Topics, R
Racah Coefficients / Formula > s.a. SU(2).
@ References: Krasnov CQG(05)gq/04 [for
the Lorentz group]; Krasnov & Louko JMP(06)mp/05 [for
SO(1, d+1), AdS-cft methods].
Radial Gauge > see gauge.
Radiation > s.a. thermal
radiation.
Radiation Reaction > see self-force.
Radiative Corrections > see quantum
field theory techniques; quantum gravity
phenomenology.
Radioactivity > see nuclear physics.
Radion Field > see tests of newtonian
gravity [constraints].
Radon Transform
* Applications: CAT, PET
and NMR scanning, found by A Cormack (Nobel prize in medicine 1979).
Radon-Nikodym Derivative
Rainbow > see light.
Raising Operator > Essentially the same as creation
operator.
@ References: Boyer
and Miller JMP(74) [second-order, for two-variable Hamiltonians].
Raman
Scattering
* Idea: Inelastic scattering
of photons off molecules, in which the internal state of the molecules change.
Ramsey Numbers / Theory
Randall-Sundrum Model > see branes;
brane cosmology.
Random Process [including
random walk]
Random Surfaces > s.a. dynamical
triangulations.
@ References: Ambjørn in(96)ht/94 [and
quantum gravity, lecture notes]; Wheater JPA(94)
[rev].
Rank of a Group > s.a. Group
Presentation.
* Idea: For a discrete
group, the number of generators in a presentation; For a Lie group, the
maximum number of generators that can be simultaneously diagonalized; It is
meaningful
(invariant) for Abelian and free groups.
Rapidity > s.a. [kinematics
of special relativity]; velocity.
* Idea: An angle-like
quantity representing boosts in special relativity, with the additivity property
that speed
doesn't have.
$ Def: Defined as y =
(1/2) ln[(E+pz)/(E–pz)]
in high-energy physics, which corresponds to tanh(y)
= pL/E, or r =
cosh–1
.
$ Pseudorapidity: The
quantity –ln[tan(
/2)],
where is the angle with
respect to the beam line in high-energy collisions; For approximately massless
particles, it is aproximately equal to the rapidity.
@ References: Lévy-Leblond & Provost AJP(79)dec;
Rhodes & Semon AJP(04)jul
[description of Wigner rotation and Thomas precession].
Rarita-Schwinger Equation / Lagrangian
* Idea: A theory of particles
with spin k + 
,
with k an
integer, that are described by a spinor field with k spacetime indices 
a...
c,
and obey a generalized Dirac equation
(a 
a +
) m...
p
= 0 ,
where a ab...
c =
0.
* k = 0: There is a gauge symmetry a →
a + a 
,
with a a =
0.
* k = 1: The Lagrangian is L = 
a(m m
+ ) a – 
a (a b +
ba) b
+ a a (b b – )
c c .
* Applications: Its main application is in supergravity to describe
gravitinos.
@ References: Rarita & Schwinger PR(41);
Rashkov MPLA(99)
[and AdS/conformal field theory]; Szereszewski & Tafel CQG(01)
[stationary axisymmetric spacetime]; Pilling IJMPA(05)
[massive, symmetries]; Kaloshin et al ht/05 [Lagrangian];
Bizdadea et al AdP(06)-a0704 [and
Weyl graviton, no cross-couplings].
Rational Numbers > see numbers.
Raumproblem
@ References: Loinger RNC(88); Urbantke IJTP(89).
Ray Emanating from a Point in Rn > see
lines.
Raychaudhuri Equation
@ References: Dadhich gq/05 [derivation],
comment Mitra gq/05;
Kar & SenGupta gq/06-in
[rev]; Dadhich gq/07/Pra
[analog for quantum gravity?].
Rayleigh Jets
* Idea: The phenomenon
by which a charged droplet of liquid becomes unstable and explodes,
ejecting a microscopic jet of liquid from each end before returning to equilibrium;
This limit relates the maximum charge the droplet can bear to its surface
tension and radius; Suggested
by Rayleigh, visualized by Thomas Leisner and colleagues at Ilmenau Technical
University [@ Duft et al Nat(03)jan].
Rayleigh Limit / Criterion > see diffraction.
Rayleigh Scattering > s.a. interaction.
* Idea: Elastic scattering
of photons; Responsible
for
the blue color of the sky.
@ References: Chakraborti AJP(07)sep
[simple experiment].
Real Numbers > see numbers.
Realism
Realization of a Group > see group
action.
Reciprocity > see number
theory [quadratic reciprocity
theorem].
Recoupling Theory > see angular
momentum.
Recueil (Souriau).
$ Pre-recueil: A set R
whose elements are regular operators such that for all A, B 
R, A–1 R and AB R.
Recurrence Paradox
@ References: Steckline AJP(83)oct
[Boltzmann & Zermelo].
Recursion Theory
@ References: Smullyan 93.
Redshift
* Idea: A frequency / wavelength
change for electromagnetic waves / photons; May be due to the relative velocity
between source and observer (> see doppler),
expansion of the background spacetime (>
see doppler, galaxies),
or
a gravitational field (> see doppler).
Reduction of a Dynamical System with Constraints > see constrained
systems.
Reduction of a Fiber Bundle > see fiber
bundles.
Reductionism > see paradigms
in physics.
Reeh-Schlieder Property
@ References: Strohmaier CMP(00)mp [quantum
field theory on a stationary spacetime]; Sanders a0801 [existence
of Reeh-Schlieder].
Reference Frame > s.a. Covariance; Frame [more
mathematical]; Observer;
Relativity.
* Idea: A smooth atlas
on the spacetime manifold; In classical non-relativistic mechanics, a reference
frame
can be seen as a connection
on
a configuration space fibered over the time axis.
* Inertial: One in which
the components of the spacetime metric are constants, usually taken to be an
orthonorml set of coordinates, for which the metric is diag(–1, 1, ..., 1);
The cornerstone of Newtonian mechanics;
> s.a. inertia, mach's
principle.
* Accelerated: Fictitious
inertial forces appear; > s.a. rindler space.
@
Inertial: Stephens FPL(96)
[in quantum field theory]; Rodrigues & Sharif FP(01)
[in general relativity, and local Lorentz invariance].
@ Accelerated: Padmanabhan ASS(82)
[definition of particle]; Desloge AJP(89)
[uniformly, non-equivalent to uniform gravitational field]; Mashhoon PRA(93)
[general theory]; Marzlin PLA(96);
Lynden-Bell et al AP(99)
[gravity and electromagnetism]; Chicone & Mashhoon AdP(02)gq/01 [kinetic
and dynamic memory]; Mashhoon in(03)gq,
IJMPD(05)
[non-locality]; Semay EJP(06)
[constant proper acceleration]; Lusanna a0707-in
[in general relativity, observables and constraints]; Mashhoon a0805-AdP
[non-local]; > s.a. Aberration, quantum
field theory effects in curved spacetime [Unruh effect], types
of field theories [non-local].
@ Arbitrary reference frames: Sardanashvily a0708 [non-relativistic
mechanics, inertial forces, etc].
@ Related topics: Meli HSPS(93) [history];
Bel gq/00 [rotation
along a world-line]; Dickson
SHPMP(04)
[and uncertainty relations]; Llosa & Soler CQG(04)
[geometric structure,
and rigid motion]; Rosinger qp/05 [covariance
of physical laws, general relativity and unification]; Marmo & Preziosi IJGMP(06)
[coordinate-free
formulation].
@ In Newtonian spacetime: Coll et al a0707 [four
causal classes]; > s.a. (post-)newtonian gravity.
> In other specific theories:
see canonical general relativity [material
reference systems]; electromagnetism; kinematics
of special relativity; relativistic
quantum mechanics.
Refined Algebraic Quantization > see dirac
quantization.
Refinement of an Open Cover > see cover.
Reflection > see groups [reflection
groups]; mirrors [moving
mirrors]; wave
phenomena; Wigner Delay.
Reflexive Banach Space > see Banach
Space.
Refraction > s.a. dispersion;
electricity;
Fermat's Principle; light.
* Idea: The
bending of light when propagating in a medium with non-constant index of
refraction n; When it occurs at the boundary between two homogeneous
media with different values of n, described by Snell's
Law.
* Index of refraction:
Defined by n(
):= c/vp = k/kvacuum;
For light, k = /c and n = c/v.
* Negative index of
refraction:
1968, Existence of materials which
bend
light in the opposite direction to conventional materials (also known as left-handed
materials) first proposed; 2000, Demonstrated for the first time in experiments;
2003, Several experiments and simulations demonstrated that negative refraction
is real and that causality is not violated; 2005, New material discovered made
from layers of superconducting and ferromagnetic thin films (until now, negative
refraction
had only been achieved in metamaterials and photonic crystals); Associated phenomena
are a reverse Doppler shift and backward Cerenkov radiation and pressure.
@ General references: Champenois et al a0711 [matter waves in gases].
@ Negative refraction:
Veselago SPU(68); Shelby et al Sci(01)apr
[experiment]; Smith et al PRL(00);
Smith
&
Kroll PRL(00);
Smith pw(03)may;
Zhang et al PRL(03)
+ pw(03)oct;
Pendry CP(04)
[rev]; Pendry & Smith PT(04)jun;
Lakhtakia & Mackay JPA(04)
[vacuum, gravitationally assisted]; Lakhtakia & McCall NJP(05)
[focus issue];
Ward
et
al NJP(05)
[physical origin]; news pw(05)dec,
SFN(06)may;
McCall PRL(07)
[not with gravity]; news pw(07)may
[natural material]; Ramm PLA(08)-a0710 [in
acoustics]; > s.a.
[meta-materials], electromagnetism
in curved spacetime.
Regge Calculus > s.a. quantum
version.
Regge Trajectory
* Idea: Regge trajectories
are straight lines connecting groups of particles in the J 2-m
plane.
* Applications: They were
important for the formulation of the phenomenological dual string models.
Regge-Wheeler Equation > see perturbations
of schwarzschild; models
in numerical general relativity.
Regular Topological Space > see types
of topologies.
Regularization > see quantum
field theory techniques; regularization
schemes.
Reidemeister Moves > see knot theory.
Reissner-Nordström Solution (no,
the
spelling is not "Reissner-Nordstrøm")
Relation > for examples, see graphs and posets.
$ Def: A relation in
a set X is a subset R of X 
X.
* Transitive relation:
A binary relation R on a set X such that
for all
x, y, z in X, x R y & y R z
implies x R z.
* Equivalence relation: One which is reflexive, symmetric, and transitive.
* k-homogeneous:
A relational structure 
is
called k-homogeneous if each isomorphism between
two k-element
substructures
of extends to an automorphism
of .
@ General references: Fraïssé 86; Kornyak mp/05-in
[compatibility
of systems of relations].
@ Types: Droste & Macpherson JCTA(91)
[k-homogeneous]; Droste & Kuske JCTA(03)
[random].
Relational Blockworld > s.a. classical
limit; relativistic
quantum mechanics.
@ References: Stuckey & Silberstein a0712 [and
quantum gravity].
Relationalism > see canonical
general relativity; modified
quantum mechanics; parametrized
theories; quantum
gravity; time; spacetime.
Relativistic Theory of Gravitation > s.a. gravity.
@ General references: Logunov & Loskutov TMP(86);
Vlasov & Logunov TMP(86)
[cosmology]; Vlasov & Logunov TMP(86), TMP(87)
[gravitational collapse]; Logunov & Loskutov TMP(87)
[test bodies]; Logunov et al IJMPA(88),
SPU(88); Logunov TMP(89), SPU(90);
Logunov TMP(90);
Logunov & Mestvirishvili TMP(91),
et al TMP(91),
TMP(94);
Logunov et al TMP(94)
[equivalence principle]; Logunov TMP(95);
Logunov TMP(95)
[post-newtonian approximation]; Logunov SPU(95),
01-gq/02-book.
@ Criticism of general relativity:
Logunov et al TMP(86);
Loskutov TMP(90).
@ Support for general relativity: Zel'dovich & Grishchuk SPU(86), SPU(88);
Chermyanin
SPU(90); Ferrari TMP(90);
Grishchuk SPU(90); Ohta & Kimura NCB(91).
Relativity Principle > s.a. [general
relativity; special
relativity]; Covariance; Reference
Frames.
* Idea: The statement
that physical laws should be invariant under a change of coordinate system;
Different versions differ by which laws are included, and which coordinate
systems (e.g., all smoothly related ones for general relativity); The
latter basically amounts to specifying which group of transformations transforms
one frame into another (e.g., the Galilean Group for
Newtonian mechanics).
* Issue: Kretschmann's
well-known criticism stated that general covariance of the Einstein equation
is not sufficient to express the principle of general relativity, the failure
being rooted in the fact that the metric in the Einstein equation is not
uniquely determined by the matter distribution.
* Issue: One would
expect the principle to be valid if the dynamical laws in question include
all the dynamical quantities they should include; It might by violated, for
example, at scales at which quantum gravity starts becoming important, if
all the relevant degrees of freedom (or dimensions) are not properly taken
into account.
@ References: Budden BJPS(97)
[Galilean vs modern definition]; Barbour gq/03-in
[new formulation];
Szabó FPL(04)
[vs Lorentz covariance]; Leston & Ferraro gq/04 [and
the
Einstein equation]; Grøn a0708 [with rotation and inertial dragging].
Relator
$ Def: One of the words
that define a group presentation.
Relaxation Phenomena
* Relaxation time: The
time taken by a system away from equilibrium to approach the equilibrium state;
For molecules
in a gas, it amounts to another name for the mean collision time.
@ Systems:
Chimonidou & Sudarshan a0705 [two interacting harmonic oscillators].
@ Relaxation time: Hod PRD(07)gq/06 [universal
bound, 
> 
/
T];
Lemos & Pereira PhyA(07)
[temperature effects].
Renormalization > s.a. renormalization
group; for quantum gravity and other
theories.
Rényi Entropy > see entropy.
Replica Symmetry > s.a. ising model.
* Idea: The "replica method"
proposed by Mark Kac is an aproach to the study of quenched disordered systems.
@ References: Dotsenko 01 [r PT(02)jan].
Representation > s.a. in
quantum
theory.
Residue of a Function > s.a. analytic
function [Cauchy
theorem].
$ Def: For a function with
a pole of order n at z0, f(z)
= F(z)/(z–z0)n,
where F(z) is regular at z0,
the residue at z0 is F(n)(z)/n!
Resistor > see electricity.
Resolution > s.a. metric spaces.
$ Def: A local homotopy equivalence M → X, when X
is a manifold.
Resolvent of a Matrix M > see matrices.
Resonance
Response Theory > see fluctuations [fluctuation-dissipation].
Resummation
@ References: DeWitt PRL(64);
Isham, Salam & Strathdee PRD(71).
Retarded Green Function > see green
functions.
Retarded Potential > see potential.
Retraction
$ Def: Given an inclusion i: A → X between
topological spaces, a retraction is a continuous map r: X → A such
that r 
i =
idA, i.e., a continuous deformation
of X onto one of its subsets; A is
called a retract of X.
* Deformation retract:
A retraction such that r 
id
rel A.
* Example: S1 is
a retract of the disk minus a point, D2 \
{0}, but not of D2.
Retrocausation > see quantum effects.
Reversibility > see arrow of
time.
Reynolds Number > see turbulence.
Rheology
* Idea: The study of
deformations and flows of unusual materials.
@ References: TIP 2004 article;
Society of Rheology site.
Ribbon
* Calugareanu's
theorem: A result which enables the integer linking number of the two edges
of a ribbon to be written as the sum of the ribbon twist (the rate of rotation
of the
ribbon
about its axis) and its writhe.
@ References: Dennis & Hannay PRS(05)mp [Calugareanu's
theorem].
Riccati Equation > see ordinary
differential equations.
Ricci Collineations
@ References: Flores et al JMP(04)gq [type-B
warped spacetimes].
Ricci Flow > see riemannian
geometry.
Ricci Rotation Coefficients > see tetrads.
Ricci Tensor > see riemann tensor;
Segre Classification.
Riemann Equation
* Idea: The first-order,
non-linear
partial
differential
equation ut
+ uux = 0,
which describes a one-dimensional accelerationless perfect fluid; It is PT-symmetric,
and possesses
solutions
that typically develop shocks in a finite time.
@ References:
Bender & Feinberg a0709 [complex deformation].
Riemann Hypothesis / Conjecture > see conjectures.
Riemann Integral > see integration.
Riemann Normal Coordinates > see coordinates.
Riemann Surface > see 2D manifold.
Riemann Tensor
Riemann Zeta Function > see Zeta
Function.
Riemann-Cartan Structure > s.a. differential
geometry; electroweak theory;
gödel spacetime;
teleparallel.
* Idea: A differentiable
manifold with a vielbein eai and
a Lorentz connection ija,
in terms of which the curvature is R ij :=
d ij, R = eai ebj R ijab,
and the torsion T i :=
dei +
ij 
e j; > s.a.
affine connection [structure equations].
@ References: Vacaru et al gq/05-book
[Riemann-Finsler]; Minkevich gq/07-in
[and cosmology]; > s.a. Defects.
Riemann-Hilbert Problem > s.a. axisymmetry [Ernst
equation].
* Idea: A boundary
value problem for partial differential equations.
Riemann-Roch Theorem
* Relationships: Can
be considered as a special case of the Atiyah-Singer index
theorem.
Riemannian Connection > see affine
connection.
Riemannian Geometry / Manifold / Metric / Structure > see riemannian geometry; metric
tensor.
Riesz Space
* Idea: A vector lattice.
@ References: Zaanen & Luxemburg 71; Zaanen 83.
Rigged Hilbert Space > see hilbert
space.
Right Action of a Group > s.a. group
action.
$ Right translation in
a Lie group: The right action of G on itself given by
Rg(h):= hg.
Rigid Body
* In classical mechanics:
Its dynamics is described by Euler's equations.
* And special relativity: An ideal
rigid body cannot exist, since otherwise its ends would move simultaneously
in all frames, and could be used to establish a "universal time".
@ In classical mechanics: Modugno & Vitolo mp/05 [geometrical];
Bender et al a0705 [complex
and PT-symmetric solutions].
@ In quantum mechanics: Modugno et al mp/05 [covariant
approach].
@ In special relativity: Eriksen et al PS(82);
Kim & Jo JPA(04)
[rigid rotation impossible]; Soler FP(06)gq/05;
Sauer a0704-in
[Einstein-Varicak letters].
Rigidity > s.a. differential
geometry [rigidity of a geometrical structure]; horizons;
classical particles.
* Idea:
For a theory, stability is a way of expressing its structural stability.
* Rigidity theorem:
For a stationary black hole solution or one with a Cauchy horizon, it establishes,
under weak assumptions, the existence of a Killing vector field
in a one-sided neighborhood of the horizon which is normal to the horizon;
If (M, g) is a pseudostationary, axisymmetric
asymptotically flat spacetime, with a simply connected domain of outer communications,
where the causality and circularity conditions are satisfied, then one can
normalize
the null tangent
to the
event horizon such that a = ka +
H ma,
where
H is
a constant.
@ Rigidity theorems: Hawking CMP(72)
[black hole]; Isenberg & Moncrief
[Cauchy horizon]; Chrusciel CMP(97)gq/96;
Friedrich et al CMP(99)gq/98,
Rácz CQG(00)gq/99 [general],
gq/07-wd
[generalization]; Chrusciel & Tod a0712.
@ In cosmology: Lidsey GRG(93); Aguirregabiria & Lazkoz MPLA(04)gq [of tachyonic
inflation].
Rindler Space
Ring, Ring of Subsets, Ring Space > see ring.
Ripple
* Idea: A connection
at spatial infinity, induced by the conformally rescaled metric on an asymptotically
flat spacetime used for compactification; Equivalently, an equivalence class
of connections on spacetime, two of them being equivalent if they induce
the same structure at spatial infinity.
Rishons > see composite models.
Ritz Theory of Electrodynamics > see alternative
formulations of electrodynamics.
Robbins Conjecture > see conjectures.
Robertson-Walker Universe
Robin Problem
* Idea: A boundary value
problem for partial differential equations.
@ Solutions: Bondurant & Fulling JPA(05)mp/04 [and
Dirichlet problem solutions].
Robinson-Bertotti Solution > see kantowski-sachs.
Robinson-Trautman Spacetimes >
s.a. schwarzschild.
* Idea: Metrics that describe gravitational radiation in the exterior of bounded sources.
@ General references: Bicák & Podolsky PRD(97)gq/99 [
0];
Moreschi & Pérez
CQG(01)gq [perturbations
with angular momentum]; Griffiths et al CQG(02)gq [type
N]; Natorf gq/05-wd
[solutions, type II = c-metric]; Bakas gq/05 [vacuum,
integrability and analog of Liénard-Wiechert fields]; Natorf & Tafel JMP(06)gq [symmetries
and reductions]; Kozameh et al CQG(06)gq [Robinson-Trautman-Maxwell,
meaning].
@ In higher dimensions: Podolsky & Ortaggio CQG(06)gq;
Ortaggio
et al CQG(08)-a0708 [with
electromagnetic field].
Roche Lobe
* Idea: An hourglass-shaped
surface surrounding the stars in a binary system, beyond which matter is not
in the gravitational sway of either star.
Rodrigues Formula > see legendre
polynomials.
Rokhsar-Kivelson Point
* Idea:
A construction initially developed in condensed matter
for the quantum dimer model, in which a quantum Hamiltonian is constructed,
on the same state space as that of a classical statistical-mechanics model
(with a discrete state space, and endowed with a dynamics satisfying detailed
balance), such that the ground state wavefunction coincides with the classical
equilibrium distribution; The word "point" refers to the fine-tuning
of a parameter in the quantum Hamiltonian.
@
References: Henley JPC(04)cm/01-in;
Fradkin et al PRB(04)cm/03 [bipartite];
Castelnovo et al AP(05)cm;
Syljuåsen IJMPB(05)cm [diffusion
Montecarlo method].
Root Lattice
@ References: Baake et al JPA(90)cm/00 [quasicrystals].
Rosen's Bimetric Theory > see bimetric.
Rotating Discs Argument
* Idea: An argument
against perdurantism.
@ References: Butterfield BJPS(06)
[defeat].
Rotation
Roton
* Idea: A kind of quasiparticle
present e.g. in liquid 4He, which is the quantum "ghost" of a
vanishingly small vortex ring, or an extra He atom moving and leaving a swirling
disturbance; Minimum energy 9 K.
Rotor [> s.a. constrained
systems].
* Configuration space:
Given by C =
SO(3) / H, where H is the symmetry group of the shape,
with 1(C) = H*.
@ Second-class constraints: Falck & Hirshfeld EJP(83);
Foerster et al PLA(94);
Kleinert & Shabanov PLA(97);
Saa CQG(97).
@ Quantum: DeWitt PR(52)
[path integral]; Falck & Hirshfeld EJP(83);
Ammann et al PRL(98)
[
-kicked,
decoherence]; Abdalla & Banerjee BJP(01)qp/98;
Brouard & Plata
JPA(03)
[-kicked].
Rotosurface > see spacetime
subsets.
Runaway Solutions > see self
force.
Runge-Kutta Method > see differential
equations.
Runge-Lenz Vector > s.a. orbits
in newtonian gravity.
* History: Invented by Hamilton; or by Laplace?
* Idea: A conserved quantity
in classical mechanics for a potential V = –a/r,
related to the fact that there is no orbit precession.
$ Def: For the Kepler problem, the vector A := p ×
L – mk (r/r) (where k is the constant appearing in V(r)
= –k/r),
whose direction is parallel to the vector from the center of attraction
to
the perihelion, and whose magnitude is proportional to the orbital eccentricity.
* Poisson brackets:
If we modify the vector A and define D := A /(2m |E|)1/2,
then
[Ai, Lj]
= 
ijk Ak , [A1, A2]
= –[p2 – (2mk/r)] L3 , but [D1, D2]
= L3 .
@ General references: in Goldstein 80; Kaplan AJP(86)
[as constant of the motion]; Dahl JPA(97)
[physical origin]; O'Connell & Jagannathan
AJP(03);
Morehead AJP(05)
[corresponding symmetry].
@ And general relativity: Brill & Goel AJP(99)gq/97 [deriving
light deflection and perihelion
precession].
@ Generalizations: Leach & Flessas JNMP(03)mp/04;
Kamath JMP(02)
[planar,
Chern-Simons electrodynamics].
Ruppeiner Metric / Theory > s.a. 3D
black holes; black hole
thermodynamics.
* Idea: A metric determined by the Hessian of the Gibbs surface; Provides a geometric
description of thermodynamic systems in equilibrium.
Rutherford Scattering
@ References: in Das & Ferbel 03 [II].
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
11 jul 2008