Topics, S

S Theorem > see lie algebra.

S-Matrix > s.a. [quantum field theory techniques]; Coleman-Mandula Theorem; LSZ Formalism.
* History: Introduced by J Wheeler in the context of nuclear physics.
$ Def: In quantum field theory, the operator S:= lim U(t, t₀) for t → , t₀ → –, where U is the time evolution operator.
* Assumptions: Causality, unitarity, analiticity.
* Properties: Its unitarity, to first order, is (formally) equivalent to conservation of probability; To second order it is guaranteed by completeness of the Hilbert space and self-adjointness of the (interaction) Hamiltonian.
* Transition matrix: The matrix T related to the S-matrix by S_fi = _fi – 2i (_f – _i) T_fi.
@ References: Stern PT(64) [criticism]; White hp/00-in [review]; Kummer EPJC(01)ht [gauge invariance]; Colosi & Oeckl a0710 [new approach].

Sachs-Wolfe Effect > see CMB anisotropies.

Saddle-Point Approximation / Method > another name for the Stationary Phase Approximation.

Sagnac Effect > s.a. atomic physics; kinematics of special relativity; tests of newtonian gravity.
* Idea: The fact that, if we send two light rays in opposite directions around a rotating ring (say, on the surface of the Earth), they return with a time difference proportional to and the enclosed area, and the interference depends on .
* Applications: Laser gyroscope, used for inertial guidance, based on beats between the two rays.
@ Early work: Michelson PM(04); Sagnac CR(13); Michelson et al ApJ(25) [experiment].
@ General references: Logunov & Chugreev SPU(88); Anderson et al AJP(94); Rizzi & Tartaglia gq/98; Klauber FPL(03)gq/02 [general case].
@ With matter waves: Gustavson et al PRL(97) [atom interferometer and Earth's rotation]; Lenef et al PRL(97) + pn(97)feb; Rizzi & Ruggiero GRG(03)gq, gq/03-in [and Aharonov-Bohm effect], GRG(03)gq.
@ In general relativity, curved spacetime: Ashtekar & Magnon JMP(75); Tartaglia PRD(98)gq; Gogberashvili FPL(02)gq/01; Sivasubramanian et al gq/03 [and gravitational waves]; Camacho GRG(04)gq/03 [non-Newtonian]; Ruggiero GRG(05) [and Aharonov-Bohm effect].
@ Related topics: Wucknitz gq/04/FP [and closed Minkowski spacetime]; > s.a. Galilean Group [boosts and Sagnac phase].
> Online resources: MathPages page; Wikipedia page.

Saha Equation
* Idea: Relates energies of states to temperature and number densities; It allows us to infer densities of various ions from spectral line intensities.

Salpeter Equation > see modified quantum mechanics.

Sampling > s.a. information.
@ Shannon sampling: Kempf PRL(00)ht/99 [generalization, unsharp coordinates]; Smale & Zhou BAMS(04).

Sand Pile > see critical phenomena.

Sandwich Conjecture
* Idea: The conjecture that, given two spatial metrics q and q' on two hypersurfaces in spacetime, there is unique see that will interpolate between them, up to gauge.
* Thin sandwich: The hypersurfaces are infinitesimally close; One specifies the spatial field configurations and their t-derivatives.
* Thick sandwich: The hypersurfaces are a finite distance apart.
@ General references: Bergmann in(70); Christodoulou & Francaviglia in(79), RPMP(77); Teitelboim in(82).
@ Thin sandwich: Bartnik & Fodor PRD(93)gq; Giulini JMP(99)gq/98 [Einstein + gauge theory + scalar]; York PRL(99)gq/98 [and initial-value problem]; A Komar; Bartnik & Isenberg gq/04-in; Pfeiffer & York PRL(05)gq [conformal, uniqueness].

Satellites > see solar planets.

Scalar Fields > s.a. klein-gordon fields.

Scalar Product > see vectors.

Scalar Theory of Gravitation > s.a. matter phenomenology; scalar-tensor.
* History: Started with Nordström's attempt at developing a special relativistic theory of gravity.
@ Nordström's theory: Nordström AdP(13); Einstein & Fokker AdP(14); in Pauli 58; Wellner & Sandri AJP(64); Harvey AJP(65); Norton AHES(92) [history]; Bauer mp/04 [self-gravitating particles].
@ Equations of motion: Arminjon RJP(00)ap [with preferred frame]; Kaniel & Itin gq/99; Beig et al PRL(07)gq/06 [helically symmetric N-particle solutions].
@ PN approximation: Arminjon gq/01-in, gq/03-in.
@ As model: Watt & Misner gq/99 [for numerical gravity]; Sundrum ht/03; > s.a. modified general relativity [analog].
@ Related topics: Bezerra et al MPLA(02) [2+1, including black hole]; Giulini SHPMP(08)gq/06 [history and assessment].

Scalar-Tensor Theories of Gravity

Scalar-Vector Theories of Gravity > see theories of gravity.

Scalar-Vector-Tensor Theories of Gravity > see MOND; theories of gravity.

Scale Relativity
* Idea; A theory based on the idea that physics must apply to coordinate systems in all "states of scale"; Spacetime is described as a non-differentiable continuum, a fractal which depends explicitly on internal scale variables.
@ General references: Nottale IJMPA(92); Nottale 93; Nottale CSF(94) [fractal spacetime]; Célérier & Nottale JPA(04)qp/06 [quantum mechanics and fields].
@ Applications to various theories: Castro ht/96 [strings]; Nottale et al JMP(06)ht [gauge theory]; Célérier & Nottale JPA(06)qp [Pauli equation]; Hammad JPA(08) [derivation of Pauli and Dirac equations].

Scale Symmetry > see conformal symmetry.

Scaling > s.a. Critical Phenomena [scale-free networks]; entropy; fractal; phase transition; renormalization group.
* Idea: The p-point correlation functions can be written in terms of the 2-point correlation function or variance.
* Scale-free distribution: One given by a power law, as opposed to an exponential with a scale in the exponent; Power laws seem to be prevalent in nature, and may signal an underlying universality.
* In galaxy distribution: Expected if an initially Gaussian distribution of density fluctuations evolves under the action of gravitational instability.
@ General references: Wiesenfeld AJP(01)RL; Henkel NPB(02) [in statistical mechanics]; West CSF(04) [renormalization group, complexity].
@ In biological systems: Brown & West 00 [in biology]; West & Brown PT(04)sep.
@ In other areas: Peterson AJP(02)phy/01 [Galileo and the geography of Dante's Inferno]; > s.a. galaxy distribution, turbulence.

Scarring > see quantum chaos.

Scattering

Scharnhorst Effect > see casimir.

Schemes > s.a. Algebraic Geometry.
* Applications: Used in algebraic topology, number theory, ...
@ References: Eisenbud & Harris 92, 00.

Schläfli Formula
* Idea: A formula relating the variations of the dihedral angles of a smooth family of polyhedra in a space form to the variation of the enclosed volume.
@ References: Souam DG&A(04) [for immersed piecewise smooth hypersurfaces in Einstein manifolds].

Schouten-Nijenhuis Bracket > see killing tensors [Killing-Yano].

Schreier's Conjecture
$ Def: The outer automorphism group of any finite simple group is solvable; Has been proved.

Schrödinger Equation

Schrödinger Representation of Quantum Theory > see representations of quantum theory.

Schrödinger's Cat > see experiments in quantum mechanics; quantum states.

Schubert Cell > see grassmann.

Schubert Symbol
$ Def: Any non-decreasing finite sequence of integers {p_i}, i = 1,..., n, i.e.p_i in N, with 1 p₁ ... p_n m.

Schur's Lemma
$ Def: In a finite-dimensional irr of a group G, the only elements which commute with all others are multiples of the identity.

Schwarz Inequality > see inequalities.

Schwarz Space > see distribution.

Schwarz Transformation > see analytic functions.

Schwarzschild Spacetime > s.a. coordinate expressions.

Schwarzschild-de Sitter Spacetime

Schwinger Effect > see particle effects.

Schwinger Function > see green functions in quantum field theory.

Schwinger-Dyson Equation > s.a. [Wikipedia page]; quantum gravity and renormalization.
@ References: Lyakhovich & Sharapov JHEP(06) [for non-Lagrangian field theory].

Schwinger Model > see dirac fields; modified QED.

Scri ("Penrose script I") > see asymptotic flatness and null infinity.

Second Countable Topological Space > see types of topologies.

Second Fundamental Form > see extrinsic curvature.

Second Law of Thermodynamics > see thermodynamics.

Second Order Equations > see elementary algebra.

Second Quantization > s.a. quantum field theory.
* Idea: It is a field quantization, not really a second quantization.
* Motivation: Seems necessary in order to obtain a consistent Lorentz-covariant quantum theory of particles.
* Commutation relations: The commutation relations between creation and annihilation operators corresponding to a given set of modes in a classical field theory are related to properties of the classical modes by [a(), a(')] = | '.

Sectional Curvature > see riemann tensor.

Seebeck Effect > see electricity [thermoelectricity].

Seesaw Mechanism > s.a. neutrinos; cosmological constant.
* Idea: A mechanism by which a phenomenon with very high characteristic energy scales can be seen at much lower energies.

Segal-Bargmann Transform > see coherent states; Holomorphic Functions.

Segre Classification of Traceless Ricci Tensors
@ References: Zachary & Carminati GRG(04) [algorithm].

Seiberg-Witten Map, Theory > s.a. non-commutative gravity.
@ References: Marcolli dg/95-ln; Flume et al NPB(97) [L_eff uniqueness], ht/96 [rev]; Morgan 96; Adam et al JMP(00) [solutions]; Ghosh JPA(03) [map, interpretation].

Seifert Forms

Seifert Manifolds
* Idea: Quotient manifolds, for example of the form S³/G, where G is a finite subgroup of SU(2); > s.a. Wikipedia.
@ References: Hikami CMP(06) [quantum invariants].

Seifert-Van Kampen Theorem > see fundamental group.

Selberg's Trace Formula > see Trace Formulas

Self-Adjoint Operator > see operators.

Self-Dual Fields > s.a. self-dual solutions in general relativity.

Self-Energy > see classical field theory; energy.

Self-Force > s.a. [semiclassical general relativity (back-reaction)], energy-momentum tensor [pN].

Self-Organization > s.a. critical phenomena.
@ References: Nicolis & Prigogine 77 [non-equilibrium systems]; Olemskoi et al PhyA(04) [with order-parameter field].

Self-Similarity
* For solutions of Einstein's equation: In the spherically symmetric case, a spacetime in which all dimensionless variables depend only on z:= r/t.
@ General references: Embrechts & Maejima 02 [self-similar processes].
@ For spacetime metrics, kinematical: Coley CQG(97)gq/96; Carr & Coley CQG(99) [rev]; > s.a. spherical symmetry.
@ For spacetime metrics, in general relativity: Carot & Sintes in(97)gq/00 [fluid]; Harada CQG(01) [pfluid, stability criterion]; Martín-García & Gundlach PRD(03)gq [scalar]; Harada & Maeda CQG(04) [scalar, stiff fluid, stability]; Maeda & Harada gq/04-in; Carr & Coley GRG(05)gq [similarity hypothesis]; > s.a. bianchi IX; bianchi models; critical collapse; spherical symmetry.

Semialgebraic Geometry > see rings [partially ordered].

Semiclassical Field Theory > see QED; semiclassical general relativity; states in quantum field theory.

Semiclassical Quantum Mechanics

Semiconductors > see electricity.

Semicontinuity, Upper / Lower
$ Def: A function is said to upper/lower semicontinuous at a point x if...

Semidirect Product of Groups
$ Def: Given a group G and an Abelian group V, with a G-action on V, their semidirect product G _s V is the set G × V with the composition law (g₁, v₁) (g₂, v₂):= (g₁g₂, v₁+g₁v₂).
* Remark: We can thus get a new group from every representation of G, with V _s G/V = G.
@ References: Geroch & Newman JMP(71).
> Examples: see the poincaré group and the BMS Group.

Semigroup > s.a. poincaré group.
$ Def: A set with an associative composition law (an associative groupoid).
* Special types: Additive or Abelian if commutative; Cancellative if a + c = b + c implies a = b; > s.a. Monoid; Semiring.
* Topological semigroup: Theory created by A D Wallace.
* Applications: Irreversible dynamics, both in classical mechanics (> see Transport) and in quantum mechanics (& Prigogine, Bohm, > see dissipation, modified quantum mechanics); Non-deterministic dynamics (Blanchard & Jadczyk); > s.a. arrow of time.
@ General references: Wallace BAMS(55); Carruth, Hildebrant & Koch 83; Steinberg JCTA(06) [representations, and Möbius functions].
@ Quantum dynamical semigroups: Davies JFA(79) [generators]; Alicki qp/02-in; Antoniou et al OSID(02) [implementability]; Courbage IJTP(07) [unstable states]; Harshman IJTP(07) [from underlying Poincaré symmetry]; Bohm et al IJTP(07) [from causal symmetries]; > s.a. neutrons [interferometry].

Seminorm > see norm.

Semiorder > see posets [generalizations].

Semiring > s.a. Burnside Ring.
$ Def: A semigroup with distributive multiplication.
* Of subsets of a set: A collection R of subsets of a set X such that Ø, X in R, and R is closed under intersection.

Semisimple Group > s.a. lie groups and representations.
$ Def: One with no (proper, non-trivial) invariant Abelian subgroup.
* And other structure: A n.s.c. for them to have a non-degenerate metric is that k_ab:= C_amⁿ C_bn^m be non-singular.
* Semisimple Lie groups: They are locally isomorphic to products of simple groups; These groups have a very rich structure and have been completely classified early.

Separable Hilbert Space > see hilbert space.

Separable Topological Space > see types of topologies.

Separation of Variables > see hamilton-jacobi; schrödinger equation.

Separatrix
* Separatrix mapping: The mapping that gives the energy and phase of a perturbed non-linear pendulum near the separatrix after a velocity pulse (swing), in terms of their values before; It shows that the reason for the emergence of local instability is the sensitivity of the variation in phase on the orbit.
@ References: in Zaslavskii et al 91, p39; Wiesenfeld JPA(04) [Hamiltonians with symmetries, existence].

Sequence

Sequence Transformation
@ References: Wimp 81.

Sequential Dynamical Systems
* Idea: a class of discrete dynamical systems which significantly generalize many aspects of systems such as cellular automata, and provide a framework for studying dynamical processes over graphs.
@ References: Mortveit & Reidys 07.

Series > s.a. summations.

Serret-Frenet Equations > see minkowski space.

Sesquilinear Form > see Quadratic Form.

Set Theory

Sextic Equation > see elementary algebra.

Shannon Coding, Information, Sampling > see information; Sampling.

Sheaf, Sheaf Cohomology

Shear of a Congruence of World-Lines
$ Def: If u^a is the unit timelike tangent vector to the congruence, one defines the traceless shear tensor and the shear scalar as

_ab:= _ab –   q_ab ,      := ( _ab ^ab)^1/2 .

where _ab is the expansion tensor and the expansion scalar of the congruence, and q_ab the spatial metric q_ab = g_ab + u_a u_b.

Shell, Gravitating > see gravitating matter; metric matching; models in canonical gravity; semiclassical general relativity; spherical symmetry.

Shift Vector > see initial value formulation of general relativity.

Shimura-Taniyama-Weil Conjecture > see number theory.

Shock Waves > see Gastrophysics; foliations, numerical general relativity [gauge shocks]; gravitational radiation; phenomenology of higher-order gravity; velocity.

Short Exact Sequence > see exact sequence.

Shot Noise > see Noise.

Sigma-Algebra (-Algebra)
$ Def: A collection of subsets of a set X with three properties: (a) The empty set is in the collection; (b) The complement X\A of any set A in is also in ; (c) The union of countably many sets in is also in .
* Relationships: A -algebra is a -ring with the added requirement of property (a).
* Generating a sigma algebra: Given any collection of subsets of X, there exists a unique -algebra generated by it, defined as the intersection of all -algebras that contain (this set is not empty, since the power set of X is in it, for example); It is easy to verify that this object is in fact a -algebra, and it is also clearly minimal.

Sigma-Field (-Field) > see ring.

Sigma Models

Sigma Ring (-Ring) > see ring.

Signature of a Metric > see metric; modifications of general relativity [signature change]; spacetime models and dynamical metric models.

Silent Universe
@ References: Bruni et al ApJ(95)ap/94, gq/96-in [Bianchi I with B field, singularities], Mars CQG(99)gq [3+1 description]; Van den Bergh & Wylleman CQG(04)gq [Petrov I with cosmological constant].

Silver Mean
* Value: The number + 1 = 2 + 1/(2 + 1/(2 + ...)).

Simon-Mars Tensor
* Idea: A tensor on the manifold of trajectories in spacetime.
@ References: Bini et al CQG(01)gq [congruence approach]; Bini & Jantzen NCB(04)gq-in [stationary spacetimes].

Simple Group > see group types.

Simplex

Simplicial Complex > see cell complex.

Simply and Multiply Connected Spaces > see connectedness.

Simply Transitive Action > see group action.

Simultaneity > s.a. kinematics of special relativity; hidden variables; types of gauge theories [fiber bundle formulation].
@ References: Jammer 06 [history; r PT(07)aug, JPA(07)#40].

Sinai's Theorem
* Idea: A box of hard spheres is a chaotic system.
@ References: Sinai UMN(70).

Sine-Gordon Equation > s.a. partial differential equations.
* Idea: An equation for a (1+1)-dimensional field with solitonic solutions.
@ General references: Schief PRS(97) [2+1, integrable]; Dorey & Miramontes NPB(04) [homogeneous, mass scales and crossover].
@ Solitons: Gegenberg & Kunstatter PLB(97)ht, ht/97-in [and dilaton gravity]; Christov & Christov PLA(08) [description as point particles, and quantization].

Singletons
* Idea: Unitary non-decomposable reps of the (3+2) de Sitter group; They have strange gauge transformation properties and can be gauged away to zero on any compact set, so they really live at infinity; Spin 0 or 1/2.
* Uses: Fronsdal has proposed that leptons are made of a Fermi singleton ("Di") and a Bose one ("Rac").
@ References: Flato & Frønsdal CMP(87), JGP(88); Flato et al ht/99-in [rev]; Frønsdal LMP(00)ht/99 [and neutrinos].

Singular Values
* Idea: A Generalization of the concept of eigenvalues.

Singularities for Differential Equations > see partial differential equations; wave phenomena.

Singularities for Mappings > s.a. Catastrophe; Cusp; Fold.
@ General references: Whitney AM(55); Arnold 91.
@ Surface singularities: Kiyek & Vicente 04 [resolution, in characteristic zero].

Singularities in Spacetime > see censorship; types of singularities.

Sinh-Gordon Equation
@ References: Xie & Tang NCB(06) [solution method].

6j-Symbols > see SU(2).

Skein Relations > see knot theory and physics.

Skein Space > see spin structures.

Skeleton of a Simplicial Complex
$ Def: Given a simplicial complex K in Rⁿ, its p-skeleton K^(p) is the set of all in K of dimension p.
* Example: The elements of K⁽⁰⁾ are the vertices of K.

Sky > see geodesics.

Skyrmion Model > s.a. [QCD phenomenology]; astronomical objects [skyrmion stars].
* Idea: A phenomenological model for QCD that contains the fields as basic fields, and constructs the nucleons as solitonic solutions in the pion fields, corresponding to bound states of pions; A "Skyrme term" has to be present in the Lagrangian for stability, and the collective coordinate method is used for quantization; > s.a. black hole solutions, black hole hair.
@ General references: Gisiger & Paranjape PRP(98); Cho et al ht/99; Abbas PLB(01) [and hadrons]; Wong hp/02, hp/02, hp/02; Cho et al IJMPA(08)ht/04 [interpretation]; Rajeev a0801 [relativistic wave equation].
@ Quantization: Jurciukonis et al JMP(05)nt [SU(3) model, canonical quantization]; Krusch ht/06 [overview].
@ Skyrme black holes: Zaslavskii PLA(92) [first law of thermodynamics]; Shiiki & Sawado CQG(05)gq [ < 0]; Brihaye & Delsate MPLA(06)ht/05 [in de Sitter]; Nielsen PRD(06)gq [isolated horizons]; > s.a. black hole hair.
@ And gravity: Ioannidou et al PLB(06)gq [gravitating], PLB(06)gq [spinning]; > s.a. topology change.

Slice
$ Def: A closed achronal subset of spacetime without edge.

Slingshot Effect > see orbits in newtonian gravity.

Smale Conjecture > see diffeomorphisms.

Smarr Formula
* Idea: A formula that gives the mass of a stationary black hole in terms of quantities defined on its horizon, such as area and surface gravity; For Kerr-Newman black holes,

M = (/4) A + · J + Q .

* Remark: It looks like the "integrated version" of the first law, but the latter holds for any perturbation, not just stationary ones.
@ References: Smarr PRL(73) [Kerr]; Breton GRG(05)gq/04-in [in non-linear electromagnetism]; Barnich & Compere PRD(05)gq/04 [higher-dimensional Kerr-AdS].

Smith Conjecture / Theorem > see spheres.

Smooth Particle Hydrodynamics > see fluid.

Smoothing > see Coarse-Graining; relativistic cosmology.

Snell's Law > s.a. Refraction.
@ References: Heller AJP(48) [teaching]; Drosdoff & Widom AJP(05) [photon beam point of view].

Sobolev Space
$ Def: The Sobolev space W_p^m(U) is the space of all functions which belong, together with their derivatives up to the m-th order, to L^p(U):

W_p^m(U):= {f | D^j f L^p(U) for all j such that | j | m} .

* Special case: For p = 2, we call H^m(U):= W₂^m(U).
@ References: Adams 75.

Solar System

Soldering Form > s.a. spin structure.
* Idea: A "disguised identity", also called Infeld-Van der Waerden Symbol, that establishes an isomorphism between spin tensors and spacetime tensors.
* SL(2, C) spinors: The objects that correspond to spacetime vectors are the self-conjugate spinorial 2-tensors, and the soldering form takes

V^a → V^AA',   with    V^a = ^a_AA' V^AA',    or    V^AA' = _a^AA' V^a ;

With the right choice of basis, these 's can be thought of as the unit 2 × 2 matrix and the Pauli matrices.
* SU(2) spinors: Objects corresponding to spacetime vectors are symmetric spinorial 2-tensors, and the soldering form takes

V^a → V^AB,    with    V^a = ^a_AB V^AB,    or    V^AB = _a^AB V^a ;

With the right choice of basis, these 's can be thought of as the three Pauli matrices.
* 4-spinors: The soldering form corresponds to the Dirac matrices.
* Applications: The (complexified) SU(2) soldering form has been used as a variable for gravity.

Solenoidal Vector Field > see vector field.

Solid State Physics > see condensed matter.

Solitons

Solutions of Einstein's Equation

Solvability, Solvable Equation > s.a. classical systems; wave equation [exactly solvable].
@ References: Pesic 03 [Abel and the quintic].

Solvable Group
$ Def: G is solvable if it has a normal series whose factors are Abelian (solvable series); Or, if the chain G = Q₀ Q₁ Q₂ ..., where Q_i is the commutant of Q_i–1, has Q_m = {e} for some m (the height of G).
* Properties: A solvable group always has a commutative invariant subgroup (the Q_m–1 above).
* Examples:
- The 2D Euclidean group, of height 2, E₂ = T_1,1 ×_s SO(2) T₂ {e}.
- The 2D Poincaré group: P₂ = T_1,1 ×_s SO(1,1) T_1,1 {e}.
- The Heisenberg group.

Sonoluminescence

Sound > s.a. music.

Space in Mathematics
$ Def: (Souriau) A set E is a space if there is a recueil R (of "glissements") acting on E.
* And other structure: A space has a natural topology, in which F E is open if id_F in R.

Space in Physics > s.a. spacetime models [absolute space]; tensor decomposition [for spacetime metric].
* Idea: Given a spacetime manifold (M, g) and a time function f on M, space is a level set for f.
@ References: Lachièze-Rey A&A(01) [for an arbitrary observer].

Spacetime > s.a. decomposition; important subsets; models in general and discrete models; topology; types.

Spacetime Algebra > see Geometric Algebra.

Sparking of the Vacuum > see vacuum [QED effect].

Sparling Forms > s.a. stress-energy pseudotensors.
* Real 2-forms: The set of four 2-forms given by

_I := – _IJKL ^JK e^L ,

where e^L is a tetrad field, and ^J_Ka = e^J_ba e^b_K its Levi-Civita connection.
* Complex 2-forms: The two sets of forms

^(+/–)_I := –_IJKL ^{(+/–) JK} e^L ,

where ^{(+/–) JK}:= (^JK i ^JKLM _LM).
* 3-form: A tetrad-dependent 3-form _I or ^(+/–)_I on the bundle of orthonormal frames over spacetime, which is a potential for a local energy-momentum density _I for the gravitational field; If e*_J is a basis of 3-forms, and G_IJ the Einstein tensor,

d_I = d^(+/–)_I = _I + G_I^J e*_J .

@ References: Dubois-Violette & Madore CMP(87); Goldberg PRD(88); Frauendiener CQG(89), GRG(90).

Special Functions > s.a. Integral Transforms; representations of lie groups.
* Idea: Usually, complete orthonormal sets of functions on some set X (typically, an interval X = [a, b]), with which we approximate a function by a finite sum f(x) _n=1^N a_nU_n(x), where the coefficients are calculated by a_n = _X dx U_n*(x) f(x) and the finite sum minimizes _X dx |f(x) – _n a_nU_n|².
* Group theoretic approach: Most special functions are connected with the representation of Lie groups; The action of elements D of the associated Lie algebras as linear differential operators gives relations among the functions in a class – for example, their differential recurrence relations; & Gelfand, Naimark, N Ya Vilenkin.
* Bochner's problem: The characterization of classical orthogonal polynomial systems as solutions of second-order eigenvalue equations.
@ General references: Rainville 63; Etingof & Kirillov Jr ht/93 [and representation theory]; Temme 96 [intro]; Lorente JCAM(03)mp/04 [rev of applications]; Batterman BJPS(07) [what makes them special].
@ Related topics: Lucquiaud JMP(90) [in curved space]; Peherstorfer mp/02 [zeros]; Gurappa et al mp/02 [new approach]; Eynard mp/05-in [asymptotics]; Giraud JPA(05)mp [vanishing average]; Simon BAMS(05) [on S¹]; Alhaidari AML(07)mp/05 [integrals]; Coftas CEJP(04)mp/06 [from hypergeometric equations]; Bruschi et al JPA(07) [from Diophantine conjectures]; Gómez-Ullate et al a0805 [generalized Bochner problem].
@ Specific functions: Raposo et al CEJP(07)a0706 [Romanovski polynomials]; > s.a. Airy; bessel; Elliptic; Gamma; Hypergeometric; Jost; Mathieu; Struve; Whittaker; Zeta Function; spherical harmonics; other functions.
> Other polynomials: see Chebyshev, Gegenbauer, Hermite, integral equations, Jack, Laguerre and legendre polynomials; graph and knot invariants.

Special Relativity > s.a. doubly special relativity; kinematics.

Specific Heat

Spectral Action > see non-commutative physics.

Spectral Decomposition > see hilbert space.

Spectral Function
@ References: Kirsten ht/00-wd [review].

Spectral Geometry

Spectral Sequence
@ References: in Spanier 66.

Spectral Theory > see operator.

Spectrometer > see experiments in physics.

Spectrum of an Algebra
$ Def: The set of its characters.

Spectrum of an Algebra Element
$ Def: The spectrum of an element a of an algebra A over K is the set of K such that a–I is not invertible,

(a):= {(a) | a character of a} .

Spectrum of an Operator > see operator theory.

Speed > see velocity; constants [speed of light]; tests of general relativity [speed of gravity].

Sphaleron > see solutions of gauge theories.

Sphere (including Sphere Packings).

Spherical Harmonics

Spherical Symmetry > s.a. spherical symmetry in general relativity; gauge theory solutions.

Spi > see asymptotic flatness.

Spin Echo Experiment
@ References: Ainsworth FPL(05) [and approaches to statistical mechanics].

Spin Foam Models

Spin Glasses and Models

Spin Networks > s.a. connection representation of quantum gravity, and other spin models.

Spin Structure

Spin-Coefficient Formalism

Spin-Statistics Theorem > s.a. particle statistics.

Spinors > s.a. 2-spinors; 4-spinors; in field theory.

Spintessence > see quintessence.

Spiral, Logarithmic {# s.a. Bernoulli.}
* Examples in nature: Galaxies, Nautilus.
@ References: in Thompson; in Maor ThSc(94)jul.

Spline
@ References: de Boor 78.

Splitting of Spacetime > see decomposition.

Splitting Sequence > see exact sequence.

Splitting Theorem
@ Lorentzian geometry: Yau 82; Galloway CMP(84), JDG(89); Ehrlich & Galloway CQG(90); Newman JDG(90); Galloway AHP(00)m.DG/99, gq/02-in [null].

Spontaneous Emission
@ General references: Crisp & Jaynes PR(69), Leiter PRA(70) [in semiclassical radiation theory]; Cray et al AJP(82) [ito interference]; Milonni AJP(84) [and fluctuation dissipation]; Olsen et al qp/05 [2-level bosonic atom, phase space approach]; Kleppner PT(05)feb [and stimulated, Einstein's 1917 paper].
@ Based on electron self-energy, without field quantization: Barut & Van Huele PRA(85), & Dowling PRA(87), & Salamin PRA(88).

Sprinkling of Points in a Manifold > see statistical geometry.

Square (magic square, ...) > see number theory.

Square Roots > see elementary algebra.

Squeezed States > s.a. distance; QED; symplectic structure [squeezing].
* Idea: A quantum minimum-uncertainty (x p = /2) state of an oscillator/field in which the complementary operators do not have the same variance; The product of the variances of course satisfies the uncertainty relation, but one of them is lower than the coherent state value, the one predicted by semiclassical models.
* Examples: Squeezed light may be applied in data transmission and high-precision metrology.
@ General references: Yuen PRA(76); Yuen & Shapiro OL(79); Caves PRD(81); Henry & Glotzer AJP(88); Muñoz-Tapia AJP(93) [properties]; Nieto qp/97-in [history]; Beckers et al PLA(98) [new sets]; Trifonov PS(98) [for n observables]; Saxena JPA(02) [eigenvalue equation]; Honegger & Rieckers PhyA(04) [non-classicality and coherence]; Sträng JPA(08)-a0708 [semiclassical evolution].
@ On S¹: Kowalski & Rembielinski JPA(02)qp, JPA(03)qp; Trifonov JPA(03)qp/02.
@ For QED, light: Loudon & Knight JMO(87) [light]; Slusher & Yurke SA(88)may [light]; Putz & Svozil NCB(04)ht/01 [vacuum, e mass shift]; Popp et al PLA(02) [in biological systems]; Petersen et al PRA(05)qp; Bachor et al CP(05); Biswas & Agarwal PRA(07) [photon-subtracted, non-classicality].
@ Other systems: Burgess PRD(97) [non-equilibrium quantum field theory]; Tavassoly JPA(06) [solvable]; Marchiolli et al PRA(07)qp [discrete].
@ Squeezed number states: Nieto PLA(97)qp/96; Albano et al JOB(02)qp/01.
@ Generalized: Marchiolli & Galetti a0709; Shchukin et al a0712.
> Relalated states and generalizations: see coherent states; fock space; Kerr State; vacuum.

Stability > for matter, see condensed matter; for solutions of dynamics, see classical systems; for theories, see physical theories.
> In gravitation: see black hole perturbations; cosmological perturbations; perturbations in general relativity.

Stability Theory in Mathematics > s.a. Bifurcation Theory.
@ References: Yoshizawa 75; Rouche et al 77.

Stabilizer of a Group Element > see group action.

Stacks > see categories.

Standard Map > s.a. chaotic systems.
* Idea: A chaotic, area preserving discrete map of the unit square map onto itself used to model a kicked rotator; Also called Taylor-Greene-Chirikov map; Defined by

p_n+1 = p_n + K sin(_n) ,   _n+1= _n + p_n+1.

@ References: Shevchenko PhyA(07).
> Online resources: MathWorld page; Wikipedia page.

Standard Model > see in cosmology and particle physics.

Star-Algebra > see abstract algebra.

Star-Convex Subset of an Affine Space > see affine structures.

Star Product > s.a. non-commutative field theory; non-commutative geometry; types of quantum field theories.
* Idea: An antisymmetric tensor ^mn used to define non-commutative geometrical structures, such that for two functions f and g,

(f *g)(x):= exp( i ^mn {/y^m} {/zⁿ}) f(y) g(z)|_y=z=x = f(x) g(x) + i ^mn _m f(x) _n g(x) + h.o.t.

* Remark: This structure is not Lorentz-invariant.
@ References: Zachos JMP(00)ht/99 [evaluation]; Gammella LMP(00) [tangential]; Freidel & Krasnov JMP(02) [and spin networks]; Man'ko et al PLA(05)ht/04 [dualities]; Pinzul & Stern NPB(08) [gauging]; Kupriyanov & Vassilevich a0806 [friendlier approach].

Stark Effect > see atomic physics.

Stars > s.a. star types.

State of a System > s.a. quantum state.

State Sum Models > see spin foam.

Static Spacetime > see general relativity solutions with symmetries.

Stationary Phase Approximation > s.a. Steepest Descent Approximation.
* Idea: An approximation used to calculate the leading order behavior of integrals of the type _–infty^infty dx f(x) exp{i(x)/} in the limit of small ; Consists in taking into account only the contribution from the critical points of (x); Related to the steepest descent approximation.
* In path integrals: The approximation of writing the field as the classical solution plus a small perturbation; Sometimes known as WKB or one-loop approximation.
@ References: Kamvissis mp/07 [and steepest descent].

Stationary Spacetime > see general relativity solutions with symmetries; types of spacetimes.

Statistics > s.a. error analysis in physics; particle statistics; probability.

Statistical Mechanics > s.a. non-equilibrium.

Steady State > see states of a system.

Steady State Cosmology > s.a. cosmological models and general relativistic models.
* History: First proposed in 1948 by H Bondi, then T Gold and F Hoyle (and Littleton?); Despite its loss of mainstream favor, to some extent the idea has been incorporated into some versions of inflation.
* Idea: It postulates that the universe is always expanding, and matter is created at precisely the rate required to maintain a constant spatial density; A steady-state universe has no beginning or end, and its overall properties are constant in time.
* And observation: They don't have the singularity and flatness problems of the standard model, but they were ruled out by observations on radio souces by M Ryle et al at Cambridge in the 1950's and early 1960's, and especially by the discovery of the microwave background.
@ General references: Hoyle in(58); Arp et al Nat(90)aug; Andrews ap/01.
@ Quasi-steady state: Hoyle et al PRS(95) [comment Wright MNRAS(95)], 00; Burbidge et al PT(99)apr [and reply by Albrecht PT(99)apr]; Burbidge ap/01-in; Narlikar et al PASP(02)ap [acceleration], ApJ(03)ap/02 [and cmb]; Vishwakarma & Narlikar JAA(07)-a0705 [and repulsive gravity]; Narlikar et al JAA(07)-a0801 [and cyclic universe].
@ Criticism of Big Bang: Arp & Van Flandern PLA(92); Arp ap/98-in; Lopez-Corredoira ap/03-in.

Steady State Equation > see partial differential equations.

Steepest Descent Approximation > see integration.

Stefan-Boltzmann Law > see thermal radiation.

Stein Structure > see 4D manifolds.

Stem > see posets.

Stephani Universe / Model
* Idea: A spherically symmetric, inhomogeneous cosmological model, recently used as a possible explanation of the cosmic acceleration.
@ General references: Stelmach & Jakacka CQG(06) [angular sizes]; Pedram JCAP-a0806 [+ scalar, classical and quantum].
@ And acceleration: Stelmach & Jakacka CQG(01)-a0802; Godlowski et al CQG(04)ap.

Stern-Gerlach Experiment > see experiments in quantum mechanics.

Stiefel Manifold of k-Frames > see differentiable manifolds.

Stiefel-Whitney Classes / Numbers

Stieltjes Constants
* Idea: The expansion coefficients in the Laurent series for the Hurwitz zeta function about s = 1.
@ References: Coffey JMAA(06)mp/05 [evaluation], PRS(06) [summation relations], a0706 [_j coefficients, Hurwitz zeta function], a0706 [series representations]; > s.a. MathWorld page.

Stieltjes Integral > see integration.

Stieltjes Transform
@ References: Schwarz JMP(05)mp/04 [generalized]; > s.a. MathWorld page.

Stimulated Emission > see quantum field theory in curved backgrounds [black holes]; Spontaneous Emission.

Stirling Formula
* Idea: For n → , n! (n/e)ⁿ (2n)^1/2, or ln n! (n+) ln n – n + ln(2).

Stirling Numbers
@ References: Branson DM(06) [representation ito recurrence relations].

Stochastic Electrodynamics > see modified electromagnetism.

Stochastic Gravity > s.a. Induced Gravity.
* Idea: Based on the Einstein-Langevin equation, which has in addition sources due to the noise kernel, the expectation value of the stress-energy bi-tensor which describes the quantum matter fluctuations.
@ References: Ross & Moreau GRG(95); Moffat PRD(97)gq/96; Zakir ht/98-in; Hu IJTP(99)gq; Cole et al PRA(01) [as residual van der Waals force]; Hu & Verdaguer gq/01-ln, CQG(03)gq/02, LRR(04)gq/03 + LRR(08)-a0802, et al gq/03; Verdaguer gq/06-in [and applications].

Stochastic Processes

Stochastic Layer / Region in Phase Space > see phase space.

Stochastic Quantization

Stokes' Law
* Idea: The friction force on a small sphere of radius r moving with terminal speed v in a homogeneous fluid of viscosity coefficient is F = 6rv.

Stokes Parameters > see polarization.

Stokes' Theorem > see integration on manifolds.

Stone Space > see types of topologies.

Stone's Theorem
* Idea: It says or implies that exp( i tH/) is unitary if H is self-adjoint, even if densely defined unbounded, on an infinite-dimensional space.

Stone-von Neumann Theorem > see representations of quantum mechanics.

Strain Tensor
@ References: de Prunelé AJP(07) [in spherical coordinates].

Strange Star > see star types.

Strangelet / Strange Quark Nugget > see astronomical objects; experimental particle physics; QCD phenomenology.

Stratified Manifold > see types of manifolds.

Stratum (Plural: Strata)
* Idea: The set of all orbits of the same topological type for the action of a group on a manifold.
@ References: Sartori & Valente JPA(03) [compact linear G on Rⁿ].

Stress > s.a. Elasticity; stress-energy pseudo-tensors.
@ In mechanics and relativistic field theory: Gronwald & Hehl gq/97-in; Medina AJP(06) [contribution to energy and momentum].

Stress-Energy Tensors > see energy-momentum.

String Field Theory
@ Reviews: Kaku IJMPA(87); Berkovits ht/01 [open superstrings]; Siegel 88-ht/01; Thorn PRP(89); Rastelli ht/05-in; Taylor ht/06-in.
@ General references: Green & Schwarz PLB(84); Hata et al PRD(86) [covariant]; Witten NPB(86) [and non-commutative geometry], NPB(96) [open], pr(87); Horowitz et al PRL(86) [cubic action]; Bowick & Rajeev PRL(87), NPB(87); Strominger PRL(87); Horowitz & Witt PLB(87); Bordes & Lizzi IJMPA(90); Hashimoto & Itzhaki JHEP(02) [observables]; Kling et al PLB(03)ht/02 [non-perturbative solutions]; Bars ht/02 [Moyal star formulation]; Drukker JHEP(03)ht [actions]; Okawa & Zwiebach JHEP(04) [heterotic]; Taylor ht/04-ln [perturbative computations].

String Theory > s.a. phenomenology; or under cosmic strings.

Strong Coupling Limit > see modified versions of general relativity.

Strong Interactions > see particle physics; QCD.

Strong Rigidity Theorem > see Rigidity.

Strongly Asymptotically Predictable Spacetime > see types of spacetimes.

Structural Realism, Structuralism
@ References: van Fraassen BJPS(06).

Structure Equations > see affine connection.

Structure Formation in Cosmology > see early universe.

Structure of Matter > see matter.

Structure of Physical Theories > see physical theories.

Structure Sheaf > see sheaf.

Struve Function
* Idea: The function H_n(z) which satisfies the inhomogeneous Bessel equation z² H_n''(z) + z H_n'(z) + (z – n) H_n = (2/) z/(2n–1)!!

Stückelberg Mechanism / Model > s.a. classical particles [and Lorentz force]; particle physics [standard model extension].
* Idea: A mechanism, proposed in 1938 by Stückelberg, for making an abelian gauge theory massive while preserving gauge invariance, by introducing an additional scalar field; 2004, Numerous generalizations have been proposed for the non-abelian case, but the Higgs mechanism in spontaneous symmetry breaking remains the only known way to give masses to non-abelian vector fields in a renormalizable and unitary theory.
@ General references: Dragon et al NPPS(97)ht [variation – BRS-invariant polynomial form]; Ruegg & Ruíz-Altaba IJMPA(04); Cianfrani & Lecian a0803-in-MPLA [historical].
@ Quantization: Horwitz ht/98; Oron & Horwitz FP(03)gq; McKeon & Marshall ht/06 [renormalization and gauge invariance].

Sturm-Liouville Theory > s.a. [ordinary differential equations]; matrices [determinants].
* History: Started in the 1830's with Sturm and Liouville's generalization of the Fourier sine series to expansions in terms of eigenfunctions of some ordinary differential equations; The hardest questions were those of convergence, resolved after 1900.

SU(2) Group

Subbase for a Topology
$ Def: A set of subsets of X from which all open sets can obtained as arbitrary unions of finite intersections.

Subgroup > see group theory.

Sublimation > see phase transition.

Submanifold > s.a. embedding; extrinsic curvature [including extremal surface]; Hypersurface; manifolds; spacetime subsets.

Submarine Paradox > see special relativity.

Submersion
$ Def: A smooth mapping f : M → B which is onto, with f_* onto for all p in M.

Subnormal Matrix / Operator
$ Def: (Halmos) A non-square matrix A is subnormal if it can be completed to a (square) normal matrix.
* Topology: The set of such A's is not closed (can give example of A(t) subnormal for all t > 0 but not for t = 0).
* Problem: Is there an intrinsic characterization of such matrices?

Subobject of an Object A
$ Def: An object A' in the same category, with a monomorphism f : A' → A.

Subspace of a Vector Space
* Idea: A subset which is closed under the vector space operations; It can be characterized by a multivector.

Substance > see Ontology.

Substantialism > see spacetime.

Subsystem > see quantum field theory formalism; quantum systems.

Sullivan-Baas Singularities > see riemannian geometry.

Sum Rules > see lattice gauge theories; standard model of particle physics.

Summations > s.a. series.

Sunyaev-Zeldovich Effect > see cosmic microwave background.

Superalgebras > see poincaré algebra.

Superbradyons > see causality violations.

Superconductivity > s.a. types of superconductors.

Superenergy Tensor > see stress-energy pseudotensors.

Superfields > see BRST; supersymmetric field theory.

Superfluids > s.a. Bose-Einstein Condensation; particle statistics; Quasiparticles; sound; turbulence.
* Examples: In ⁴He, pairs of atoms condense into a macroscopically coherent quantum state (Bose condensation), which manifests itself as a frictionless fluid; in ³He, the situation is not so simple; He II (0 to 2.172 K) is a superfluid, highly heat-conductive by friction-free convection; He I (2.172 to 4.2 K) is an ordinary fluid; 2005, Evidence seen in solid hydrogen [@ news pn(05)mar].
* Method: Study using second-waves, regions with different concentrations of ordinary/superfluid components.
* Properties: They exhibit quantized vortices when rotated or subject to a T gradient.
@ General references: Feynman RMP(57); Khalatnikov 65; SA(76)dec; Collins PT(92)jun; news pn(96)oct; Guénault 03; Adams & Bry PhyA(04); Annett 04 [intro]; Brandão NJP(05) [order parameter and entanglement]; Balibar CP(07); Pilati et al PRL(08) [critical T, 2D and 3D].
@ ³He: Bunkov et al PRL(00) [sets of 4 atoms?]; Finne et al Nat(03)aug + pn(03)aug [criterion for the onset of turbulence]; Volovik cm/07 [history].
@ In general relativity and cosmology: Carter gq/99-in [vortex dynamics], G&C(00)ap [neutron stars]; Casini & Montemayor gq/99 [covariant]; Volovik PRP(01)gq/00 [analogs]; Garcia de Andrade gq/05 [with torsion].
@ Examples: Donnelly pw(97)feb [rotons]; Kapusta PRL(04)ht [for Dirac neutrinos]; Bulgac et al PRL(06) [spin-1/2 fermions].
@ History: Andronikashvili 90; Donnelly PT(95)jul; Balibar phy/06 [discovery].

Supergravity

Superluminal Communication / Propagation > see causality violations; wave phenomena.

Supermanifold > see manifolds.

Supermetric > see geometrodynamics.

Supernova > see star types.

Superoscillations > s.a. schrödinger equation; wave phenomena.
* Idea: The phenomenon by which differentiable functions can locally oscillate on length scales that are much smaller than the smallest wavelength contained in their Fourier spectrum.

Superparticle > see quantum particles.

Superposition Principle > related to Linearity.
* In classical field theory: Holds when the field equations are linear, so that a linear combination of solutions is a solution.
* In quantum mechanics: The space of states of quantum theory is a vector space; Linear superpositions of states are also allowed states.
@ In classical field theory: Notte-Cuello & Rodrigues mp/06 [and energy-momentum conservation].
@ In quantum mechanics: Károlyházy in(90) [breakdown]; Greenberger et al PT(93)aug [and interferometry]; Cirelli et al JGP(99) [extension]; Bassi & Ghirardi PLA(00)qp [against], d'Espagnat PLA(01)qp [reply]; Peacock qp/02 [suggested explanation]; Lan qp/03 [superposition mixture]; Corichi GRG(06)qp/04 [and geometrical formulation]; Lynn & Caponigro qp/06 [epistemological].
@ In quantum mechanics, systems / states: Morimae & Shimizu PRA(06) [macroscopically distinct states]; Dowling et al PRA(06) [atom and molecule].

Superpotential > see conservation laws.

Superradiant Scattering > s.a. black hole analogs; black hole radiation.
* Idea: The amplification of a wave scattering off a black hole, a wave analog of the Penrose process for energy extraction, which can be interpreted as stimulated emission.
* Conditions: It occurs only for bosonic fields.
@ References: Zeldovich JETP(72); Starobinskii JETP(73); Bekenstein PRD(73); Wald PRD(76); & Misner; Bekenstein & Schiffer PRD(98)gq; Winstanley PRD(01)gq [scalar in Kerr-Newman-AdS black holes].

Superscattering Matrix

Superselection Rules

Supersolids > see condensed matter.

Superspace > for space of geometries, see geometrodynamics; for bosonic + fermionic coordinates, see manifolds [supermanifolds].

Superstatistics > see statistics.

Supersymmetry > s.a. lie algebras [superalgebras]; modified quantum mechanics.
* Supersymmetry group: An extension of the Poincaré group of flat spacetime isometries to symmetry transformations between integer and half-integer spin fields; Its generators Q change the spin by 1/2, and the number N that classifies susy theories is like a "degree of kinship" between bosons and fermions.
* Supersymmetry algebra: A graded Lie algebra, with generators {Q_i^A, Q*_j'B, P_a}, with i, j ' = 1, 2 (spinor indices), a, b = 1, ..., 4 (spacetime indices), and A, B = 1, ..., N, with commutation relations

{Q_i^A, Q*_j'B} = 2 _ij'^a P_a ^A_B ,   {Q_i^A, Q_j^B} = {Q*_i'A, Q*_j'B} = 0 ,   [P_a, Q_i^A] = [P_a, Q*_i'A] = 0 ,   [P_a, P_b] = 0 .

@ References: Cornwell 92; Jolie SA(02)jul; Ichinose ht/06, ht/06 [graphical representation].
> In field theory: see supersymmetry in field theory; supersymmetry phenomenology; supersymmetric theories.

Supertranslation > see asymptotic flatness.

Surface > s.a. Area; dynamical triangulations [random]; Singularities.
* Flexible: A surface in a smooth manifold M is called flexible if, for any diffeomorphism on the surface, there is a diffeomorphism on M whose restriction on the surface is and which is isotopic to the identity.
@ In 3D euclidean space: Guzzardi & Virga PRS(07) [constant mean curvature].
@ In 4D manifolds: Hirose & Yasuhara Top(08) [flexible surfaces].
@ Deformations: Capovilla & Guven CQG(95).

Surface Gravity > s.a. laws of black hole dynamics.
* In Newtonian gravity: The quantity g = GM/r², for a spherical body of mass M and radius r.
$ For a black hole: If l is the stationary Killing vector field of a black hole, normalized at infinity, then is defined by l ^b _b l ^a = l ^a; It is constant over the horizon surface.
* Schwarzschild black hole: Given by = GM/(2G