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A rotating gravitational pattern resonates with a stellar orbit when their azimuthal frequencies agree. In action-angle variables, this is the $k_R=0$ resonance of an azimuthal harmonic. Its torque changes azimuthal angular momentum without changing radial action to leading order, allowing stellar radial migration with little heating. The circular-orbit location is the corotation radius.

The $k_R=-1$ orbital resonance of an $m$-fold perturbation of a galactic disk. For near-circular stellar orbits, ${\Omega }_R$ is the radial epicyclic frequency. The harmonic changes radial action and azimuthal angular momentum in the ratio $\Delta J_R/\Delta J_{\phi }=-1/m$, so an orbit losing angular momentum gains random radial motion.

The $k_R=+1$ orbital resonance of an $m$-fold perturbation of a galactic disk. For near-circular stellar orbits, ${\Omega }_R$ is the radial epicyclic frequency. The harmonic changes radial action and azimuthal angular momentum in the ratio $\Delta J_R/\Delta J_{\phi }=1/m$, so an orbit gaining angular momentum gains random radial motion.

For an integrable or approximately integrable background Hamiltonian, action-angle variables turn an otherwise complicated stellar orbit into $\dot{\mathbf{J}}=0$ and $\dot{\mathbf{\theta }}=\mathbf{\Omega }(\mathbf{J})$. This has two useful consequences. First, a phase-mixed equilibrium can be represented directly as $f_0(\mathbf{J})$: the action variables label the unperturbed orbits, while angle variables describe position along them. Second, the angles are periodic, so Fourier series reduce unperturbed transport to multiplication by $i\mathbf{n}\cdot \mathbf{\Omega }$ and make orbital resonances explicit. The action variables also separate rapid orbital motion from slow secular changes. These advantages apply on regular orbital regions; a globally chaotic background need not admit global action-angle variables.

The transformation is canonical, preserving the phase space measure and the stated total mass $M=\int f{d}^3\theta d^{3}J$. The collisionless Vlasov equation isHere $H$ is the single-star specific Hamiltonian; the normalization of $f$ by mass changes neither the Poisson bracket nor the transport equation.

For a regular integrable Hamiltonian system, action-angle variables offer two related simplifications.

These advantages assume regular or approximately integrable mean motion; globally defined action-angle variables need not exist in chaotic regions.

The restriction $k_{\phi }=\pm m$ describes an $m$-fold rotating perturbation of a galactic disk, for example a transient spiral density wave or, for $m=2$, a galactic bar. Its pattern speed is ${\Omega }_{\mathrm{p}}$, its amplitude peaks at $t_{\mathrm{peak}}$, and its Gaussian function envelope has duration of order $\tau$. The coefficients $u_{\mathbf{k}}(\mathbf{J})$ describe its strength along different stellar orbits. A real perturbing potential requires $u_{-\mathbf{k}}=u_{\mathbf{k}}^{\ast }$.

Write $g(t)=\exp [-(t-t_{\mathrm{peak}}{)}^2/(2{\tau }^2)]$. The product entering c isThe phase therefore measures detuning from the rotating pattern:Substitution givesThe strongest long-lived response occurs near an orbital resonance, ${\omega }_{\mathbf{k}}=0$, where successive force contributions do not rapidly cancel in phase.

For $k_R=\pm 1$, the selected orbital resonances satisfyFor near-circular stellar orbits, ${\Omega }_R$ is the radial epicyclic frequency. Thus $k_R=-1$ is the inner Lindblad resonance, $m({\Omega }_{\phi }-{\Omega }_{\mathrm{p}})={\Omega }_R$, and $k_R=+1$ is the outer Lindblad resonance, $m({\Omega }_{\phi }-{\Omega }_{\mathrm{p}})=-{\Omega }_R$.

The flux is driven by $m{\partial }_{J_{\phi }}{f}_0\pm {\partial }_{J_R}{f}_0$, and each resonant harmonic exchanges both angular momentum and radial action in the ratioThese terms can therefore produce galactic disk heating as well as angular momentum transport. In the usual absorbing response, stars losing angular momentum at the inner Lindblad resonance gain radial action, and stars gaining angular momentum at the outer Lindblad resonance also gain radial action. The actual direction of flux follows from the displayed derivatives of $f_0$; it is not fixed by the resonance condition alone. A finite pulse replaces exact Lindblad resonance selection by a band of detunings of width of order $1/\tau$.

The Penrose singularity theorem states that a globally hyperbolic spacetime with a noncompact Cauchy hypersurface, a closed trapped surface, and the null convergence condition is future null-geodesically incomplete. Time reversal replaces the trapped surface by an anti-trapped surface and gives past incompleteness.

In his 1965 paper Gravitational Collapse and Space-Time Singularities, Roger Penrose used trapped surfaces and global causal structure to address whether nonspherical perturbations could avert a breakdown during gravitational collapse. Quasars supplied the opening astrophysical motivation. The paper used metric signature $(+,-,-,-)$ and a stronger timelike local-energy assumption; the statement above uses the modern null convergence condition.^[ref]

For this version, take a smooth connected four-dimensional spacetime with a time orientation. A Cauchy hypersurface meets every inextendible causal curve exactly once; its existence expresses global hyperbolicity. The hypersurface must be noncompact, meaning not compact; this is a global spatial condition, not an assumption about the size of the collapsing body. The null convergence condition is $R_{ab}{k}^a{k}^b\ge 0$ for every null vector $k$. Through the Einstein field equations it follows from the null energy condition; a cosmological constant drops out of the null contraction.^[ref]

The conclusion is that at least one future-inextendible null geodesic has a finite upper endpoint of its affine parameter. It neither asserts infinite mass density or divergent curvature at every endpoint, nor says that such an endpoint is visible. Locating an event horizon requires additional global information; the theorem is not a horizon-finding rule for arbitrary spacetimes. Weak cosmic censorship conjecture is an additional issue.^[ref]

The supermassive black hole account describes the motivating observational evidence from stellar orbits at the Galactic centre. Such measurements infer a compact gravitating mass. They do not observe a trapped surface, measure a geodesic's complete future, or experimentally prove the Penrose singularity theorem.

The diffusive evolution of an averaged phase-space distribution function caused by weak fluctuating fields in the quasilinear approximation. In a stationary homogeneous one-dimensional electrostatic problem, if the potential's temporal power spectral density is $S_{\phi ,k}(\omega )$ with covariance $\int S_{\phi ,k}(\omega )e^{-i\omega \tau }d\omega /(2\pi )$, thenThis follows from the real part of the one-sided time correlation along an unperturbed particle trajectory. Both signs of the spatial wavenumber are included. The action-space counterpart for regular stellar orbits is quasilinear diffusion in action space.

A stellar orbit is a star's trajectory in the gravitational field of its surroundings. A close binary star can often be approximated by a Kepler orbit, while a star moving through a galaxy responds to an extended distribution of stars, gas and dark matter. Its path need not close into an ellipse.

Observed positions, proper motion and line-of-sight velocity constrain different components of the motion. Reconstructing an orbit also requires a distance and a model of the gravitational potential. Comparing many stellar orbits can therefore constrain an unseen mass distribution, but one projected track need not uniquely determine the full three-dimensional trajectory or potential.

The useful idea is to weigh an unseen object through its effect on a visible tracer. A curved stellar orbit carries more information than a single fast-moving star: its changing position and velocity constrain a common centre, a length scale and a clock. The Galactic-centre stellar-orbit experiments combine astrometry with Doppler spectroscopy; the Galactic-centre point-mass orbit model below shows how the two measurements complement one another.