Foundations

A classical black-hole event horizon is a causal boundary: signals from its interior cannot reach distant future observers in the exterior.

A point mass in the weak-field geometric-optics limit deflects light by magnitude α̂=4GM/(c²b).

Before recombination, scattering tightly coupled photons and baryons.

For a locally planar bed with true dip δ, choose horizontal x toward its down-dip direction and elevation z upward.

Within a conformable depositional succession, vertically successive facies can represent environments that were laterally adjacent as the depositional system migrated.

A depositional sequence is a relatively conformable succession bounded by unconformities and their correlative conformities under the stated sequence model.

Accommodation is space available for sediment accumulation.

On a spherical rigid-plate model, displacement relative to a specified reference plate is represented by a finite rotation about an Euler pole.

Conjugate continental geology and fossils, ocean-floor magnetic reversal sequences, spreading geometry and paleomagnetic directions provide complementary constraints on past plate positions.

For a geocentric axial dipole with normal-polarity directions expressed northward and downward, tanI=2tanλ relates field inclination to latitude.

The Phanerozoic comprises the Paleozoic, Mesozoic and Cenozoic eras.

The end-Ordovician, Late Devonian, end-Permian, end-Triassic and end-Cretaceous intervals mark especially prominent losses in the Phanerozoic fossil record.

A petrographic microscope combines observations in plane-polarized and crossed-polarized light: relief, colour or pleochroism, cleavage, extinction behaviour and interference response help distinguish minerals.

In a suitable transmitted-light grain–medium boundary test, increasing the objective-to-specimen separation moves the Becke bright line into the material of higher refractive index.

For a homogeneous plane-parallel transparent plate at normal incidence with two effective indices for the common wave-normal direction, optical path retardation is Γ=d(n_slow−n_fast).

The optical indicatrix represents directional refractive behaviour in an ideal transparent anisotropic crystal.

For two equilibrium phases α and β with compositions x_α<x_β and bulk composition x₀ between them, conservation gives f_α=(x_β−x₀)/(x_β−x_α) and f_β=1−f_α.

In a simple binary eutectic system at fixed pressure, liquid can coexist with two distinct solid phases at the eutectic temperature and liquid composition.

In a binary peritectic equilibrium at fixed pressure, an earlier solid and liquid react to form a different solid at an invariant temperature.

A solid solution accommodates component substitution in one crystal phase.

Three-component compositions occupy a normalized triangle: x_A+x_B+x_C=1 with nonnegative fractions.

For closed mixing by mass of two endmembers, concentration is C_mix=fC₁+(1−f)C₂.

For parent P decaying into daughter D at constant λ, normalization to a stable isotope S gives D/S=(D/S)₀+(P/S)(e^(λt)−1).

For a closed system with corrected common lead and retained decay-chain products, radiogenic ratios satisfy ²⁰⁶Pb*/²³⁸U=e^(λ₂₃₈t)−1 and ²⁰⁷Pb*/²³⁵U=e^(λ₂₃₅t)−1.

Specific storage S_s is water volume released per aquifer bulk volume per unit decline in hydraulic head.

For a vertically uniform confined aquifer of fixed thickness b, transmissivity is T=Kb and storativity S=S_sb.

Combining classical elastic storage with Darcy flux yields ∇·(K∇h)=S_s∂h/∂t in a saturated aquifer without internal sources.

Unsaturated water content θ and conductivity K depend on pressure head ψ and the medium's pore structure.

In a rigid porous medium with effectively incompressible water and fixed air pressure, local water conservation with unsaturated Darcy flux gives θ_t=∇·[K(ψ)(∇ψ+e_z)].

For a one-dimensional incompressible reach, ∂(A+A_s)/∂t+∂Q/∂x=q_L balances conveying area A, additional nonconveying storage area A_s and discharge Q.