**Topics, T**

**T-Duality** > see M-theory;
principal fiber bundles.

**T _{0}, T_{1},
T_{2}, T_{3},
T_{4} Spaces** > see
types of topological spaces.

**T Symmetry / Violation** > see CPT symmetry.

**Tables of Integrals** > see integration.

**Tails** > see gravitational phenomenology;
huygens' principle; quantum localization
[Hegerfeldt theorem and infinite tails]; wave phenomena.

**Tajmar Effect**

* __Idea__: An unexplained
acceleration observed by accelerometers and laser gyroscopes close to rotating
supercooled rings; The observed ratio between the gyroscope and ring accelerations
was 3 ± 1.2 × 10^{–8} (for clockwise rotations, and about half this size for anticlockwise ones).

@ __References__: Tajmar et al AIP(07)gq/06 [rotating superconductors];
McCulloch EPL(10)-a0912,
EPL(11)-a1106 [and modification of inertia].

**Takagi Factorization** > see metric tensor.

**Takagi Function**

* __Idea__: A continuous, nowhere-differentiable
function whose graph is a fractal, related to the Weierstrass functions.

@ __References__: Allaart & Kawamura RAE-a1110.

> __Online resources__:
see Wikipedia page [Blancmange curve].

**Talbot Effect** > s.a. light.

* __Idea__: A diffraction
phenomenon by which a wave imprinted with transverse periodicity, such as in
a diffraction grating, reconstructs itself at regular intervals; It occurs
in many physical systems.

@ __References__: McMorran & Cronin NJP(09)-a0812 [for
electrons].

**Tangent Bundle, Map, Vector** > see tangent
structures.

**Tangherlini Solutions** > see higher-dimensionsional black holes [Schwarzschild-Tangherlini].

**Tangle** > s.a. knot
theory; lattice
field theory [random].

@
__References__: Baez & Langford LMP(98)
[2-tangles]; Zinn-Justin & Zuber
JKTR(00)mp,
Zinn-Justin CMP(03)mp/01 [counting,
and matrix models]; Kravchenko & Polyak LMP(11) [Milnor's μ-invariants and diassociative algebras].

**Tau Lepton** > see particle
types.

**Taub Function** > see initial-value
formulation
of general relativity [densitized lapse].

**Taub Numbers**

* __Idea__: A set of tensorial
conservation laws derivable from curves of solutions to the Einstein equation,
i.e., solutions of the linearized equation; For asymptotically flat backgrounds
with Killing symmetries, when the field equations and the linearized field
equations for a metric perturbation are solved, such perturbed space-times
admit 0th-, 1st-, and 2nd-order Taub numbers; 0th-order Taub numbers
are Komar constants of the background; For each Killing symmetry of the background,
first-order Taub numbers give the contribution of the perturbation to the associated
Komar constant, such as the perturbing mass; second-order Taub numbers give
the rate of gravitational radiative loss of the background conserved quantity.

@ __Asymptotically flat backgrounds with Killing vectors__:
Glass PRD(93);
Glass & Naber JMP(94)
[extended to stationary electrovac];
Naber & Glass JMP(94)
[at future null infinity];
Glass & Naber CQG(97) +
Glass gq/97-MG8.

**Taub Solution**

* __Idea__: A locally rotationally-symmetric
homogeneous solution of Bianchi type IX; Resembles a radiation filled Robertson-Walker
solution which contains long-wavelength gravitational radiation
rather than electromagnetic radiation.

@ __General references__: Matzner JMP(68);
Jantzen GRG(05)
[comment on rediscovery], Meléndez & Chauvet GRG(05)
[reply].

@ __Quantum__: Battisti & Montani PRD(08)-a0707
[with gup]; Battisti et al PRD(08)-a0806 [polymer
approach]; Battisti et al a0903-proc
[fate of singularity in gup and polymer approaches]; > s.a. path-integral
approach to quantum gravity.

**Taub-Bolt Solution** > see gravitational
instantons.

**Taub-NUT Solution** > s.a. dirac
fields in curved spacetime; modified kerr
solutions [Kerr-Taub-NUT-de
Sitter]; wormhole solutions.

* __Idea__: A spatially homogeneous vacuum solution of Bianchi type IX with
topology S^{3} × \(\mathbb R\); It describes a black hole which, in addition to the mass parameter (gravito-electric charge) as in the Schwarzschild solution, depends on a "gravito-magnetic charge", also known as NUT parameter; The NUT metric does not violate the uniqueness and no-hair theorems because it is not asymptotically flat.

* __History__: Initially proposed
by Taub, with coordinate singularities; Newman, Unti & Tamburino gave an extension outside the horizon; It was once called by Charles Misner "a counterexample to almost anything"; Its physical interpretation is challenging and it has often been considered as unphysical.

* __Line element__: If we define *U*(*t*):= –1 + 2 (*mt* +
*l*^{ 2})/(*t*^{ 2} + *l*^{ 2}),

d*s*^{2} = –*U*^{ –1} d*t*^{2}
+ (2*l*)^{2}*U* (d*ψ* +
cos*θ* d*φ*^{})^{2} +
(*t*^{ 2 }+ *l*^{ 2})
(d*θ*^{2} +
sin^{2}*θ* d*φ*^{2})
.

@ __General references__: Taub AM(51); Newman,
Tamburino & Unti JMP(63);
Misner JMP(63);
Misner in(67) [has the 2D example];
Misner & Taub JETP(69);
Baleanu & Codoban
GRG(99)gq/98 [dual,
symmetries]; Chanda et al a1503; Clément et al PLB(15)-a1508 [physical relevance, version without time periodicity condition, geodesically complete]; Jefremov & Perlick CQG+(17).

@ __Higher-dimensional__: Astefanesei et al JHEP(05)ht/04 [Taub-NUT-AdS],
JHEP(06)ht/05 [nuttier
bubbles]; Awad CQG(06).

@ __Other generalizations__: Moncrief JGP(84);
Van Holten PLB(95)
[supersymmetric extension]; Beyer CQG(08)-a0804 [with
cosmological constant]; Beyer & Hennig CQG(14)-a1401 [exact, smooth Gowdy-symmetric generalized Taub-NUT solution].

@ __Thermodynamics__:
Mann & Stelea PRD(05); Holzegel CQG(06)gq [and instability].

@ __Related topics__: Visinescu JPA(00)
[Killing-Yano
tensors]; Catren & Ferraro PRD(01)gq/00 [quantization];
Bini et al CQG(02)gq [holonomy];
González gq/02 [thin disks as sources]; Jezierski & Lukasik CQG(07)
[conformal Killing-Yano tensors].

**Tavis-Cummings Model** > s.a. Dicke Model.

* __Idea__: A model which
describes *N* two-level atoms interacting with one mode of the quantized
electromagnetic field containing an arbitrary number of excitations; It is the integrable approximation of the Dicke Model.

@ __References__: Fujii et al IJGMP(04)qp [evolution
operator], IJGMP(05);
Castaños et al PS(09)-a0909 [ground state, coherent-state description].

**Taylor Series** > see series.

**Teaching** > see physics teaching.

**Technicolor** > see composite models.

**Technology** > s.a. electronic
technology; microscopes;
nuclear technology;
optical technology;
quantum technology.

**Teichmüller Space** > see 2D
manifolds.

**Teleology** > s.a. causality; Final Cause.

* __Idea__: The notion that there is a purpose or ultimate end of physical laws.

* __Example__: Used as motivation for the strong anthropic principle.

@ __References__: Carlin HSPSA(12) [Robert Boyle and immanent teleology]; Sols a1301-in [the presence or absence of finality in nature is fundamentally outside the scope of the scientific method]; Visser a1707-FQXi.

> __Online resources__: see Wikipedia page.

**Teleparallel Structures** > see teleparallel structures and gravity; Elasticity.

**Teleportation** > see quantum technology.

**Temperley-Lieb Algebra**

@ __References__: Bytsko JMP(15)-a1505 [tensor space representations].

> __Online references__:
see Wikipedia page.

**10 j Symbols** > s.a. SU(2).

@

**Tendex Lines** > see spacetime; spacetime structure.

**Tensile Strength**

* __Idea__: The force required
to pull something such as rope, wire, or a structural beam to the point where it breaks.

* __Example__: Spider
silk has higher tensile strength than steel.

> __Online references__:
see ScienceDaily page;
Wikipedia page.

**Tension** > see force [gravitational vacuum tension or maximum tension];
matter dynamics in gravity; strings.

**Tensor** (including tensor product) > s.a. tensor fields.

**Tensor Density ** > see tensor fields.

**Tensor Models** > a generalization of Matrix Models.

@ __ General references__: Sasakura JMP(11)-a1104; Gurau a1209-proc;
Narain et al JHEP(15)-a1410 [physical states]; Bonzom et al NPB(15)-a1502 [mixing melonic and planar maps];
Sasakura & Sato a1506 [canonical, formal continuum limit and general relativity constraint algebra]; Dartois a1512-PhD.

@ __ Tensor field theories__: Ben Geloun a1601-conf [renormalizable]; Rivasseau Sigma(16)-a1603 [informal introduction, and quantum gravity], Sigma(16)-a1603 [constructive]; Banks & Fischler a1606 [and quantum gravity in *d*-dimensional Minkowski space]; Eichhorn & Koslowski a1701 [3D quantum gravity, functional renormalization group and continuum limit]; > s.a. approaches to quantum gravity [tensor track].

> __Related topics__: see causal dynamical triangulations.

**Tensor Networks**

* __Idea__:
A collection of tensors with indices connected according to a network pattern,
used to efficiently represent quantum many-body states of matter based on their
local entanglement structure; A tool used to approximate ground states of local
Hamiltonians on a lattice.

* __MERA__: Multiscale entanglement renormalization ansatz.

@ __General references__: Sachdev Phy(09);
Biamonte et al AIP(11)-a1012 [tensor network factorization];
Orús AP(14)-a1306 [introduction];
Bridgeman & Chubb JPA(17)-a1603-ln [with problems];
Nezami & Walter a1608 [multipartite entanglement structure];
Wille et al PRB(17)-a1609;
Al-Assam et al JSM-a1610 [software library];
Biamonte & Bergholm CP-a1708 [intro];
Robeva & Seigal a1710 [on hypergraphs];
Fishman et al a1711 [contracting];
Bhattacharyya et al a1805 [Lorentzian, and causal structures];
Glasser et al a1806 [and probabilistic graphical models].

@ __Renormalization__:
Dittrich et al NJP(16)-a1409 [decorated];
Evenbly & Vidal PRL(15)-a1412;
Sasakura & Sato PTEP(15)-a1501 [for random tensor networks];
Hauru et al PRB(18)-a1709 [using graph-independent local truncations];
> s.a. renormalization group.

@ __For gauge theories__: Tagliacozzo et al PRX(14)-a1405 [lattice];
Buyens et al PoS-a1511;
Silvi et al Quant(17)-a1606 [finite-density phase diagram];
Bañuls et al EPJwc(17)-a1611 [lattice, overcoming the Monte Carlo sign problem].

@ __Specific types of systems__: Alsina & Latorre a1312 [frustrated anti-ferromagnetic systems];
Orús AP(14) [introduction];
Orús EPJB(14)-a1407 [for strongly correlated systems];
Wahl PhD(15)-a1509,
Schrodi et al PRB(17)-a1703 [quantum many-body systems];
Bao et al a1709 [de Sitter spacetime and MERA].

@ __And gravity__: Chen et al PRD(16)-a1601 [emergent geometries];
Han & Hung PRD(17)-a1610 [and lqg];
May JHEP(17)-a1611,
a1709-MSc [for dynamic spacetimes];
Chirco et al a1701,
a1711 [and group field theory, Ryu-Takayanagi formula];
Han & Huang JHEP(17)-a1705 [discrete gravity and Regge calculus];
Dong & Zhou a1804 [entanglement,
spacetime as generative network of quantum states, and Susskind's QM = GR];
> s.a. Mike Zaletel talk.

@ __And holography__:
Orús EPJB(14)-a1407 [fermionic TNs, entanglement, MERA];
Ouellette Quanta(15);
Bao et al PRD(15)-a1504
+ Carroll blog(15)may [AdS/MERA correspondence, consistency conditions];
Bhattacharyya et al JHEP(16)-a1606 [perturbations and Coxeter construction];
Czech et al JHEP(17)-a1612 [with defects].

@ __Variants__: Rader & Läuchli PRX(18) [iPEPS, infinite projected entangled pair states];
Tilloy & Cirac a1808 [continuous].

> __Online references__:
see John Baez page;
Perimeter Institute tensor networks initiative
page.

> __Related topics__: see quantum phase transitions.

**Tensor-Vector-Scalar Theory of Gravity ** (TeVeS)
> s.a. MOG (scalar-vector-tensor MOdified Gravity theory); MOND.

* __Idea__:
A tensor-vector-scalar theory of gravitation proposed by Bekenstein in 2004;
It can be written as a bimetric theory, with the two metrics related by a disformal
transformation defined by a dynamical vector field and a scalar field; 2009, It seems
to have trouble simultaneously fitting lensing and rotation curves without any dark matter.

@ __General references__: Bekenstein PoS-ap/04;
Skordis et al PRL(06)ap/05 [large-scale structure];
Bekenstein & Sanders EAS(06)ap/05-proc [intro];
Zlosnik et al PRD(06) [equivalence to Einstein-ether-type theories];
Contaldi et al PRD(08)-a0802 [caustic problem];
Tamaki PRD(08)-a0803 [PN parameters];
Sagi PRD(10)-a1001 [gravitational waves]; Bekenstein PTRS(11)-a1201 [review]; Chaichian et al PLB(14)-a1402 [as a ghost-free, viable theory of gravity].

@ __And cosmology__: Skordis PRD(06)ap/05 [evolution and perturbations];
Díaz-Rivera et al PRD(06)ap [inflation];
news pw(06)jun;
Bourliot et al PRD(07)ap/06;
Dodelson & Liguori PRL(06)
+ pw(06)dec;
Zhao IJMPD(07),
MPLA(08)-a0802 [dark matter and dark energy];
Skordis PRD(08)-a0801, CQG(09)-a0903; Xu et al PRD(15)-a1412 [and observations, ΛCDM].

> __Other phenomenology__:
see gravitomagnetism.

**Tensorgluons** > see QCD.

**Ternary Operations**

@ __References__: Kerner IJGMP(08)
[in theoretical and mathematical physics, including
non-associative]; Curtright et al PLB(09)-a0903 [ternary
algebras and physics, classical and quantal]; Bocharov et al QIC-a1512 [quantum ternary arithmetics].

**TESS (Transiting Exoplanet Survey Satellite)** > see extrasolar systems.

**Tessellations** > see tilings.

**Tesseract**

* __Idea__: A hypercube, or 4D cube.

@ __References__: Aravind AJP(01)mar [toy and Bell's theorem].

**Test Body / Particle** > s.a. geodesic;
particle models; test-body motion.

* __Idea__: An idealized object used to define
and study properties of physical systems, usually fields; The test body couples to the system
(through its mass, electric charge, or other coupling parameter) and its response measures properties
of the system, but the effect of the test body itself on the system is considered to be negligible.

> __Online resources__:
see Wikipedia page.

**Test Space**

$ __Def__: A test space (*X*, Σ) consists of a set *X* together with a set of subsets Σ ⊆ 2^{X} such that the members of Σ cover X,
i.e., ∑ _{T }_{∈ Σ} *T* = *X*.

* __Related concepts__: A test is an element of Σ, while an element of *X* is an outcome; A subset of a test *T* is an event.

@ __References__: in Fritz & Leifer a1505/QPL.

**Tetrahedron** > s.a. Packings;
simplex; spin-foam models [quantum tetrahedron];
SU(2) [6*j*-symbol].

* __For tiling__: There
exist three kinds of monogenic filling of \(\mathbb R\)^{3}
by tetrahedra, but only one has a dual lattice with monogenic filling (truncated octahedra).

@ __References__: Korepanov n.SI/00 [5 tetrahedra on 5 points, and curvature].

> __Online resources__:
see Wikipedia page.

**Tetraquarks** > see hadrons.

**Teukolsky Equation** > see black-hole
perturbations [around the Kerr solution].

**TeVeS** > see under Tensor-Vector-Scalar Theory.

**Textures** > see topological defects.

**Theia** > s.a. moon.

* __Idea__: The proto-planet which collided with
the Earth in the Giant Impact that formed the Moon according to the most widely accepted scenario.

@ __References__: Meier et al Icarus(14)-a1410 [origin and composition]; Quarles & Lissauer Icarus(15)-a1410 [dynamical evolution]; Kaib & Cowan Icarus(15)-a1506.

**Theorem** > see proof theory;
also MathWorld page.

**Theory** > s.a. physical theories [including Theories of Everything];
physics and theory of physical theories.

@ __References__: Mielczarek & Trześniewski PLB(16)-a1601 [phase spaces of field values that are not affine spaces].

> __Theory space__: see types of distances;
types of quantum theories; symplectic structures in physics
[space of quantum field theories].

**Thermal Bath / Reservoir**
> s.a. heat; thermal radiation.

* __Idea__: (A.k.a. heat bath) A thermodynamic system with
a heat capacity so large that can be considered to be infinite, so that when it is coupled to a system of
interest the two will reach thermal equilibrium without affecting the temperature of the bath/reservoir.

@ __Finite thermal baths__: Potiguar & Costa PhyA(04) [thermodynamic relations];
Gemmer & Michel EPL(06)qp/05 [thermalization];
Ford & O'Connell PhyE(05)qp/06 [different models, and quantum oscillator];
Nechita & Pellegrini CM(09)-a0908 [statistical model]; Brody & Hughston JPA(16)-a1406 [quantum].

@ __Coupled to quantum system__: Ghosh et al PLA(05) [noise-induced transitions]; Hilt et al PRE(11)-a1106 [of harmonic oscillators, Hamiltonian of mean force];
Sergi JChemP(13)-a1306 [classical spin bath];
Ghesquière & Dorlas PLA(13)
[effect on the entanglement of a bipartite Gaussian state].

> __Online resources__: see Wikipedia page.

**Thermal Conductivity, Diffusivity ** > see Heat Flow.

**Thermal Dimension ** > see dimensionality of quantum spacetime.

**Thermal Expansion** > s.a. heat.

* __Idea__: The phenomenon by which
matter changes size in response to a change in temperature through heat transfer.

* __Thermal expansion coefficient__:
The volume and linear expansion coefficients are defined by *α*
= *V*^{–1} ∂*V*/∂*T*|_{P} and *β* = *P*^{–1} ∂*P*/∂*T*|_{V}, respectively.

@ __Thermal expansion coefficient__: Dounas-Frazer et al AJP(13)may [simple experiment, and measurement uncertainty].

@ __Negative thermal expansion__: Mary et al Sci(96)apr [Zirconium Tungstate ZrW_{2}O_{8}];
news livesci(04)dec;
Li et al PRL(11)
+ focus;
Handunkanda et al PRB(15)
+ news ea(15)oct [scandium trifluoride];
Wang et al PRL(16)
+ focus [lightweight metamaterials].

> __Online resources__:
see Wikipedia page.

**Thermal Field Theory** > see finite-temperature field theory.

**Thermalization** > see statistical mechanical equilibrium.

**Thermoacoustics** > see sound.

**Thermodynamic Limit**

* __Idea__: The thermodynamic
limit of a statistical mechanical system is the large-size limit of the system,
in which the values of the extensive variables of the system tend to infinity
with the intensive ones remaining constant, so the fractional fluctuations of
statistical quantities go to zero and are ignored.

@ __General references__: Compagner AJP(89)feb [continuum limit];
Styer AJP(04)jan [paradoxes];
Batterman SHPMP(05) [validity of idealized limit];
Huang et al PhyA(09) [extensive and non-extensive systems, MaxEnt approach];
Snoke et al AP(12)-a1112 [from quantum field theory];
Kuzemsky IJMPB(14)-a1402 [rev];
De Pasquale et al nComm(16)-a1504 [local quantum thermal susceptibility].

@ __Specific types of systems__: Kalogeropoulos EPJB(14)-a1312 [systems with long-range interactions].

> __Online resources__:
see Wikipedia page.

**Thermodynamics** > s.a. laws of thermodynamics;
thermodynamic systems; generalized versions.

**Thermoelectric Effect** > see electricity.

**Thermofield Dynamics** > Equivalent to a restricted version
of quantum statistical mechanics.

**Thermoluminescence**

* __Idea__: The phenomenon by which
certain materials that had previously absorbed energy that is trapped in electronic
excited states, emit light when heated.

@ __References__: Martini & Meinardi RNC(97);
Furetta RNC(98).

> __Online resources__:
see Wikipedia page.

**Theta Function**

* __Idea__: Theta functions are
special functions of several complex variables, with many applications in the
theories of abelian varieties and moduli spaces, quadratic forms, soliton theory and,
when generalized to a Grassmann algebra, they also appear in quantum field theory.

@ __References__: Gelca 14 [and knots].

> __Online resources__:
see Wikipedia page.

**Thin Sandwich** > see constraint equations
in general relativity [solution method]; Sandwich Conjecture.

**Third Degree Algebraic Equation** > see elementary algebra.

**Third Quantization**

@ __General references__: Das in(87);
Prosen & Seligman a1007,
Seligman & Prosen AIP(10)-a1011 [for open many-body bosonic systems].

@ __Of gravity__: Gielen & Oriti proc(12)-a1102 [matrix models and group field theory];
Ohkuwa & Ezawa CQG(12)-a1203, CQG(13) [*f*(*R*) theories]; Faizal CTP-a1407 [and the multiverse]; > s.a. multiverse; quantum cosmology formalism and techniques.

**Thirring Model** > s.a. renormalization group.

* __Idea__: A theory, originally
formulated in 2 spacetime dimensions, of a Dirac field with a quartic self-interaction.

@ __General references__: Thirring AP(58);
Ilieva & Thirring ht/98-conf [rev];
Faber & Ivanov ht/03 [dynamical conformal symmetry breaking];
Benfatto et al ht/06 [rigorous];
Korenblit & Semenov JNMP(11)-a1108 [massless, canonical quantization]; Bufalo & Pimentel IJMPA(14)-a1408 [massive, gauged, non-perturbative aspects].

@ __3D theory__: Gies & Janssen PRD(10)-a1006 [UV fixed-point structure]; Janssen & Gies PRD(12)-a1208 [chiral symmetry breaking, critical behavior].

> __Online resources__:
see Wikipedia page.

**Thomas Precession** > s.a. atomic
physics; geometric phase;
Rapidity; special-relativistic kinematics.

* __Idea__: The spin precession for a particle in electromagnetic fields.

* __Rem__: Thomas precession is a Berry phase that the particle's wavefunction picks up when subjected to cyclic changes of reference frames [Oblak CQG+].

@ __General references__: Rindler & Perlick GRG(90);
Urbantke AJP(90)aug;
Samuel PRL(96);
Philpott AJP(96)may;
Hamilton AJP(96)sep;
Muñoz AJP(01)may [in lab frame];
Bordovitsyn & Myagkii proc(01)qp [in classical and quantum theory];
Herrera & Di Prisco FPL(02)gq [approaches];
Behera ap/03,
ap/03,
ap/03 [gravitational];
Matolcsi & Matolcsi IJTP(05)mp;
Chrysos EJP(06) [the 1/2 factor];
Matolcsi et al GRG(07) [and rotating reference frames];
Klioner a0803 [explicit exact expression];
Rębilas FP(11)
[for the evolution of any vector quantity, and the Bargmann-Michel-Telegdi equation];
Dragan & Odrzygóźdź a1211 [quick derivation].

@ __In curved spacetime__: Jonsson AJP(07)may;
Silenko PRD(16)-a1606;
Oblak CQG(18) [gravitational generalization, BMS asymptotic symmetries].

**Thomas Scattering**

* __Idea__: The capture of a particle by a fast projectile.

@ __References__: Bitensky AJP(98)jul.

**Thomas-Fermi Statistical Model / Theory** > s.a. atomic physics.

@ __References__: Solovej MP(15)-a1601.

**Thomas-Wigner Rotation**

* __Idea__: Tghe spatial rotation involved in the composition of two non-collinear boosts.

@ __References__: Beyerle a1706 [visualization].

**Thomson Scattering**

* __Idea__: Scattering
of an electromagnetic wave by a free electron.

@ __References__: in Jackson 75;
Anderson gq/97
[in an Einstein-de Sitter universe].

> __Online resources__:
see Wikipedia page.

**Thought Experiment** > see under Gedankenexperiment.

**Threading** > s.a. decomposition.

* __Idea__: An alternative
approach to the decomposition of spacetime tensors, in which spacetime is
decomposed not using a foliation by spacelike hypersurfaces, but a congruence
of timelike curves representing a family of observers; It leads to an alternative evolution formulation for covariant field equations.

@ __References__: Ehlers in(59);
Roy a1405.

> __Related topics__: see velocity [3-velocity].

> __In gravity theory__:
see canonical formulation of general relativity and modified approaches.

> __Other applications__: see casimir
effect; QED in curved spacetimes.

**Three-Body Physics**

@ __Three-body forces__:
Hammer et al RMP(13) [colloquium].

> __Three-body problem in classical mechanics__:
see classical systems; orbits in newtonian gravity.

> __Three-body states in quantum mechanics__:
see composite quantum systems; Efimov Effect.

**3 j-Symbols** > s.a. SU(2) group [6

*

@

>

**Throttling** > see Joule-Thomson Effect.

**Thurston Norm**

@ __References__: Friedl & Kim Top(06)
[fibered manifolds and twisted Alexander polynomials].

**Tidal Acceleration / Force** > s.a. geodesics [geodesic deviation];
gravitation.

@ __References__: Asenbaum et al PRL(17)-a1610 + Jaffe & Müller Phy(17) [on an individual particle's wave function].

> __Specific spacetimes__:
see dynamics of gravitating bodies; reissner-nordström spacetime.

> __In specific theories__: see Metric-Affine
Theories; newtonian gravity.

**Tidal Tensor** > see gravitomagnetism.

**Tikhonov Theorem** > see under Tychonoff Theorem.

**Tiling** > s.a. random
tiling; voronoi tiling.

**Tilt in Cosmology** > s.a. bianchi models.

@ __References__: Coley et al PLB(06) [and phantom matter behavior].

**Time**
> s.a. time in physical theories [and in classical gravity,
quantum theory, quantum gravity].

> __Online resources__:
see arrow of time; CPT symmetry [time reversal].

**Time Crystal**
> see classical systems; crystals [quantum time crystals].

**Time Delay** > see tests of general relativity.

**Time Dilation** > see kinematics of special relativity;
tests of gravity with light [gravitational time dilation].

**Time Machine / Travel** > see causality violations.

**Time-Ordered Product**

* __Idea__: A tool in quantum field theory introduced by Wick.

$ __Def__: In the algebra
of position-dependent (or often just time-dependent) boson and fermion operators
on Fock space, it is defined on products by the time-ordering

T(*A*_{1}(*x*_{1})
*A*_{2}(*x*_{2})
...* A*_{n}(*x*_{n}))
:= (–1)^{p} *A*_{p1}(*x*_{p1})* A*_{p2}(*x*_{p2})
... *A*_{pn}(*x*_{pn})
,

where {*p*_{i}} is a permutation
of {1, 2, ..., *n*} such that *t*_{pi} ≥ *t*_{pi+1},
or *x _{pi}* ∈

T(*A*+*B*) = T(*A*) + T(*B*) , T(*cA*) = *c* T(*A*) .

* __Wick's theorem__: The
time-ordered product of *n* operators is equal to the sum of all possible
normal products formed with all possible contractions.

@ __General references__: Wick PR(50);
Hollands & Wald CMP(01)gq,
CMP(02)gq/01 [in curved spacetime, covariant].

@ __Time-ordered exponentials__: Ebrahimi-Fard & Patras LMP(14) [and enveloping algebras of pre-Lie algebras].

**Time Orientability** > see orientation.

**Time Reversal **> see CPT theorem.

**Time-Slice Axiom in Quantum Field Theory**

* __Idea__: The observables
which can be measured within an arbitrarily small time interval suffice to
predict all other observables; Well known for free field theories, where
the validity of the axiom is an immediate consequence of the field equations,
and for certain superrenormalizable models in 2 dimensions; 2008, Extended
to scalar field theories in globally hyperbolic spacetimes within formal renormalized
perturbation theory.

@ __References__: Chilian & Fredenhagen CMP(09)-a0802
[for interacting scalar fields on globally hyperbolic spacetimes].

**Time-Symmetric Quantum Theory** > see modified formulations of quantum mechanics.

**Time-Translation Invariance **> see symmetries in quantum theory.

**Timelike Curve** > see
lines; Worldlines.

**Timescape** > see cosmological acceleration and inhomogeneities.

**Tipping Pencil / Rod Problem** > see quantum effects.

**Tired Light**

* __Idea__: A 1929 proposed explanation
by Zwicky for the cosmological redshift, based on the loss of energy by light to
intergalactic gases rather than cosmological expansion; One objection to it is that
it does not explain the time-dilation of supernova light curves.

**Titius-Bode Law**

* __Idea__: The most stable
situation for a planetary system is achieved when each planetary orbit
is roughly twice as far from the Sun as the preceding one; Empirically
observed by Titius (1766) and Bode (1778).

@ __And resonances__: Bass ap/02
+ Bass & Del Popolo IJMPD(05)ap/04;
Christodoulou & Kazanas a1507 [physical interpretation].

**Titchmarsh's Theorem** > see dispersion.

**TOE (Theory of Everything)**
> see physical theories; unified theories.

**Toeplitz Operators**

@ __References__: Boutet de Monvel & Guillemin 80.

**Tokens**

* __Idea__: Tokens allow
to express a semigroup on one set via a semigroup convolution on another set;
They are similar to intertwining operators, but more flexible.

@ __References__: Kisil m.FA/02-in.

**Tolman Solutions / Wormholes** > s.a. cosmological
models in general relativity; cosmological perturbations; regge
calculus; wormholes.

* __History__: Introduced by Tolman in 1934.

@ __References__: Tolman PNAS(34)
– reprinted GRG(97);
Ribeiro ApJ(92)-a0807,
ApJ(92)-a0807 [and
self-similar cosmology]; Hellaby PRD(94)gq/99 [Vaidya
limit]; > s.a. Lemaître-Tolman-Bondi
Solutions.

**Tolman-Ehrenfest Effect**

* __Idea__: The variation of temperature in space
so that *T* (g_{00})^{1/2}
remains constant.

@ __References__: Rovelli & Smerlak CQG(11)-a1005 [and the "speed of thermal time"].

**Tolman-Oppenheimer-Volkoff Equations / Solutions**
> s.a. gravitating bodies / semiclassical
general relativity; torsion phenomenology.

* __Idea__: The general relativistic equation
for hydrostatic equilibrium in a static spherically symmetric spacetime supported by an
isotropic perfect fluid, used to model stars.

@ __References__: Semiz SHPMP(16)-a1610;
Carballo-Rubio PRL(18)-a1706 [semiclassical gravity corrections].

**Tomboulis-Yaffe Inequality**

* __Idea__: An inequality
in lattice gauge theory stating that a system in a box that is sufficiently
insensitive to boundary conditions has a non-zero mass gap.

@ __References__: Kanzawa AP(09) [generalized from SU(2) to SU(*N*)].

**Tomimatsu-Sato Solutions** > s.a. solutions in general relativity.

@ __References__: Manko et al G&C(99) [ring singularities];
Hikida & Kodama gq/03-proc;
Kodama & Hikida CQG(03)gq;
Gegenberg et al CQG(11)-a1010 [holography and quantum gravity].

**Tomita Representation**

@ __References__: in Chew et al a1703.

**Tomography / Tomographic Representation**

* __Idea__: An approach to the
reconstruction/estimation of the state of a physical system based on a possibly
incomplete set of data.

@ __General references__: Asorey et al PS(15)-a1510 [rev, classical and quantum].

> __Classical systems__: see earth;
dirac fields; scalar fields.

> __And quantum theory__: see representations
of quantum mechanics and quantum field theory; canonical
quantum gravity; quantum states and mixed states.

**Tonks Gas** > see Cluster Expansion; VirialExpansion.

**Tonks-Girardeau Gas** > see Fermionization.

**Topological Algebra** > see algebra / functional analysis.

**Topological Censorship** > see models of topology change.

**Topological Defects** > s.a. cosmic strings.

**Topological Entropy / Order**
> s.a. mixed quantum states.

* __Idea__: A measure of the
complexity of a dynamical system or mapping, giving the exponential growth
rate of the number of distinguishable orbits when the map is iterated.

@ __References__: Dikranjan et al T&A(12) [for mappings of uniform spaces and topological groups];
Giordano Bruno T&A(12),
Dikranjan & Giordano Bruno T&A(12) [vs adjoint algebraic entropy].

> __Online resources__:
see MathWorld page;
PlanetMath page;
Scholarpedia page;
Wikipedia page.

**Topological Glass**

* __Idea__: A disordered
system in which the configurations are topological; For example, a model
whose state space is given by all triangulations of a sphere with *N*
nodes, half of which are red and half are blue.

@ __References__: Eckmann & Younan PMB(12)-a1104 [decay of correlations].

**Topological Gravity** > s.a. BF theory;
black holes in 3D and in modified theories;
3D gravity.

* __Idea__: The reduction of
general relativity to spacetimes with vanishing curvature, but with global
topological degrees of freedom.

* __Action__: Given by Horowitz
in terms of a Lie-algebra-valued 2-form *B*_{ab
ij}, and a connection *ω*_{aij}
in the *BF* form

*S*[*ω*,* B*]
= ∫_{M}* B*_{ij} ∧ *R*^{ ij} ;

Variation of the action with respect to *B*_{ij} gives
that *ω*_{ij} is
flat, so there are local degrees of freedom only if one admits torsion.

@ __General references__: Myers & Periwal NPB(91);
Nakamichi et al PRD(91);
Oda & Sugamoto PLB(91);
Wu JGP(93);
Waelbroeck & Zapata CQG(94)gq/93 [canonical];
Zapata CQG(96)gq [self-dual, lattice];
Thuillier JGP(98);
Mignemi PRD(99)gq/98 [4D];
Constantinidis et al CQG(04)gq [approaches]; Salgado et al PLB(14)-a1311 [action in even dimensions as gauged Wess-Zumino-Witten term].

@ __And general relativity__: Montesinos CQG(01)gq [self-dual
general relativity with topological term]; Smolin & Starodubtsev
ht/03 [larger
theory].

@ __Variations__: Mitskievich et al a0706-MGXI
[with topological invariants for coefficients]; > s.a. 3D
gravity; black holes in modified theories [quasi-topological gravity]; Topologically Massive Gravity.

**Topological Group**

$ __Def__: A group *G* such
that the map *m*: *G* × *G* → *G* by *m*(*g*,* h*)
= *gh* is continuous (this combines requirements on the product and the inverse).

* __Properties__: Its fundamental group is always commutative.

* __Amenable topological group__: *G* is
amenable if there is a positive *G*-invariant
functional *A* (a "measure") on the Banach space of bounded
Borel measurable functions on *G* such that *A*(1_{G})
= 1; For example, SO(3) is not amenable with respect to the discrete topology,
but it is as a compact Lie group;
SO(*n*,1) is not amenable; Local gauge groups
also have topologies different from their natural ones with respect to which
they are amenable.

@ __General references__: Pontrjagin 66;
Greenleaf 69 [means];
Arhangel'skii & Tkachenko 08.

@ __Amenable groups__: Pier 84;
Carey & Grundling LMP(04)mp [gauge groups];
Seiler a1011 [theories with non-amenable symmetry groups].

@ __Related topics and results__: Megrelishvili T&A(08) [every Hausdorff topological group is a group retract of a minimal topological group].

@ __Quasitopological groups__:
Hernández & Tkachenko T&A(06) [pseudocompact]; Li & Mou T&IA(14) [semimetrizability].

@ __Other generalizations__: Arhangel'skii & Reznichenko T&A(05)
[paratopological and semitopological groups].

**Topological Insulators** > see under Insulators.

**Topological Manifold** > see types of manifolds.

**Topological Materials** > s.a. Topological Glass.

* __Idea__: A topological material
is one with a property that is robust and insensitive to perturbations and impurities.

@ __References__: collection Phy(16) [topological phases of matter].

**Topological Particle Theory**

@ __References__: Finkler & Jones PRD(85);
Jones & Finkler PRD(85).

**Topological Polariton** > see Quasiparticles.

**Topological Quantization** > see formulations
of quantum theory.

**Topological Recursion**

* __Idea__: A relation appearing in the asymptotic expansion of many integrable systems and in enumerative problems.

@ __References__: Eynard a1412-proc [rev].

**Topological Semigroup** > see Semigroup.

**Topological Space** > s.a. types
of topological spaces; topology in physics.

**Topological Tangent Bundle**

$ __Def__: A neighborhood
of the diagonal in *M* × *M*.

@ __References__: Milnor Top(64).

**Topological Transitivity**

* __Idea__: A property of dynamical
systems that is a form of mixing, defined using just the topology of the underlying
space and not a measure.

@ __References__: Akin & Carlson T&A(12)-a1108 [relationships among definitions].

> __Online resources__: see Encyclopedia of Mathematics page; MathWorld page; Scholarpedia page; Wikipedia page on mixing systems.

**Topological Vector Space** > see vectors.

**Topologically Massive
Gravity** > see 3D massive gravity theories.

**Topologically Massive Gauge / Yang-Mills Theory
** > see types of gauge theories; types of yang-mills theories.

**Topology** > s.a. topology
of the universe; topology in physics.

**Topology Change** > s.a. dynamical
models.

**Topos Theory** > s.a. category theory in physics;
history of mathematics.

* __Idea__: A branch of category
theory, which generalizes set theory.

* __In physics__:
Topos theory has been suggested first by Isham and Butterfield, and then by
Isham and Döring, as an alternative mathematical structure within which
to formulate physical theories; In this view, constructing a theory is equivalent
to finding a representation in a topos of a certain formal language that is attached
to the system; For example, classical physics uses the topos of sets, while other
theories involve a different topos; Used to assign values to quantities in quantum
theory (Isham & Butterfield).

* __And quantum mechanics__: There
are two topos-theoretic approaches to quantum mechanics in
the literature, the "contravariant approach", proposed by Isham and Butterfield,
and later extended by Döring and Isham, and the "covariant approach", developed
by Heunen, Landsman and Spitters.

@ __General references__: Bell 88; Wyler 91; Mac Lane & Moerdijk 92;
Nishimura IJTP(96)
[quantization]; Nakayama JMP(13) [topologies induced by quantization]; Eva EPTCS(15)-a1511 [connection with topos quantum theory]; Gauthier a1802 [Segal topos used to model natural phenomena].

@ __In physics__: Döring & Isham in(11)-a0803
["what is a thing?"]; Tsatsos
a0803-dipl; Isham
a1004-in [intro];
Döring a1004-in [daseinisation]; Flori a1110, a1207-ln; Wolters JMP(14)-a1309; Flori 13 [textbook]; > s.a. poisson structures [Poisson algebras for non-linear field theories in the Cahiers topos].

@ __And gravity / geometry__: Grinkevich gq/96;
Guts & Grinkevich gq/96;
Isham & Butterfield
FP(00)gq/99-in;
Raptis gq/01-conf
[review]; Zafiris FP(04),
IJGMP(06)
[quantum events]; Król FP(06),
IJGMP(07);
Döring
& Isham JMP(08)qp/07,
JMP(08)qp/07,
JMP(08)qp/07,
JMP(08)qp/07;
Raptis
IJTP(07);
Król
a0804/FP [and
the cosmological constant]; > s.a. approaches
to quantum gravity;
causal sets; quantum
spacetime.

@ __And quantum theory__: Isham & Butterfield IJTP(98),
Butterfield & Isham IJTP(99)qp/98,
Hamilton et al IJTP(00)qp/99,
Butterfield & Isham IJTP(02)qp/01 [Kochen-Specker];
Isham & Butterfield FP(00)
[and quantum gravity]; Fearns qp/02|IJTP;
Heunen et al CMP(09)-a0709 [algebraic
quantum mechanics]; Döring a0712-proc
[rev]; Wolters CMP(13)-a1010 [comparison of two approaches]; Flori a1106 [rev]; Döring & Soares Barbosa a1107-proc [and 'unsharp values' of physical quantities]; Nakayama a1109; Brenna & Flori a1206; Corbett EPTCS(12)-a1210; Nakayama JMP(14)-a1404; Freytes et al FP(14)-a1412 [physical properties as modal operators]; Nakayama JMP(16)-a1507 [reduced sheaf-based quantum theory]; Miyazaki a1605; > s.a. histories
formulations;
interpretations [modal]; logic; quantum computing; state-vector
reduction.

> __Online resources__: see Wikipedia page.

**Toronto Space**

* __Idea__: A
topological space that is homeomorphic to every one of its full-cardinality subspaces.

@ __References__: Brian T&IA(14) [Toronto problem].

**Torsion Balance** > s.a. Cavendish Experiment.

@ __References__: Adelberger a1308-conf [and probes of fundamental physics].

**Torsion Invariant of a Manifold** > see 3D
manifolds.

**Torsion Pendulum** > see Pendulum.

**Torsion Subgroup of an Abelian Group** > see
group types.

**Torsion Tensor** > s.a. torsion
in physical theories; torsion phenomenology.

**Tortoise Coordinate** > s.a. schwarzschild
geometry [Eddington–Finkelstein coordinates].

$ __Def__: The coordinate *r**:=
*r* + 2*GM* log(*r*/2*GM* – 1) used for Schwarzschild
spacetime as part of the Eddington-Finkelstein coordinates, or in the Kruskal
Extension.

@ __References__: Li & Zhao NCB(95) [in a non-static spacetime].

**Torus** > see differentiable manifolds.

**Totally Disconnected Space** > see connectedness.

**Totally Geodesic Mapping** > s.a. harmonic
map [especially, condition
on *f*].

$ __Def__: A mapping *f* : *M* → *N* between
metric manifolds, such that all *M*-geodesics are mapped to *N*-geodesics.

**Totally Geodesic Submanifold**

$ __Def__: A submanifold *M* of
a manifold *N* such that a geodesic which is tangent to *M* at
a point lies entirely in *M*.

**Totally Vicious Spacetime** > see types
of spacetimes.

**Trace-Class Operator** > see operator
theory.

**Trace Dynamics** > s.a. origin of quantum mechanics.

* __Idea__: A classical-like theory of non-commutating variables proposed as a model by Stephen Adler for the emergence of quantum mechanics; In it, statistical averages give the Schrödinger equation and operator algebra of quantum theory, while Brownian motion corrections give the low level noise on which phenomenological reduction models are based.

@ __References__: Adler 04 + JPCS(12); Adler JPCS(14)-a1401 [the future of quantum mechanics].

**Trace Formulas** > s.a. quantum
chaos.

* __Idea__: Formulas that
establish links (dualities) between the energy spectra in the quantum description
of a system and the spectrum
of actions of periodic orbits in the Newtonian description; The dualities
hold for chaotic as well as for integrable systems.

@
__General references__: Cohen et al AP(98)
[and spectrum statistics]; Primack & Smilansky JPA(98),
Smilansky JPA(00)
[semiclassical]; Sugita AP(01)
[and phase space path integrals].

@ __Selberg's__: Marklof LNP-m.SP/04
[intro].

**Tractor Beam** > s.a. sound [acoustic tractor beam].

@ __References__: Gorlach et al PRL(17) [pulling force from a quantum-mechanical matter wave].

**Tractor Calculus** > see Conformal Geometry.

**Trajectories in Classical Mechanics**
> s.a. orbits of gravitating bodies, in newtonian
gravity, of test particles.

* __Rem__:
They can be obtained from the least action variational principle
(> s.a. classical mechanics).

@ __General references__: Holland IJTP(12) [constructing field evolution using a continuum of trajectories].

@ __From wave equations__: Orefice et al a0705 [Helmholtz-like equations in electromagnetism and quantum mechanics];
> s.a. wave mechanics.

**Trajectories in Qantum Mechanics**
> s.a. classical limit; histories-based quantum theory;
pilot-wave interpretation.

* __Idea__:
Trajectory-based approaches include the de Broglie-Bohm
and Nelson's stochastic interpretations.

@ __ General references__: Galehouse IJTP(81); Cufaro Petroni & Vigier FP(92)
[and interference]; Griffiths PRL(93);
Yamada PRA(96) [probabilities];
Holland FP(98),
PRA(99);
Floyd in(02)qp/00;
Hiley et al qp/00;
Bouda & Djama PS(02)qp/01;
Brun AJP(02)jul [models];
Hall JPA(04) [incompleteness of trajectory-based interpretations];
Moser AP(08);
John AP(10)-a1007,
a1007-conf [complex trajectories, and dynamical origin of quantum probability];
Olivares et al PhyA(11) [from classical phase space];
Błaszak & Domański PLA(12) [based on Moyal description of quantum theory];
Hiley et al a1610 [Dirac, Moyal and Bohm];
Vink a1711 [trajectories and quantum field theory].

@ __And measurement__: Wiseman QSO(96)qp/03; Steinberg qp/03-ch; Bauer et al JPA(16)-a1512.

@ __Past of a quantum particle__: Vaidman PRA(14)-a1312; Tsang a1403 [inferring the past of quantum observables]; Svensson a1407; Englert et al PRA(17)-a1704; Paneru & Cohen a1709 [in an entangled state]; Hashmi et al a1710;
Bhati & Arvind a1807 [and weak measurement];
> s.a. photons.

@ __Quantum states as ensembles of trajectories__:
Schiff & Poirier JChemP(12);
Poirier a1208 [relativistic particles];
Oaknin a1306.

@ __Modified quantum theory__: Wiseman & Gambetta PRL(08) [none for non-Markovian stochastic Schrödinger equations].

> __Related topics__: see ergodic theory;
experiments in quantum theory; quantum fluctuations
[fermions and classical paths].

**Transcendental Functions**

@ __References__: Yukalov & Yukalova PLA(07) [method of self-similar factor approximants].

**Transcendental Number** > see types of numbers.

**Transfer-Matrix Method**

* __Idea__: A technique for finding the partition function of certain types of systems in statistical mechanics.

@ __References__: Sánchez-Soto et al PRP(12) [1D lossless systems];
Pujol et al EJP(14)
[1D quantum particle in a stationary potential].

> __Online resources__:
see Wikipedia page.

**Transition Amplitude** > see path-integral quantum mechanics; vacuum.

**Transition Function** > see fiber bundles.

**Transition Matrix** > see S-Matrix.

**Transition Radiation** > see radiation.

**Transitive Action of a Group on a Manifold** > see group action.

**Transitive Relation** > see Relation.

**Transitivity, Topological** > see under Topological Transitivity.

**Translation**

* __Idea__: An element
of the translation subgroup of the Euclidean or Lorentz group.

> __Translational symmetry__:
see gauge theories; Homogeneity;
Position [tests of position invariance].

**Transmission** > see quantum systems [potential steps];
wave phenomena.

**Transport Phenomena** > s.a. chaos [anomalous];
graphs; networks;
non-equilibrium statistical mechanics.

* __Idea__: Non-equilibrium
(but possibly steady-state) statistical mechanics processes in which there is
a net flow of some quantity inside a system; Examples are heat flow (by conduction),
electric conduction, convection (related to viscosity), particle transport or diffusion.

@ __Books__: Lundstrom & Jeong 13 [near-equilibrium, and application to nano-devices];
Soto 16 [and kinetic theory]; Venerus & Öttinger 18.

@ __Methods__: Desloge AJP(64)oct,
AJP(64)oct [coefficients from Boltzmann equation];
Mackay et al PhyD(84) [in Hamiltonian systems];
Arlotti et al m.AP/06 [semigroups];
Keanini PRS(07) [based on random walk];
Rossani PhyA(07)
[electrons and phonons, integral equations for transport coefficients];
Cohen JSP(09) [failure of density expansion of transport coefficients];
Malyshev & Zamyatin a1110 [stochastic models];
Berkolaiko & Kuipers JMP(13)-a1307 [semiclassical treatment, combinatorial theory];
Engibaryan & Barseghyan TMP(14)
[general theory, homogenous half-space and plane layer].

@ __Turbulent__: Bakunin PhyA(05)
[weak compressibility, percolation approach].

@ __Random medium__: Panasyuk et al JPA(06)mp/05 [homogeneous];
Corngold JSP(05) [and diffusion].

@ __In other media__: Marklof a0909-proc
[in crystals, failure of linear Boltzmann equation]; Halperin & Bergman
PhyB(10)-a0910-conf
[in inhomogeneous and disordered media, contributions by Landauer]; Denicol et al PLB(12) [relativistic fluid, viscous transport coefficients].

@ __Quantum__: Rau & Müller PRP(96)
[irreversible]; Rammer 98 [disordered
conductors, localization, ...]; Nazarov & Blanter 09; Buot 09 [in nanosystems]; Mahajan et al a1608 [for interacting quantum systems, based on entanglement structure].

@ __Related topics__:
Suzuki PhyA(11), PhyA(12) [irreversibility and entropy production].

> __Related topics__: see Boltzmann
Equation; Kadanoff-Baym
Equations.

> __Specific phenomena__: see Conductivity;
diffusion; Heat Flow; Percolation;
viscosity.

**Transversality Theory**

@ __References__: in Banyaga & Hurtubise 04.

**Transverse Gauge in General Relativity** > see gauge.

**Transverse Gravity** > see unimodular relativity.

**Trapped Surface** > s.a. horizons [apparent horizon].

* __Idea__: A compact spacelike
2-surface without boundary, such that both families of null geodesics orthogonal
to it (incoming and outgoing) converge,
for each point on the surface; Used in singularity theorems.

@ __General references__: Clarke CQG(88);
Beig & Ó Murchadha PRL(91),
CQG(94);
Malec & Ó Murchadha PRD(96)gq/95 [cosmological];
Senovilla JHEP(03)ht [not
with vanishing curvature scalars], gq/03-conf,
m.DG/04-proc
[trapped submanifold]; Mitra gq/05 [occurrence];
Krishnan CQG(08)-a0712
[review, isolated-dynamical horizons and applications]; Andersson AIP(09)-a0901;
Hayward IJMPD(11)-a0906 [types]; Senovilla IJMPD(11)-a1107-ln [rev and applications]; Carrasco a1201-PhD [in spacetimes with symmetries, and applications to uniqueness theorems].

@ __Spherical symmetry__: Malec & Ó Murchadha PRD(94)gq;
Iriondo et al PRD(96) [asymptotically flat].

@ __Marginally trapped surfaces__:
Carrasco & Mars CQG(09)
[stability]; > s.a. hawking radiation.

@ __Other special cases__: Galloway et al CMP(12)-a1005 [2+1, non-existence result]; Bengtsson a1112-ch [simple examples]; Jakobsson CQG(13)-a1208 [2+1].

@ __Criteria for formation__: Bizoń et al CQG(89)
[spherical symmetry]; Mitra
gq/98 [spherical];
Andersson et al PRL(05); Klainerman et al IM(14)-a1302 [fully anisotropic]; Bini & Esposito a1705 [intro].

@ __And event horizons__: Claudel gq/00;
Dafermos CQG(05)gq/04 [spherical, and
\(\cal I\)]; Åman et al JPCS(10)-a0912
[and apparent horizons].

@ __Related topics__: Ellis GRG(03)gq [compact
cosmology]; Dain CQG(04)gq/03 [as inner boundary for constraints].

> __Examples__: see Vaidya Spacetime.

**Trapping Horizon** > see horizons.

**Traveling Waves** > see types of waves.

**Tree** > see graph.

**Tree Diagram** > see quantum field theory formalism.

**Triality**

* __Idea__: A relationship among three objects, often vector spaces.

@ __References__: Smolin IJTP(17)-a1503 [trialities and the foundations of dynamics].

**Triangle** > see simplex.

**Triangle Inequality** > see inequalities.

**Triangulable Topological Space**

$ __Def__: A topological
space *X* homeomorphic to some polyhedron *K*.

* __Remark__: The polyhedron *K* (which
is not unique) is a triangulation of *X*.

* __Example__: Every paracompact
manifold is triangulable [@ Whitehead AM(40)].

**Triangular Number** > see number theory.

**Triangulation**
> s.a. simplices [triangles]; tiling [including physics].

$ __Def__: A simplicial
complex that covers (is homeomorphic to) a manifold; A special case of tiling.

* __Irreducible__: A triangulation
of a surface is irreducible if there is no edge whose contraction produces
another triangulation of the surface.

* __Special cases__: The triangulations
of the disc with *n* + 2 vertices on its boundary are counted by the *n*th
Catalan number [@ in Benardi & Rué EJC(12)].

* __Pseudo-triangulation__:
A partition of (the convex hull of a point set *S* in) the manifold into
interior-disjoint pseudo-triangles (whose vertices are points of *S*).

@ __General references__: Lackenby G&T(00)m.GT [3D,
taut]; Jaco & Rubinstein JDG(03)
[3D, 0-efficient]; Effenberger JCTA(11) [any dimensionality, tight triangulations]; Alagic & Bering a1108 [quantum algorithms for invariants].

@ __Statistics, enumeration__:
Aste JPA(98);
Dumitrescu et al CG(01)
[enumerating paths]; Bespamyatnikh CG(02);
Aichholzer et al CG(04)
[number of triangulations on a set of points in flat 2D]; King & Pelsmajer DM(10)
[size of dominating sets]; Rivasseau EPL(13)-a1303 [number of triangulations of 3D homology spheres]; Adiprasito & Benedetti a1710 [Cheeger-type exponential bound].

@ __Computational aspects__: Castelli Aleardi et al IJCGA(11) [efficient representations of 2D triangulations].

@ __Of spheres__: King G&T(01)
[of S^{3} with
given link];
Brinkmann & McKay DM(05)
[of
S^{2}, minimum degree 5]; Benedetti NPB(17)-a1608 [not all 3-balls are Mogami].

@ __Of other spaces__: Nakamoto & Tsuchiya DM(12) [of the Möbius band].

@ __Different triangulations on the same space__:
Ontaneda Top(02)
[space with non-equivalent triangulations]; Pournin & Liebling CG(07)
[paths in the flip-graph of triangulations]; Wilson T&A(07)
[convergence of cochain
algebra under
refinement]; Joret & Wood JCTB(10) [irreducible]; Jia et al DM(13) [balanced triangulations].

@ __As a dynamical system__:
Collet & Eckmann JSP(05)mp/04,
Eckmann JSP(07)
[triangulations of S^{2}];
> s.a. dynamical triangulations.

@ __Causal triangulations__:
Durhuus et al JSP(10) [spectral dimension];
Sisko et al JSP(13) [growth process];
> s.a. dynamical triangulations.

@ __Pseudo-triangulations__:
Aichholzer et al CG(04);
Bereg CG(05) [of the plane];
Orden et al DM(07) [and rigid planar graphs].

@ __Combinatorial aspects__: Bagchi & Datta DM(05);
> s.a. cell complex; graphs
and graph types.

@ __Random triangulations__: Guitter AHP(17)-a1511 [distance-dependent two-point function];
> s.a. statistical geometry [and computational].

@ __Other types__: Datta & Singh JCTA(13)
[vertex-transitive tight triangulations of manifolds].

> __Types of triangulations__: voronoi tiling
[Delaunay triangulations]; Whitehead Triangulation.

> __Related structures__:
see connection; distances
[on the set of triangulations of *M*].

**Tribology** > see Friction.

**Trinion**

$ __Def__: A 3-punctured sphere.

**Trion**

* __Idea__: An object with
three types of charges; 2011, For example, in semiconductors, trions consisting
of one electron bound to two holes via Coulomb forces have been observed in a
variety of systems, most recently in the optical spectra of doped carbon nanotubes.

@ __References__: Eto a1001;
Hohenester & Goldoni Phy(11) [in carbon nanotubes].

**Triple Point** > see spin models [XXZ model].

**Triple-Product Rule**

$ __Def__: The fact that, for three variables *x*,
*y* and *z* with a functional dependence between them,

(∂*z*/∂*y*)_{x} = –(∂*x/∂y*)_{z} / (∂*x/∂z*)_{y} .

**Triple-Slit Interference** > see
interference; pilot-wave
interpretation.

**Triplectic Quantization** > see quantization
of constrained systems.

**Triplet** > see hilbert space [rigged].

**Trispectrum**

* __Of cosmological perturbations__: The connected four-point correlation function; One motivation for studying it is that the trispectrum has weaker dependence on non-linear clustering than the bispectrum, and its measurement may provide useful additional constraints on primordial non-Gaussianity; > see cosmological matter distribution.

@ __References__: Verde & Heavens ApJ(01)ap [as a Gaussian test].

**Trivialization, Local** (Of a fiber bundle)

$ __Def__: Given a fiber
bundle with base space *B* and fiber *F*, and a covering {*U*_{i}}
of *B*, a trivialization is an assignment, for each *j*, of
a map *φ*_{j}: *π*^{–1}(*U*_{j}) → *U*_{j} × *F*,
which is a homeomorphism.

**Trojan State** > see quantum states.

**Tropes**

@ __References__: Livanios JGPS(07) [and spacetime].

**Tropical Geometry / Mathematics**

@ __References__: Litvinov a1005 [introduction and applications];
Angelelli a1701 [and the micro-macro correspondence].

> __Online resources__:
see Wikipedia page.

**Tropical Limit of Statistical Mechanics**

* __Idea__: The theory one gets
in the limit *k*_{B} → 0.

@ __References__: Angelelli & Konopelchenko PLA(15)-a1502.

**Trouser Spacetime** > see models of topology change.

**Trouton-Noble Paradox**

* __Idea__: There is a 3D
torque *T* in an inertial frame *S* in which a thin parallel-plate
capacitor is moving, but there is no 3D torque *T'* in *S'*,
the rest frame of the capacitor.

@ __References__: in Panofsky & Phillips 62;
Ivezić FP(07).

**Truth** > s.a. Explanation;
logic [truth values in quantum theory]; physical theories.

* __In science__: We discover individual facts
that are objectively true, but is our entire view of the universe, based on our current scientific
theories, true? We can't even assume that we are making substantial progress toward knowing the
truth about the universe, because we don't know how far our current theories are from the truth;
However, we can measure the extent to which our present theories explain what we can currently
examine [@ from Winkler letter PT(17)apr].

* __About the universe__: Can you arrive at the truth by a method other than science?

@ __References__: Margenau PhSc(34)oct [flexibility];
Breuer & Springer GRG(09) [in science].

**Tsallis Non-Extensive Entropy, Statistics** > see critical
phenomena; non-extensive statistics.

**Tsirelson Bound** (a.k.a. Cirel'son Bound)
> see types of quantum correlations.

**Tsirelson's Problem**

* __Idea__: The question
whether all quantum correlation functions between two independent observers
represented as commuting observables on a joint Hilbert space can also be expressed
using observables defined on a Hilbert space of tensor product form; Tsirelson
showed that the distinction is irrelevant if the ambient Hilbert
space is finite-dimensional.

@ __References__: Scholz & Werner a0812 [finite vs infinite dimensionality];
Avis et al IEICE(09)-a0812 [Tsirelson's theorem, for non-specialists].

**Tube Formula** > see under Weyl Tube Formula.

**Tulczyjew Triples** > s.a. geometric formulations of quantum theory.

@ __References__: Grabowska & Grabowski JGM(13)-a1306;
Grabowska & Vitagliano JGM(15)-a1406 [in higher-derivative field theories].

**Tully-Fisher Relation**
> s.a. galaxies / cosmological expansion; dark matter;
Modified Gravity; MOND and astrophsyics/cosmology.

* __Idea__: The more massive (and therefore brighter) a spiral galaxy is, the faster it spins.

@ __References__: Tully & Fisher A&A(77); Giovanelli ap/96-proc;
Kannappan et al AJ(02)ap [scatter]; Russell ApJ(04)ap [type
dependence]; Cattaneo et al a1706
+ Lincoln livesci(17)jul [reproducing the Tully-Fisher relation with dark matter];
O'Brien et al PLB(18)-a1704 [from first principles];
Alexander & Smolin a1804
[and flat rotation curves, from equivalence principle and assumptions].

**Turaev-Viro Theory**
> s.a. 3-manifolds; spin-foam
models; topology-change models.

* __Idea__: 3D Riemannian
quantum gravity with Λ > 0; A spin coupling theory, quantum-group
generalization of the Ponzano-Regge model, giving a non-perturbative definition
of the path integral for 3D gravity; The Turaev-Viro state sum invariant is
known to give the transition amplitude for the 3D BF theory with a cosmological
constant, related to the deformation parameter \(\hbar\) by \(\hbar = \Lambda^{1/2}\).

@ __References__: Turaev & Viro Top(92);
Ionicioiu gq/96 [3-manifold partition function];
Girelli et al CQG(02)gq/01 [topological invariance];
García-Islas CQG(04)gq,
Barrett et al JMP(07)m.QA/04 [observables];
Pranzetti PRD(14)-a1404 [amplitudes from 2+1 lqg].

**Turing Machine** > s.a. computation.

@ __References__: Turing PLMS(36);
Fouché et al qp/07,
Iriyama et al PLA(08) [quantum].

> __Online resources__:
see Wikipedia page.

**Turing's Thesis** > see under Church-Turing Thesis.

**Tutte Polynomial** > see graph invariants.

**12 j Symbols** > see SU(2); spin-foam models.

**Twin Paradox**
> see special-relativistic kinematics [including curved-spacetime version].

**Twist of a Vector Field** > see vector calculus.

**Twisted Geometry** > s.a. discrete geometry;
semiclassical quantum gravity.

@ __References__: Charles & Livine CQG(15)-a1501
[generalization to a *q*-deformed gauge group].

**Twisted Light** > see photons.

**Two-Body Problem** > see classical
systems; orbits of gravitating bodies.

**2dF Galaxy Redshift Survey** > see galaxy distribution.

**Two-Point Function** > see correlations [correlation
function]; green function.

**Two-Slit Experiment** > see interference.

**Tychonoff Space** > see types of topological spaces.

**Tychonoff Theorem**
> see compactness.

**Type Theory**

@ __References__: Kamareddine et al 04 [overview].

> __Online resources__:
see nLab page;
Wikipedia page.

**Typicality (in Cosmology)** > see multiverse cosmology.

main page
– abbreviations
– journals – comments
– other sites – acknowledgements

send feedback and suggestions to bombelli at olemiss.edu – modified 25 aug 2018