partial }_x^2\psi +{\partial }_y^2\psi =0$. The two coefficient functions describe the two gravitational wave polarizations.

Choose coordinates $x^{\mu }=(ct,\mathbf{x})$ and Minkowski metric $\eta =\mathrm{diag}(1,-1,-1,-1)$. With the convention $F^{0i}=-E_i/c$ and $F^{ij}=-{ϵ}_{ijk}{B}_k$, lowering both tensor indices gives $F_{0i}=+E_i/c$ while leaving $F_{ij}=F^{ij}$. An array with all upper indices cannot be used unchanged as the corresponding array with all lower indices. The propagation Lorentz four-vector $n^{\mu }=(1,\mathbf{s})$ for unit spatial direction $\mathbf{s}$ similarly has $n_{\mu }=(1,-\mathbf{s})$. These signs matter in contractions that encode Maxwell equations.

Why add the daughter four-momenta? Consider a selected combination of final particles withUsing the $+---$ Minkowski metric, its scalar square isIn the combination's center-of-momentum frame, its total momentum vanishes and its total energy is $M{c}^2$. Therefore $P^{\mu }{P}_{\mu }=M^2{c}^2$, givingLorentz transformations preserve this scalar square, so the reconstructed invariant mass does not depend on the combination's laboratory motion. It includes relative kinetic energy and need not equal the sum of the daughter rest masses. A random pair has an invariant mass too; that number does not itself establish a common parent.

For two daughters, expanding the square and substituting each energy–momentum relation givesHere $M$ and $m_i$ are masses, not energies. Reporting $M$ in $\mathrm{GeV}/c^2$ and momenta in $\mathrm{GeV}/c$ is equivalent to calculating numerically with $M{c}^2$ and $c{\mathbf{p}}_i$ in $\mathrm{GeV}$. Setting $c=1$ in natural units suppresses these conversion factors; it does not change the measured quantity.

Alvarez's lecture illustrates invariant mass searches with $K\pi$ combinations and contrasts resonance lifetimes near $10^{-23}\mathrm{s}$ with the much longer lived states the chambers originally targeted. These resonances decayed before producing a resolvable parent track.^[ref]

Operationally, choose a reaction hypothesis, require an acceptable bubble-chamber kinematic fitting result, and form specified daughter combinations. Accumulate their invariant masses in a histogram across events. A particle resonance is inferred from the ensemble's dynamics; its short-lived parent is not directly photographed. Daughter tracks can be visible even when the production and decay positions of the parent cannot be separated. Wrong combinations create background, and several combinations from one event are statistically correlated.

The source-free Linearized Einstein equations. For a metric perturbation about the Minkowski metric, imposing the Lorenz gauge in linearized gravity reduces them to $\square h_{\mu \nu }=0$ together with ${\partial }^{\mu }{\bar{h}}_{\mu \nu }=0$, where $\bar{h}$ is the trace-reversed metric perturbation. A linearized coordinate gauge transformation preserves $G^{(1)}[h]=0$, while preserving the imposed gauge additionally requires its generator to satisfy the wave equation.

For an on-shell massive vector with $k^{\mu }=(E,|\mathbf{k}|\hat{\mathbf{k}})$ in Minkowski metric signature $+---$, the displayed vector satisfies $k\cdot {ϵ}_L=0$ and ${ϵ}_L^2=-1$. It specifies the longitudinal one of the three physical polarization states. It is not a replacement for a final polarization sum when other states are unobserved.

The Lorentz group consists of real matrices preserving the Minkowski metric: ${\Lambda }^T\eta \Lambda =\eta$.