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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.

Across a thin timelike matter layer, the induced metric is continuous while the jump in its extrinsic curvature is determined by the surface stress-energy tensor. Choose one common spacelike normal and $K_{ab}=\frac{1}{2}{\partial }_y{h}_{ab}$ in Gaussian normal coordinates. The singular tangential Einstein tensor is $-([K_{ab}]-h_{ab}[K])\delta (y)$; equating it to ${\kappa }_D^2{S}_{ab}\delta (y)$ gives the displayed conditions. Reversing the normal or the sign convention for extrinsic curvature changes the corresponding jump signs consistently. These conditions determine the cosmological expansion law of a thin gravitating brane.

An exchange economy specifies consumers, their preference relations, their consumption sets and their initial endowments. A consumption allocation is feasible when aggregate consumption equals aggregate endowment; free-disposal models instead allow a coordinatewise inequality. These conventions must be distinguished when specifying supporting prices.

At any price vector, the total value of aggregate excess demand is zero when consumers exhaust their budgets. After normalization to a price simplex, choosing a price that maximizes the value of a candidate excess-demand vector is a useful Kakutani fixed-point theorem construction.

A Daniell cell uses the zinc/${\mathrm{Zn}}^{2+}$ and ${\mathrm{Cu}}^{2+}$/copper pairs. In the separated-compartment teaching version, a zinc electrode contacts an aqueous chemical solution containing ${\mathrm{Zn}}^{2+}$, and a copper electrode contacts one containing ${\mathrm{Cu}}^{2+}$. A salt bridge supplies an ionic connection; a wire and load supply an electronic connection. During discharge,The zinc electrode loses material and copper is deposited at the cathode. Each half-reaction conserves chemical atom counts and electric charge: the first has net charge zero on each side, the second $+2e-2e=0$ on the left and zero on the right. The net reaction has $+2e$ on both sides, with no electrons left in its chemical equation. The Zn/Cu teaching cell and these half-reactions are described in Florida State University, Redox Reactions in Voltaic Cells: Construction of a Potential Series, lab background.

Creating one mole of ${\mathrm{Zn}}^{2+}$ adds $2F$ of positive electric charge to the anode compartment's ionic inventory; consuming one mole of ${\mathrm{Cu}}^{2+}$ removes $2F$ of positive electric charge from the cathode compartment. Here $F$ is the Faraday constant. In an ideal bookkeeping example using a potassium nitrate salt bridge, two moles of nitrate anions entering the anode compartment compensate the first change, and two moles of potassium cations entering the cathode compartment compensate the second. Other ions in the cell can share the actual transport. The two compensating inventory changes do not mean that $4F$ flowed through the load: the same series electric current transports $2F$ through each cross-section of the circuit per mole of net reaction. See salt bridge.

Under standard-state conditions near $298\mathrm{K}$, tabulated reduction potentials give a cell electromotive force of approximately $1.10\mathrm{V}$. This is a reversible reference value, not a promised loaded reading; actual composition matters through thermodynamic activity. The tabulated copper and zinc values and their subtraction are given in Purdue University, Electrochemical Reactions: standard-state reduction potentials. For the net chemical equation as written, the molar Gibbs free energy change obeysThus $\mathcal{E}=1.10\mathrm{V}$ corresponds to about $-212\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ of net reaction. At fixed temperature and pressure, $-{\Delta }_{r}G$ is the maximum reversible electrical work per mole for this cell reaction. It is not automatically all the heat released by a finite-current cell. See 17.4 Potential, Free Energy, and Equilibrium (§17.4: cell potential and free energy).