An antikink is a scalar-field kink with the opposite orientation of its vacuum boundary values. If ${\varphi }_K(x)$ joins vacua ${\varphi }_{-}$ to ${\varphi }_{+}$, then ${\varphi }_K(-x)$ joins ${\varphi }_{+}$ to ${\varphi }_{-}$. For an even double-well potential and an odd centered phi-four kink, the centered antikink is $-{\varphi }_K(x)$. With $Q=[\varphi (+\infty )-\varphi (-\infty )]/({\varphi }_{+}-{\varphi }_{-})$, a kink has $Q=1$ and an antikink has $Q=-1$. Their localized energies coincide by spatial reflection.

The mass of a canonically normalized scalar field is determined by its quadratic fluctuation potential at a stationary vacuum. With several real fields, physical squared masses are the eigenvalues of the scalar mass matrix. A negative squared mass signals an unstable direction rather than a stable massive particle.

A real scalar field is minimally coupled to a metric tensor when its kinetic term uses the metric contraction of first derivatives and its scalar potential depends only on the field, without an explicit scalar curvature coupling. For metric signature $(-,+,+,+)$ the displayed Lagrangian and volume density $\sqrt{-g}$ give $\square \varphi -V^{\prime }(\varphi )=0$. Its stress-energy tensor is $T_{ab}={\nabla }_a\varphi {\nabla }_b\varphi -g_{ab}[\frac{1}{2}(\nabla \varphi {)}^2+V]$. The scalar stress-energy divergence identity explains its conservation on the field equation.

With negative extrinsic curvature convention, the background trace is $\bar{K}=-3H$, where $H=\dot{a}/(a\bar{N})$. The scalar expansion perturbation is $\kappa =K+3H$. Use the explicit ADM conventionwith ${\gamma }_{ij}=a^2[(1-2\Phi ){\delta }_{ij}+2{\partial }_i{\partial }_{j}E]$. Substituting into $K_{ij}=-({\dot{\gamma }}_{ij}-D_i{\beta }_j-D_j{\beta }_i)/(2N)$ giveswhere $\chi =a^2(\dot{E}-B)/\bar{N}$ is the scalar shear potential and $\Delta =a^{-2}{\nabla }^2$.

Thus the positive metric cross term and ${\beta }_i=+a^2{\partial }_{i}B$ accompany this choice of $\chi$; changing the shift sign requires changing its definition too. For the future unit-normal congruence, the expansion is $-K=3H-\kappa$, so positive $\kappa$ means reduced expansion. A homogeneous perturbation with $\Psi =0$, $B=E=0$ checks the sign: increasing $\Phi$ decreases the local spatial scale and gives $\kappa =3\dot{\Phi }/\bar{N}$.

Choose the ADM shift vector ${\beta }^i$ throughwith $N=\bar{N}(1+\Psi )$ and ${\gamma }_{ij}=a^2[(1-2\Phi ){\delta }_{ij}+2{\partial }_i{\partial }_{j}E]$. For negative extrinsic curvature, $K_{ij}=-({\dot{\gamma }}_{ij}-D_i{\beta }_j-D_j{\beta }_i)/(2N)$. The scalar shear potential isRaised spatial derivatives use $a^{-2}{\delta }^{ij}$, so $\Delta =a^{-2}{\nabla }^2$. The expression follows by keeping the trace-free part of ${\dot{\gamma }}_{ij}$ and the symmetrized shift gradient in the extrinsic-curvature formula. The sign of $B$ is therefore tied to the metric cross term; it is not determined by the name “shift” alone.

The positive-shift scalar shear convention instead writes ${\chi }_{+}=-\chi$ for this same explicit metric. Its trace-free and trace shear terms both reverse their displayed sign. Stating the line element prevents a mistaken combination of a minus-cross-term shift definition with a plus-cross-term extrinsic-curvature formula.

For $N=\bar{N}(1+\Psi )$, $g_{0i}=a^2{\partial }_{i}B$ and spatial scalar convention $h_{ij}=a^2[(1-2\Phi ){\delta }_{ij}-2{\partial }_i{\partial }_{j}E]$, a passive displacement $({\xi }^0,{\partial }^i\lambda )$ gives $\tilde{\Psi }=\Psi -{\dot{\xi }}^0-(\dot{\bar{N}}/\bar{N}){\xi }^0$, $\tilde{B}=B+{\bar{N}}^2{\xi }^0/a^2-\dot{\lambda }$, $\tilde{\Phi }=\Phi +(\dot{a}/a){\xi }^0$ and $\tilde{E}=E+\lambda$. These follow from subtracting the background metric Lie derivative.

The uniform-density curvature perturbation is a dimensionless spatial-curvature amplitude evaluated on hypersurfaces where the density perturbation vanishes. Fix the scalar spatial metric conventionHere primes denote conformal time, $\mathcal{H}=a^{\prime }/a$ and ${\bar{\rho }}^{\prime }\ne 0$. Under the passive cosmological gauge transformation $\tilde{\eta }=\eta +T$, $\tilde{\Phi }=\Phi +\mathcal{H}T$ and $\delta \tilde{\rho }=\delta \rho -{\bar{\rho }}^{\prime }T$, so the two changes cancel in $\zeta$. Choosing $T=\delta \rho /{\bar{\rho }}^{\prime }$ reaches uniform-density slicing and gives $\zeta =-{\Phi }_{\delta \rho =0}$.

Using background energy conservation, the same convention isIt agrees with the uniform-density curvature sign test. For a spatially flat background, the intrinsic three-dimensional Ricci perturbation on these slices is $\delta {}^{(3)}R=-4a^{-2}{\nabla }^2\zeta$: $\zeta$ is its scalar amplitude, rather than the Ricci scalar itself. The full-Hessian $E$ term is a spatial-coordinate contribution and cancels from that Ricci scalar.

An overall reversal of the definition of $\zeta$ is also used, including in the explicitly signed superhorizon conservation of uniform-density curvature equation. Signs must follow the declared metric and curvature definition together. Uniform-density slicing also differs from comoving gauge in cosmology; their regular adiabatic large-scale relation needs additional dynamical assumptions.^[ref]

A comoving gauge in cosmology chooses hypersurfaces with zero scalar momentum density for the specified fluid or total system. For cold pressureless matter it can be implemented by taking the matter velocity potential to vanish. The choice of component must be stated in a multi-fluid system; density contrasts on this slicing can differ from those in a Newtonian or spatially flat gauge.

Let $\mathcal{D}={\bar{N}}^{-1}{\partial }_t$ and $\bar{\Pi }=\mathcal{D}\bar{\varphi }$. For negative-expansion extrinsic curvature of a spatial hypersurface, its trace perturbation is $\kappa$. With $\delta \Pi =\mathcal{D}\delta \varphi -\bar{\Pi }\Psi$, scalar evolution is $\mathcal{D}\delta \Pi -\Psi \mathcal{D}\bar{\Pi }+3H\delta \Pi -\kappa \bar{\Pi }-\Delta \delta \varphi +V_{,\varphi \varphi }\delta \varphi =0$. Combining background acceleration terms produces the standard lapse source $-\bar{\Pi }(\kappa +\mathcal{D}\Psi -3H\Psi )+2\Psi V_{,\varphi }$.

For the Newtonian gauge in cosmology with potentials $\psi ,\varphi$, photon comoving momentum $ϵ$ obeys $d\ln ϵ/d\eta ={\varphi }^{\prime }-\mathbf{e}\cdot \nabla \psi =-d\psi /d\eta +{\varphi }^{\prime }+{\psi }^{\prime }$. The homogeneous physical-energy redshift is separated by $E=ϵ/a$. The total derivative follows the unperturbed photon path at linear order.

For a one-sided normal-mean test of size $\alpha$ with known standard deviation $\sigma$, a positive alternative ${\delta }^{\ast }$ and target statistical power $1-\beta >\alpha$, the required sample size is the ceiling of ${\sigma }^2(z_{1-\alpha }+z_{1-\beta }{)}^2/({\delta }^{\ast }{)}^2$. This follows by shifting the standard normal quantile in the alternative rejection probability.

In a symmetric initial population of particles and antiparticles, exact CP symmetry equates conjugate decay rates after summing final momenta and helicities. Since parity does not reverse baryon number, this equality prevents a total baryon asymmetry even if charge conjugation alone is violated. Different same-helicity rates can cancel against the opposite-helicity rates.

In a real species basis, the two-particle S-matrix satisfies $S(-{\theta }^{\ast })=S(\theta {)}^{†}$ under the usual scattering analyticity assumptions. On the real rapidity axis this relates inverse-rapidity amplitudes to complex conjugates. Combined with the algebraic inverse relation $S(\theta )S(-\theta )=I$, it gives physical unitarity.

For relativistic two-body scattering in one spatial dimension, this strip is the standard domain containing bound-state and crossed-channel poles. A stable-particle interpretation requires appropriate pole kinematics and residues; not every singularity automatically represents a new particle. Crossing relates rapidity $\vartheta$ to $i\pi -\vartheta$.

A strong stability preserving Runge-Kutta method writes its stages as convex combinations of previously computed states and suitable Forward Euler method steps. It transfers any convex-functional nonincrease property of the forward step, such as a norm bound, under a proportionally scaled time-step restriction.

For example, if $E_k(U)=U+kF(U)$ is nonexpansive in a norm, then $Y=E_k(U)$ and $U_{\mathrm{new}}=(U+E_k(Y))/2$ is also nonexpansive: the triangle inequality bounds the new difference by one half of the initial difference plus one half of the twice-advanced difference. Both are at most the initial difference. A Taylor expansion yields $U_{\mathrm{new}}=U+kF(U)+k^2{F}^{\prime }(U)F(U)/2+O(k^3)$, so this is a second-order Runge-Kutta method. The forward-step hypothesis must hold on the stage states; preservation of one chosen convex bound is not the same as unconditional B-stability.

For $J=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$, the real matrix solutions of $AJ=JA$ are preciselyA nonzero such matrix equals $\sqrt{u^2+v^2}{R}_{\theta }$, where $R_{\theta }$ is a planar rotation matrix with $\cos \theta =u/\sqrt{u^2+v^2}$ and $\sin \theta =v/\sqrt{u^2+v^2}$. It acts like multiplication by the complex number $u+iv$. The zero matrix is the zero scalar multiple of any rotation.

For a layer rotating at angular speed $\Omega$, of depth $d$ and kinematic viscosity $\nu$, the Taylor number compares the Coriolis force to viscous damping through the squared ratio $(2\Omega d^2/\nu {)}^2$. Definitions in other geometries may include different geometric factors; this is the plane-layer convention.

Linearizing shallow-water quasi-geostrophic potential vorticity about a zonal current $U(y)$ gives $Q_y=\beta -U^{\prime \prime }+U/R_D^2$. A normal mode $\hat{\varphi }(y)e^{i(kx-\omega t)}$ obeys $(U-\omega /k)[{\hat{\varphi }}^{\prime \prime }-(k^2+R_D^{-2})\hat{\varphi }]+Q_y\hat{\varphi }=0$. A global transverse plane wave is generally unavailable for an arbitrary nonuniform jet; boundary conditions or a local short-wavelength approximation are required.

The lowest common ancestor of two vertices is the common ancestor farthest from the root. In the ancestor partial order with the root least, it is their infimum, also called their greatest common ancestor.

For finite rooted trees in the ancestor partial order, a rooted-tree homeomorphic embedding is an injective vertex map preserving lowest common ancestors: $h(u\wedge v)=h(u)\wedge h(v)$. The image of the source root need not be the host root. Edge paths and branching are preserved, while paths may be subdivided in the larger tree.