**Topics, V**

**Vacuum**
> s.a. vacuum phenomenology [including energy sequestering].

**Vaidya Metric**
> s.a. black-hole solutions and thermodynamics;
censorship [Vaidya-de Sitter]; gravitational collapse;
spinning particles.

* __Idea__: A metric describing
the gravitational collapse of a finite shell of incoherent radiation falling
into flat spacetime and giving rise to a Schwarzschild black hole.

* __Line element__: In usual
radiation coordinates, given by [in the generalized one,
*M* = *M*(*v*, *r*)]

d*s*^{2} = 2*c*
d*r* d*v* − [1−2*M*(*v*)/*r*]
d*v*^{2}
+ *r*^{2}
dΩ^{2} .

@ __General references__:
Denisova & Zubrilo G&C(00) [radial geodesics];
Singh & Vaz PLB(00) [quantum scalar field];
Girotto & Saa PRD(04)gq [double null coordinates];
Krisch & Glass JMP(05)gq [energy transport];
Ben-Dov PRD(07)gq/06 [outer trapped surfaces];
Bengtsson & Senovilla PRD(09)-a0809 [closed trapped surfaces];
Chirenti & Saa CQG(12)-a1201 [double-null formulation];
Berezin et al CQG(16)-a1603 [maximal analytical extension].

@ __Quasinormal modes__: Shao et al PRD(05)gq/04;
Abdalla et al PRD(06)gq;
Chirenti & Saa JPCS(11)-a1012.

@ __Other perturbations__: Nolan & Waters PRD(05)gq [self-similar, even perturbations];
Nolan CQG(07)gq/04 [self-similar];
> s.a. metric matching.

@ __Particle motion__: Bini et al CQG(11)-a1408 [effect of radiation flux].

@ __Charged__: Ibohal & Kapil IJMPD(10)-a1001 [Reissner-Nordström-Vaidya and Kerr-Newman-Vaidya solutions];
Chatterjee et al GRG(16)-a1512 [and the weak energy condition].

@ __More general solutions__: Hughston IJTP(71) [including vacuum metrics without symmetry];
Wang & Wu GRG(99)gq/98;
in Joshi & Dwivedi CQG(99)gq/98;
Saa PRD(07)gq [*N*-dimensional, + cosmological constant];
Mkenyeleye et al PRD(14)-a1407 [gravitational collapse];
Böhmer & Hogan MPLA(17)-a1710 [rotating, Kerr-like];
> s.a. Tolman Solutions.

@ __More general matter__: Gair CQG(02) [anisotropic pressure].

@ __In Lovelock gravity__: Cai et al PRD(08)-a0810;
Rudra et al ApSS(11)-a1101 [gravitational collapse].

@ __In other modified gravity theories__: Mehdipour CJP(12)-a1212 [non-commutative];
Ruan PRD(16)-a1512 [cubic gravity].

@ __Related topics__: Venditti & Dyer CQG(13)-a1201 [particle detector response].

**Vainshtein Mechanism**
> s.a. Horndeski Action;
massive gravity; Screening.

* __Idea__: An effect
which plays a crucial role in massive gravities and related theories such
as Galileons and their extensions; It is conjectured that this mechanism,
also known as k-mouflage, allows to hide via non linear effects (typically
for source distances smaller than a so-called Vainshtein radius which
depends on the source and on the theory considered) some degrees of
freedom whose effects are then only left important at large distances,
e.g., for cosmology; It has been shown to work, e.g., for spherically
symmetric, stationary situations.

@ __General references__:
Babichev & Deffayet CQG(13)-a1304 [introduction];
Koyama & Sakstein PRD(15)-a1502 [astrophysical probes];
Winther & Ferreira PRD(15)-a1505 [beyond the quasi-static approximation];
Karwan et al a1606 [in general disformal gravity theory].

@ __Cosmology and astrophysics__:
de Rham et al PRD(13)-a1208 [in binary pulsars];
Falck et al JCAP(14)-a1404 [cosmic web morphology];
Koyama & Sakstein PRD(15)-a1502 [astrophysical probes];
Kase et al PRD(16)-a1510 [theories beyond Horndeski].

**Valence of a Tensor Field**

* __Idea__: Its index structure,
or ther number of vector and covector fields it acts on to produce a scalar.

**van Dam-Veltman-Zakharov
Discontinuity** > s.a. Pauli-Fierz Theory.

* __Idea__: A theoretical argument
requiring that the mass of the graviton be exactly zero, since otherwise
measurements of the deflection of starlight by the Sun and the precession
of Mercury's perihelion would conflict with their theoretical values.

@ __References__: Carrera & Giulini gq/01 [*m* > 0, with electromagnetism];
Sato ht/05 [formulation with smooth *m* = 0 limit];
in Eckhardt et al NA(10)-a0909.

**van der Waals Fluid / Force / Equation
of State** > s.a. Induced
Gravity; QCD phenomenology [for gluonic forces].**
***

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**Van Hove's Theorem**
> see representations of quantum mechanics.

**Van Kampen's Theorem**
> see under Seifert-Van Kampen Theorem.

**Van Leeuwen's Theorem**

* __Idea__: The statement that the
phenomenon of diamagnetism cannot occur in classical statistical mechanics.

**Van Vleck Determinant**

**Variational Methods / Principles**
> s.a. variational principles in physics;
schrödinger equation.

@ __References__: Giusti 03 [direct methods];
Pasicki T&A(12)
[and a stronger form of the Caristi and Takahashi fixed point theorems];
Adamyan & Sushko a1306-text;
Voicu & Krupka JMP(15)-a1406
[turning a non-variational system of differential equations into a variational one];
Dacorogna 14 [introduction];
Chang 16 [lecture notes].

**Variational Quantum Tomography (VQT)**
> see quantum states.

**Variational Tricomplex**
> see symmetries.

**Variety**
> see Wikipedia page.

**Varifold**
> see types of manifolds [generalizations].

**Varying Speed of Light**
> see varying constants.

**Vassiliev Invariants**
> see knot invariants.

**Vectors** [including vector space, vector algebra]

**Vector / Vector-Tensor Theories of Gravity**
> see theories of gravity.

**Veiled General Relatvity**
> see modifications of general relativity.

**Venn Diagram**
> s.a. set theory; thermodynamics.

* __Idea__: A type of
diagram used in set theory to show the inclusion and intersection
properties among elements of a finite collection of sets.

@ __References__:
news cosmos(19)aug [John Venn].

> __Online resources__:
see Wikipedia page.

**Verifiability of a Theory**
> see criteria for physical theories.

**Veronese Web**
> see Web.

**Vertex Algebras**

@ __References__: Bakalov & Nikolov JMP(06) [in higher dimensions].

**Vertical Tensor Field**
> see vector field.

**Very Special Relativity**
> s.a. finsler geometry and physics.

* __Idea__: A theory of
relativity based on a small subgroup of the Lorentz group (the subgroup
T(2) of parabolic transformations within the Special Euclidean group SE(2)
obtained as the stabilizer of a null vector in Minkowski spacetime,
extended to include either parity or time reversal) [from Wikipedia]; It
maintains the main features of special relativity but breaks rotational
invariance, so it breaks Lorentz symmetry in a very mild way.

@ __General references__: Cohen & Glashow PRL(06) [proposal];
Sheikh-Jabbari & Tureanu PRL(08) [non-commutative spacetime realization];
Ahluwalia & Horvath JHEP(10)-a1008 [Lorentz symmetries for the standard model, VSR for the dark sector];
Das et al PLA(11)-a1004 [and non-commutative spacetime].

@ __Curved spacetime version__: Mück PLB(08)-a0806;
Kouretsis et al PRD(09)-a0810
[Finsler geometry and cosmology].

@ __Phenomenology__: Das & Mohanty MPLA(11)-a1007 [incompatibility with Thomas precession];
Romero et al MPLA(13)-a1203 [equivalence to relativistic particle in a gauge field];
Alfaro & Rivelles PRD(13)
[non-abelian fields], PLB(14)-a1306
[spinning particle]; Lee PRD(16)-a1512
[effect on quantum field theory]; > s.a. born-infeld
electrodynamics.

> __Online resources__:
see Wikipedia page.

**Vibrations** > see oscillator.

**Vielbein, Vierbein** > see tetrad.

**Viète Formulas** > see algebra.

**Vietoris Topology** > see types of topologies.

**Vietoris-Rips Complex**
> a type of abstract simplicial complex.

@ __References__: Attali et al CG(13) [and shape reconstruction].

> __Online resources__:
see Intelligent Perception page;
Wikipedia page.

**VIP (Violation of the Pauli Exclusion Principle) Experiment**
> see spin-statistics.

**Virasoro Algebra** > s.a. diffeomorphisms [Witt algebra].

* __Idea__: The central
extension of the Lie algebra of Diff(S^{1}).

$ __Def__: A Lie
algebra with generators which, for the open string, is the Witt
algebra with an anomalous term (depending on spacetime dimension),

[*L*_{n},*
L*_{m}] = (*n*−*m*)
*L*_{n+m} + *c
*(*n*^{2} −1)

δ_{n+m, 0} .

* __Representations__:
One is interested in finding representations for the quantization of
string theory; One normally demands that all generators be represented as
positive-definite Hermitian operators, in which case consistency then
restricts the dimensionality, but there are many anomaly-free
representations in which *L*_{0} is
not positive-definite; This is not a problem, since *L*_{0} is a constraint anyway.

@ __General references__: Goddard & Olive ed-88;
Germoni LMP(01) [classification of representations];
Wassermann a1004-ln;
Iohara & Koga 11 [representations];
Schlichenmaier a1111 [second cohomology];
Gómez et al JGP(12) [geometric approach];
Zhdanov & Huang a1310
[inequivalent realizations].

@ __Deformations__: Bouwknegt & Pilch CMP(98);
Gieseker m.QA/99 [from Toda and KdV],
JDG(03).

@ __Generalized to higher dimensions__:
Larsson in(04)-a0709 [and quantum gravity];
Gurau NPB(11).

@ __Related topics__: Milas CMP(04)m.QA/03 [and Ramanujan's "Lost Notebook"];
Kitölä & Rideout JMP(09)-a0905 [staggered indecomposable Virasoro modules].

**Virasoro Group**

@ __References__: Kontsevich & Suhov mp/06 [Malliavin measures].

**Virial Coefficients / Expansion**
> s.a. Equation of State; Mayer
Cluster Expansion; statistical-mechanical systems.

* __Idea__: The coefficients
and series one gets expanding the pressure equation of state for any gas in powers
of the number density *n*, of the form

*p* = *kT* [*n* + *B*_{2}(*T*)
*n*^{2} + *B*_{3}(*T*)
*n*^{3} + ...] .

* __Values__: For an ideal
gas all virial coefficients *B*_{2}
= *B*_{2} = ... = 0, and at
sufficiently low densities any gas behaves that way.

* __Method__: The virial
coefficients can be calculated as sums over labeled 2-connected graphs.

@ __General references__:
Bhaduri et al JPA(10);
Kaplan & Sun PRL(11) [new field-theoretic computation method];
Jansen JSP(12) [at low temperatures, interpretation of the radius of convergence];
Tate JSP(13)-a1303 [cluster expansion bounds on the virial expansion];
Procacci & Yuhjtman LMP(17)-a1508,
Procacci JSP(17)-a1705 [bounds for the radius of convergence].

@ __Expansion in terms of 2-connected graphs__:
Androulaki et al JPA(10) [method for graph construction];
Tate AIHP(15)-a1402.

@ __For ideal gases__: Liu PRP(13) [cold fermion gas];
Olaussen & Sudbø RNSSL-a1502 [ideal quantum gases in arbitrary dimensions].

@ __Other specific systems__:
Gavrilik & Mishchenko UJP-a1312-proc,
PRE(14)-a1409 [composite, interacting bosonic particles];
Brydges & Marchetti a1403 [gas of particles with uniformly repulsive pair interaction];
Jansen JSP(15)-a1503 [multi-species Tonks gas];
> s.a. gas [cold fermion gas].

**Virial Theorem**
> s.a. energy.

* __In classical physics__:
The relationship between the time average of the kinetic energy of a
system of mass points and the *virial of Clausius*; When averages
are taken over a period (or, for non-periodic motion, over a long time),

avg{*K*} = −\(1\over2\)avg{∑_{i}
**F**_{i} ·
**r**_{i}} ;

It is very useful in the kinetic theory of gases, e.g., to derive Boyle's law.

* __In relativistic physics__:
The tensor virial theorem states that, for a system with vanishing
stresss-energy outside a bounded *V*,

∂_{t}^{2}
∫ d^{3}*x*
*T*^{ 00}
*x*^{ i}
*x*^{ j}
= 2 ∫
d^{3}*x*
*T*^{ ij} .

@ __General references__: in Goldstein 80;
Milgrom PLA(94)ap [from action principle];
Böhmer et al JCAP(08)-a0710,
Sefiedgar et al PRD(09)-a0908 [in *f*(*R*) gravity];
Amore & Fernández EJP(09)-a0904
[in non-linear problems]; > s.a. MOND.

@ __In relativistic physics__: in Schutz 85,
ex.4.23; Lucha & Schöberl PRL(90);
Gourgoulhon & Bonazzola CQG(94);
Bonazzola & Gourgoulhon CQG(94);
Nowakowski et al PRD(02) [with a cosmological constant];
Georgiou CQG(03) [rotating charged pfluids in general relativity];
Tan AP(08) [generalized, for two-component Fermi gas];
Sefidgar et al PRD(09)-a0908
[for *f*(*R*) gravity];
Roshan CQG(12)-a1208 [parametrized post-Newtonian];
Javadinezhad et al PRD(16)-a1510 [for cosmological structures];
Mashhoon Univ-a1512
[in non-local Newtonian gravity].

@ __Generalizations__: Georgescu & Gérard CMP(99) [in quantum physics];
Cariñena et al JPA(12)-a1209 [geometric approach];
Sukumar a1410.

**Virtual Knot Theory** > see knot theory.

**Virtual Particles** >
s.a. photons; vacuum [zero-point fluctuation].

* __Idea__: Points in
momentum space that are included in a sum over 4-momenta for intermediate
states in a Feynman diagram / quantum amplitude; They do not need to lie
on the mass shell, but they must be related to other 4-momenta in the
process by conservation laws.

* __Analogy__: Like
spray around a turbulent waterfall; A more precise classical analog are
evanescent wave solutions of wave equations.

@ __General references__: in Rubbia & Jacob AS(90) [a description as good as most];
de la Torre a1505
[and the quantum field theory interpretation of quantum mechanics];
Malyshev PIT(16)-a1605
[and the Newtonian trajectories of classical physics].

@ __Related topics__: Nimtz FP(09)-a0907
= FP(10) [macroscopic analog];
> s.a. wave phenomena [evanescent waves].

**Virtual Work**
> see noether's theorem [generalized].

**Viscoelasticity**
> s.a. Elasticity; fluids.

* __Examples__:
Viscoelastic fluids include "complex" (non-Newtonian) fluids, "squishy" or
"gooey" ones like pudding, blood, the Earth's mantle, toothpaste, ketchup
and gelatin, for which viscoelasticity depends, in a mathematically
complex way, on the magnitude of the applied force.

@ __References__: Pipkin 86;
news pn(07)may [viscoelastic fluid that tears];
Khair & Squires PRL(10) [technique to measure normal stress coefficients].

> __Online resources__:
see Wikipedia page.

**Visscher Basis**

* __Idea__: The
functions Ψ_{lmn}:=
(*x*/*a*)^{l}
(*y*/*b*)^{m}
(*z*/*c*)^{n},
where *l*, *m*, and *n* are positive integers,
which are not orthonormal, but in terms of which one can decompose
functions on \(\mathbb R\)^{3} – The
expansion is related to a Taylor series expansion around *x*
= *y* = *z* = 0.

**Visualization** > s.a. computational physics.

@ __References__: Goodwin PhSc(09)jul [visual representations and truth-bearing role].

**Vlasov Equation** > s.a. stochastic
processes; magnetism [plasma physics]; solutions
of einstein's equation with matter [Einstein-Vlasov system].

* __Idea__: A
differential equation describing the time evolution of the distribution
function of a plasma consisting of charged particles with long-range
interactions.

> __Online resources__:
see Wikipedia page.

**Vlasov-Poisson Equations / System**

@ __References__: Esen & Gümral a1203 [and differential geometry];
Lazarovici & Pickl ARMA(17)-a1502 [mean-field limit].

**Void (in Cosmology)** > see galaxy
distribution [the void phenomenon]; milky
way galaxy [Local Void]; theories
of cosmological acceleration.

**Volume **> s.a. form [volume form];
geometrical operators in quantum gravity.

@ __Spacetime volume__: Gao IJMPCS(12)-a1108 [as a scalar field, and holographic dark energy].

**"Von Freud Pseudotensor"**
> There is no such thing; The correct name is Freud Pseudotensor.

**Von Neumann Algebra** > see observable algebras.

**Von Neumann Entropy** > see entropy in quantum theory.

**Von Neumann's Theorem**
> see representations of quantum mechanics [Stone-von Neumann].

**Voronoi Complex / Polyhedron / Tiling**
> see voronoi tiling.

**Voros Bracket / Product**
> see poisson brackets.

**Vortex**
> s.a. gauge theories [vortices in condensed-matter physics]; QCD;
topological defects; topology in physics.

* __In cosmology__: Another
name for a global cosmic string, due to spontaneous breaking of a global
symmetry; The energy density is not concentrated on a narrow tube, but
diverges if integrated over all distances from the string.

@ __References__: Falconer a1906
[history, Maxwell's vortex analogy for the electromagnetic field and Kelvin's vortex theory of atoms].

**Vortex Lines**
> see spacetime structure.

**Vorticity of a Congruence of World-Lines**

$ __Def__: If
*u*^{a} is the unit
timelike tangent vector to the congruence, one defines the vorticity
tensor and its trace as

*ω*_{ab}:=
*q*_{[a}^{m}
*q*_{b]}^{n}
∇_{m} *u*_{n}
, *ω*:=
(\(1\over2\)*ω*_{ab}
*ω*^{ab})^{1/2} .

**Vorticity of a Cosmological Model**
> see bianchi models; cmb.

**VSL Theories** > see varying constants [speed of light].

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

send feedback and suggestions to bombelli at olemiss.edu – modified 25 aug 2019