For affected offspring and heterozygous parents at a diallelic locus, let $b$ count transmissions of allele 1 with allele 2 untransmitted and $c$ count the reverse. Under the fair-transmission null, conditional on $b+c$, $b$ is binomial with probability $1/2$. The asymptotic statistic is $(b-c{)}^2/(b+c)$, with a one-degree-of-freedom chi-squared null distribution. Homozygous-parent transmissions give no contrast. Conditioning within families protects this test from ordinary population stratification.

Association mapping localizes genetic influences on a phenotype by testing biological population genetic association with markers. Linkage disequilibrium allows a typed marker to tag a nearby causal variant; historical genetic recombination can narrow the associated interval. Population stratification can produce association without a local causal relation.

// Target: biology.bigb

The haplotype-relative-risk design compares transmitted alleles to the corresponding untransmitted parental copies, used as internal controls for an affected offspring. If allele-1/allele-2 transmitted counts are $w,x$ and untransmitted counts are $y,z$, the allele-1 odds ratio estimate is $wz/(xy)$. An unmatched Pearson chi-squared test of independence treats this as a two-by-two table. The transmission disequilibrium test instead keeps parent-specific pairing and uses only discordant transmissions. The sampling assumptions of the unmatched calculation and any missing-parent reconstruction must be justified separately.

The genotype relative risk compares the penetrance for genotype $g$ with that of a chosen baseline $g_0$. The common baseline factor cancels when computing conditional probabilities of transmitted genotypes given an affected offspring. It is a ratio of disease probabilities, unlike an odds ratio.

An ergosphere is the region outside a rotating black hole horizon in which the stationary time-translation Killing vector is spacelike. Observers cannot remain stationary relative to infinity there. The Kerr ergoregion is its Kerr-metric instance.

The Kerr metric is the stationary axisymmetric vacuum metric of a rotating mass. Parameters $M$ and $a$ determine whether its geometry has the horizons of a Kerr black hole; the metric also admits over-extremal parameters with no horizon.

A (pseudo-)Riemannian manifold is conformally flat if every point has a neighborhood on which the metric is $g={\Omega }^2\eta$ for a flat metric $\eta$ and a positive function $\Omega$. This is a local property and does not imply global flatness.

The advanced time $v=t+r_{\ast }$ or retarded time $u=t-r_{\ast }$ combines Schwarzschild time with the Schwarzschild tortoise coordinate to remove one horizon-coordinate singularity. The advanced choice gives ingoing Eddington-Finkelstein coordinates; the retarded choice is outgoing.

A perfect fluid has no viscosity or heat flux in its local rest frame; its stress is specified by energy density and isotropic pressure. Its relativistic realization is the perfect fluid in general relativity.

Nordstrom's scalar theory restricts the spacetime metric to the conformally flat form $g_{\mu \nu }={\phi }^2{\eta }_{\mu \nu }$. Its null geodesics have the same unparametrized paths as those of the Minkowski metric, so the theory predicts no gravitational bending of light.

In a static spacetime with $g_{tt}=-N^2$, a photon has conserved Killing energy $E=-p_t$. A static observer with four-velocity $N^{-1}{\partial }_t$ measures frequency proportional to $E/N$. Hence light emitted at one lapse and received at another obeys the displayed ratio. For $N=1+\Phi /c^2+O({\Phi }^2/c^4)$ it gives the weak-field shift $({\nu }_{\mathrm{r}}-{\nu }_{\mathrm{e}})/{\nu }_{\mathrm{e}}=({\Phi }_{\mathrm{e}}-{\Phi }_{\mathrm{r}})/c^2$. This local clock effect does not require light-ray bending.

An upward photon has lower received frequency when compared with the receiver's local clock: ${\Phi }_{\mathrm{r}}>{\Phi }_{\mathrm{e}}$ makes the shift negative. The corresponding upper clock runs faster in a comparison using the same static time coordinate. These are reciprocal descriptions of gravitational redshift and gravitational time dilation, not opposite predictions. A locally measured speed of light remains $c$.

An analytic extension embeds a spacetime isometrically as a proper open region in a larger analytic Lorentzian manifold with a nondegenerate metric tensor. A divergence of coordinate components does not obstruct extension if a new chart makes all metric coefficients analytic and the determinant nonzero. For a static spherical metric with an analytic lapse $f$ and a positive simple horizon radius, ingoing coordinates replace $-fd{t}^2+d{r}^2/f$ by $-fd{v}^2+2dvdr$. This removes the coordinate singularity at $f=0$; transverse metric coefficients must also remain regular. An extension across such a horizon is distinct from extending through a curvature singularity.

Twistor theory encodes the conformal geometry of four-dimensional spacetime using incidence geometry in a complex vector space. In the flat correspondence, a spacetime point is a two-plane in a four-dimensional twistor space, and hence a projective line in projective twistor space. Null separation becomes intersection of these lines. Choosing a spacetime signature is an additional reality condition on the complex geometry.

The Einstein-Maxwell system couples the metric to an electromagnetic field. In geometrized units its source-free equations are $G_{ab}=8\pi T_{ab}^{\mathrm{EM}}$, ${\nabla }_a{F}^{ab}=0$ and $dF=0$, with $T_{ab}^{\mathrm{EM}}=(F_{ac}{F}_b{}^c-\frac{1}{4}{g}_{ab}{F}_{cd}{F}^{cd})/(4\pi )$. Electrovacuum need not have vanishing Ricci tensor, so a Komar integral need not be unchanged between different enclosing surfaces.

A limit retaining the geometry near a horizon while discarding terms describing its distant asymptotic region. For a regular extremal brane throat, removing the constant from its harmonic function can yield an Anti-de Sitter spacetime factor. Local symmetries and supersymmetries of the limiting geometry need not extend to the full original spacetime.

The single-centre Euclidean Taub–NUT geometry is a gravitational instanton with a circle fibre over three-dimensional space. Its fibre shrinks smoothly at the centre when its period is chosen appropriately. Adding flat worldvolume directions gives the local geometry of a Kaluza-Klein monopole.

// Target: algebra.bigb

This covariant scalar-force model uses the Minkowski metric and the displayed momentum law, with $\square \Phi =-4\pi G{T}^i{}_i$. For a massive test particle, write $p^i=m{u}^i$, $u^i{u}_i=1$. Contracting the force law with $p_i$ gives constant rest mass; the acceleration is $d{u}^i/d\tau ={\partial }^i\Phi -u^i(u^j{\partial }_j\Phi )$, independent of the mass. A static weak field obeys the Poisson equation and yields Newtonian acceleration $-\nabla \Phi$. For a photon, $p\cdot dx=0$ and $dx$ is parallel to $p$, so $d{p}^i=-p^{i}d\Phi$: photon energies change but their unparametrized paths remain straight. The motion law has the same unparametrized curves as the conformal metric $e^{2\Phi }\eta$, but the field equation must not be silently identified with a different scalar-gravity theory.

The causal structure of a spacetime classifies separations and tangent directions as timelike, null or spacelike through its light cones. It determines which events can communicate by causal curves. A null curve is one member of this broader geometric structure.

A congruence in general relativity is a family of curves through an open spacetime region such that each point lies on exactly one member. Its tangent field describes the family locally. A geodesic congruence is the special case in which every member is a geodesic.

A trapping horizon is a hypersurface foliated by marginal two-surfaces with one null expansion zero and the other nonzero. A future outer trapping horizon has ${\theta }_{+}=0$, ${\theta }_{-}<0$ and ${\mathcal{L}}_n{\theta }_{+}<0$ for a future ingoing null normal $n$. Unlike an event horizon, this is a quasi-local definition. Under the null energy condition, a regular future outer horizon is spacelike or null. The ingoing derivative condition distinguishes its outer type from an inner trapping horizon.