Topics, E
e
$ Def: The number
e:= limn→∞(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].
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 astrophysics [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.
@ References: Bañados et al PRD(09)-a0811 [and
large-scale structure]; Bañados & Ferreira PRL(10)-a1006 +
news po(10)jul
[minimal length and cosmology]; Pani et al PRL-a1106 [collapse and compact stars]; Avelino PRD(12)-a1201 [astrophysical and cosmological constraints]; Pani et al PRD-a1201 [matter coupling and dust collapse]; De Felice et al PRD(12)-a1205 [cosmological constraints]; Scargill et al PRD(12)-a1210 [cosmology, singularity avoidance and expansion rate]; Pani & Sotiriou PRL(12) [surface-singularity pathologies].
Eddington-Finkelstein Coordinates > see coordinates for schwarzschild spacetime.
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.
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 Theories > s.a. effective
quantum field theory; interaction.
@ References: 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].
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 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.
@ References: Esry et al PRL(99);
Bulgac PRL(02);
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 et al PRL(09)-a0903 +
Esry Phy(09)
[four-body states]; Ferlaino & Grimm Phy(10);
Bhaduri et al AJP(11)mar-a1009 [elementary]; Wang et al PRL(11) [for three interacting dipolar molecules]; Gridnev a1204 [on the N-body Efimov effect], a1210 [not for 4 bosons]; 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]
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⟨A⟩ /
dt = ⟨[A, H]⟩ /
i
.
@ General references: Friesecke & Schmidt PRS(10) [sharp version,
for general self-adjoint operators].
@ 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].
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].
Eigenvalues > see for ordinary differential equations and matrices; Quaternions [for quaternion operators]; s.a. Antieigenvalues.
Eightfold Way
@ References: Gell-Mann & Ne'eman 64.
Eikonal Approximation > see optics.
Einstein Algebras > see models of quantum spacetime; types of manifolds; in Hole Argument.
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 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:= 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 Gab :=
Rab – Rgab,
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-Aether Theory > s.a.
hořava-lifshitz gravity; modified general relativity.
* Idea: General relativity
coupled to a dynamical, unit timelike vector field u; A generally covariant theory with
a preferred vector field, or a preferred frame.
* Dynamics: The general form of the action is a sum of an Einstein-Hilbert term for the metric and terms involving u; The leading ones are
S[g, u] = (16πG)–1 ∫ d4x (–g)1/2 [R + c1 (∇aub) (∇aub) + c2 (∇aua)2 + c3 (∇aub) (∇bua) + c4 (ua ∇aub)2] ,
but one can add terms with higher derivatives of u and/or higher-order terms in the curvature; Stability and energy positivity imply some inequalities for the coefficients ci.
* Propagating modes: There are 5, two spin-2 gravity modes (traveling at speed v2 = 1/[1–(c1+c3)], which leads to very strong bounds on the coefficients from the absence of vacuum Cerenkov radiation), two spin-1 modes and one spin-0 mode (each of which has to travel at or above the speed of light, or possibly just a little bit below, without this leading to causality violations); Gravitational waves can have a longitudinal component.
* Newtonian limit: The slow-motion, weak-field approximation camn be made to agree with that of general relativity, and in the limit one gets GN = G/[1–(c1+c4)/2].
@ General References: Jacobson & Mattingly PRD(01)gq/00, PRD(01)gq/00, gq/01-conf
[in general relativity], PRD(04)gq;
Eling et al gq/04-proc;
Foster PRD(05);
Foster & Jacobson PRD(06)
[PN parameters]; Eling & Jacobson PRD(06)gq [2D
theory]; Bonvin et al PRD(08)
[solar system]; Jacobson a0711-proc,
PoS-a0801 [rev];
Withers CQG(09)-a0905 [quantum
effective field theory]; Donnelly & Jacobson PRD(10)-a1008 [stability]; Garfinkle et al a1207 [constraint equations and the weak gravitational field limit]; > s.a. Ether; positive-energy theorem.
@ Cosmology: Tartaglia & Radicella PRD(07);
Li et al PRD(08)-a0709 [primordial
perturbations]; Zuntz et al PRL(08)-a0808; Carruthers & Jacobson PRD(11)-a1011 [deviations from isotropy and alignment]; Meng & Du PLB(12) [as an alternative to dark energy]; Barrow PRD(12)-a1201 [accelerating-universe solutions]; Saga et al a1302 [generation of magnetic fields]; > s.a. cmb polarization.
@ Other solutions:
Bertolami & Páramos PRD(05)ht [vacuum
solutions]; Eling & Jacobson CQG(06)
[spherical]; Garfinkle et al PRD(07)gq [gravitational
collapse]; Eling et al PRD(07)-a0705 [neutron
stars]; Foster PRD(07)-a0706 [binaries,
strong field effects]; > s.a. black
holes; gödel
solutions.
@ Variations: Pujolàs & Sibiryakov a1109 [supersymmetric extension, and emergent Lorentz symmetry].
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.
Eisenhart Theorem
@ References: in Cordani 03; Minguzzi CQG(07)gq/06 [and
causal simplicity].
Ekpyrotic Scenario > see brane cosmology.
Elasticity > s.a. Hooke's
Law; 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 = cε;
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.
* 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; Fokas & Yang a1010,
a1010 [analytic solutions].
@ Relativistic: Bel gq/96 [deformations];
Beig & Schmidt CQG(03)gq/02;
Beig gq/04-proc;
Beig & Schmidt CQG(05)gq/04 [rigid
rotation].
@ Related topics: Delph PRA(05)
[atomic level stresses]; Marder et al PT(07)feb
[thin sheet crumpling and buckling]; Fülöp & Ván a1007 [kinematics
of finite
elastic and plastic deformations]; Böhmer et al QJM(11)-a1008 [rotational model]; > s.a. Tensile
Strength.
> Applications in gravity and field theory: see 2-spinors [Cosserat
model]; FRW 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 / 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)
Elements of Reality > see realism.
Eliezer's Theorem > see self-force.
ELKO Spinors > see 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
@ 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 [and 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 > s.a. paradigms
in physics.
* History: The earliest
representatives of emergent theories were Plato, Aristotle (anti-atomistic,
but not a process philosophy); In the 20th century, G H Mead, H Bergson,
A N Whitehead, J Margolis,
P Teilhard de Chardin.
* Idea, I: Emergent
properties of complex
physical systems are those that cannot be understood solely in terms of the
laws governing their microscopic constituents, i.e., taking a reductionist
approach.
* Idea, II: In determining
the macroscopic or classical properties of a system, the process by which
an effective description appears from a more fundamental
system is more fundamental than the substance of the
fundamental system itself.
@ References: Kronz & Tiehen PhSc(02)jun
[and quantum mechanics]; Rossberg phy/05 [general formalism]; Batterman PhSc(06)dec
[hydrodynamics vs molecular dynamics]; Juarrero & Rubino ed-10 [complexity and self-organization,
essays]; Carroll 10; Butterfield FP(11)-a1106, FP(11)-a1106 [and reduction and supervenience]; Batterman FP(11) [renormalization group and symmetry breaking]; Morrison PhSc(12) [ontological and dynamical aspects]; Hu JPCS(12)-a1204 [key issues, including coarse-graining and persistent structures].
> In gravity and spacetime: see cosmological models [emergent universe]; Einstein-Aether Theories; emergent-gravity theories; lorentzian geometry; spacetime models [emergent]; special relativity; time.
> Other examples: see bose-einstein
condensates; gauge
theories; ising
model; origin of quantum mechanics; particle
models; supersymmetry.
Empiricism > see philosophy of science.
Empty Waves > see pilot-wave interpretation.
End > see compact set.
Endomorphism > see category.
Enhancon > see string phenomenology.
Ensemble > see entropy; quantum states.
Entanglement > s.a. entanglement entropy; examples of entangled systems; 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 > s.a. electricity [Coulomb's law]; emergent
gravity; formalisms
for chaos; formulations of electrodynamics; origin of quantum theory.
@ 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].
Entropy > s.a. entropy bound; quantum entropy.
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 > see composite quantum systems.
Envelope of a Family of Curves > see lines.
Environment > s.a. 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 (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].
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)
= 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; philosophy of physics; physics
teaching; science.
* Idea: "What we know",
as opposed to "what is" (ontology).
@ References: Mugur-Schächter FS(02).
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.
> Examples: see dark
energy or observational
cosmology for the cosmological one; van der
Waals Gas.
Equicontinuity > see distance.
Equilibrium
>
Thermal equilibrium: see statistical mechanics [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 bpi2 (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]; > s.a. specific
heat.
> Online resources: see Hyperphysics page; Wikipedia page.
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 
f =
idX and f 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.
Erasure (Quantum) > see information; interference; Landauer's Principle.
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-in
[and instability]
Erlangen Programme > see geometry.
Ermakov Invariant > see quantum states [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 FRW spacetimes [3D].
Ernst Equation, Spacetime > see axisymmetry.
Errors > s.a. statistics.
@ In scientific practice: Schickore SHPSA(05) [epistemic roles].
Eschatology > see cosmology [future of the universe].
Eschenburg Space
@ References: Dickinson DG&A(04) [positively curved].
Essential Extension
* Example: Q is
an essential extension of Z.
Essential Monomorphism between R-Modules > see Monomorphism.
Etale Cohomology > see types of cohomology.
Eternalism > see time.
Ether > s.a. Einstein-Aether
Gravity; 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),
ASS(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]; Dupre & Tipler a1007 [Einstein's equation from ether theory]; > s.a. cosmological-constant problem [gravitational aether].
> Online resources:
see Arminjon's page [ether-based
gravity].
Euclidean Group, Metric, Theories
Euler Angles > see lie groups; rotation.
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 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-Mascheroni Constant
* Value: The number
γ:= –∫0∞
e–u (ln u) du.
@ References: Derbyshire 03 [I].
Euler-Rodrigues Formula > see examples of lie groups [SO(3)].
Euler's Theorem
$ Def: For a homogeneous
function f of degree 1, f(x1,
..., xn) = ∇f · x.
Eulerian Observers > see Observers.
Evanescent Waves > see wave phenomena.
Evenly Covered Neighborhood
Event
* Idea: Mathematically,
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-in
[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 a1303 [sharpness of localization]; > s.a. Coevent; quantum
spacetime; Topos.
Event Horizon > s.a. horizons [other types].
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; wave equation.
Exceptional Structures > see Octonions.
Exchange Force / Interaction > see force.
Exclusion Principle > see spin-statistics.
Existence
@ References: Heinrich a1202 [relativity of existence]; Lynds a1205 ["why there is something rather than nothing"].
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 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)n ∇m 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.
Experimental Physics > s.a. experiments in particle physics; experiments in quantum mechanics.
Explanation in Mathematics
@ References: Cellucci SHPSA(08).
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].
@ 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: R → 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; 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: Te G → G that
takes γ 
exp
γ:=
gγ(1).
$ In a manifold: The
map exp: Tp M → M that
takes V a X(1),
with X ·a(0)
= V a and X(t)
an affinely parametrized geodesic.
Exponential Metric > see Yilmaz Theory.
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]; Avron & Kenneth NJP(06)mp [swimming
in curved spacetime]; Bower 10 [mechanics of solids].
@ Order: Mazenko 02 [fluctuations and defects]; Olemskoi PhyA(05) [long-range,
theory].
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.
Extensors > see tensors.
Exterior Algebra > s.a. grassmann.
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.
Extremely Disconnected > see connectedness.
main page – abbreviations – journals – comments – other
sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified
12 apr 2013