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The direct channel combines the two incoming four-momentum vectors. A one-particle pole in this channel has the intermediate particle's squared mass equal to $s$. Imaginary rapidity differences can satisfy this mass relation for a bound state even though there is no pole at real physical scattering rapidity.

For a Schwarzschild black hole of mass $M$ in geometrized units, $f=1-2M/r$ gives $\kappa =f\prime (2M)/2=1/(4M)$. The Killing vector field is normalized at infinity; changing that normalization changes $\kappa$.

In units $\hslash =1$, periodic-angle quantization gives canonical charge $Q_{\theta }=\ell \in \mathbb{Z}$. The mechanical charge is $Q_0=\ell +\theta T/(2\pi )$ and the rotating-kink mass gives the displayed leading semiclassical spectrum. The $T=1$ and $T=-1$ families are related by reversing the kink. Theta periodicity relabels the integer; fluctuation corrections are not excluded by this leading calculation.

Semiclassical quantization uses classical trajectories and actions to approximate quantum states in a regime where relevant actions are large compared with $\hslash$. Phase consistency on a periodic orbit leads to Bohr-Sommerfeld quantization. Soliton collective coordinates can be quantized this way, while fluctuations supply additional corrections to the leading classical mass.

At renormalized coupling $\gamma =8\pi /n$, the kink-antikink transmission poles lie at $\vartheta =i\pi (1-k/n)$. The relativistic bound-state mass from a rapidity pole gives the displayed increasing breather masses. The kink mass is $M=mn/\pi$, and $k=n$ is an excluded threshold state. No breathers occur at $n=1$; scattering involving a physical second breather requires $n\ge 3$.

The static kink ${\varphi }_K(x)=4\arctan e^{x-a}$ joins adjacent scalar-field vacua $0$ and $2\pi$. It obeys ${\varphi }_K^{\prime }=2\mathrm{sech}(x-a)$ and has topological charge $Q=[\varphi (+\infty )-\varphi (-\infty )]/(2\pi )=1$. In the Sine-Gordon theory normalization with physical mass scale $m$ and coupling $\beta$, its classical mass is $8m/{\beta }^2$. A Lorentz boost gives ${\varphi }_K=4\arctan \exp [(x-vt-a)/\sqrt{1-v^2}]$. Spatial reflection gives an antikink.

For a mass $m$ passing at relative speed $u$ and impact parameter $b$, integration of $Gmb/(b^2+u^2{t}^2{)}^{3/2}$ gives the transverse kick $2Gm/(bu)$. Weak deflection requires $b\gg Gm/u^2$, up to the equal-mass order-one factor. This scale supplies a small-impact cutoff for the weak-scattering diffusion approximation, not a physical exclusion of stronger encounters.

The product $\mu =GM$ of the gravitational constant and a central mass. In the exact relative two-body problem, the central parameter is $G(M_1+M_2)$. The appropriate conservative parameter can change when inverse-square radiation pressure is included in a dust Kepler orbit.

For $T=-[\cos \varphi {]}_{-\infty }^{+\infty }/2$, complete the static energy into squares $({\varphi }_x\mp m\sin \varphi {)}^2$ and ${\sin }^2\varphi {\alpha }_x^2$. The boundary term is $\pm rmT$. A $T=1$ saturating kink is $\varphi =2\arctan e^{m(x-x_0)}$, with constant $\alpha$ and mass $rm$. This is a model-specific Bogomolny bound.

The stellar age is the elapsed time since a star formed. A constant number formation rate on a finite interval gives a uniform distribution of present ages if every formed object remains counted, including stellar remnants. An age distribution inferred only from surviving main sequence stars generally differs because their lifetimes depend on mass.

A stellar population is an ensemble of stars described by its formation times, initial masses, composition and present stellar evolution states. Number fractions must be distinguished from mass-weighted fractions. Shared formation times can correlate the states of components of a binary star even when their initial masses have independence.

A negative squared mass in a relativistic field spectrum signals instability of the background about which the field is expanded. The bosonic string mass spectrum contains such an unexcited state.

A sheet of mass per area $m$ and elastic-sheet tension $T$ backed by an identical outgoing fluid half-space presents the stated surface acoustic impedance to the other half-space, using upward sheet velocity $i\omega \eta =-V$. The fluid term is its normal acoustic impedance; the remaining term is structural. Infinite elastic-sheet tension suppresses every fixed nonzero spatial wavenumber, but not $k=0$, whereas infinite mass suppresses even spatially uniform motion. A massless untensioned sheet between identical fluids is transparent.

For the unit-modulus hyperbolic scattering block with elementary mass $m$, a pair bound at rapidity difference $iu$ has mass $m_2=2m\cos (u/2)$. Bootstrap fusion with another elementary particle gives a further direct pole at $3iu/2$. It is inside the physical rapidity strip with positive imaginary residue only for $0<u<2\pi /3$. In that range the new bound state has rest-frame constituent rapidities $iu,0,-iu$, so its energy is $m(1+2\cos u)=m\sin (3u/2)/\sin (u/2)$. Charges add at the bound-state vertex. At $u=2\pi /3$ the apparent pole cancels; beyond that point it lies outside the physical strip. The formula is conditional on this particular scattering block and its direct-pole interpretation, not a universal three-body binding law.

Disruption of a self-gravitating body when differential gravitational acceleration across it exceeds its binding acceleration. For a body of radius $R$, mass $m$ at distance $r$ from a much more massive body $M$, the competing scales are $GMR/r^3$ and $Gm/R^2$; their equality gives $r\sim R(M/m{)}^{1/3}$, up to geometry and internal-structure factors. The tidal tensor gives the directional differential acceleration. A tidal disruption event near a black hole is a specific astronomical application.

Differential gravitational acceleration, multiplied by the affected mass, across a spatially extended body. In the linear approximation the acceleration difference is $T\Delta \mathbf{r}$, where $T$ is the tidal tensor. Stellar tidal forces limit the domain of a circumplanetary orbit; sufficiently strong ones can produce tidal disruption.

A low-mean molecular weight hydrogen-helium envelope can add a large radius while adding a small fraction of the mass. The exoplanet transit photometry radius is the altitude where tangential optical depth becomes order unity, so temperature, opacity, and exoplanet clouds can move the measured radius above the deep material radius. Atmospheric loss and cooling change this contribution over time.