Topics, K

K-Causality > see causality conditions.

k-Essence > s.a. causality; quintessence; time in gravity.
* Idea: (Kinetic-energy-driven quintessence) A scalar field with a Lagrangian of a special form, that in cosmology causes its energy density to track that of radiation when the universe is radiation-dominated, and to follow its own evolution (first around a cosmological-constant-like value, then a different attracting behavior, when the universe becomes matter-dominated; The motivation is to solve the coincidence problem; & Mukhanov.
@ References: Rendall CQG(06)gq/05 [dynamics]; Bonvin et al PRL(06) [no-go theorem]; Yang & Gao CQG(11)-a1006 [phase-space analysis].
> In cosmology: see bianchi-I models; cosmological acceleration; dark energy.

K-System > see under Kolmogorov System.

K-Theory > s.a. bundle [gerbes]; KK-Theory; tiling.
* Idea: The Abelian group constructed from the space Vect(M) of equivalence classes of vector bundles over M, using the Grothendieck construction; A generalized cohomology theory (does not satisfy the dimension axiom for cohomology, and the K-theory of a point is not trivial), used to classify vector bundles; Notice however that it does not fully classify them, but only up to stable equivalence.
* Remark: Its dual homology theory does not seem to be useful.
@ General references: Atiyah 67; Milnor 74; Bak 81; Wegge-Olsen 93 [also C*-algebra]; Husemoller 94; Blackadar 98 [operator algebras].
@ And physics: Witten JHEP(98)ht [D-branes], IJMPA(01)ht/00 [strings]; Braun ht/00 [K-torsion]; Freed mp/02-ln, Woit ht/02 [and quantum field theory].
@ Generalizations: Mickelsson LMP(05) [twisted, invariants].

Kac's Lemma > see Recurrence.

Kac-Moody Algebra
* Idea: An infinite-dimensional Lie algebra, with generators satisfying

[Tni, Tmj] = i f ijk Tm+nk + K δ ij m δm+n,0 ,

where K is an operator such that [K, Tni] = 0 (it is effectively a c-number for the algebra).
* Example: If K = 0, we get a loop algebra.
* Remark: It is based on some compact simple Lie algebra.
* Scalar product: One with Lorentzian signature can be defined, improperly denoted by \(\langle\)A, B\(\rangle\) = tr AB, requiring that: tr AB = tr BA, and tr[A, B]C + tr B[A, C] = tr[A, BC] = 0 (to guarantee group invariance).
@ General references: Kac Izv(68); Moody JA(68); Zhe-Xian 91; Ray BAMS(01) [generalized]; Wassermann a1004-ln; Gómez et al JGP(12) [geometric approach].
@ And physics: Dolan PRL(81) [2D chiral models], PLB(82) [4D self-dual Yang-Mills]; Goddard & Olive ed-88; Fuchs LNP(97)ht [and conformal field theory]; > s.a. types of spacetime singularities.

Kac-van Moerbeke Lattice > see toda lattice.

Kadanoff-Baym Equations > s.a. Boltzmann Equation; early-universe cosmology.
* Idea: The (non-)equilibrium real-time Green's function description (or "closed-time-path Green's function" – CTPGF) of transport equations.
@ References: Greiner & Leupold AP(98)hp, hp/98-conf [stochastic interpretation].

Kadomtsev-Petviashvili Equation
* Idea: A completely integrable partial differential equation used to describe non-linear wave motion; It generalizes the 1D Korteweg–de Vries (KdV) equation.
@ References: in Xu a1205-ch [algebraic approach].
> Online resources: see MathWorld page; Scholarpedia page; Wikipedia page.

Kähler Metric, Structure > see symplectic structures.

Kalb-Ramond Field > see types of gauge theories.

Kalman Filter
@ References: in Casti 00.

Kaluza-Klein Theories > s.a. models and phenomenology.

KAM Theorem (Kolmogorov, Arnold, Moser) > s.a. Arnold Diffusion.
* Idea: When perturbing a completely integrable, non-degenerate (det |∂2H0 / ∂Ii Ij| ≠ 0) Hamiltonian system, "most" of the invariant tori, on which motion is quasi-periodic, persist for small perturbations; The Lebesgue measure of the complement of their union is small.
* Remark: The set of invariant tori is Lebesgue-measurable, although probably not Riemann-measurable, but it may be that there is an R-measurable set of points that move close to the unperturbed tori (but not quasi-periodically) – true for 2D autonomous systems.
@ General references: in Gallavotti 83, p466; in Arnold 89; Bricmont et al CMP(99)cd/98 [and quantum field theory]; Pöschel a0908-ln [detailed]; Dumas 14 [friendly introduction, mathematical]; most books on chaos.
@ Related topics: Gallavotti & Gentile CMP(02)mp/01 [invariant tori]; Broer BAMS(04) [Kolmogorov's 1954 paper]; De Simone RVMP(07) [renormalization proof]; Yuan CMP(07) [nearly integrable Hamiltonian systems of infinite dimensions].
@ Generalizations: Evans CMP(04) [quantum analog]; Jo & Jong a1505 [for generalized Hamiltonian systems without action-angle variables].
> Online resources: see MathWorld page; Wikipedia page.

Kaniadakis Framework / Statistics > see non-extensive statistical mechanics.

Kantowski-Sachs Models

Kaons > see hadrons.

Kapitza-Dirac Effect / Diffraction > see diffraction.

Kardashev Civilizations > see civilizations.

Karlhede Classification > see petrov classification; types of lorentzian geometries [classification].

Karlhede Invariant > see riemann tensor.

Kasner Solution > see bianchi I models.

Kauffman Bracket Polynomial > see knot invariants.

Kawai-Lewellen-Tye Relations > see covariant quantum gravity.

KdV Equation / System > see integrable systems.

Kelvin's Circulation Theorem > see under Circulation.

Kemmer Equation
* Idea: A relativistic (first-order, Dirac-like) field equation describing spin-0 and spin-1 particles.
@ References: Struyve et al PLA(04)qp/03 [paths and Bohm interpretation].

Kennedy-Thorndike Test > s.a. special relativity.
* Idea: A test of the velocity independence of the speed of light.
@ References: Hils & Hall PRL(90) [improved experiment]; Lipa et al a1203 [prospects for an experiment in low Earth orbit].

Kepler Conjecture > see sphere [packings].

Kepler Laws, Problem > see orbits in newtonian gravity; Runge-Lenz Vector.

Kerr Solution > s.a. particles in kerr spacetimes.

Kerr State
* Idea: A type of squeezed state.
@ References: Stobińska et al PRA(08)qp/06 [Wigner function].

Kerr-Bolt, Kerr-de Sitter, Kerr-NUT, Kerr-Sen Solutions > see modified kerr solutions.

Kerr-CFT Correspondence > see fields in kerr spacetimes.

Kerr-Newman Solution

Kerr-Schild Metric / Solution > s.a. generation of solutions; kerr-newman [boosted]; lorentzian geometry [flat deformation theorem].
* Idea: A spacetime metric written as a linear superposition of the flat spacetime (or de Sitter / anti-de Sitter spacetime) metric ηab and a squared null vector ka, multiplied by some scalar function H, or

gab = ηab + H kakb .

* Examples: The class includes the Kerr and Kerr-(A)dS solutions.
@ General references: Kerr & Schild in(65); Gergely & Perjés PLA(93)gq/02, JMP(94)gq/02, JMP(94)gq/02, AdP(94)gq/02 [vacuum]; Sopuerta JMP(98) [generalized stationary]; Coll et al GRG(01) [generalized transformations]; Hildebrandt GRG(02)gq, GRG(02)gq; Ivanov PRD(05)gq/04, Natorf GRG(05)gq/04 [and news, gravitational radiation]; Burinskii G&C(05) [multi-particle]; Kerr in(09)-a0706 [historical]; Bini et al IJGMP(10)-a1408.
@ Generalizations: Vaidya & Bhatt Pra(74); Málek & Pravda CQG(11)-a1009 [with (A)dS background]; Málek CQG(14)-a1401 [extended Kerr-Schild spacetimes]; Gürses et al CQG(17)-a1603 [Kerr-Schild-Kundt metrics as universal metrics].
@ Higher-dimensional: Ortaggio et al CQG(09)-a0808, AIP(09)-a0901.
@ In modified theories: Macías & Camacho GRG(05) [2+1, topologically massive]; Ett & Kastor JHEP(11)-a1103 [in Lovelock gravity].

Kervaire Problem > see differentiable manifolds [classification of exotic spheres].

Killing Fields / Vectors > s.a. killing tensors / forms.

Killing Horizon > see horizons.

* Idea: The result of the mergers of binary compact objects; So called because their transients peak at a luminosity that is a factor approximately \(10^3\) higher than a typical nova; The observation of gravitational waves from a binary neutron star merger in 2017 proved the theory that the nuclear reactions happening within the kilonova, called r-process (rapid neutron capture process), were the source of most, if not all, of the universe's heavy metals such as gold, platinum and silver.
@ References: Metzger et al MNRAS(10)-a1001 [proposal].

Kinematics > s.a. Configuration Space; special-relativistic kinematics.
* Idea: The study of the possible configurations or states of a system and relationships between them, including possible motions and transformation laws under changes of reference frame, independently of the dynamics (e.g., of the Hamiltonian).
* Rem: One example in which there have been different opinions on whether a certain phenomenon is kinematical or dynamical is the interpretation of the length contraction and time dilation [> see kinematics of special relativity]; There are also situations in which structures that are normally considered part of the dynamics are treated as part of the kinematics [> see covariant symplectic structures], or structures that are normally considered part of the kinematics are treated as dynamical [> see quantum gravity].
@ References: Martínez 09 [I]; in Janssen SHPMP(09); Spekkens a1209-FQXi [kinematics and dynamics must yield to causal structure]; Curiel a1603 [and the structure of a physical theory].
> Related topics: see Dynamics; Motion.

Kinetic Energy > see energy.

Kinetic Focus > see lagrangian dynamics.

Kinetic Theory > s.a. statistical mechanics; thermodynamics.
* Idea: The interpretation of thermodynamics in terms of which T corresponds to the average kinetic energy of molecules; Gave rise to statistical mechanics, and allows to derive properties such as viscosity, thermal conduction, and diffusion in non-uniform gases based on the solution of the Maxwell-Boltzmann equations.
@ Books: Kennard 38; Jeans 40; Chapman & Cowley 91; Brush 03; Loeb 04.
@ General references: Beck JSP(10) [deterministic approach].
@ Conceptual: Brush 76 [history]; de Regt BJPS(96) [and philosophy].
@ Relativistic: García-Perciante et al JNT(12)-a1007 [and microscopic description of dissipation–heat flow and viscosity]; Sarbach & Zannias AIP(13)-a1303 [intro], CQG(14)-a1309 [geometric perspective]; Ringström 13 [CQG+(15)].
@ Related topics: Latyshev & Yushkanov TMP(10)-a1001 [for degenerate quantum gases]; > s.a. Boltzmann Equation; gas; Maxwell-Boltzmann Distribution; Transport Phenomena.
> Online resources: see Wikipedia page.

Kink > s.a. geons; topological defects.
* Idea: A solution of a field theory (with non-simply-connected target space) which cannot be smoothly deformed to a constant field.
@ Gravitational: Shastri & Zvengrowski RVMP(91).
@ Topological fermions: Williams & Zvengrowski IJTP(77), Faber FBS(01)ht/99; > s.a. particle statistics; spinors in field theory.
@ Other examples: Alonso-Izquierdo & Mateos-Guilarte AP(12)-a1205, Alonso-Izquierdo PhyD-a1711 [(1+1)-dimensional scalar field models, and dynamics].

Kinnersley Black Hole > see black-hole thermodynamics; generating solutions to einstein's equation.

Kirby Calculus > see 4D manifolds.

Kirby-Siebenmann Invariant
* Idea: An object in H4(M; \(\mathbb Z\)2), which equals (index ω)mod 8 when the intersection form is even.

Kitaev Chain > see geometric phase.

KK-Theory > s.a. K-Theory.
* Idea: A bivariant version of topological K-theory, useful in the index theory for elliptic pseudo-differential operators.
@ References: Jensen & Thomsen 91.

Klein Bottle > see 2D manifolds.

Klein Geometry > s.a. geometry [history, relationships]; Cartan Geometry.
* Idea: A conception of geometry proposed in Felix Klein in 1872 with his Erlangen Programme, in which a geometry is characterized by an underlying set X and a group G of transformations acting on it, that are to be considered as equivalences; In modern terminology, if Euclidean geometry describes flat Euclidean space, Klein geometry describes general homogeneous manifolds.
* Examples: If the stabilizer group of an (arbitrary) element of X is denoted by H, one can express X = G/H, and some examples are
With positive-definite metrics, S2 = SO(3)/SO(2), E2 = ISO(2)/SO(2), and H2 = SO(2,1)/SO(2);
With Lorentzian-signature metrics, dS3,1 = SO(4,1)/SO(3,1), 4D Minkowski = E3,1 = ISO(3,1)/SO(3,1), and AdS3,1 = SO(3,2)/SO(3,1).
> Online resources: see Wikipedia page.

Klein Paradox > s.a. dirac field theory.
* Idea: In relativistic quantum mechanics, the surprising result obtained by Oskar Klein in 1929, applying the Dirac equation to electron scattering by a potential barrier (or well), that if its height V0 is of the order of the electron mass the barrier is nearly transparent, and as V0 approaches infinity the reflection diminishes and the electron is always transmitted (the particle can effectively continue on by transforming into its antiparticle); In quantum field theory, the phenomenon by which if the potential is strong enough it becomes supercritical and emits positrons or electrons spontaneously.
@ General references: Klein ZP(29); Bongaarts & Ruijsenaars AP(76) [as many-particle problem]; Bakke & Wergeland PS(82); Su et al JPA(93); Holstein AJP(98)jun; Calogeracos & Dombey IJMPA(99)qp/98, CP(99)qp, Dombey & Calogeracos PRP(99) [rev]; Nitta et al AJP(99)nov [simulations]; Bounames & Chetouani PLA(01)-a0712; Krekora et al PRL(04) [numerical solutions]; Dragoman PS(09)-qp/07 [experiment with graphene, phenomenon does not occur]; Alhaidari PS(11)-a0907 [resolution]; Kononets FP(10); Gerritsma et al PRL(11)-a1007 [quantum simulation using trapped ions]; Payandeh et al ChPC(13)-a1305 [Krein quantization approach]; Truebenbacher EJP-a1704 [new approach].
@ Variations: Grübl et al JPA(01)qp/02 [and Bohmian trajectories]; Ghose et al PLA(03)qp [not found for bosons]; De Leo & Rotelli PRA(06) [and potential barrier]; Cardoso et al CJP(09)-a0905 [not for massive bosons with non-minimal interactions]; Wagner et al PRA(10) [bosonic analog]; De Leo & Rotelli JPA(11)-a1202 [tests in graphene]; Ghosh IJTP(14)-a1202 [with generalized uncertainty principle]; Dodaro a1312 [in the pilot-wave interpretation]; > s.a. Refraction [classical analog in metamaterials].
> Online resources: see Wikipedia page.

Klein-Gordon Fields > s.a. klein-gordon fields in curved spacetime; quantum klein-gordon fields.

KLS Model (Katz-Lebowitz-Spohn) > see non-equilibrium thermodynamics.

KLT Relations (Kawai-Lewellen-Tye) > s.a. unimodular gravity.
@ References: Kawai et al NPB(86).

Klyachko Inequality > see Contextuality.

KMS States > see spin models.

Knee > see cosmic rays.

Knot Theory > s.a. knots in physics; knot invariants.

Knowledge > s.a. Epistemology; Explanation; Understanding.
@ General references: Josephson in(03)-a1307 [are scientific theories the result of the particular mathematical and experimental tools we use?]; Cottey a1102 [knowledge-inquiry and wisdom-inquiry in nuclear-physics textbooks]; Alexanian a1506 [William Oliver Martin's The Order and Integration of Knowledge]; Wolpert a1711 [constraints on physical reality].
@ Knowledge and beliefs: Andrews a1205 [knowledge and justification of beliefs]; Martins a1508 [beliefs about the real world and probabilistic knowledge].

Knudsen Number > s.a. Maxwell-Boltzmann Distribution.
* Idea: A dimensionless number defined as the ratio of the molecular mean free path length to a representative physical length scale; It is used to distinguish situations in which statistical mechanics or the continuum approximation are better descriptions for a fluid.
> Online resources: see Wikipedia page.

Kobayashi-Maskawa Matrix > see Cabibbo-Kobayashi-Maskawa Matrix.

Koch Curve > see fractals.

Kochen-Specker Experiment / Theorem > s.a. experiments in quantum mechanics / realism; Topos.
* Idea: Usually interpreted to imply that predictions of non-contextual hidden variable theories cannot agree with Copenhagen quantum mechanics.
@ General references: Kochen & Specker JMM(67); Lenard in(74); Peres JPA(91); Gill & Keane JPA(96)qp/03 [geometric]; Hamilton JPA(00) [obstruction-based approach]; Cabello et al PLA(05)qp [proof in any D > 3]; Nagata JMP(05) [inequalities]; Malley PLA(06)qp [implication]; Rudolph qp/06 [and ontological models]; Brunet PLA(07) [and a priori knowledge]; Straumann a0801 [simple proof]; Lisoněk et al PRA(14)-a1308 [simplest set of contexts]; Calude et al TMMP-a1402 [two geometric proofs]; Malley & Fine PLA(14)-a1407 [simplified]; Loveridge & Dridi a1511 [mathematical aspects of Mermin's proof]; Rajan & Visser a1708 [simplified geometrical proof]; Ramanathan et al a1806 [01-gadgets, subgraphs of a Kochen-Specker graph].
@ Experimental precision / nullification: Meyer PRL(99)qp + Mermin qp/99, Clifton & Kent PRS(00)qp/99 ["nullification"]; Appleby PRA(02)qp/00, qp/01 ["nullification" of "nullification"]; Cabello PRA(02)qp/01; Breuer PRL(02)qp; Appleby SHPMP(05)qp/03; Peres qp/03/PRL; Barrett & Kent SHPMP(04).
@ Single particle: Simon et al PRL(00)qp; Cabello PRL(03) [qubit]; Huang et al PRL(03) [photons, test]; D'Ambrosio et al PRX(13) [single-photon experiment].
@ Generalized: Cabello et al PLA(96), IJMPA(00)qp/99; Peres FP(96)qp/95; Aravind PRA(03)qp; Hrushovski & Pitowsky SHPMP(04)qp/03-conf [and Gleason's theorem]; Döring IJTP(05)qp/04 [for von Neumann algebras]; Dowker & Ghazi-Tabatabai JPA(08)-a0711 [for quantum measure theory]; Lisoněk et al a1401 [generalized parity proofs]; de Ronde et al SHPMP-a1404 [modal Kochen-Specker theorem, physical interpretation].

Kodama State > see Chern-Simons Function; loop quantum gravity; quantum gauge theory.

Kolmogorov Backward / Forward Equation > see fokker-planck equation.

Kolmogorov Probability > see probability in physics.

Kolmogorov System or K-System > s.a. ergodic theory [ergodic hierarchy]; lyapunov exponents; Mixing.
* Idea: A dynamical system in which trajectories mix due to local instabilities.
$ Def: A dynamical system (X, μ, φ) with positive Kolmogorov-Sinai entropy h.
* Relationships: It implies mixing and local instability (positive Lyapunov exponents), and h ~ h0 ~ 1 / τc.
* Examples: Bernoulli shift; Discretized Bianchi IX.
@ References: in Zaslavsky et al 91.

Kolmogorov-Sinai Entropy
* Idea: The growth rate h of the phase-space volume of a phase drop with time; By Liouville's theorem, for a Hamiltonian system we have h = 0 if there is no coarse-graining; If V is a coarse-grained phase-space volume, we estimate V(t) = V0 eht, and define

h:= limV0 → 0 limt → ∞ (1/t) ln V(t) .

* Remark: Notice that h is not actually an entropy but the time derivative of the entropy S ~ ln V; It is related to the stability or instability (and random behavior) of the system, and characterizes the rate of entropy production in a classical dynamical system.
* And Lyapunov exponents: Related by h = ∑i=1n λi [@ Pesin UMN 77].
@ General references: see Klimek & Lesniewski AP(96) [non-commutative Connes-Størmer entropy].
@ And chaos: Frigg BJPS(04); Kamizawa et al JMP(14) [relationship with entropic chaos degree and Lyapunov exponents].
@ Related topics: Bianchi et al a1709 [and growth of entanglement entropy for a quantum system].

Komar Integral > see energy in general relativity.

Kondo Effect
* Idea: A cooperative many-body phenomenon where electrons in a metal interact via spin-exchange with magnetic impurity atoms; The impurity increases the scattering of electrons at the Fermi level, causing an anomalous increase in resistance below a certain temperature; First observed in the 1930s and understood only three decades later, the Kondo effect attracted renewed interest with its realization in quantum dots.
@ References: news pw(13)aug [ferromagnetic].
> Online resources: see Wikipedia page.

Kondo Problem
* Idea: A single magnetic impurity in a non-magnetic material.
@ References: Rajeev AP(10) [Lie-algebraic approach].

Kontsevich Integral > see integration.

Koopman-von Neumann Formalism > s.a. approaches to classical mechanics.
* Idea: A Hilbert space/operator approach to classical mechanics proposed by Koopman and von Neumann in the 1930s; It was later shown that this formulation could also be written in a path-integral form.
@ References: Abrikosov et al AP(05)qp/04 [path-integral version, from dequantization]; Gozzi FP(10)-a0910-proc [and supermetrics in time]; Gozzi & Pagani PRL(10)-a1006; Bondar et al PRL(12)-a1105 [Ehrenfest quantization and unification of quantum and classical mechanics]; Klein QS:MF(17)-a1705 [proposal of new phase space function].
> Types of systems: see dissipative systems [Koopman operator]; electrodynamics; macroscopic quantum systems [hybrid]; yang-mills theories.
> And quantization: see approaches to quantum mechanics; canonical quantum mechanics; geometric quantization.

Korteweg-de Vries Equation > see types of integrable systems.

Kottler Metric / Solution > s.a. schwarzschild-de sitter; solutions with symmetries.
* Idea: The static form of the Schwarzschild-de Sitter metric, when written using Schwarzschild type coordinates.

Kovalevskaya Top > see systems in classical mechanics.

KP Equation / Hierarchy > see integrable systems.

Krajewski Diagrams
@ References: Stephan JMP(09)-a0809 [and the standard model].

Kramers Degeneracy Theorem
* Idea: The energy levels of systems with an odd total number of fermions remain at least doubly degenerate in the presence of purely electric fields.
@ References: Roberts PRA(12)-a1208 [without appealing to eigenvectors of the Hamiltonian].
> Online resources: see Wikipedia page.

Kramers Equation
* Idea: A partial differential equation, arising as a special form of the Fokker-Planck equation used to describe Browian motion in a potential.
@ Methods: Zhdanov & Zhalij JPA(99)mp [separation of variables].
> Online resources: see MathWorld page.

Kramers-Kronig Relations > see dispersion.

Krasnikov Tube > see wormholes.

Kraus Representation > see quantum open systems.

Krein Quantization > see Klein Paradox.

Krein Space
* Idea: An indefinite inner product space (K, \(\langle\cdot,\cdot\rangle\), J), in which the inner product (x, y):= \(\langle x, Jy\rangle\) is positive-definite and K possesses a majorant topology.
* Physical motivation: Krein spaces appear in the study of unitary irreducible representations of the de Sitter group, which can be used to classify elementary particles when modeled by fields propagating on a de Sitter background; In Krein-space quantization, the negative norm states are unphysical, and are used as mathematical tools for regularizing the theory.
@ General references: Gazeau et al Sigma(10)-a1001 [in de Sitter quantum theories].
@ Krein-space quantization: Sojasi & Mohsenzadeh IJTP(12)-a1202 [and ultraviolet divergences of Green functions]; Pejhan et al AP(14)-a1204 [and Casimir effect].
> Applications: see fock space [generalized]; approaches to quantum gravity; black-hole radiation; Klein Paradox; modified approaches to QED; non-commutative geometry; regularization schemes; Weyl Algebra.
> Online resources: see Encyclopedia of Mathematics page; Wikipedia page.

Kretschmann Scalar Invariant > s.a. riemann tensor; schwarzschild geometry.
$ Def: The curvature scalar quantity K:= Rabcd Rabcd.
* Motivation: It is used as a convenient rough measure of how relativistic a system is, because it increases with curvature and does not automatically vanish for a vacuum solution of general relativity (the diagnostic power of the Ricci scalar is limited for this reason).
@ References: Gkigkitzis & Haranas PhInt(14)-a1406 [for black holes, and singularities, entropy and information].
> Online resources: see Wikipedia page.

Kron Reduction > see graphs.

Kronecker Delta
* Expansion: The Kronecker delta δnm, where n and m vary over N possible values, can be expanded as δnm = N–1 k=1N exp{2πi k(nm)/N}; Proof: For n = m, the exponential is 1 and the sum equals N; For nm, the sum is equal to (sum of all N-th roots of unity)nm = 0.

Kronecker Index > see cohomology.

Kronecker Power / Product > see matrices.

Kruskal Extension > s.a. schwarzschild spacetime.
* Idea: The maximally extended Schwarzschild solution, obtained by introducing coordinates that extend across the horizon.
$ Def: The Schwarzschild metric, with line element written in the form

ds2 = – (2M / r) er/2M du dv + r2(dθ2 + sin2θ dφ2) ,

where u:= tr*, v:= t + r*, and r*:= r + 2 M ln(r/2M–1) is the tortoise coordinate.
@ General references: Kruskal PR(60); in Birrell & Davies 82; Boersma PRD(97) [identification]; Lake CQG(10), a1202 [Kruskal–Szekeres completion].
@ Related topics: Gibbons NPB(86) [elliptic interpretation, and quantum mechanics]; Gautreau IJMPA(99) [Kruskal-Szekeres incompleteness??]; Qin gq/00 [causal structure]; Varadarajan PRD(01)gq/00 [as canonical variables]; Augousti et al EJP(12) [use for infalling observers, pedagogical]; Sbierski a1507, a1711-proc [C0 inextendibility].

Kuiper Belt > see solar system.

Kullback-Leibler Distance > see entropy [relative entropy].

Kummer Tensor Density
@ References: Baekler et al AP(14)-a1403 [introduction, in electrodynamics and gravity].

Kundt Spacetimes / Waves > s.a. chaotic motion.
* Idea: Spacetimes with a non-expanding, shear-free, twist-free, geodesic principal null congruence.
* Result: Degenerate Kundt spacetimes (the ones in which the preferred kinematic and curvature null frames are all aligned) are the only spacetimes in 4 dimensions that are not \(\cal I\)-non-degenerate, so that they are not determined by their scalar polynomial curvature invariants.
@ General references: Griffiths et al CQG(04) [type III, non-zero cosmological constant, generalized]; Jezierski CQG(09) [and degenerate Killing horizons]; Coley et al CQG(09)-a0901 [degenerate]; McNutt et al CQG(13) [invariant classification].
@ Types of matter: Fuster AIP(06)gq/05 [type III, with null Yang-Mills field]; Tahamtan & Svítek EPJC(17)-a1505 [with minimally-coupled scalar field].
@ In higher dimensions: Podolský & Žofka CQG(09)-a0812; Podolský & Švarc CQG(13)-a1303 [explicit algebraic classification], CQG(13)-a1306 [physical interpretation using geodesic deviation]; Podolský & Švarc CQG(15)-a1406 [Weyl tensor algebraic structure].
@ In other theories: Brännlund et al CQG(08)-a0807 [and supersymmetry]; Chow et al CQG(10)-a0912 [topologically massive gravity].

Kunneth Formula / Theorem > see homology.

Kuratowski Lemma > see axiom of choice.

Kustaanheimo-Stiefel Transformation
* Idea: A transformation that maps the non-linear and singular equations of motion of the 3D Kepler problem to the linear and regular equations of a 4D harmonic oscillator; It is used extensively in studies of the perturbed Kepler problem in celestial mechanics and atomic physics.
@ References: Kustaanheimo & Stiefel JRAM(65); Bartsch JPA(03)-phy/03 [geometric Clifford algebra approach]; Saha MNRAS(09)-a0803 [interpretation, and quaternion form].
> Online resources: see Encyclopedia of Mathematics page.

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