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Algebraic geometry studies geometric objects defined by polynomial equations and the algebra of functions on them. For example, $y=x^2$ defines a parabola, while $xy=0$ defines the union of two coordinate lines. The coordinate ring $k[x,y]/(xy)$ records the second example algebraically: the images of $x$ and $y$ are nonzero but their product is zero, reflecting two components. Modern schemes extend this dictionary beyond ordinary point sets and permit geometric reasoning over general commutative rings. Applications range from elliptic curves in number theory to families of solutions depending on parameters.^[ref]

Algebraic topology translates questions about topological spaces and continuous maps into questions about groups and other structures from algebra. A topological invariant compresses the enormous amount of data in a space into something one can calculate. For example, the fundamental group records which based loops can be continuously deformed into one another. A filled disk has trivial fundamental group, whereas a circle has fundamental group $\mathbb{Z}$: an integer remembers how many times a loop goes around.

This compression also respects maps. By functoriality of the fundamental group, a continuous map preserving the chosen basepoints induces a group homomorphism; successive maps induce successive group homomorphisms. Thus geometry supplies arithmetic constraints. A proposed homotopy equivalence must induce an isomorphism, so incompatible fundamental groups rule it out before anyone attempts to construct the deformation.

The beauty is that a whole annulus can retain exactly the same homotopy type as its inner circle. Its thickness disappears under a deformation retraction, while the obstruction to shrinking a loop survives. Equal fundamental groups alone do not establish a homeomorphism or a homotopy equivalence between arbitrary topological spaces.

This turns a visual question into a reusable test. Suppose a continuous map sends the inner circle of an annulus once around another circle. Could it extend over a filled disk spanning that boundary? Such an extension would supply a homotopy shrinking the image loop to a point. Its winding number would have to change from one to zero, which homotopy invariance of winding number forbids. Algebra rules out the extension even when no particular formula for it has been proposed.

The Bondi-Sachs formalism uses outgoing null hypersurfaces to describe gravitational radiation. A null coordinate $u$ labels wavefronts, areal radius $r$ labels cross-sections, and angular coordinates label null generators.

The Einstein field equations split into radial relations and evolution equations for the angular geometry. At future null infinity, changing Bondi shear gives the Bondi news function, linking radiation to loss of Bondi mass in nonlinear general relativity.^[ref]

Numerical relativity computes solutions of the Einstein field equations when an analytic solution or a small-perturbation approximation is insufficient. A 3+1 decomposition of spacetime separates initial spatial geometry and matter from their subsequent evolution. Initial data must satisfy the Hamiltonian constraint and momentum constraint, while the lapse function and shift vector specify the coordinate evolution.

The calculation also needs boundary conditions and a numerical evolution formulation. Resolution studies, constraint monitoring and comparison with known solutions test whether the computed result approximates the intended spacetime. Binary black-hole mergers provide an observational application: numerical waveforms connect strongly dynamical geometry with interferometer signals.^[ref] Extracting a physical wave requires controlling the frame and propagation model as well as solving grid equations.

The Arnowitt-Deser-Misner energy is the Hamiltonian boundary charge generating an asymptotic time translation in the ADM formalism. For a spatial metric tensor $h_{ij}$ in asymptotically Cartesian coordinates, use geometrized units $G=c=1$:Here $n^i$ is the outward Euclidean unit normal vector. Standard differentiable asymptotic flatness gives $h_{ij}={\delta }_{ij}+O(r^{-1})$ and ${\partial }_k{h}_{ij}=O(r^{-2})$; suitable additional decay and integrability conditions ensure that the limit exists. Together with ADM momentum, this charge forms the asymptotic energy-momentum four-vector.^[ref]

For a time-symmetric Schwarzschild spacetime slice in isotropic coordinates, $h_{ij}={\psi }^4{\delta }_{ij}$ with $\psi =1+M/(2r)$. Its surface integrand is $-2{\partial }_r{\psi }^4=4M/r^2+O(r^{-3})$. Integrating over $S_r$ therefore gives $E_{ADM}=M$. The corresponding slice of Minkowski spacetime has constant $h_{ij}={\delta }_{ij}$ and zero ADM energy. In a moving asymptotic frame, the energy component need not equal the invariant ADM mass.

A stationary spacetime admits a Killing vector field that is timelike in the region representing stationary observers, in particular near infinity for an asymptotically flat stationary black hole.

A black hole is a region from whose events no future-directed causal curve can reach the distant exterior. In an asymptotically flat spacetime with future null infinity ${\mathcal{I}}^{+}$, the standard definition isHere $J^{-}$ denotes the causal past and ${\mathcal{H}}^{+}$ is the future event horizon. This is a statement about the complete causal geometry, rather than a material surface or a local force threshold.

For a Schwarzschild black hole, ingoing Eddington-Finkelstein coordinates give outgoing radial null geodesics the slope $dr/dv=(1-2M/r)/2$ in units $G=c=1$. Inside $r=2M$ even this outgoing family moves toward smaller areal radius; the other future radial null family does too. A regular horizon can therefore prevent escape without being a curvature singularity.^[ref]

Rotating examples are described by a Kerr black hole. A dynamically forming event horizon need not coincide with a slice's apparent horizon; locating the former requires global escape information.

If $D={\bigcap }_{i=1}^q\ker {\alpha }^i$ has constant rank, the Frobenius theorem says that $D$ is integrable exactly when its annihilator is closed under the exterior derivative: equivalently,This is also equivalent to involutivity of $D$.

For a nowhere-zero 1-form $\alpha$, the hyperplane distribution $\ker \alpha$ is tangent to hypersurfaces exactly when $\alpha \wedge d\alpha =0$. In general relativity this characterizes hypersurface-orthogonal vector fields and implies vanishing twist.

A globally hyperbolic spacetime has no causal pathologies and admits a Cauchy hypersurface intersected exactly once by every inextendible causal curve.

A spacetime is null-geodesically complete when every inextendible affinely parametrized null geodesic has an affine parameter extending without a finite endpoint in both directions.

A complete manifold requires completeness for every geodesic, whereas this condition only tests null ones.

A null hypersurface has a degenerate induced metric. Its normal covector raises to a null vector tangent to the hypersurface, whose integral curves are its null generators.

A geodesic congruence is a family of nonintersecting geodesics filling an open spacetime region. Its derivative tensor describes the relative motion of neighboring geodesics.

Requiring each member to be a geodesic specializes a congruence of curves in general relativity.

The Ricci scalar is the full metric contraction of the Ricci tensor:

This metric is the hyperbolic plane with curvature scale $a$. Its nonzero connection coefficients are

A metric tensor is a nondegenerate symmetric covariant tensor that defines squared line elements by $d{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$.

After using Killing constants and four-velocity normalization, radial timelike geodesic motion can often be written as kinetic energy plus an effective potential equal to a constant.

Rotational symmetry conserves the angular-momentum direction, so every nonradial geodesic lies in a plane through the symmetry centre; coordinates may place that plane at $\theta =\pi /2$.

Anti-de Sitter spacetime is a maximally symmetric spacetime with a Lorentzian metric and constant negative sectional curvature. In four dimensions, curvature radius $L$ corresponds to cosmological constant $\Lambda =-3/L^2$. Its global static geometry has a timelike conformal boundary.

A radial light ray approaches that boundary in finite static-coordinate time but infinite physical affine parameter. Consequently, specifying evolution of a field generally requires suitable boundary conditions as well as initial data. One commonly uses the universal cover to remove the closed timelike time circles of the embedded hyperboloid. This geometry is central to AdS/CFT correspondence, where bulk fields encode observables of a boundary conformal field theory.^[ref]

De Sitter spacetime is the maximally symmetric Lorentzian spacetime of constant positive curvature. Its expanding flat slicing has scale factor $a(\tau )=-1/(H\tau )$ in conformal time $\tau <0$.

The Schwarzschild spacetime describes the spherically symmetric vacuum spacetime with zero cosmological constant. Its exterior chart, in units $G=c=1$ and metric signature $(-+++)$, isThe areal radius $r$ makes each symmetry sphere have area $4\pi r^2$; the mass parameter $M$ has length units in this convention. Birkhoff's theorem makes this the local vacuum geometry outside a spherical body, including when its interior changes spherically. A material star need not extend down to the Schwarzschild radius $2M$.

For the black-hole extension, $r=2M$ is an event horizon. The singular metric coefficients of this chart can be removed with Eddington-Finkelstein coordinates or Painlevé–Gullstrand coordinates. The invariant $R_{abcd}{R}^{abcd}=48M^2/r^6$ is finite there and diverges at $r=0$ in the extension. This Ricci flat metric has nonzero tidal forces. The $M=0$ limit recovers Minkowski spacetime in spherical coordinates.^[ref]