Topics, E

e > s.a. Euler's Equation.
$ Def: The number e:= \(\lim_{n\to\infty}(1+1/n)^n = 2.7182,8182,8459...\) (slow convergence); It is transcendental (Hermite 1873).
* History: The concept might have appeared in connection with the formula for compound interest.
@ References: Maor 94, ThSc(94)jul; Mohazzabi AJP(98)feb [Monte Carlo calculation]; Adrian 06 [I].

E Modes > see gravitational radiation.

E Theory
* Idea: A low-energy effective theory obtained from string theory, which has an \(E_{11}\) symmetry.
@ References: West IJMPA(19)-a1905, Glennon & West a2102 [irreducible representations].

Early-Universe Cosmology

Earnshaw's Theorem
* Idea: An electric charge cannot be in stable equilibrium under electrostatic forces alone.
* Proof: Use the fact that a harmonic function has no maximum or minimum.


Eccentricity > see conical sections.

Echoes (Gravitational) > see black-hole phenomenology; quantum gravity and cosmology [echoes of the early universe].

Eddington Limit > see star formation and evolution [accretion disk].

Eddington-Born-Infeld / Eddington-Inspired Gravity > s.a. cosmological models [bouncing alternative to inflation].
* Idea: A theory proposed as an alternative to general relativity that offers a resolution of spacetime singularities.
@ General references: Bañados et al PRD(09)-a0811 [and large-scale structure]; Avelino PRD(12)-a1201 [astrophysical and cosmological constraints]; Pani et al PRD(12)-a1201 [matter coupling and dust collapse]; Pani & Sotiriou PRL(12) [surface-singularity pathologies]; Bouhmadi-López & Chen JCAP(16)-a1609 [quantization, and cosmology].
@ Isolated objects: Pani et al PRL(11)-a1106 [collapse and compact stars]; Sotani PRD(14)-a1404 [distinguishing it from general relativity using neutron stars].
@ Cosmology: Bañados & Ferreira PRL(10)-a1006 + news po(10)jul [and minimal length]; De Felice et al PRD(12)-a1205 [constraints]; Scargill et al PRD(12)-a1210 [singularity avoidance and expansion rate]; Harko et al Gal(14)-a1410 [Bianchi-I models]; Li & Wei PRD(17)-a1705 [stability of Einstein static universe].

Eddington-Finkelstein Coordinates > see coordinates for schwarzschild spacetime.

Eddy Currents > see physics teaching.

Edgar-Ludwig Metric > s.a. conformal structures [conformal Killing vector].
* Idea: A conformally flat, pure radiation solution of Einstein's equation.
@ References: in Pollney et al CQG(00) [classification]; Barnes CQG(01)gq [symmetries].

* Of an achronal set S: @ in Wald 84, p200.

Edge State > s.a. boundaries in field theory.
* Idea: A time-harmonic solution of a conservative wave system, e.g. Schrödinger, Maxwell, which is propagating (plane-wave-like) parallel to, and localized transverse to, a line-defect or "edge".
@ References: Fefferman et al APDE(16)-a1506 [in honeycomb structures].

Edth Operator
@ References: Geroch, Held & Penrose JMP(73); Bartnik CQG(97)gq/96 [null quasi-spherical gauge for general relativity]; Frauendiener & Szabados CQG(01)gq/00 [kernel, on higher-genus surfaces].

Effect Algebra > see algebra.

Effective Action of a Group on a Manifold > see group action.

Effective Dynamics > see classical mechanics [quantum effects].

Effective Field Theory > s.a. effective quantum field theory; interactions / beyond the standard model.
* Idea: A field theory that arises as an approximation to a more fundamental (quantum) one, in which one looks at the dynamics above a certain length scale by summarizing the effect of the coupling to the smaller-length-scale degrees of freedom into averaged quantities that enter the action or Hamiltonian.
* Example: The cosmological constant has been considered as a low-energy effect of the zero-point energy from small-scale fluctuations in matter fields.
@ General references: Polchinski ht/92-ln [intro]; Wetterich PLB(93)-a1803 [exact evolution equation]; Pich ht/98-ln; Burgess ht/98-proc [non-equilibrium physics], hp/98-conf [effective Lagrangians, intro], ARNPS(07)ht [intro]; Wells a1106-ln, 12 [as tools to predict new physics]; Carroll blog(13)jun; Gripaios a1506-ln [intro and examples]; Cheung et al PRL(16)-a1509 [on-shell recursion relations]; Petrov & Blechman 16; Cheung et al JHEP(17)-a1611 [4-parameter classification]; Reall & Warnick a2021 [rigorous justification in classical field theory].
@ Gravity and cosmology: Donoghue AIP(12)-a1209 [quantum general relativity as an effective field theory, intro]; Cardoso & Porto GRG(14)-a1401-GR20 [gravity]; Agarwal et al JCAP(14)-a1311 [inflation]; Kase & Tsujikawa IJMPD(14)-a1409 [modified gravity including Horndeski theory and Hořava-Lifshitz gravity]; Barceló et al IJMPD(15)-a1505-GRF [from a minimal modification of the structure of general relativity]; Bartolo et al JCAP(16)-a1511 [signatures of spacetime diffeomorphism-invariance breaking]; Porto PRP(16)-a1601 [rev]; Levi a1811 [and post-Newtonian gravity], a1901 [for gravity at all scales]; > s.a. dark-energy models; matter distribution in cosmology; motion of gravitating bodies [spin-orbit coupling].
@ Non-equilibrium processes at finite temperature: Glorioso & Liu a1805 [rev, fluctuating hydrodynamics and the second law of thermodynamics].
> Online resources: see Wikipedia page.

Effective Mass / Effective Mass Tensor > see mass.

> In astronomy: see anomalous acceleration [Pioneer effect].
> In physics: see aharonov-bohm effect; Aharonov-Casher Effect; Hall Effect; Hanbury Brown-Twiss Effect.

Effectus Theory > see category theory [categorical logic, and quantum mechanics].

Efficient (or Moving) Cause
* Idea: The efficient cause for some change or movement in a thing is an object, person, ... (apart from the thing itself), which interacts so as to be an agency of the change or movement.
> Online resources: see Wikipedia page.

Efficiency > see thermodynamic systems.

Efimov Effect / State > s.a. atomic physics; composite quantum systems; Phases of Matter; Three-Body Forces.
* Idea: A quantum phenomenon in which the atoms in a cloud attract each other when considered two at a time, but repel each other when considered three at a time; It could lead to an incompressible, undilutable liquid 1013 times thinner than water!
* Other version: A purely quantum phenomenon whereby particles, such as neutral atoms, which ordinarily do not interact strongly enough to form 2-way bound states can join together with a third one and form three-way bound states (mainly through the van der Waals effect, in which rearrangements of electrical charge in one atom form an electric dipole whose electric field can induce dipoles in, and thereby attract, neighboring atoms); First predicted around 1970 by Vitaly Efimov, then a PhD candidate, but originally considered "too strange to be true"; For decades, experimenters tried in vain to create these "Efimov trimers"; 1999, Chris Greene and collaborators predicted that gases of ultracold atoms might provide the right conditions; 2005, The team led by Rudi Grimm of the University of Innsbruck confirmed the Efimov state in an ultracold Cs gas cooled to 10 nK.
@ General references: Esry et al PRL(99); Bulgac PRL(02); Wang JFA(04); Kraemer et al Nat(06)mar; news pw(06)mar; Day PT(06)apr; news pn(07)may; Macek PS(07); Rau a0706 [pedagogical]; Ferlaino & Grimm Phy(10); Bhaduri et al AJP(11)mar-a1009 [elementary]; Modugno Phy(14) [giant states observed]; news wired(14)may; Naidon et al PRA(14) [physical origin of the universal three-body parameter]; Naidon & Endo RPP(17)-a1610 [rev].
@ Extensions: Ferlaino et al PRL(09)-a0903 + Esry Phy(09) [4-body states]; Gridnev JFA(12)-a1204 [N-body effect], JMP(13)-a1210 [not for 4 bosons]; Moroz et al PRL(15)-a1506 [generalized, in 1D]; Bazak proc(20)-a1812 [beyond three particles].
@ Special cases: Wang et al PRL(11) [for 3 interacting dipolar molecules]; news pw(12)may [for fermions with long-range interactions];

Egorov's Theorem
* Idea: A result establishing a condition for the uniform convergence of a pointwise convergent sequence of measurable functions.
> Online resources: see Wikipedia page.

Eguchi-Hanson Metric
@ References: Mahapatra ht/99 [obtaining it as a limit]; Ishihara et al PRD(06)ht [+ black hole, in 5D Einstein-Maxwell].

Ehlers Group > see types of spacetimes [stationary].

Ehlers-Geren-Sachs Theorem
@ References: Faraoni a0811-fs [extended to modified gravity].

Ehrenfest Classification of Phase Transitions > see phase transition.

Ehrenfest Dynamics
* Idea: The dynamics satisfied by mean values of observables in quantum theory.
@ References: Alonso et al a1010 [geometry and statistics]; Clemente-Gallardo & Marmo NCC(13)-a1306-ln [and the geometry of quantum mechanics].

Ehrenfest Equations
* Idea: Equations which describe changes in specific heat capacity and derivatives of specific volume in second-order phase transitions.
> Online resources: see Wikipedia page.

Ehrenfest Paradox > see Reference Frame [rotating frames].

Ehrenfest Theorem
$ Def: For a non-explicitly-time-dependent observable A, d\(\langle\)A\(\rangle\) / dt = \(\langle\)[A, H]\(\rangle\) / i\(\hbar\).
@ General references: Friesecke & Schmidt PRS(10) [sharp version, for general self-adjoint operators]; Bondar et al PRA(13)-a1307 [violation in finite-dimensional quantum and classical mechanics]; Lin a1609 [infinite square well]; Li et al a1802 [violation for particle constrained on a hypersurface]; Renziehausen & Barth a1904 [generalization]; Arodz a1907 [and generalizations]; Giordano & Amodio EJP-a2011 [incompleteness].
@ For non-linear Schrödinger equation: Bodurov IJTP(98); Kälbermann JPA(04)qp/03 [and Galilean invariance]; Friesecke & Koppen JMP(09) [rigorous derivation].
@ For other theories: Parthasarathy a0911 [quantum field theory]; Kanatchikov JGSP(15)-a1501 [in precanonical quantization of fields].
> Online resources: see Wikipedia page.

Ehrenfest Time > s.a. quantum-mechanical effects [wave-packet spreading]
* Idea: The time characterizing the departure of quantum dynamics for observables from classical dynamics.

EHT > see under Event Horizon Telescope.

@ References: Kauffman a1109 [and the foundations of physics].

Eigenstate Thermalization Hypothesis > see states in quantum statistical mechanics; statistical mechanical equilibrium.

Eigenvalues > see for ordinary differential equations and matrices; quaternions [for quaternion operators]; s.a. Antieigenvalues.

Eightfold Way
* Idea: A precursor theory to the quark model, in which hadrons were organized into octets.
@ References: Gell-Mann & Ne'eman 64.
> Online resources: see Wikipedia page.

Eikonal Approximation > see optics.

Einstein Algebra > s.a. models of quantum spacetime; in Hole Argument.
* Idea: An algebraic structure generalizing the concept of spacetime satisfying Einstein's equation.
@ References: Geroch CMP(72); Heller & Sasin IJTP(95); Rosenstock et al a1506 [equivalence of the theory to general relativity].

Einstein Boxes > s.a. energy-momentum [electromagnetic].
* Idea: A thought experiment intended to demonstrate the incompleteness of the quantum description of reality, developed by Einstein, de Broglie, and several others; It involves the splitting in half of the wave function of a single particle in a box.
@ General references: Norsen AJP(05)feb-qp/04; Marcella qp/06/AJP.
@ Einstein-Bohr photon box: Dieks & Lam AJP(08)sep-a0705 [complementarity].

Einstein Equation > s.a. solutions.

Einstein Frame > see scalar-tensor theories of gravity.

Einstein Manifold / Metric / Space > see types of spacetimes.

Einstein Model > s.a. specific heat.
* Idea: A simple model for a crystalline solid.
@ References: Bertoldi et al EJP(11) [exact microcanonical treatment].

Einstein Relation > see diffusion.

Einstein Ring > see lensing.

Einstein Telescope > see gravitational-wave interferometers.

Einstein Temperature
* Idea: The quantity ΘE:= \(\hbar\)ω/kB, where ω is the frequency of oscillation of a mode in a solid, which defines the temperatures around which the behavior of that mode (in terms of its contribution to the cV, e.g.) switches between quantum and classical.

Einstein Tensor > s.a. einstein equation.
$ Def: The tensor \(G_{ab}:= R_{ab} - Rg_{ab}\), constructed from the Riemann tensor, appearing in the left-hand side of the Einstein equation.
@ References: Lamey & Obermair BJP(05)gq [re physical significance].

Einstein-Æther Theories

Einstein-Cartan Theory (Einstein-Cartan-Sciama-Kibble Theory)

Einstein-Dirac Theory

Einstein-Hilbert Action > see action for general relativity.

Einstein-Hopf Model > see statistical mechanical models.

Einstein-Infeld-Hoffmann Approximation > see einstein equation.

Einstein-Jordan Conundrum > see quantum field theory.

Einstein-Rosen Bridge / Wormhole Throat > see wormholes and wormhole solutions.

Einstein-Sasaki Spaces > s.a. kerr spacetime [Kerr-de Sitter].
@ References: Lu et al PRD(06)ht/05 [in D ≥ 7 dimensions]; Gauntlett et al CMP(07)ht/06 [obstructions].

Einstein-Smoluchowki Equation > see diffusion.

Einstein-Straus Model > see under Swiss-Cheese Model.

Einstein-Vlasov System > see solutions of einstein's equation with matter.

Einstein@home Computing Network > see gravitational-wave analysis.

Eisenhart Theorem
@ References: in Cordani 03; Minguzzi CQG(07)gq/06 [and causal simplicity].

Ekpyrotic Scenario > see brane cosmology.

Elasticity > s.a. Continuous Media; Plasticity; Viscoelasticity.
* Idea: The property of many materials of returning to their original shape after a deformation.
* Quantitatively: The elastic constant c of a material is related to the stress σ and strain ε by the defining equation σ = ; It is frequency-dependent, and can be studied with RUS; In an inhomogeneous material the elastic constant is a function of the point, and in an anisotropic material it becomes an elastic tensor cij, related to the stress and strain by σi = cij εj; > s.a. sound.
* Stiffness: Some indicative numbers are rubber (0.1), wood (10), steel (200), diamond (1200), spacetime (\(10^{24}\)).
* And damping: The elastic tensor cij can be seen as the real part of a damping tensor Dij, whose imaginary part is related to the Q of an object.
@ General references: Landau & Lifshitz 86; Lebedev & Cloud 09 [mathematical]; Fokas & Yang a1010, a1010 [analytic solutions]; in Thorne & Blandford 15; Pommaret JModP(16)-a1512 [potentials and variational calculus]; Balluffi 16 [for crystal defects]; Capovilla a1709 [variational approach]; in Hentschke 17 [numerical, with Mathematica].
@ Relativistic: Bel gq/96 [deformations]; Beig & Schmidt CQG(03)gq/02; Beig gq/04-proc; Beig & Schmidt CQG(05)gq/04 [rigid rotation]; Natário GRG(14)-a1406 [rigid 1D elastic body in 2D]; Capoferri & Vassiliev a1805 [including curved spacetimes]; Natário a1912; Brown CQG-a2004 [Lagrangian perspective].
@ Non-linear materials: Coulais et al PRL(14) + Daniels Phy(14) [experiments with a model soil near jamming].
@ Cosserat theory: Delphenich a1305 [teleparallelism and the Cosserat approach to deformable bodies]; Delphenich a1510 [relativistic theory].
@ Related topics: Delph PRA(05) [atomic level stresses]; Marder et al PT(07)feb [thin sheet crumpling and buckling]; Fülöp & Ván MMAS(12)-a1007 [kinematics of finite elastic and plastic deformations]; Böhmer et al QJM(11)-a1008 [rotational model]; Norris PRS(14) [elastic networks]; news pw(19)jul [colossal elastocaloric effect, and applications]; > s.a. exterior calculus [discrete]; Hooke's Law; physics labs [Young'smodulus]; Stress Tensor; Tensile Strength.
> Applications in gravity and field theory: see 2-spinors [Cosserat model]; FLRW models; general relativity; spacetime [as an elastic continuum].
> Online resources: see Wikipedia page.

Electric Charge > see charge.

Electric Dipole Moment > see electromagnetism; types of particles.

Electric Field > see electricity.

Electric Part of Weyl Curvature > see weyl tensor.

Electricity > s.a. electronic technology.

Electrodynamics > s.a. Maxwell-Lorentz Equations.
* Idea: A theory of coupled electromagnetic fields and charged particles; In practice, the classical theory most often consists of Maxwell's equations for electromagnetism coupled to the Lorentz force equation, and the quantum theory of Quantum Electrodynamics (QED).
@ General references: Marino AP(02)phy/01; Gabrielov et al mp/04 [equilibrium points]; Frolov a1111 [interaction]; Hadad et al JPCS(15)-a1503 [and hidden geometrical structure of electromagnetic field-lines]; Deckert & Hartenstein JPA(16)-a1602 [initial-value formulation]; > s.a. distributions; self-force.
@ Specific types of sources: Chen et al ChPL(03)ht/01 [magnetic sources]; Moulin NCB(01)mp/02 [monopoles]; Silbergleit et al JMP(03)mp [surface point charge singularities]; Singal AJP(11)oct-a1101 [accelerated charge]; Tolish & Wald PRD(14)-a1401 [particle moving on a null geodesic, retarded solution].
@ Related topics: Boyer FP(02) [and Aharonov-Bohm phase]; Levin & Johnson AJP(11)aug [repulsion between a point charge and a neutral metallic object].

Electromagnetism > s.a. alternative formulations; field equations; in curved spacetime; with particles and media and modified theories.

Electron > s.a. particle types; particle models; locality.

Electroweak Interactions > s.a. Leptons.

Elegance of a Theory > see physical theories.

Elements (chemical elements)

Elementarity > see composite quantum systems [condition]; particles [elementary vs composite].

Elements of Reality > see realism.

Eliezer's Theorem > see self-force.

ELKO Spinors > see types of spinors.

Ellipse > see conical sections.

Ellipsoid > see euclidean geometry.

Elliptic Curves > s.a. number theory.
* History: Pioneered in the XIX century by Abel, Gauss, Jacobi, Legendre, became one of the century's jewels.
* Example: y2 = (1 − x2) (1 − k2 x2), with k2 ≠ 0, 1.

Elliptic Functions
* Idea: Inverses of functions obtained from elliptic integrals.
$ Jacobi elliptic functions: Given a modulus k, they are given by sn u = x = sin φ, cn u = (1 − x2)1/2 = cos φ, tn u = tan φ, where

\[ \def\dd{{\rm d}}
u = \int_0^\infty {\dd\xi\over\sqrt{(1-\xi^2)(1-k^2\xi^2)}} = \int_0^\phi {\dd\Phi\over\sqrt{1-k^2\sin^2\Phi}}\;,
\qquad {\dd x\over\dd u} = \sqrt{(1-x^2)(1-k^2x^2)} \;.\]

@ Jacobi elliptic functions: Erdös AJP(00)oct [geometrical view]; Khare & Sukhatme JMP(02)mp, mp/03, Khare et al JMP(03)mp/02, Pra(04)mp/03 [identities, Landen transformations]; Chouikha JNMP(05)mp [applications]; Brizard EJP(09) [applications, and Weierstrass elliptic functions]; Bagis a0907.

Elliptic Genera
@ And quantum field theory: Witten CMP(87).

Elliptic Integrals
$ Def: Integrals of the form ∫ R(x,y(x)) dx, with R a rational function, and y2(x) a cubic or quartic polynomial.
* Result: They can all be expressed in terms of the three standard kinds of Legendre-Jacobi elliptic integrals.
* Remark: Mathematica can calculate them.
@ References: CRC tables, 26th ed, p408; in Wolfram 91.

Elliptic Space
$ Def: A compact 3-manifold covered by the 3-sphere, i.e., SU(2)/H, where H is some subgroup of SU(2).


Emergence, Emergent Systems / Theories
* Emergent gravity: A theory in which gravity is an effective interaction arising from some other microscopic degrees of freedom; > see emergent gravity.
* Emergent spacetime: A spacetime manifold and/or metric originating from a non-Lorentzian structure; > see emergence; time.
* Emergent universe: A non-singular universe that starts expanding from an Einstein static universe; > see early-universe models.

Empiricism > see philosophy of science.

Empty Set > see set theory.

Empty Waves > see Epistemology; pilot-wave phenomenology.

Emulations of Physical Systems > see Simulations.

Emulsion > see fluids [complex fluids].

End > see compact set.

Endomorphism > see category.


Energy Conditions

Energy-Momentum Tensor

Engines > see Heat Engine; thermodynamic systems.

Enhanced Quantization > see canonical quantum theory.

Enhancon > see string phenomenology.

Ensemble > see entropy; history of physics; mixed quantum states.

* Idea: The quantity \(\epsilon = {\frac12}\,(\nabla\times{\bf v})^2\), a measure of the magnitude of vorticity for a fluid.
@ References: Wittor et al MNRAS(17)-a1707 [in the intracluster medium]; Marjieh et al a2009 [from symmetry].

Entanglement > s.a. entanglement entropy; examples of entangled systems; measures and phenomenology of entanglement.

Entanglement Hamiltonian > see canonical quantum theory.

Enthalpy > s.a. black-hole thermodynamics.
$ Def: The thermodynamical quantity H:= E + pV, defined for a homogeneous substance.
* Idea: The total energy stored in a system, including the work needed against the environment at pressure p to put the system in place.

Entourage > s.a. uniformity.
$ Def: A subset U of X × X among the ones defining a uniformity on X.

Entropic Dynamics / Force > s.a. formalisms for chaos; origin of quantum theory.
* Idea: A framework in which dynamical laws / quantum theory are derived as an application of entropic methods of inference; There is no underlying action principle, and the dynamics is derived by maximizing an entropy subject to constraints that represent the physically relevant information.
@ General references: Duncan et al PLB(11)-a1103 [derivation of F = ma for circular motion]; Nozari et al IJTP(12)-a1111 [effects of a minimal length]; Mehdipour EPJP(12)-a1111 [and the equivalence principle]; Plastino et al PhyA(15)-a1403 [3D effects]; Caticha et al AIP(15)-a1412, Ent(15)-a1509 [rev]; Demme & Caticha AIP(17)-a1612 [classical limit]; Vanslette QS:MF(17)-a1704 [as a hybrid-contextual theory of quantum mechanics]; Caticha a1704-in, a1711-AdP [rev].
@ Quantum theory: Ipek a1711; Ipek et al a1803-conf [manifestly covariant]; Ipek et al CQG(19)-a1803 [quantum field theory in curved space]; Vanchurin FP(20)-a1901; Caticha Ent(19)-a1908; Ipek PhD(21)-a2105 [quantum fields in curved spacetime].
@ Examples: Ipek & Caticha AIP(15)-a1412 [quantum scalar field]; Nawaz et al a1601-conf [N particles on a curved space]; Ipek & Caticha a1910-conf [geometrodynamics]; Caticha & Carrara a2007 [spin-1/2 point particle].
> Examples: see electricity [Coulomb's law]; entropic gravity; formulations of electrodynamics; Relational Dynamics.

Entropy > s.a. entropy bound; quantum entropy.

Entwinement > see measures of entanglement.

Enumeration Principle > s.a. collapse of the wave function.
* Idea: If marble 1 is in the box and marble 2 is in the box and so on through marble n, then all n marbles are in the box.

Enumeration Theory > see combinatorics.

Envariance > s.a. composite quantum systems.
* Idea: Entanglement assisted invariance, a symmetry of composite quantum systems.
@ References: Vermeyden et al a1408 [experimental test with entangled photons]; Deffner & Zurek NJP(16)-a1504, comment Alicki a1504 [and the characterization of thermodynamic equilibrium states].

Envelope of a Family of Curves > see lines.

Envelope Theory > see many-body quantum systems; schrödinger equation.

Environment > s.a. Bath; Open System.
* Examples: A heat bath; Boundary conditions on a field.
@ References: in Eckstein & Horodecki a1904 [role of environment in the 'experiment paradox'].

Eötvös Experiment > s.a. equivalence principle; fifth force; Hyperphoton / tests of general relativity.
* Idea: An experiment to test differences in the gravitational acceleration of different materials.
* History: It found that (δg)/g < 10−8, comparing accelerations towards the Earth; Repeated by Dicke and others, comparing accelerations towards the Sun.
@ References: Eötvös, Pekár & Fekete AdP(22); Gibbons & Whiting ?; Fischbach et al PRL(86); Nieto et al AJP(89)may; Kraiselburd & Vucetich PLB(12) [and constraints on the fundamental interactions]; Fischbach & Krause a1901-PoS.

$ Def: An element f of Hom(A, B) is an epimorphism if for all g and g' in Hom(B, C), g f = g'f implies that g = g'.
* Special cases: For some categories (e.g., sets), it coincides with an onto morphism.
* For groups: An onto homomorphism f : GH with cokernel Cok(f = eH; Or G / Ker(f) = H.
* Properties: The composition of epimorphisms is an epimorphism.

Epistemology > s.a. computation; histories quantum theory; history of relativistic physics; Ontology; philosophy of physics; physics teaching; science.
* Idea: "What we know", as opposed to "what is" (ontology).
@ General references: Mugur-Schächter FS(02).
@ And quantum theory: Mansfield a1306 [ontic and epistemic interpretations]; Holland a1409-in [empty waves and wave-function collapse]; Mohrhoff a1410; > s.a. ψ-Epistemic Theories.
@ And cosmology: Page a1412-proc; > s.a. references on cosmology.

Epoch Function > see time in gravity.

EPR Paradox

Epstein-Glaser Approach to Renormalization > see renormalization.

Equation of State > s.a. fluid; perfect fluid; Virial Expansion.
* Idea: A relationship between external (macroscopic) parameters, their conjugate generalized forces, and temperature for a system in thermodynamics.
* Example: A common example is that of a relationship φ(p, ρ) = 0 between the pressure and density of a fluid; More generally, the relationship can be temperature-dependent, and it can be written in the form p = kT ρ G(ρ, T); Here, G = 1 is the perfect-fluid case or low-density limit, and a series expansion of G in powers of ρ for small densities gives the virial expansion.
@ References: Fortov 16 [from ideal gas to quark-gluon plasma].
> Examples: see dark energy or observational cosmology for the cosmological one; van der Waals Gas.

Equicontinuity > see distance.

> Thermal equilibrium: see statistical mechanical equilibrium [many-body, approach to equilibrium]
> Phase equilibrium: see condensed matter.
> Mechanical equilibrium: see hamiltonian dynamics [stability of equilibria/orbits].
> Diffusive equilibrium: see diffusion.

Equipartition of Energy > s.a. holography [holographic equipartition].
* Idea: In a classical theory, every canonical variable which appears in the action or Hamiltonian only quadratically, in a term of the form \(b\,p_i^2\) (or similar for qi), contributes an amount kT/2 to the mean energy in a canonical ensemble at temperature T; (But if energy levels are quantized, the way energy is distributed in a system will depend on T ); Applications: The Dulong-Petit law on specific heat of solids.
@ References: Patrascioiu pr(81); Komar GRG(96) [relativistic]; Berchialla et al PLA(04) [time, Fermi-Pasta-Ulam model]; Mello & Rodríguez AJP(10)aug [corrections with confining potentials]; Masi a1109 [bound]; Bialas et al JPA(19)-a1805, Łuczka JSP(20)-a1911 [quantum version]; > s.a. specific heat.
> Online resources: see Hyperphysics page; Wikipedia page.

Equivalence (between Physical Theories) > see theory of physical theories.

Equivalence (in a Category)
$ Def: Two objects X and Y in a category C are equivalent if there exist f in Hom(X, Y) and g in Hom(Y, X) such that g \(\circ\) f = idX and f \(\circ\) g = idX.
$ Equivalence: A morphism that realizes the condition for two objects in a category to be equivalent.

Equivalence Class > see Equivalence Relation.

Equivalence Principle > see also quantum equivalence principle and tests.

Equivalence Relation > see Relation.

Equivalence Theorem > see Chisholm's Theorem.

ER = EPR Conjecture > s.a. entanglement and spacetime.
* Idea: The conjecture by Maldacena and Susskind that maximally entangled states of two black holes that form a complex EPR pair are connected through the interior via a wormhole, or Einstein-Rosen bridge.
@ References: Maldacena & Susskind FdP(13)-a1306 [proposal]; Susskind a1412 [consistency with quantum mechanics]; Chen et al JCAP(17)-a1608 [counterexample]; Patrascu JHEP(17)-a1703 [and the Mayer-Vietoris theorem]; Susskind & Zhao PRD(18)-a1707 + news sn(17)aug [proposal for test in the lab]; Dai et al a2002 [tests of the proposal].

Erasure (Quantum) > s.a. information; interference; Landauer's Principle.
@ References: Cardoso et al JAdvP(16)-a1503 [the quantum eraser does not always erase]; Benoist et al a1602 [full statistics]; Salih a1606 [quantum eraser that works counterfactually]; Kastner FP(19)-a1905 ['quantum eraser' experiments do not erase any information].

* Idea: Dark-matter particles of mass around \(10^{-5}\) g, predicted by Penrose's scheme of conformal cyclic cosmology (CCC).
@ References: Penrose a1707 [erebon decay and correlated noise in LIGO].

Ergodic Systems / Theory [including ergodic hierarchy]

Ergoregion / Ergosphere / Ergosurface > s.a. kerr and kerr-newman solutions.
* Idea: For a stationary spacetime (with a black hole), the ergoregion or ergosphere is the part of the external region in which the stationary Killing vector field becomes spacelike; The ergosurface is its boundary, where the Killing vector field is null; If t is the coordinate adapted to the stationary symmetry, the ergosurface is at gtt = 0; The outer ergosurface is physically the staticity limit, the boundary of the outer connected region in which a timelike particle can be static.
* Instability: Every spacetime with an ergosphere but no horizon is unstable; Physically, the instability is triggered by energy extraction processes lke the Penrose process for particles and superradiance for waves; For regular stars, which may have an ergoregion, the instability time scale may be very large (many years), but it is short (seconds to days) for compact objects and hypothetical objects that have been advocated as black hole substitutes, such as gravastars or wormholes.
* In quantum gravity: One sees "at the back-reaction level" that even spherical holes have an ergosphere [@ York in(84)].
@ General references: Hájíček PRD(73); Butterworth & Ipser ApJ(76) [toroidal]; Pelavas et al CQG(01)gq/00 [Kerr].
@ Horizonless (mimickers), instability: Friedman CMP(78); Cardoso et al PRD(08)-a0709, CQG(08)-a0808; Pani et al PoS-a0901.
@ Radiation: Ashtekar & Magnon CRAS(75); Kang PRD(97)gq, gq/97-proc [and instability].
@ And observation: Kinugawa et al PTEP(16)-a1601 [possible confirmation].

Ergostar > see types of neutron stars.

Erlangen Programme > see geometry.

Ermakov Invariant > see quantum state evolution.

Ermakov System
@ References: Haas & Goedert JPA(96)mp/02, JPA(99)mp/02; Goedert & Haas PLA(98)mp/02 [generalized, Lie symmetries]; Haas JPA(02)mp [Poisson structures]; Cariñena et al in(07)-a0810 [superposition rules for solutions].

Ermakov Transformation > see quantum integrable systems.

Ermakov-Pinney Equation > see FLRW spacetimes [3D].

Ernst Equation, Spacetime > see axisymmetry.

Errors > s.a. statistics.
@ In scientific practice: Schickore SHPSA(05) [epistemic roles].
> In computation: see quantum computers; thermodynamic systems [error correction].

Error-Disturbance Relations > see uncertainty.

Eschatology > see cosmology [future of the universe].

Eschenburg Space
@ References: Dickinson DG&A(04) [positively curved].

Essential Extension
* Example: \(\mathbb Q\) is an essential extension of \(\mathbb Z\).
> Online resources: see Wikipedia page.

Essential Monomorphism between R-Modules > see Monomorphism.

Etale Cohomology > see types of cohomology.

Eternalism (or Block Universe) > see time.

Ether > s.a. einstein-æther gravity; history of special relativity; theories of gravity [with preferred frame].
* History: 1810, Arago attempted to detect the absolute motion of the Earth by measuring the deflection of starlight passing through a prism fixed to the Earth; Idea abandoned after the negative results of the Michelson-Morley experiment; 2004, Revived in the context of Lorentz symmetry violation; If the violation is rotationally symmetric in some frame, then it is characterized by an ''aether'', i.e., a unit timelike vector field.
* Ether-based gravitation theory: A preferred-frame bimetric theory in which the gravitational field both influences the metric and has direct dynamical effects.
@ History: Kostro 01 [Einstein, special relativity and general relativity]; Dirac Nat(51)nov; Saatsi SHPSA(05) [shift to Maxwell's theory, truth and scientific realism]; Auffray phy/06 [Preston's 1875 postulates]; Solari & Natiello a2002 [persistence of the concept].
@ Theory: Sinha et al FP(76), FP(76) [fermion-antifermion pair superfluid].
@ Ether-drift experiments: Consoli & Costanzo NCB(04)gq [reanalysis and proposal], PLA(04) [modern versions], gq/05 [evidence for preferred frame]; Ferraro & Sforza EJP(05)phy/04 [Arago]; > s.a. Michelson-Morley Experiment.
@ And gravitational theory: Petry GRG(81), GRG(81), ApSS(97), in(02); Schmelzer gq/00, AACA-gq/02 [tensor theory]; Szondy gq/03 [Janossy's theory]; Arminjon IJMPA(02)gq, in(08)gq/04, BJP(06)gq/04; Zlosnik et al PRD(07)ap/06 [as dark matter alternative]; Afshordi a0807 [and thermodynamic solution to cosmological constant problem]; Afshordi PiC-a1004 [motivation, non-technical]; Dupré & Tipler IJMPD(12)-a1007 [Einstein's equation from ether theory]; > s.a. cosmological-constant problem [gravitational aether].
> Related topics: see bose-einstein condensates; Lorentz-FitzGerald Contraction.
> Online resources: see Arminjon's page [ether-based gravity].

Euclidean Group, Metric, Theories

Euler Angles > see lie groups; rotation.

Euler Classes and Number / Characteristic

Euler or Euler-Mascheroni Constant > s.a. Coupon Collector's Problem.
* Value: The irrational number \(\gamma:= -\int_0^\infty\) eu (ln u) du = −Γ'(1) = ... = 0.5772156649...
@ References: in Derbyshire 03 [I]; Lagarias BAMS(13).
> Online resources: see MathWorld page; Wikipedia page.

Euler Equations > see fluid; Navier-Stokes Equation.
* Idea: The equations describing fluid flow without viscosity and heat conduction; They are a special case of the Navier-Stokes equation.
@ References: Euler NCASP(1761)-a0804; Nachtergaele & Yau CMP(03) [from quantum dynamics]; Frauendiener CQG(03) [relativistic]; Golse CMA(13)-a1111-conf [as fluid dynamic limit of the Boltzmann equation].
> Online resources: see Wikipedia page.

Euler Function > see Wikipedia page.

Euler's Totient Function > see Wikipedia page.

Euler-Calogero-Sutherland Model > see bianchi I models.

Euler-Lagrange Equations > s.a. lagrangian dynamics.
* Idea: The equations of motion one obtains from the Lagrangian for a physical system using a variational principle.
@ References: Gamboa Saraví & Solomin JPA(03) [global version].

Euler-Rodrigues Formula > see examples of lie groups [SO(3)].

Euler's Equation
$ Def: The relationship e + 1 = 0 between the five most important numbers and the three basic operations in mathematics.

Euler's Theorem
$ Def: For a homogeneous function f of degree 1, f(x1, ..., xn) = ∇f · x.

Eulerian Observers > see Observers.

Evanescent Waves / Fields / Particles > s.a. wave phenomena; approaches to quantum field theory [evanescent particles]; scattering / Heat Transport.
* Idea: Fields which do not propagate but whose intensity decays exponentially with distance from the surface where they are formed.
@ Quantum theory: Bern et al PRL(15)-a1507 [in quantum gravity]; Colosi & Oeckl a2105 [using the general boundary framework].
> Online resources: see Wikipedia page.

Evaporation > see water.

Evenly Covered Neighborhood

Event (Probability Theory) > see Test Space.

Event (Spacetime)
* Idea: Mathematically, a classical event is an element of a spacetime manifold.
* Philosophical issue: The fact that spacetime events are invariant, as opposed to coordinate- or gauge-dependent, makes them real (ontological point of view) or just the things we can experience (epistemological point of view)?
@ Classical: Lusanna & Pauri gq/05-conf [objectivity, and Dirac observables].
@ Quantum: Ruebenbauer IJTP(80) [non-localizability]; Giannitrapani JMP(98) [quantum coordinates in local algebras]; Zafiris FPL(01) [categorical scheme], FPL(01) [interpretation], FPL(04) [object of truth values]; Haag FP(13)-a1303 [sharpness of localization]; Zafiris & Karakostas FP(13) [category-theory representation]; Dorato Topoi(15)-a1503 [ontology]; Blanchard et al NPB(16)-a1603 ["ETH approach"]; > s.a. Coevent; quantum spacetime; Topos.
@ Related topics: Chajda & Länger IJTP(13) [spaces of abstract events].

Event Horizon > s.a. horizons [other types].

Event Horizon Telescope (EHT) > s.a. black-hole observation.
* Idea: A project composed of many radio telescope facilities around the world to produce a high-sensitivity, high-angular-resolution telescope.
@ References: Psaltis GRG(19) [and tests of general relativity]; Rummel & Burgess a2001 [constraining deviations from general relativity].
> Online resources: see EHT website; Wikipedia page.

Evolution > s.a. time.
@ In quantum theory: Myrvold BJPS(03) [(special) relativistic].
@ Evolutionary processes: Padmanabhan ApJ(02)ap [in cosmology, with non-local self-replication].

Evolving Set > see sets.

Ewald Construction / Sphere
@ References: Foadi & Evans EJP(08) [and reciprocal lattice, pedagogical].

Exact Sequence

Exactly Solvable > see classical systems; types of wave equations.

Exceptional Field Theory > see types of field theories.

Exceptional Groups > see examples of lie groups.

Exceptional Structures > see Octonions.

Exchange Force / Interaction > see force.

Exclusion Principle > see spin-statistics.

Exclusivity Principle > see quantum correlations.

* Idea: An early form of the modern thermodynamic concept of exergy, which is the generic name for the amount of work obtainable when some matter is brought to a state of equilibrium with its surroundings by means of reversible processes.
@ References: Marquet QJRMS(91)-a1402 [exergy and available enthalpy].

@ General references: Heinrich a1202 [relativity of existence]; Heinrich a1306 [physical relativism].
@ Why there is something rather than nothing? Lynds a1205; Carroll a1802-in, and references therein.

Exotic Differentiable / Smoothness Structure > see differentiable manifolds.

Expansion Mapping
* Idea: A mapping f : XX from a metric space to itself is an expansion if there is a positive constant c > 1 such that for all x1 and x2 in X,

d(f(x1), f(x2) ≥ c d(x1, x2) .

Expansion of a Congruence of World-Lines
$ Def: If ua is the unit timelike tangent vector to the congruence, one defines the expansion tensor and its trace as

θab := q(am qb)nm un ,      θ:= θaa = ∇a ua ,

where qab is the projection operator normal to ua; The tensor can be decomposed into trace + symmetric traceless (= longitudinal + transverse traceless) + antisymmetric parts.
* Special case: If ua is tangent to (affinely parametrized) geodesics, then one can simply write θab := ∇a ub .

Expansion of a Function > see fourier analysis; Special Functions; Taylor Series.

Expansion of the Universe > s.a. expansion effects and expansion rate.

Experimental Physics [including important experiments] > s.a. experiments in particle physics; experiments in quantum mechanics.

Explanation in Mathematics > s.a. mathematics.
@ References: Cellucci SHPSA(08); Bangu BJPS(13) [mathematical explanations of physical phenomena and indispensability argument for mathematical realism].

Explanation in Physics / Science > s.a. causality; philosophy of science; Physical Laws; physical theories; Understanding.
* Types of explanations: Given a set of observations, an explanation describing those observations can be a structural one or a dynamical one within a model or theory; In principle, a dynamical explanation can be a teleological or a causal one (a theory and a set of causes within that theory that explain the observations), but in practice in physics dynamical explanations are causal.
* Question: The concept of explanation in physics is different from what it is in mathematics, for example; Does an explanation have to be a dynamical one? That would seem to exclude anthropic explanations.
@ General references: Salmon 97 [and causality]; Glennan PhSc(02)sep [mechanistic explanations]; McGrew BJPS(03); Kelly PhSc(07)dec [truth and simplicity, puzzle of simplicity]; Grimm BJPS(08) [understanding the need for explanation]; Douglas PhSc(09)oct [and prediction]; Potochnik PhSc(10)jan; Weslake PhSc(10)apr [explanatory depth]; Grimm SHPSA(10) [understanding as the goal of explanation]; Andersen PhSc(11)apr [mechanisms, laws, and regularities]; Deutsch 11 [I]; Bangu BJPS(13), Lange BJPS(13) [mathematical explanations].
@ Examples: Bokulich BJPS(08) [of quantum phenomena in terms of classical structures]; Weatherall PhSc(11)-a1106, a1206-ch; > s.a. quantum foundations.
@ Historical references: Leunissen 10 [explanation and teleology in Aristotle's science; r Isis(11)#4]; Chalmers SHPSA(12) [intermediate causes and explanations, and the scientific revolution].
> Related topics: see Knowledge; Occam's Razor; probability in physics.

"Explanations exist; they have existed for all time; there is always a well-known solution to every human problem – neat, plausible, and wrong." — H.L. Mencken

Exponential Family
* Idea: An exponential family is an important class of probability distributions sharing a certain form; Exponential families include many of the most common distributions, including the normal, exponential, gamma, chi-squared, beta, Dirichlet, Bernoulli, binomial, multinomial, Poisson, Wishart, Inverse Wishart and many others.
> Online resources: see Wikipedia page.

Exponential Function > s.a. matrices [and the Zassenhaus Formula].
$ Def: The function exp: \(\mathbb R\) → \(\mathbb R\) defined by exp(x) = ex; Up to an arbitrary multiplicative constant, it is the only function that equals its own derivative.
* Stretched exponential: A function of the form f(t) = exp{−(t/τ)b}, where the stretching parameter b is between 0 and 1; It is used in the description of relaxation phenomena.
@ Stretched exponential: Cardona et al AdP(07)-a0710, Berberan-Santos et al AdP(08)-a0804 [history]; > s.a. Wikipedia page.
@ Related topics: Childs et al a1912 [Lie-Trotter formula for the exponential of a sum of operators].
> Online resources: see Wikipedia page.

Exponential Hilbert Space / Representation > see fock space.

Exponential Mapping
$ In a group: The map exp: Te GG that takes γ \(\mapsto\) exp γ:= gγ(1).
$ In a manifold: The map exp: Tp MM that takes V a \(\mapsto\) X(1), with \(\dot X\)a(0) = V a and X(t) an affinely parametrized geodesic.

Exponential Metric > see Yilmaz Theory.

Extended Field Theories > s.a. Relative Field Theory; types of field theories [exceptional].
* Idea: Theories that unify the local symmetries of supergravity fields into a single symmetry manifest on a higher-dimensional space (in the spirit of Kaluza-Klein construction); Depending on whether one starts with Type II or 11-dimensional supergravity, one obtains double or exceptional field theory, respectively.
@ References: Chaemjumrus & Hull PRD(16)-a1512 [finite gauge transformations and geometry]; Otsuki a2008-PhD [exotic aspects].

Extended Objects > s.a. Continuous Media; fluid; gravitating matter.
@ Dynamics: Collet & Eckmann 90 [instabilities and fronts]; Capovilla et al CQG(04)ht [Hamiltonian, in Minkowski space]; Bower 10 [mechanics of solids].
@ Swimming in curved spacetime: Avron & Kenneth NJP(06)mp; Mendes & Poisson a1707 [the view from a Fermi observer]; > s.a. test particle orbits.
@ Order: Mazenko 02 [fluctuations and defects]; Olemskoi PhyA(05) [long-range, theory].

Extended Real Numbers > see types of numbers.

Extended Relativity Theory > see clifford spaces.

Extended Theories of Gravity > a.k.a. f(R) theories.

Extension of a Group > see group theory.

Extension of a Topological Space > see topological space.

Extensive Quantities / Properties
* Idea: A physical quantity is extensive if it is additive for subsystems; Note that it is not always proportional to the mass of a system.
@ References: Mannaerts EJP(14) [definition].

Extensors > see tensors.

Exterior Algebra > s.a. grassmann structures.

Exterior Calculus > s.a. forms.

* Idea: A concept that generalizes that of a conventional function as well as the concept of a distribution; Extrafunctions have been used for a rigorous mathematical definition of the Feynman path integral, and for solving some problems in probability theory.
@ References: Burgin 12.

Extrasolar Systems

Extremal Surface > see extrinsic curvature.

Extreme Value Statistics > see statistics in physics.

Extremely Disconnected > see connectedness.

Extrinsic Curvature

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