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 PLB(08)-a0710 [new approach]; Giddings & Porto a0908 [gravitational].

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)nov; 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 theory.
* 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)jan; Harvey AJP(65)feb; 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 in(02)gq/01, in(04)gq/03.
@ 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)sep-RL; Henkel NPB(02) [in statistical mechanics]; West CSF(04) [renormalization group, complexity]; Gupta et al PhyA(08) [power law scaling and limitations in Tsallis statistics].
@ In biological systems: Brown & West 00 [in biology]; West & Brown PT(04)sep.
@ In other areas: Peterson AJP(02)jun-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 irreducible representation 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]; Tanasa & Kreimer a0907 [for non-commutative 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 [post-Newtonian]; gravitational self-force.

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
* For linear maps: 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 AP(08)-a0801 [relativistic wave equation]; Ioannidou & Kevrekidis PLA(08)-a0807 [2+1 and 3+1 lattice versions].
@ 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)sep [teaching]; Drosdoff & Widom AJP(05)oct, comment Pérez AJP(06)sep [photon beam point of view].

Sobolev Space > s.a. p-Adic Numbers.
$ 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.

Sommerfeld Paradox
* Idea: Mathematically, the Couette linear flow is linearly stable for all Reynolds numbers, but experimentally arbitrarily small perturbations can induce the transition from the linear shear to turbulence when the Reynolds number is large enough.
@ References: Li & Lin a0904 [proposed resolution]; Lan et al a0905.

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.

Spacetime Diagrams > see kinematics of special relativity.

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]; Dunkl SIGMA(08)-a0812 [in four variables].
@ 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.

Spectral Triple > s.a. holonomy; non-commutative geometry.
@ References: Aastrup et al a0807 [over a holonomy algebra].

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].

Speed of State Evolution > see quantum effects.

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.

Spinon > see Luttinger Liquid.

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, in(02)gq [null].

Spontaneous Emission
@ General references: Crisp & Jaynes PR(69), Leiter PRA(70) [in semiclassical radiation theory]; Cray et al AJP(82)nov [in terms of interference]; Milonni AJP(84)apr [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)apr; Muñoz-Tapia AJP(93)nov [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]; > s.a. types of coherent states.
@ 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 PS(08)-a0709; Shchukin et al a0712.
> Relalated states and generalizations: see coherent states; fock space; Kerr State; vacuum.

SQUID (Superconducting Quantum Interference Device) > see superconductivity.

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.
@ General references: Zachos JMP(00)ht/99 [evaluation]; Gammella LMP(00) [tangential]; Man'ko et al PLA(05)ht/04 [dualities]; Pinzul & Stern NPB(08) [gauging]; Kupriyanov & Vassilevich EPJC(08)-a0806 [friendlier approach]; Aniello a0902 [group-theoretical point of view].
@ Special types: Aniello et al PLA(09) [on finite and compact groups]; Vassilevich CQG(09)-a0904 [diffeomorphism-covariant, and non-commutative gravity].
@ Special contexts: Freidel & Krasnov JMP(02) [and spin networks]; McCurdy et al a0809 [differential forms on symplectic manifolds].

Stark Effect > see atomic physics.

Stars > s.a. star types.

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

State Sum Models > see spin-foam models.

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 CM(08)mp/07 [and steepest descent]; Sorkin a0911-in [saddle-point approximations and tunneling].

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, systems.

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 1950s and early 1960s, and especially by the discovery of the microwave background.
@ General references: Hoyle in(58); Arp et al Nat(90)aug; Andrews ap/01; Altaie a0907 [from back-reaction effect of quantum fields].
@ 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(08)-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 Moment Problem > see types of coherent states.

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 in terms of 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.
@ General 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.
@ Applications: Verdaguer JPCS(07)gq/06; > s.a. cosmological perturbations.

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)oct [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 pseudotensors.
@ In mechanics and relativistic field theory: Gronwald & Hehl gq/97-in; Medina AJP(06)nov [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 cosmology.

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 IJMPA(08)-a0803-in [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 1830s 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.
@ References: Azad & Mustafa a0906 [and orthogonal functions].

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.

Subfactor Theory > see topological field theories.

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.

Sub-Riemannian Geometry / Manifold
* Idea: A sub-Riemannian manifold is a generalization of a Riemannian manifold, in which to measure distances you are allowed to go only along curves tangent to so-called horizontal subspaces.
* Properties: Sub-Riemannian manifolds carry a natural intrinsic metric called the Carnot-Carathéodory metric; Their Hausdorff dimension is always an integer and larger than their topological dimension (except in the case of a Riemannian manifold).
* Applications: Found in the study of constrained systems such as the motion of vehicles on a surface and the orbital dynamics of satellites in classical mechanics, and geometric quantities such as the Berry phase; The Heisenberg group, carries a natural sub-Riemannian structure.
@ References: Calin & Chang JDG(08); Calin & Chang 09.
> Online resources: see the Wikipedia page.

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.

Suicide, Quantum > see many-worlds interpretation; types of measurements.

Sullivan-Baas Singularities > see riemannian geometry.

Sum Rules > s.a. lattice gauge theories; [standard model of particle physics].
* Idea: Relationships between structure functions for different particles, or expressions for them derived or guessed on the basis of their constitution (hadrons in terms of quarks); Examples are the Bjorken sum rules (no evidence of any violation, but if found, could be serious) and Ellis-Jaffe sum rules (seem to be violated; no big deal); To verify them, use deep inelastic scattering.
@ References: Adler a0905-in [Adler sum rule].

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]; Yu AP(08) [as a Bose exchange effect]; Sewell & Wreszinski JPA(09) [mathematical theory]; Dupuis PRL(09) [unified picture]; Roberts CP(09) [drag forces on moving objects].
@ 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]; Ma & Wang PhyA(08) [new models].
@ ⁴He: Pollet et al PRL(08), comment Balibar Phy(08) [solid]; > s.a. condensed matter [supersolid].
@ In general relativity and cosmology: Zurek Nat(85) [and superfluidity]; 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]; Kastrinakis a0901 [new states].
@ History: Andronikashvili 90; Donnelly PT(95)jul; Balibar phy/06, Griffin pw(08)aug [discovery].

Supergravity

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

Supermanifold > see manifolds.

Supermassive Objects > see black holes [alternatives].

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 RPMP(08)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 IJTP(08)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]; Day PT(09)sep [chiral molecule, and quantum-to-classical transition].

Superpotential > see conservation laws.

Superradiance / 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]; Finster et al CMP(09) [rigorous treatment].

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 supersymmetric 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/(2GM/c²)² = c⁴/4GM .
* Kerr black hole: Given by = (r₊ + r_–)/4, where := A/4, r_+/–:= M (M²–Q²–a²)^1/2 and a:= L/M; It vanishes only in the extreme case M² = Q² + a² (which does not mean A = 0).
> Other situations: see horizons [isolated horizons]; killing horizons.

Surface Physics > see condensed matter.

Surface Tension > s.a. metric matching; thermodynamics; Water.
@ References: Callaway PRE(96) [using black-hole analogy].

Surgery > see algebraic topology.

Surreal Numbers > see numbers.

Susceptibility
* Idea: The susceptibility of a material is parameter characterizing its response to a small variation in an applied field, an example of linear response function; For example, the magnetic susceptibility = M/B.
@ Topological: Del Debbio et al PRL(05)ht/04 [SU(3) gauge theory], JHEP(04)ht [SU(N) for large N, finite T].

Suspension of a Topological Space > see topology.

Sutherland Model > see integrable system.

Swiss Cheese (Einstein-Straus) Cosmological Models
> Models: see brane cosmology, cosmological models in general relativity; perturbations in general relativity.
> Effects: see theory of cosmological acceleration; cosmological expansion; lensing; light [propagation in curved spacetime].

Sylow Subgroup, Theorems > see finite group.

Sylvester Graph > see group theory.

Sylvester's Theorem > see laplacian.

Symbolic Logic > see logic.

Symmetric Criticality Principle > see lagrangian dynamics.

Symmetric Group > see finite groups.

Symmetric Operator or Matrix > s.a. operator theory.
* Remark: An operator is usually an object of the type A^a_b, so we need a metric in order to ask whether it is symmetric, or A_ab = A_ba; In expressions like f | Av = Af | v, we are implicitly using the metric given by the Hilbert-space inner product.

Symmetric Space > s.a. matrices [random].
* Idea: A Riemannian manifold whose curvature is invariant under all parallel translations.
* History: The theory was developed by Cartan in the 1920s.
* As coset space: A symmetric space is diffeomorphic to G/H, where H is associated with a given involutive inner automorphism of G, as the subgroup generated by all the Lie-algebra elements which are eigenvectors with eigenvalue +1 of the differential of the inner automorphism, considered as an operator on the Lie algebra.
@ References: Helgason 78; Anker & Orsted ed-05 [reductive, Plancherel theorems]; Borel & Li JDG(07) [compactifications].

Symmetrization Operator > see tensors.

Symmetry

Symmetry Breaking

Symmetry Properties of a Tensor > see tensors.

Symplectic Group > see examples of lie groups.

Symplectic Integrators > s.a. Perturbation Methods.
* Idea: A method to evolve dynamical systems according to modified Hamiltonians whose error terms are also well-defined Hamiltonians
@ General references: Donnelly & Rogers AJP(05)oct [intro]; Brown PRD(06) [and midpoint rule for Hamiltonian systems]; Chin PLA(06) [theorem]; Kobayashi PLA(07).
@ Applications: Chin PRE(07)mp/06 [and perihelion advance in Kepler problem]; Frauendiener a0805, Richter & Lubich CQG(08)-a0807 [in numerical relativity].

Symplectic Structure > s.a. symplectic geometry; in physics; variations.

Synchronization > s.a. chaos; clocks; special relativistic kinematics.
@ Non-chaotic dynamical systems: Bagnoli & Cecconi PLA(01).

Synchrotron Radiation > see radiation.

Synge's Theorem > see orientation.

System Theory > s.a. classical and quantum systems; state of a system.

Syzygies
@ References: Evans & Griffith 85.

Szekeres Model / Spacetime > s.a. cosmological acceleration; types of singularities.
* Idea: The quasispherical Szekeres model is an exact solution of the Einstein field equations, which represents a time-dependent mass dipole superposed on a monopole and therefore is suitable for modelling double structures such as voids and adjourning galaxy superclusters.
@ References: Bolejko ap/06-in [and cosmology]; Krasinski PRD(08)-a0805 [properties of the quasi-plane model].

Szilard's Demon > see thermodynamics.

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