Topics, V

Vacuum

Vaidya Metric > s.a. black hole solutions; censorship [Vaidya-de Sitter]; collapse; Tolman.
* Idea: A metric representing the high-frequency (eikonal) approximation to unidirectional radial flow of unpolarized radiation (e.g., from a black hole).
* Line element: In usual radiation coordinates, given by [in the generalized one, M = M(v, r)]

ds² = 2c dr dv – [1–2M(v)/r] dv² + r² d² .

@ 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].
@ Perturbations: Shao et al PRD(05)gq/04 [charged, quasinormal modes]; Nolan & Waters PRD(05)gq [self-similar, even perturbations]; Abdalla et al PRD(06)gq [quasinormal modes]; Nolan CQG(07)gq/04 [self-similar].
@ Generalizations: Hughston IJTP(71); Wang & Wu GRG(99)gq/98; in Joshi & Dwivedi CQG(99)gq/98; Gair CQG(02) [anisotropic pressure]; Saa gq/07/PRD [N-dimensional, + cosmological constant].

Valence of a Tensor Field

van der Waals Force, Equation of State > s.a. Induced Gravity; QCD phenomenology [for gluonic forces].
* Idea: A phenomenological equation of state for a non-ideal gas; Includes the effects of a force representing the interaction between neutral molecules in a gas, decreasing faster than r^–2 (higher-multipole components of the electric force); Contains a term corresponding to long-range attraction and one corresponding to short-range repulsion (> see Lennard-Jones Potential).
* Equation of state: Of the form (p + a/v²)(v–b) = RT, where v = V/n is the molar volume, and a and b two parameters.
* Applications: Apparently spiders walk upside down on any surface using van der Waals forces [@ news pw(04)apr].
@ References: in Reif 65, p173, p425; Holstein AJP(01); Wu et al PRA(02) [retarded, fluctuations]; Sambale et al PRA(07)qp/06 [local-field corrected]; Yannopapas & Vitanov PRL(07) [between macroscopic objects, first-principles theory]; Román-Velázquez & Sernelius a0710-JPA [effect of structure and geometry]; Bärwinkel & Schnack PhyA(08) [from non-ideal quantum gas].

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

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

Van Vleck Determinant

Variational Methods / Principles > s.a. schrödinger equation.

Vector, Vector Algebra, Vector Space > see vectors.

Vector Calculus

Vector Field

Velocity

Veronese Web > see Web.

Vertex Algebras
@ References: Bakalov & Nikolov JMP(06) [in higher dimensions].

Vertical Tensor Field > see vector field.

Very Special Relativity
@ References: Cohen & Glashow PRL(06) [proposal]; Mueck a0806 [in curved spacetimes].

Vibrations > see oscillator.

Vielbein, Vierbein > see tetrad.

Vietoris Topology > see types of topologies.

Virasoro Algebra > s.a. diffeomorphisms [Witt algebra].
* Idea: The central extension of the Lie algebra of Diff(S¹).
$ 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 (n² –1) _{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 L₀ is not positive-definite; This is not a problem, since L₀ is a constraint anyway.
@ General references: Goddard & Olive ed-88; Germoni LMP(01) [classification of representations].
@ Deformations: Bouwknegt & Pilch CMP(98); Gieseker m.QA/99 [from Toda and KdV], JDG(03).
@ Related topics: Milas CMP(04)m.QA/03 [and Ramanujan's "Lost Notebook"]; Larsson a0709-in [multi-dimensional generalizations, and quantum gravity].

Virasoro Group
@ References: Kontsevich & Suhov mp/06 [Malliavin measures].

Virial Coefficients / Expansion
* Idea: The coefficients and series one gets expanding the equation of state for any gas in powers of the number density; Of the form

p = kT [n + B₂(T) n² + B₃(T) n³ + ...] .

* Values: For an ideal gas all virial coefficients B₂ = B₂ = ... = 0, and at sufficiently low densities any gas behaves that way.

Virial Theorem > see energy.

Virtual Particles > s.a. vacuum [zero-point fluctuation]; wave phenomena [evanescent waves].
* Idea: Points in momentum space that are included in a sum over 4-momenta for intermediate states in a 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.
@ References: in Rubbia & Jacob AS(90) [a description as good as most].

Viscoelasticity > s.a. Elasticity.
* Examples: Viscoelastic fluids include blood clots, the earth's mantle, toothpaste, and gelatin.
@ References: Pipkin 86; news pn(07)may [viscoelastic fluid that tears].

Viscosity > s.a. dark energy; fluid; Kinetic Theory; Stokes' Law; Transport.
* Coefficient of viscosity: The constant of proportionality between the stress and the derivatives of the velocity in a fluid (when the velocity gradient is not too large, in terms of the mean free path),

P_ij = u ^j/xⁱ .

* Properties: The viscosity of a gas tends to increase with temperature, that of a liquid tends to decrease.
* Kovtun-Starinets-Son bound: The viscosity-to-entropy-density ratio is bounded by /s 1/(4).
@ General references: Desloge AJP(62) [coefficient of viscosity for a gas].
@ Lower bound: Cherman et al a0708 [re viability of conjecture]; Fouxon et al PRD(08)-a0710 [and generalized second law]; Brigante et al PRL(08) [and inconsistency from microcausality violation], PRD(08).
@ Related topics: news pn(08)may [shear thinning – as in ketchup].

Visscher Basis
* Idea: The functions _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 R³ – The expansion is related to a Taylor series expansion around x = y = z = 0.

Vlasov Equation > see stochastic processes.

Volume > see form [volume form]; geometrical operators in quantum gravity.

"Von Freud Pseudotensor" > There seems to be no such thing; Maybe you mean "Freud Pseudotensor"?

Von Neumann Algebra > see observable algebras.

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

Voronoi Complex / Polyhedron / Tiling > see voronoi tiling.

Vortex > s.a. QCD; topological defects.
* Idea: 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.

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

_ab:= q_[a^m q_b]ⁿ _m u_n ,      := ( _ab ^ab)^1/2 .

Vorticity of a Cosmological Model > see bianchi models, cmb.

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