An infinitesimal coordinate change generated by $\xi$ changes a metric perturbation about Minkowski spacetime by $h_{\mu \nu }\mapsto h_{\mu \nu }-{\partial }_{\mu }{\xi }_{\nu }-{\partial }_{\nu }{\xi }_{\mu }$, in the passive sign convention. For a passive coordinate change $x^{\mu }\mapsto x^{\mu }+{\xi }^{\mu }$, the first-order change is minus the Lie derivative of a tensor field of the background metric, equivalently the pullback by the inverse flow. The gauge invariance of the linearized Riemann tensor follows because the added terms have commuting third derivatives, which cancel in the curvature antisymmetrizations. Gauge choices alter the metric components while preserving linearized tidal effects.

A massless Fierz-Pauli action is the quadratic flat-background action for a symmetric metric perturbation that yields the vacuum Linearized Einstein equations. One normalization, up to a total derivative, isVarying the symmetric field with compact support gives the displayed linearized Einstein tensor coefficient. Its linearized coordinate-gauge symmetry is consistent with the contracted Bianchi identity. It is the massless gravitational action, distinct from imposing the massive Fierz-Pauli equations with five polarizations.

A slack variable turns an inequality constraint $a^{T}x\le b$ into the equality $a^{T}x+s=b$ with $s\ge 0$. Its value measures the unused amount of the constraint. In the simplex algorithm, slack variables often supply the initial basic feasible solution.

A linear polyhedron is an intersection of finitely many closed affine half-spaces in finite-dimensional real space. It may be empty, unbounded, or lower-dimensional. This usage differs from a three-dimensional geometric polyhedron. The strict separation of disjoint linear polyhedra follows from linear programming duality.

A vector cosmological perturbation is the transverse-vector sector of linear metric and matter perturbations. For a Fourier mode with nonzero wavevector $\mathbf{k}$, transversality removes the component parallel to $\mathbf{k}$, leaving two helicity components. In a gauge with $d{s}^2=a^2[-d{\eta }^2+2B_{i}d\eta d{x}^i+{\delta }_{ij}d{x}^{i}d{x}^j]$, ${\partial }_i{B}_i=0$ describes the vector metric perturbation.

A cosmological gauge transformation changes the identification of a perturbed spacetime with its homogeneous background. Individual field perturbations can change without changing physical observables. At first order the metric change is the linear metric gauge-transformation law; matter scalars obey the corresponding background-shift rule.

In a flat matter-dominated universe with $a\propto t^{2/3}$ and $P\propto {\rho }^{4/3}$, the squared sound speed scales as $t^{-2/3}$. Each comoving Fourier mode then satisfies an Euler-Cauchy equation, with gravitational growth at long wavelengths and logarithmic-time acoustic oscillations at short wavelengths.

The Lighthill acoustic analogy rewrites fluid mass and momentum conservation as a constant-speed wave equation driven by a nonlinear Lighthill stress tensor. It is an exact rearrangement before making source approximations. The outgoing field can be constructed with a retarded acoustic Green function.

Acoustic sources may be organized by the number of spatial derivatives in their effective wave equation forcing. A direct scalar source is an acoustic monopole, a divergence of a force density an acoustic dipole, and a double divergence of a stress tensor an acoustic quadrupole.

An acoustic source is compact when its size is much smaller than the acoustic wavelength. The propagation delay across the source can then be neglected in leading source integrals. This condition is distinct from the radiation-region requirement $k_{0}r\gg 1$.

An acoustic density perturbation is the deviation of fluid mass density from its quiescent reference value. In a homogeneous isentropic linear acoustic field, its corresponding pressure perturbation is $p^{\prime }=c_0^2{\rho }^{\prime }$. Nonlinear and entropy-dependent departures from this relation enter the Lighthill stress tensor.

Consider adiabatic perturbations of a static ideal gas, with background pressure $p_0(\mathbf{x})$, mass density ${\rho }_0(\mathbf{x})$ and constant heat-capacity ratio $\gamma$. Assume a prescribed force density $-\nabla \chi$ that does not change when the fluid is perturbed. Static balance is $\nabla (p_0+\chi )=0$. Linearized Euler equations and adiabatic compression then giveDifferentiating the second equation and substituting the first yieldsThe divergence form gives weighted acoustic Green-function reciprocity. Neither uniform mass density nor uniform entropy is required. A prescribed gravitational acceleration instead produces a force density proportional to the perturbed mass density; its extra buoyancy term cannot be omitted under this derivation. The homogeneous limit is the acoustic pressure wave equation.

Apply Haar twirling conditional expectation to the complement of a neighbourhood around the initial operator support. A Lieb-Robinson bound on commutators with arbitrary complement-supported unitaries then bounds the approximation error without summing over sites. For a bound proportional to $e^{-\mu d}(e^{2kst}-1)$, choosing $v=4ks/\mu$ gives an error at most $\mu vt|X|\Vert A_X\Vert e^{-\mu l/2}$ outside radius $vt+l$.

The interaction distance between disjoint supports can count the minimum number of interacting hyperedges in a chain connecting them. Overlapping supports have distance zero in the usual support convention. Distances between interaction terms can instead assign shell zero only to the central term and positive shells to distinct terms. The convention must be specified when using a truncated neighborhood Hamiltonian: ordinary zero support distance includes distinct overlapping terms.

Iterating the integral commutator inequality gives ordered interaction chains with factors $(2|t|{)}^n/n!$. A per-site interaction weight at most $s{e}^{-\mu }$ bounds each successive chain extension by $ks{e}^{-\mu }$. No chain shorter than the interaction distance connects separated supports. Reversing chains supplies the smaller support prefactor, producing the Lieb-Robinson bound. Overlapping supports require retaining the equal-time commutator term.

A Duhamel comparison can bound truncation error using commutators evolved under the full Hamiltonian. This avoids assuming that a black-box Lieb-Robinson bound for $H$ also holds for every truncation. Polynomial shell weights and exponential distance suppression make the sum over omitted shells convergent. Comparing two truncations through the full evolution bounds their difference.

A Lagrangian is regular when its velocity Hessian matrix is invertible. The inverse function theorem then makes the fiber derivative $p_i=L_{v^i}$ locally invertible, allowing a local Legendre transform in mechanics to a Hamiltonian. If that fiber derivative is globally invertible, the Lagrangian is called a hyperregular Lagrangian. Regularity alone does not assert global invertibility, and singular Lagrangians may give constrained rather than ordinary unconstrained Hamilton's equations.

For symmetric $L$ and antisymmetric $A$ satisfying $\dot{L}=LA-AL$, solve $\dot{Q}=-AQ$, $Q(0)=I$. Then $Q$ is orthogonal and $Q^{T}LQ$ is constant. Hence every eigenvalue and every coefficient of the characteristic polynomial of $L$ is conserved.

A Puiseux series permits fractional exponents with a fixed common denominator: ${\sum }_{n\ge n_0}{a}_n{z}^{n/q}$. These are natural local expansions of branches of algebraic functions. A fractional-power root expansion uses such powers to resolve roots that coalesce as a parameter tends to zero.

Let $0<c<1$, $A>0$, $\sigma >0$ and $p\ge 2$. For design matrix columns of norm $\sqrt{n}$ and $\epsilon \sim N_n(0,{\sigma }^{2}I)$, the event $\Vert X^T\epsilon {\Vert }_{\infty }/n\le c\lambda$ has probability at least $1-p^{1-A^2{c}^2/2}$ when $\lambda =A\sigma \sqrt{\log (p)/n}$. Apply the sharp two-sided Gaussian tail bound to each standardized score and then the union bound; independence of the scores is not required.