**Topics, R**

**Rabi Model**

* __Idea__: The simplest
model describing the coupling of light and matter; In its semiclassical
form, the model describes the coupling of a two-level system and a
classical monochromatic field, and it is the basis for understanding
nuclear magnetic resonance; The fully quantum model considers the same
situation with the light field quantized, and has been solved in 2011;
2012, There is renewed theoretical interest in the model due to Braak's
analytical solution and introduction of a new criterion for integrability.

@ __References__:
Braak PRL(11),
Solano Phy(11) [analytical solution to the quantum model];
Travěnec PRA(12)-a1201 [two-photon Rabi Hamiltonian];
Moolekamp a1201 [understanding its solvability];
Ziegler JPA(12) [spectral density, continued-fraction approach];
Batchelor & Li a1510 [re Braak's solution];
Savasta et al a2006 [gauge invariance].

**Racah Algebra**

@ __References__:
De Bie et al a2001-conf [overview].

**Racah Coefficients / Formula**
> s.a. SU(2).

* __Idea__: A formula
and coefficients used in the theory of coupling of quantum mechanical
angular momenta; Also known as 6*j*-symbols.

@ __References__: Krasnov CQG(05)gq/04 [for the Lorentz group];
Krasnov & Louko JMP(06)mp/05 [for SO(1, *d*+1), AdS-cft methods].

> __Online resources__:
see Wikipedia page.

**Radial Gauge**
> see gauge.

**Radiation**
> s.a. thermal radiation.

**Radiation Memory**
> see Gravitational Memory.

**Radiation Reaction**
> see self-force.

**Radiative Corrections**
> see quantum field theory techniques; quantum-gravity phenomenology.

**Radioactivity**
> see nuclear physics.

**Radion Field** > s.a. tests
of newtonian gravity [constraints]; types of dark matter.

* __Idea__: A field that stabilizes an
extra spatial dimension in a higher-dimensional theory of gravity/spacetime.

@ __References__: Chakraborty & SenGupta EPJC(17)-a1701 [stabilization from the higher-curvature gravitational action].

**Radon Transform**

* __Applications__:
CAT, PET and NMR scanning, found by A Cormack (Nobel prize in medicine 1979).

@ __References__: Facchi et al JMO(09)-a0905.

**Radon-Nikodym Derivative**

**Rainbow (Optics)** > s.a. light.

* __Remark__: It cannot form
in the middle of the day because the Sun has to be at most 40° above
the horizon in order for its light to refract at the right angles.

@ __References__:
Pincock SHPMP(11) [explanations];
Ford TPT(20)feb [graphical approach].

**Raindrop**

* __Idea__: An object
which plunges radially toward a black hole from rest at infinity.

**Raising Operator**
> s.a. Ladder Operators; spherical harmonics.

* __Idea__: An operator acting
on a set of states or functions labelled by a discrete parameter; >
Similar to a creation operator, but without the
particle interpretation, and the opposite of a Lowering
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.

* __Applications__: Detecting explosive powders from a distance
[@ news sn(14)aug].

**Randall-Sundrum Model**
> see branes; brane cosmology.

**Randers Space**
> see finsler geometry; Cartan-Randers
Systems / pioneer effect.

**Random Dynamics**

@ __References__: Nielsen & Kleppe a1403-proc [derivation of space];
Kleppe & Nielsen a1412-conf [derivation of diffeomorphism invariance].

**Random Matrix (and Tensor) Models**
> see Matrix Models.

**Random Medium**
> see Disordered System; electromagnetism
in matter; light [propagation]; scattering;
solid matter [amorphous solids, glass].

**Random Process**
> s.a. random walk.

**Random Surfaces**
> s.a. dynamical triangulations.

@ __References__:
Ambjørn in(96)ht/94 [and quantum gravity, lecture notes];
Wheater JPA(94) [rev];
Ambjørn et al 97.

**Random Walk** (including quantum walk)

**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*+*p*_{z})
/ (*E*−*p*_{z})]
in high-energy physics, which corresponds to tanh(*y*)
= *p*_{z}/*E*, or
*r* = cosh^{−1}*γ*, with
*p*_{z} the longitudinal momentum.

$ __Pseudorapidity__: The quantity
−ln[tan(*θ*/2)], where *θ* is the angle
with respect to the beam line in high-energy collisions; For approximately
massless particles, it is approximately equal to the rapidity.

@ __References__: Lévy-Leblond & Provost AJP(79)dec;
Rhodes & Semon AJP(04)jul
[description of Wigner rotation and Thomas precession].

> __Online resources__:
see Wikipedia page.

**Rarita-Schwinger Equation / Lagrangian / Theory**

* __Idea__: A theory of
particles with spin *k* + \(1\over2\), with *k* an
integer, that are described by a spinor field with *k* spacetime
indices *ψ*_{a... c},
and obey a generalized Dirac equation

(*γ*^{a}
∇_{a} + *κ*)
*ψ*_{m... p} = 0 ,

where *γ*^{a}
*ψ*_{ab... c} = 0.

* __k____ = 0__:
There is a gauge symmetry *ψ*_{a}
→ *ψ*_{a}
+ ∇_{a} *φ*,
with *γ*^{a}
∇_{a} *φ* = 0.

* __k____ = 1__:
The Lagrangian is *L* = *ψ**_{a}
(*γ*^{m}
∇_{m} + *κ*)
*ψ*^{a}
− \(1\over3\)*ψ**_{a}
(*γ*^{a}
∇_{b}
+ *γ*_{b}∇^{a})
*ψ*^{b}
+ \(1\over3\)*ψ**_{a}
*γ*^{a}
(*γ*^{b}
∇_{b} − *κ*)
*γ*^{c}
*ψ*_{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];
Carballo Pérez & García-Compeán FP(12)-a1104 [and deformation quantization];
Torres-Gómez & Krasnov IJMPA(13)-a1212 [and propagating spin-1/2 particles];
Adler PRD-a1508 [classical massless, gauged],
PRD-a1508 [quantized massless, gauged].

**Rastall Gravity**
> s.a. brans-dicke theory.

@ __References__: Visser PLB(18)-a1711 [equivalent to Einstein gravity];
Hansraj & Banerjee a1807 [equilibrium stellar configurations];
Cruz et al CQG(19)-a1904 [thermodynamics revision];
Bronnikov et al a2007
[static, spherically symmetric solutions with scalar field, stability].

**Rational Function**
> s.a. Padé Approximant.

* __Idea__: A function
that can be written as the quotient of two polynomials.

> __Online resources__:
see MathWorld page;
Wikipedia page.

**Rational Numbers**
> see numbers.

**Raumproblem (Problem of Space)**

@ __References__: Loinger RNC(88);
Urbantke IJTP(89);
Rau a1009-proc;
Scholz a1310-in
[the views of H Weyl and É Cartan];
Scholz a1608 [hist].

**Ray Emanating from a Point in R ^{n}**
> see lines.

**Raychaudhuri Equation**

$ __Def__: Given a
timelike geodesic congruence parametrized by proper time, with unit
tangent vector *u*^{a},
the rate of change of its expansion *θ* is given by

d*θ*/d*τ* = \(-{1\over3}\)*θ*^{
2} − *σ*_{ab}
*σ*^{ab}
+ *ω*_{ab}
*ω*^{ab}
− *R*_{ab}
*u*^{a}*u*^{b} ,

where *σ*_{ab} is the
shear and *ω*_{ab} the
vorticity of the congruence, and *R*_{ab}
the Ricci tensor ; For lines that are not geodesics, a term
∇_{a}(*u*^{b}
∇_{b} *u*^{a})
is added to the right-hand side.

@ __General references__:
Raychaudhuri PR(55);
in Hawking & Ellis 73;
in Stephani et al 03;
Dadhich gq/05 [derivation],
comment Mitra gq/05;
Kar & SenGupta Pra(07)gq/06 [rev];
in Witten a1905-ln [intro].

@ __In specific spacetimes__: Albareti et al JCAP(14) [FLRW and Bianchi-I models].

@ __Variations__:
issue Pra(07)jul,
including Dadhich Pra(07)gq [analog for quantum gravity?];
Abreu & Visser PRD(11)-a1012 [generalizations];
Mohseni GRG(15)-a1502 [for spinning test particles];
Burger et al PRD(18)-a1802 [beyond general relativity];
Iosifidis et al a1809 [with torsion and non-metricity];
Pesci a1809 [for a null congruence];
Namboothiri a1911 [in Kaluza-Klein spacetime].

@ __Quantum version, based on Bohmian trajectories__: Das PRD(14)-a1311;
Farag Ali & Das PLB(15)-a1404;
Lashin & Dou PRD(17)-a1606,
response Das PRD(17)-a1702 [problematic points];
Vagenas et al NPB(18)-a1706 [and the GUP];
Chakraborty et al PLB(19)-a1904 [with zero-point length].

> __Related topics__:
see Expansion, Shear
and Vorticity of a Congruence.

> __Online resources__:
see Wikipedia page.

**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; optical technology.

**Rayleigh Scattering**
> s.a. interaction.

* __Idea__: Elastic
scattering of photons; It is responsible for the blue color of the sky.

@ __References__: Chakraborti AJP(07)sep [simple experiment];
Vinogradov et al a2010 [quantum model].

> __Online resources__:
see Wikipedia page.

**Reaction Networks**
> see networks.

**Real Numbers**
> see numbers.

**Reality** > s.a. realism.

* __Quantum reality
problem__: Finding a mathematically precise definition of a sample
space of configurations of beables, events, histories, paths, or other
mathematical objects, and a corresponding probability distribution,
for any given closed quantum system.

@ __General references__:
Stern a2001
[Objective Cognitive Constructivism and recursive production of objective realities].

@ __In quantum theory__: Sudbery a1205
[quantum theory and dual attitudes to reality (realist vs idealist, material vs mental)
and knowledge (objective vs subjective, scientific vs romantic)];
Kent PRA(14)
[quantum reality problem, outline of solution];
Krizek a1708
[overview of conceptions, structural realism];
Bertlmann a2005 [John Bell's ideas];
> s.a. foundations of quantum theory;
*ψ*-Ontic Theories.

**Realization of a Group**
> see group action.

**Rebit**
> s.a. quantum computation.

* __Idea__: A state of a quantum
system with a 2D Hilbert space described by a real density matrix.

**Reciprocity**
> see Born's Reciprocity Hypothesis;
number theory [quadratic reciprocity theorem].

**Reconnection / 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 (for a Dynamical System)**

* __Idea__: The return of a
dynamical system to (a state arbitrarily close to) its initial state.

* __Poincaré's recurrence
theorem__: If *V* is a space with measure \(\mu\), and \(T:
V\to V\) a measure-preserving flow map, then for any open set \(A\subset
V\) there exists an orbit of *T* that intersects *A*
infinitely many times; Used with *V* the phase space for a
physical system.

* __Kac's lemma__: For
a discrete-time evolution, the average recurrence time for a point
in a space *V* to return to a subset *A* is \(T =
\mu(V)/\mu(A)\); For example, for all gas particles in a 1-liter container
at room temperature and atmospheric pressure to be in one half of the
container this time is many times larger than the current age of the
universe, and, practically speaking, one can ignore the possibility [>
see Wikipedia page].

@ __General references__: Wallace a1306 [unified account of classical and quantum recurrence theorems].

@ __Poincaré recurrence__: Burić et al JPA(03) [area-preserving maps];
Saussol RVMP(09) [quantitative results];
Varandas SD(16)-a1411 [Kac's lemma for suspension flows];
Dong & Stojković PRD(16)-a1611 [the effect of gravity].

@ __Recurrence paradox__: Steckline AJP(83)oct [Boltzmann & Zermelo].

@ __In quantum theory__: Campos Venuti a1509 [recurrence time].

> __Online resources__:
see Wikipedia page.

**Recursion Relations**
> see scattering [for quantum field theory scattering amplitudes].

**Recursion Theory**

@ __References__: Smullyan 93.

**Redshift / Blueshift**
> s.a. doppler effect;
tests of gravity with light [gravitational redshift].

* __Idea__: A frequency
/ wavelength change for electromagnetic waves / photons; It may be
due to the relative velocity between source and observer, expansion
of the background spacetime, or a gravitational field.

* __Quantitatively__:
The redshift is defined as the quantity *z*:=
(*λ*_{2}−*λ*_{1})
/ *λ*_{1}
= *ω*_{1}/*ω*_{2} − 1.

> __And cosmology__: see
cosmological expansion; geometry
of the universe [distance-redshift relation].

**Reduce**
> see programming languages.

**Reduced Density Matrix**
> see Density Matrix.

**Reduced Mass**
> see classical systems [two-body problem].

**Reduction of a Dynamical System**
> s.a. constrained systems; Emergence;
Momentum Map; types of symplectic structures
[polysymplectic].

* __Idea__: The process by
which from a dynamical system with symmetries one obtains another one
with a smaller number of degrees of freedom, by dividing out by the action
of the symmetry group; The reduction procedure can be based on the standard
momentum map (symplectic or Marsden-Weinstein reduction) or on generalized
distributions (the optimal momentum map and optimal reduction); In general
it yields a stratified structure.

* __ History__: It was first
formulated by Marsden and Weinstein for a phase space described by a
symplectic manifold, and then generalized to a Poisson manifold by
Marsden and Ratiu.

@ __ References__:
Ortega & Ratiu LMP(04) [rev],
RPMP(06) [stratified structure];
Butterfield phy/05-ch;
Laudato a1703-MS [classical and quantum interacting systems, and non-commutative geometry];
> s.a. symmetries in quantum theory.

> __Specific systems__:
see klein-gordon quantum field theory.

**Reduction of a Fiber Bundle** > see fiber bundles.

**Reduction Rule in Quantum Theory**
> same as Projection Postulate.

**Reductionism**
> see paradigms in physics.

**Reeh-Schlieder Property / Theorem**
> s.a. locality in quantum field theory.

* __Idea__: A result of
relativistic local quantum field theory, stating that the vacuum is a
cyclic vector for the field algebra of any open set in Minkowski space, or
that it is spacelike superentangled relative to local fields, meaning that
any part of the Hilbert space can be reached by operations on the vacuum
state with operators constructed within any, arbitrarily small space-time
region.

@ __General references__: Reeh & Schlieder NC(61);
Schlieder CMP(65);
Clifton et al PRA(98) [superentangled states];
Jaekel AdP(03)ht/00 [for ground states];
Sanders CMP(09)-a0801 [existence of Reeh-Schlieder].

@ __In curved spacetime__: Strohmaier CMP(00)mp [on a stationary spacetime];
Dappiaggi RVMP(11)-a1102 [for higher-spin free fields].

> __Online resources__:
see Wikipedia page.

**Refined Algebraic Quantization**
> see dirac quantization of constrained systems.

**Refinement of an Open Cover** > see cover.

**Reflection** > see groups [reflection groups];
mirrors [moving mirrors]; quantum systems [potential step];
wave phenomena; Wigner Delay.

**Reflexive Banach Space** > see Banach Space.

**Refrigerators** > see technology.

**Regge Calculus**
> s.a. quantum version.

**Regge Poles / Theory**

@ __References__: Bottino AAST-a1807,
Del Duca & Magnea a1812-in [retrospective look].

**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 in Quantum Field Theory**

**Reidemeister Moves**
> s.a. knot theory.

* __Idea__: Elementary,
local changes to knot or link diagrams preserving their topology.

* __Types__: There are
three, called the (I) twist/untwist move, (II) poke/unpoke move, and (III) slide move.

> __Online resources__:
see nLab page;
MathWorld page;
Wikipedia page.

**Reissner-Nordström Solution**
(the spelling is not "Reissner-Nordstrøm")

**Rejection Sampling**

* __Idea__: A technique for sampling
numerically from a target distribution, given the ability to sample from another
distribution; > s.a. montecarlo method.

@ __References__: Ozols et al a1103
[quantum rejection sampling].

**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; An equivalence relation
~ partitions a set *X* into disjoint equivalence classes [*x*]:=
{*y* ∈ *X* | *y* ~ *x*} whose union is the whole set
*X*; > s.a. Wikipedia page.

* __ k-homogeneous__:
A relational structure \(\cal A\) is called

@

@

@

>

**Relational Blockworld**
> s.a. classical limit; relativistic
quantum mechanics; time [block universe vs evolution].

@ __References__: Stuckey et al qp/06,
FP(08);
Stuckey & Silberstein a0712 [and quantum gravity];
Silberstein et al SHPMP(08) [and interpretation of quantum mechanics];
Miller SHPMP(08) [and quantum mechanics];
Stuckey et al a0903 [discrete path integral over graphs];
Stuckey et al a0908 [and interpretation of quantum field theory].

**Relationalism / Relational Theories**
> s.a. parametrized theories; Newton's Bucket;
time; spacetime.

* __Idea__: Relationalism can be
thought of in two different ways, in terms of background independence (as
opposed to absolutism) or in terms of processes (as opposed to substantialism).

* __In quantum theory__: In the late
1990s Mermin proposed an interpretation in which quantum correlations were
considered as elements of physical reality; The proposal encountered difficulties
after the no-go theorems proved by Cabello and Seevinck in the following years;
> s.a. quantum correlations.

@ __General references__:
Martin-Dussaud a2012 [and fundamental physics].

@ __Relational mechanics__: Barbour CQG(03)gq/02;
Anderson a1001,
CQG(11)-a1003 [of shape and scale],
a1111 [problem of time and quantum cosmology];
Anderson IJMPD(14)-a1202 [quadrilateralland];
Ferraro GRG(16)-a1410 [as a gauge theory, and dragging effects];
Ferraro FP(17)-a1801 [and the frame of fixed stars];
> s.a. modified quantum mechanics; Shape Dynamics.

@ __Quantum theory__: Anderson & Kneller IJMPD(14) [quadrilateralland];
Ipek & Caticha a1601-conf [based on entropic dynamics];
Laudisa a1710;
Rovelli PTRS(18)-a1712 [overview and update];
de Ronde & Fernández a1712 [history and present use];
Trassinelli FP(18)-a1803 [and probability];
de Ronde et al a1810
[evading Cabello's and Seevinck's no-go-theorems in Mermin's relational interpretation];
Höhn et al a1912,
a2007 [three approaches];
Oldofredi FP-a2011 [mereological bundle theory approach];
Jennings a2011 [modified version];
Muciño et al a2105 [critique].

@ __Gravity__: Anderson a0908-MG12 [quantum cosmology];
Anderson a1205-talks [quantum gravity];
Anderson a1411 [and the problem of time];
Vidotto a1508-ch [relational quantum cosmology];
Koslowski et al PLB(18)-a1607
[and the cosmological singularity];
Marchetti & Oriti a2008 [cosmology, from group field theory];
> s.a. canonical general relativity; quantum gravity.

**Relative Field Theory**
>s.a. anomalies.

**Relative Locality**
> see locality.

**Relativistic Theory of Gravitation**
> s.a. gravity theories.

@ __General references__:
Logunov & Loskutov TMP(86);
Vlasov & Logunov TMP(86) [cosmology];
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;
Gershtein et al TMP(09)-a0810 [gravitational waves];
Soloviev & Tchichikina a0812 [canonical formalism].

@ __Cosmology__: Chugreev TMP(09) [vacuum cosmological solutions];
Zhelnorovich G&C(10) [positive cosmological constant],
G&C(10).

@ __Gravitational collapse__:
Vlasov & Logunov TMP(86),
TMP(87);
Gershtein et al TMP(09) [impossible for dust ball];
Logunov & Mestvirishvili TMP(13) [impossible].

@ __Other phenomenology__: Logunov & Loskutov TMP(87) [test bodies].

@ __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. 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).

* __And transformations between
reference frames__: In 1910 von Ignatowsky showed that the relativity
principle, combined with linearity and isotropy, leads almost uniquely
to either the Lorentz transformations or Galileo's transformations.

* __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.

@ __General references__: Budden BJPS(97) [Galilean vs modern definition];
Barbour gq/03-proc [new formulation];
Szabó FPL(04) [vs Lorentz covariance];
Leston & Ferraro gq/04 [and the Einstein equation];
Grøn AJP(09)apr-a0708,
reply Hartman & Nissim-Sabat AJP(09)apr [with rotation and inertial dragging];
Bandyopadhyay EJP(09) [meaning, students' ideas];
De Angelis & Espirito Santo JAHH-a1504 [Giordano Bruno as precursor].

@ __Einstein-Kretschmann debate__:
Sorkin PSA(02);
Antoci & Liebscher a0910 [general coordinate transformations and isometries];
Pitts a0911 [adding artificial gauge freedom to a theory].

@ __Generalizations__:
Delbourgo & Stack IJMPA(14)-a1401 [general relativity of space-time-property, and unification];
Hetzroni FP(20)-a2001 [in abstract spaces].

> __Application to specific theories__:
see electrodynamics; general relativity;
special relativity.

**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, and can be estimated in the framework of the
Boltzmann equation.

@ __Systems__: Chimonidou & Sudarshan PRA(08)-a0705 [two interacting harmonic oscillators];
> s.a. gas [bosons]; magnetism.

@ __Relaxation time__: Lemos & Pereira PhyA(07) [temperature effects];
Maiocchi & Carati CMP(10) [for a Hamiltonian system, lower bound].

@ __Relaxation time, universal bound__: Hod PRD(07)gq/06 [*τ* > \(\hbar\)/(π *T*)];
Pesci IJMPD(09)-a0807 [and covariant entropy bound].

@ __In quantum theory__: Tsekov IJMS(01)-a1505 [Markov processes, phase-space description
of a mechanical system and extension of the Gibbs canonical distribution];
Santos et al PRL(12)-a1110,
García-Mata et al PRE(15)-a1408 [and transition to chaos];
> s.a. pilot-wave quantum theory.

@ __Related topics__: Shapiro PLA(08) [energy duality];
Kochubei IEOT(11)-a1105 [and fractional calculus];
Ginoux & Letellier Chaos(12)-a1408 [relaxation oscillations, history].

> __Specific types of
systems__: see spin models.

> __Online resources__:
see Wikipedia page.

**Relaxion Field**
> see Electroweak Hierarchy Problem.

* __Idea__: An axion-like field whose expectation value
determines the electroweak hierarchy as well as the QCD strong CP violating \(\bar{\theta}\) parameter.

@ __References__: Nelson & Prescod-Weinstein a1708 [and the strong CP problem].

**Relaxon** > s.a. Heat Flow.

* __Idea__: A collective phonon excitation
responsible for thermal conductivity in electrical insulators.

**Renormalization**
> s.a. renormalization group; for quantum gravity,
gauge theories, and other theories.

**Rényi Entropy** > see types of entropies.

**Reparametrization-Invariant Systems**
> see parametrized theories.

**Replica Symmetry / Theory**
> s.a. ising model.

* __Idea__: The
"replica method" proposed by Mark Kac is an approach to the study of
quenched disordered systems.

@ __General references__:
in Dotsenko 95;
Dotsenko 01 [r PT(02)jan].

@ __Dynamical replica theory__: Coolen et al PRB(96)cm/95 [for disordered spin systems];
Sakata JPA(13) [evolution of the autocorrelation function].

@ __Related topics__: Arenzon JPA(99)cm/98 [for granular systems];
Campellone et al JSP(10) [breaking, and finite-volume corrections to mean-field theory];
news gizm(17)jun [replica symmetry breaking in glasses].

> __Online resources__: see Wikipedia
page on Replica Trick.

**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 *z*_{0},
*f*(*z*) = *F*(*z*)/(*z*−*z*_{0})^{n},
where *F*(*z*) is regular at *z*_{0},
the residue at *z*_{0} is *F*^{(n)}(*z*)/*n*!

**Resistance / 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.

**Resource Theory**
> s.a. generalized and non-equilibrium thermodynamics;
probability theory in physics [in general probabilistic theories].

* __Idea__: Quantum
resource theory is an approach to the study of the connection
between quantum information theory and thermodynamics.

@ __General framework__: Brandão & Gour PRL(15)-a1502 [general structure];
Girolami a1709 [information geometry];
Bromley et al JPA(18)-a1802 [lower bounds for resource quantifiers];
Sparaciari et al Quant(20)-a1806 [first law];
Chitambar & Gour RMP(19)-a1806 [rev];
Egloff et al PRX(18)
[focus on local operations and communication channels];
Takagi & Regula a1901 [operational];
Gour & Scandolo a2101 [dynamical resource theory];
Dey a2103 [introductory review].

@ __Related topics__: Lostaglio et al NJP(17)-a1511
[non-commutativity and maximum entropy principles];
del Rio et al a1511 [resource theories of knowledge];
Sparaciari et al PRA(17)-a1607 [without a thermal reservoir];
Ben Dana et al PRA(17)-a1704 [of coherence];
Regula JPA(18)-a1707 [geometry of quantum resource quantification];
Alicki a1801 [information is not a thermodynamic resource];
Morris et al a1908
[entanglement between identical particles];
Martins et al a2004 [quantum incompatibility].

**Response Functions / Theory**

@ __References__: in Alastuey et al 16.

> __Examples, applications__:
see Compressibility; heat [thermal expansion];
fluctuations [fluctuation-dissipation theorem].

**Rest Frame**
> see reference frames.

**Rest of the Universe**
> see Feynman's Rest of the Universe.

**Resummation**

@ __References__: DeWitt PRL(64);
Isham, Salam & Strathdee PRD(71).

**Resurgence / Resurgent Functions**
> see Alien Calculus / quantum field theory techniques.

**Retarded Green Function**
> see green functions.

**Retarded Potential**
> see Potentials in Physics.

**Retraction**

$ __Def__: Given an
inclusion *i*: *A* → *X* between topological spaces,
a retraction is a continuous map *r*: *X* → *A* such
that *r* \(\circ\) *i* = id_{A},
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__: S^{1}
is a retract of the disk minus a point, D^{2} \ {0}, but not
of D^{2}.

**Retro-MACHOs**
> see black-hole phenomenology.

**Retrocausality / Retrocausation** (Backwards in Time Signaling)

* __Idea__: A phenomenon
by which an effect can precede its cause; It arises for example in
time-symmetric approaches to quantum mechanics, as the possibility of
backward influences from measurement events to the state of systems at an
earlier time; It gives rise to the "retrocausal loophole" in Bell's Theorem.

@ __General references__: Nielsen a1503-proc [arguments, including cosmological constant];
Guryanova et al Quant(19)-a1708 [no-backwards-in-time signaling conditions].

@ __In quantum mechanics__: Suarez qp/98,
qp/98,
qp/98/PLA;
Costa de Beauregard qp/98;
Wharton qp/98|AJP;
Pegg FP(08) [and quantum measurement];
Berkovitz SHPMP(08);
Price SHPMP(08) [toy models];
Hokkyo SHPMP(08) [and single-electron double slit];
Schulman SHPMP(08);
Pegg SHPMP(08);
Price SHPSB(12)-a1002 [and time symmetry];
Stapp AIP(11)-a1105,
a1111
[and the principle of sufficient reason];
Aharonov et al a1206-conf [using weak + strong measurements];
Lewis SHPMP(13);
Tan et al PRL(15) [retrodiction, for a qubit];
Aiello & Woerdman a1506/PS [in classical and quantum optics];
Evans a1506;
Price & Wharton Ent(15)-a1508 [and Costa de Beauregard's explanation for quantum correlations],
a1510
[as an alternative to quantum spooks];
Aharonov et al Quanta(16)-a1512 [and free will];
Sutherland AIP(17)-a1706;
Carmi et al a1903 [and hidden variables];
Zwirn a2009
[no retrocausation, but not entirely determined past];
Norsen & Price a2102 [dialogue];
> s.a. modified quantum theory [time-symmetric].

@ __And hidden variables__:
Price & Wharton a1307;
Drezet FP(19)-a1904 [Lorentz-invariant, deterministic framework].

@ __Models, systems__: Wharton Ent(18)-a1805 [retrocausal but not retro-signaling classical fields];
Argaman Ent(18)-a1806 [toy models with 'lenient' arrow of time].

> __And theory__:
see interpretations of quantum mechanics
[transactional]; pilot-wave quantum theory.

> __And experiment__:
see experiments in quantum mechanics
[delayed-choice]; quantum effects.

> __Online resources__:
see Wikipedia page.

**Retrodiction**
> s.a. histories formulations of quantum theory.

* __Idea__: The making
of statements about past events based on current information.

@ __In quantum mechanics__: Mermin PRL(95) [problems];
Hartle Compl(97)gq,
PS(98)gq/97 [prediction vs retrodiction];
Barnett et al Symm-a2103;
> s.a. time in quantum theory.

**Reversibility**
> see arrow of time; quantum foundations;
quantum measurements and types of quantum measurements.

**Revival**
> see Quantum Revival.

**Reynolds Number** > s.a. turbulence.

* __Idea__: A
dimensionless parameter for liquid flow in a pipe, that measures
the importance of non-linearity in the Navier-Stokes equation and
characterizes the conditions for laminar flow or turbulence in terms
of speed of flow, density, viscosity and diameter of the pipe; It
can be thought of as giving the ratio of inertial forces to viscous
forces in the liquid; The number of active degrees of freedom in
turbulent flow is about *R*^{ 9/4}
per *L*^{3}.

$ __Def__: The number *R*
:= *UL*/*ν*_{mol} = (typical
flow velocity) (typical length scale) / (kinematic molecular viscosity).

* __Values__: For small
*R*, the flow is laminar; As *R* increases, instabilities
set in; For large *R*, there is full turbulence (in real life,
*ν*_{mol} is usually of the
order of 10^{−2}–10^{−1} cm^{2}/sec).

@ __References__: Reynolds PTRS(1895),
reprint PRS(95).

**Reynolds Transport Theorem**

* __Idea__: A result giving an
expression for the rate of change of an integral over an evolving domain.

@ __References__:
Falach & Segev a1312 [generalization];
Reddiger & Poirier a1906 [for manifolds with corners].

> __Online resources__:
see Wikipedia page.

**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.

@ __General references__: Dennis & Hannay PRS(05)mp [Calugareanu's theorem].

@ __Braided ribbon networks__: Hackett a1106 [definition, evolution and invariants];
> s.a. spin networks.

**Riccati Equation**
> see ordinary differential equations.

**Ricci Collineations**

@ __References__: Flores et al JMP(04)gq [type-B warped spacetimes].

**Ricci Flow**
> s.a. Geometric Flow; riemannian geometry.

* __Idea__: The study of the
initial-value problem ∂_{t}
*g*_{ab}
= −*R*_{ab} + (2/*n*)
*r* *g*_{ab},
with *g*_{ab}(0)
= *g*_{ab}, for an
*n*-dimensional manifold *M* (usually closed), with
*g*_{ab} a Riemannian
metric, and *r *its averaged scalar curvature over *M*.

* __Results__: Solutions
always exist for finite *t*, and continue while the curvature
is bounded; Isometries and volume are preserved; Under certain
circumstances, the flow converges to a sphere.

@ __General references__: Hamilton JDG(82),
JDG(85),
in(88);
Carfora & Marzuoli CQG(88);
Carfora et al JDG(90);
Cao & Chow BAMS(99);
Samuel & Chowdhury CQG(07),
CQG(08)
[gradient formulation in terms of entropy and general relativity].

@ __Applications and special cases__: Holzegel et al a0706 [with surgery, for biaxial Bianchi IX metrics];
Woolgar CJP(08)-a0708-proc [applications in physics];
Carroll a0710 [and quantum theory];
Abraham et al IJGMP(09)-a0810 [corresponding classical mechanical system];
Carfora & Romano RPMP(09)-a0902,
Carfora a1001-conf
[and renormalization-group flow for non-linear σ-models];
> s.a. averaging in relativistic
cosmology; origin of quantum theory.

@ __Related topics__: Isidro et al IJMPA(09) [quantum-mechanical];
> s.a. cell complexes [simplicial];
types of manifolds [PL manifolds].

**Ricci Rotation Coefficients** > s.a. tetrads.

* __Idea__: The
connection coefficients defined by an orthonormal frame field;
It is named after Gregorio Ricci-Curbastro.

@ __References__: Ricci MRAL(1896);
Levy BAMS(25).

**Ricci Tensor**
> s.a. cell complex [simplicial]; riemann
tensor; types of manifolds [PL manifolds].

* __Idea__: The once-contracted
Riemann curvature tensor; Named after Gregorio Ricci-Curbastro.

@ __General references__:
Carfora & Familiari IJGMP(20)-a2102 [relationships in physics and interpretation].

@ __Classification__:
Cormack & Hall IJTP(79);
Zachary & Carminati GRG(04) [Segre classification, algorithm].

**Richardson Extrapolation** (> n630)

> __Online resources__:
see Wikipedia page.

**Riemann Equation**

* __Idea__: The first-order, non-linear
partial differential equation *u*_{t}
+ *uu*_{x} = 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 JPA(08)-a0709 [complex deformation].

**Riemann Hypothesis / Conjecture**
> see conjectures.

**Riemann Integral**
> see integration.

**Riemann Mapping Theorem**

* __Idea__: If *U*
is a non-empty simply connected open subset of the complex plane \(\mathbb
C\) which is not all of \(\mathbb C\), then there exists a Riemann mapping
(a biholomorphic mapping, i.e., a bijective holomorphic mapping whose
inverse is also holomorphic) from *U* onto the open unit disk.

**Riemann Normal Coordinates**
> see coordinates.

**Riemann Surface**
> see 2D manifold.

**Riemann Zeta Function** > see Zeta Function.

**Riemann-Cartan Structure**
> s.a. affine connection
[structure equations]; differential geometry.

* __Idea__: A differentiable manifold
with a vielbein *e*_{a}^{i} and a Lorentz connection
*ω*^{ ij}_{a},
in terms of which the curvature and the torsion are

*R*^{ ij}
:= d*ω*^{ ij}, *R*
= *e*^{a}_{i}
*e*^{b}_{j}
* R*^{ ij}_{ab}
, and
*T*^{ i}
:= d*e*^{i}
+ *ω*^{ i}_{j}
∧ *e*^{ j} .

@ __And gravity theories__: Vacaru et al gq/05-book [Riemann-Finsler];
Romero et al a0903 [higher-dimensional theories];
Sobreiro & Vasquez JGP(11)-a1003 [from metric-affine theories];
> s.a. gödel spacetime;
Metric-Affine Gravity Theories;
teleparallel gravity.

@ __And cosmology__:
Minkevich gq/07-proc;
Minkevich MPLA(13)-a1309 [regular, accelerating FLRW models].

> __Related topics__:
see Defects; electroweak theory;
thermodynamics; unified theories.

**Riemann-Hilbert Problem**
> s.a. axisymmetry [Ernst equation].

* __Idea__: A
boundary-value problem for partial differential equations.

@ __References__: Deift a1903 [method of non-linear steepest descent].

**Riemann-Lovelock Curvature Tensor**
\ s.a. riemann tensor.

* __Idea__: The *k*th-order
Riemann-Lovelock tensor is a tensor with 4*k* indices, of *k*th
order in the Riemann curvature tensor, with which it shares its basic
algebraic and differential properties; It was introduced to study the
properties of Lovelock gravity theories in low dimensions.

@ __References__: Kastor CQG(12)-a1202,
Camanho & Dadhich EPJC(16)-a1503 [*k*th-order tensor, two formulations].

**Riemann-Roch Theorem**

* __Relationships__: It can be
considered as a special case of the Atiyah-Singer index theorem.

**Riemann-Silberstein Vector**

* __Idea__: A tool that
simplifies the description of the electromagnetic field both in the
classical and the quantum domain; It has also been considered as the
best possible choice for the photon wave function.

@ __References__: Cheremisin ht/03-conf;
Białynicki-Birula & Białynicka-Birula JPA(13)-a1211 [rev].

**Riemannian Connection**
> see affine connection.

**Riemannian Geometry / Manifold / Metric / Structure**
> see riemannian geometry; metric tensor.

**Riesz Space**

* __Idea__: A vector
lattice, i.e., an ordered vector space which is also a lattice with
respect to a partial order <.

@ __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 R_{g}(*h*):=
*hg*.

**Rigid Body** > s.a. reference frames;
Chasles' Theorem; Screw Theory.

* __In classical mechanics__:
Its dynamics is described by Euler's equations; Any rigid body can be
replaced by an equivalent system of exactly four masses, located at
the vertices of an irregular tetrahedron.

* __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 JPA(07)-a0705 [complex and PT-symmetric solutions];
Rajeev a0906 [geometry of motion];
Gil et al EJP(14) [equivalent system of four masses];
Wilkinson a1506-wd
[no physical classical solutions to Euler's equations].

@ __In quantum mechanics__: Modugno et al mp/05 [covariant approach].

@ __In special relativity__:
Martínez & Gambí IJTP(81) [definitions];
Eriksen et al PS(82);
Kim & Jo JPA(04) [rigid rotation impossible];
Soler FP(06)gq/05;
Sauer a0704-MGXI [Einstein-Varicak letters];
Brotas a0810 [history];
Butler et al a1005
[geometric, and Clifford algebras];
Llosa et al GRG(12)-a1103 [radar-holonomic congruences of wordlines as weaker substitutes];
Bażański GRG(11) [rigidly rotating circle];
Franklin FP(13) [dynamics].

@ __In curved spacetime__: Combi & Romero GRG(20) [in an expanding universe].

**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 *l* to the
event horizon such that *l*^{ a}
= *k*^{a} + Ω_{H}
*m*^{a}, where Ω_{H}
is a constant.

@ __Rigidity theorems__: Hawking CMP(72) [black hole];
Isenberg & Moncrief [Cauchy horizon];
Chruściel CMP(97)gq/96;
Friedrich et al CMP(99)gq/98,
Rácz CQG(00)gq/99 [general],
gq/07-wd [generalization];
Chruściel & Tod CMP(09)-a0712;
Alexakis et al a0902 [without analyticity assumption];
Yu AHP(10)-a0903 [charged black holes];
Costa e Silva & Flores a1408 [null geodesic incompleteness];
Ionescu & Klainerman a1501 [mathematical survey];
Galloway & Vega AHP(17)-a1608
[Hausdorff closed limits and rigidity for Lorentzian horospheres].

@ __In cosmology__:
Lidsey GRG(93);
Aguirregabiria & Lazkoz MPLA(04)gq [of tachyonic inflation].

**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.

**Rips Complex**
> see under Vietoris-Rips Complex.

**Rishons**
> see composite models.

**Ritz Theory of Electrodynamics**
> see alternative formulations of electrodynamics.

**Robbins Conjecture**
> see conjectures.

**Robertson Uncertainty Principle**
> see modified uncertainty relations.

**Robin Boundary Conditions / Problem**
> s.a. boundaries in field theory
[edge states in quantum field theory].

* __Idea__: A
boundary-value problem for partial differential equations.

@ __General references__: Bondurant & Fulling JPA(05)mp/04 [and Dirichlet problem solutions];
Belchev & Walton JPA(10) [reflecting wall].

@ __Specific theories__: Krishnan et al a1702-proc [general relativity].

**Robinson-Bertotti Solution**
> see kantowski-sachs metrics.

**Robinson-Trautman Spacetimes**
> s.a. schwarzschild spacetime.

* __Idea__: Metrics
that describe gravitational radiation in the exterior of bounded sources.

@ __General references__:
Bičák & Podolský PRD(97)gq/99 [Λ ≠ 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],
CQG(08) [past quasilocal horizons];
Kozameh et al CQG(06)gq [Robinson-Trautman-Maxwell, meaning];
Podolský & Svítek PRD(09)-a0911 [with cosmological constant, past horizons];
de Oliveira et al PRD(11)-a1107 [numerical evolution];
Nashed IJTP(14) [energy and momentum in teleparallel gravity];
Davidson GRG(17).

@ __In higher dimensions__: Podolský & Ortaggio CQG(06)gq;
Ortaggio et al CQG(08)-a0708 [with electromagnetic field];
Svítek MG12(12)-a1001 [existence of horizons];
Podolský & Švarc CQG(15)-a1406 [Weyl tensor algebraic structure].

**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.

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

**Rømer Time-Delay Measurement**
> see tests of general relativity [gravitational-wave speed].

**Rokhsar-Kivelson Point** \
s.a. Dimers.**
***

@

**Root Lattice**

@ __References__: Baake et al JPA(90)cm/00 [quasicrystals].

**Rosen's Bimetric Theory** > see bimetric.

**Rössler System**

* __Idea__: A system of
three non-linear ordinary differential equations, originally studied by
Otto Rössler, that exhibits chaotic dynamics associated with the
fractal properties of its strange attractor.

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

**Rotating Discs Argument**

* __Idea__: An argument against perdurantism.

@ __References__: Butterfield BJPS(06) [defeat].

**Rotation**
> s.a. mach's principle [rotation problem].

**Rotation Curves**
> see dark matter; galaxies;
milky way galaxy; unified theories.

**Roton**

* __Idea__: A kind of
quasiparticle present, e.g., in liquid ^{4}He,
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 \(\cal C\) = SO(3) / *H*, where *H* is the symmetry
group of the shape, with π_{1}(\(\cal C\))
= *H**.

@ __Second-class constraints__:
Falck & Hirshfeld EJP(83);
Foerster et al PLA(94);
Kleinert & Shabanov PLA(97);
Saa CQG(97);
Bratek JPA(10)-a1004 [relativistic].

@ __Quantum, δ-kicked__: Ammann et al PRL(98) [decoherence];
Brouard & Plata JPA(03).

@ __Quantum, other__: DeWitt PR(52) [path integral];
Falck & Hirshfeld EJP(83);
Abdalla & Banerjee BJP(01)qp/98.

**Rotosurface**
> see spacetime subsets.

**Routh Reduction Procedure**
> see lagrangian systems [singular]; noether's theorem.

**Runaway Solutions**
> see self force.

**Runge-Kutta Method**
> see differential equations / computational physics.

**Runge-Lenz (Laplace-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** /(2*m* |*E*|)^{1/2},
then

[*A*_{i},
*L*_{j}]
= ε_{ijk}* A*_{k}
, [*A*_{1},*
A*_{2}] = −[*p*^{2}
− (2*mk*/*r*)] *L*_{3}
, but [*D*_{i},
*D*_{j}]
= *ε*_{ijk}
*L*_{k} ;

The *L*_{i} and
*D*_{i} are generators
of an SO(4) algebra, possibly related to transformations in the space
one obtains by mapping the 3D Kepler problem to a 4D harmonic oscillator
problem; > see the Kustaanheimo-Stiefel Transformation.

* __Generalization__: There
probably are Runge-Lenz type vectors for other types of central potentials.

@ __General references__: in Goldstein 80;
Kaplan AJP(86)feb [as a constant of the motion];
Dahl JPA(97) [physical origin];
O'Connell & Jagannathan AJP(03)mar;
Morehead AJP(05)mar [corresponding symmetry].

@ __In general relativity__: Brill & Goel AJP(99)apr-gq/97 [deriving light deflection and perihelion precession].

@ __In other theories__: Hakobyan & Nersessian PRA(15)-a1504 [Clogero-Coulomb problem].

@ __Generalizations__:
Leach & Flessas JNMP(03)mp/04;
Kamath JMP(02)
[planar, Chern-Simons electrodynamics];
Nikitin JMP(13)-a1308 [for arbitrary spin, and superintegrable
quantum mechanical systems with so(4) dynamical symmetry];
Nikitin JPA(14)-a1403 [with spin, in any dimension].

**Ruppeiner Metric / Theory**
> s.a. thermodynamics.

* __Idea__: A metric on the
state space of a thermodynamical system, determined by the Hessian of
the Gibbs surface; It provides a geometric description of thermodynamic
systems in equilibrium, and is conformally related to the Weinhold metric.

@ __References__: Ruppeiner PRA(79) [introduction],
RMP(95) [rev];
Ruppeiner JPCS(13)-a1210 [physical interpretation of the scalar curvature].

> __And black holes__:
see 3D black holes; black-hole
thermodynamics.

**Russell Paradox**

* __Idea__: Partition the set of all sets
into two subsets, those that contain themselves as elements, and those that don't;
Does the set of all sets that don't contain themselves contain itself? If it does,
then it's one of those sets that don't contain themselves, so it doesn't contain
itself; Likewise, if it doesn't, then it does.

* __Rem__: The upshot is that one has to
be careful when defining what a set is, if one wants to avoid logical troubles.

**Rutherford Scattering**

@ __References__: in Das & Ferbel 03 [II];
Lemasson et al PRL(09)
+ Ackermann & Simon Phy(09) [21st-century version, and nuclear structure].

**Rydberg Atoms** > see atomic physics.

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

send feedback and suggestions to bombelli at olemiss.edu – modified 27 may 2021