**Topics, E**

**e** > s.a. Euler's Equation.**
**$

*

@

** E Modes** > see gravitational radiation.

**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.

**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].

**Edge**

* __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 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.

* __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].

@ __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]; > 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.

**Effects**

> __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].

**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; Could lead to an incompressible,
undilutable liquid 10^{13} 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].

@ __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*\(\rangle\) / d*t*
= \(\langle\)[*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].

@ __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.

**Eigenforms**

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

**Eigenstate Thermalization Hypothesis** > see states in quantum statistical mechanics.

**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
Θ_{E}:= \(\hbar\)*ω*/*k*_{B},
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 *c*_{V}, 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-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* = *c**ε*;
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 *c _{ij}*,
related to the stress and strain by

*

@

@

@

@

@

>

>

**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__: *y*^{2} =
(1 – *x*^{2}) (1 – *k*^{2}* x*^{2}),
with *k*^{2} ≠ 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–*x*^{2})^{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*))
d*x*, with *R* a rational function, and *y*^{2}(*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 metric originating from a non-geometrical structure; > see emergence.

* __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.

**Engines** > see Heat Engine; thermodynamic systems.

**Enhanced Quantization** > see canonical quantum theory.

**Enhancon** > see string phenomenology.

**Ensemble** > see entropy; quantum
states.

**Enstrophy**

* __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].

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

**Enthalpy**

$ __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** = *m***a** 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];
Ipek a1711, Ipek et al a1803-conf [quantum, manifestly covariant].

@ __Examples__: Ipek & Caticha AIP(15)-a1412 [quantum scalar field];
Nawaz et al a1601-conf [*N* particles on a curved space].

> __Examples__: see electricity [Coulomb's law];
entropic gravity; formulations of electrodynamics;
Relational Dynamics.

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

**Entwinement** > see examples of entangled systems.

**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 under Auxiliary-Field Method.

**Environment** > s.a. Bath; Open System.

* __Examples__: A heat bath;
Boundary
conditions on a field.

**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 (δ*g*)/*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].

**Epimorphism**

$ __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* : *G* → *H* with
cokernel Cok(*f*)
= *e*_{H};
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.

**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.

**Equilibrium**

>
__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 *bp*_{i}^{2} (or
similar for *q*^{i}), 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 a1805 [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* =
id_{X} and *f* \(\circ\) *g*
= id_{X}.

$ __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**

* __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 a1707 + news sn(17)aug [proposal for test in the lab].

**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].

**Erebons**

* __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.**
***

*

*

@

@

@

@

**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].

@ __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.

> __Online resources__:
see Arminjon's page [ether-based
gravity].

**Euclidean Group, Metric, Theories**

**Euler Angles** > see lie groups; rotation.

**Euler or Euler-Mascheroni Constant** > s.a. Coupon Collector's Problem.**
***

@

>

**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^{iπ} + 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*(*x*_{1},
..., *x*_{n}) = ∇*f* · **x**.

**Eulerian Observers** > see Observers.

**Evanescent Fields / Operators**

@ __References__: Bern et al PRL(15)-a1507 [in quantum gravity].

**Evanescent Waves** > s.a. wave
phenomena / Heat Transport.

* __Idea__: Waves which do not propagate but whose intensity decays exponentially with distance from the surface where they are formed.

> __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**

* __Idea__: A project composed of many radio telescope facilities around the world to produce a high-sensitivity, high-angular-resolution telescope.

> __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].

**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.

**Exergy**

* __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].

**Existence**

@ __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* : *X* → *X* from a metric space to itself is an expansion if there is a positive constant *c* > 1 such that for all *x*_{1} and *x*_{2} in *X*,

*d*(*f*(*x*_{1}), *f*(*x*_{2}) ≥ *c* *d*(*x*_{1}, *x*_{2}) .

**Expansion of a Congruence of World-Lines
**$

*θ*_{ab} := *q*_{(a}^{m} *q*_{b)}^{n} ∇_{m}
*u*_{n} , *θ*:= *θ*_{a}^{a} = ∇_{a} *u*^{a} ,

where *q*_{a}^{b} is
the projection operator normal to *u*^{a};
The tensor can be decomposed into trace + symmetric traceless (= longitudinal
+ transverse traceless) + antisymmetric parts.

* __Special case__: If *u*^{a} is
tangent to (affinely parametrized) geodesics, then one can simply write *θ*_{ab} := ∇_{a} *u*_{b} .

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

**Experimental Physics** > 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; Knowledge; Occam's Razor; philosophy
of science; Physical Laws; physical theories; Understanding.

* __Types of explanations__: Given a set of observations, an explanation can be a model describing those observations; A dynamical explanation can be a teleological one, or a causal one, theory and a set of causes within that theory that produce them.

* __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].

"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*) = e^{x}; 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; Applied to 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.

> __Online resources__: see Wikipedia page.

**Exponential Hilbert Space / Representation** > see fock space.

**Exponential Mapping**

$ __In a group__: The map
exp: T_{e} *G* → *G*
that takes *γ* \(\mapsto\) exp *γ*:=
*g*_{γ}(1).

$ __In a manifold__: The
map exp: T_{p} *M* → *M* 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.

@ __References__: Chaemjumrus & Hull PRD(16)-a1512 [finite gauge transformations and geometry].

**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.

**Extrafunction**

* __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.

**Extremal Surface** > see extrinsic curvature.

**Extreme Value Statistics** > see statistics in physics.

**Extremely Disconnected** > see connectedness.

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

send feedback and suggestions to bombelli at olemiss.edu – modified 11 may 2018