Topics, E
e
$ Def: The number lim_{n→
}(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].
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 Gravity
@ References: Bañados et al a0811 [and large-scale structure].
Eddington-Finkelstein Coordinates > see coordinates for schwarzschild.
Edgar-Ludwig Metric
* 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 Field Theories > s.a. effective
quantum field theory; interaction.
@ References: Burgess ht/98-in
[non-equilibrium physics], hp/98-in
[effective Lagrangians, intro], ARNPS(07)ht [intro].
Efimov Effect > see atomic physics.
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-in [extended to modified gravity].
Ehrenfest Classification of Phase Transitions > see phase transition.
Ehrenfest Paradox > see special relativistic kinematics.
Ehrenfest Theorem
$ Def: For a non-explicitly-time-dependent
observable A, d
A
/
dt = [A, H] /
i
.
@ For non-linear Schrödinger equation: Bodurov IJTP(98);
Kälbermann JPA(04)qp/03
[and Galilean invariance]; Friesecke & Koppen JMP(09) [rigorous derivation].
Ehrenfest Time
* Idea: The time characterizing
the departure of quantum dynamics for observables from classical dynamics.
Eigenvalues > see for ordinary differential equations and matrices; Quaternions [for quaternion operators].
Eightfold Way
@ References: Gell-Mann & Ne'eman 64.
Einstein Algebras > see models of quantum spacetime; types of manifolds; in Hole Argument.
Einstein Boxes
* 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
a0705 [complementarity].
Einstein Equation > s.a. solutions.
Einstein Frame > see scalar-tensor gravity.
Einstein Ring > see lensing.
Einstein Manifold / Metric / Space > see types of spacetimes.
Einstein Model for the Specific Heat of a Solid > see specific heat.
Einstein Relation > see diffusion.
Einstein Temperature
* Idea: The quantity
E:= 
/kB,
where 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 lhs of the Einstein
equation.
@ References: Lamey & Obermair BJP(05)gq
[re physical significance].
Einstein-Cartan(-Sciama-Kibble) Theory > see einstein-cartan.
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-Rosen Bridge > see wormholes.
Einstein-Sasaki Spaces > s.a. kerr [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. Viscoelasticity.
* Idea: The elastic constant
c of a material is related to the stress 
and
strain 
by the defining
equation = c;
It is frequency-dependent, and can be studied with RUS; > s.a. sound.
@
General references: Landau & Lifshitz 86.
@ Relativistic: Bel gq/96 [deformations];
Beig & Schmidt CQG(03)gq/02;
Beig gq/04-in;
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]; > s.a. general relativity.
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, in media and modified theories.
Electron > see locality; particle models and types.
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 
, cn u
= (1–x2)1/2 =
cos ,
tn u = tan ,
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; 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 > 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]; Batterman PhSc(06)dec
[hydrodynamics vs molecular dynamics]; Juarrero & Rubino ed-08 [complexity
and
self-organization,
essays].
> Examples: see Bose-Einstein
Condensates; gauge
theories; gravitation; ising
model; origin of quantum mechanics; particle
models; spacetime models; time.
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. 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.
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 (
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.
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.
* Idea: A relationship
between external (macroscopic) parameters, their conjugate generalized forces,
and temperature for a system in thermodynamics; For example, a relationship (p, 
)
between the pressure and density of a fluid; More generally, it can be written
in the form p/kT = G(,
T).
> 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 > see energy.
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 Principle > see also quantum equivalence principle and tests.
Equivalence Relation > see Relation.
Equivalence Theorem > see Chisholm's Theorem.
Erasure > see 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 1984].
@ General references: Hájícek 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 a0901-in.
@ 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 a0810 [superposition rules for solutions].
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.
Ether > s.a. modified
lorentz symmetry [Einstein-Aether theory]; 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, proposals: 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.
@ Ether-based gravitation theory: Petry GRG(81), GRG(81),
ASS(97),
in(02); Schmelzer gq/00, gq/02 [tensor
theory]; Szondy gq/03 [Janossy's
theory]; Arminjon IJMPA(02)gq,
gq/04-in,
BJP(06)gq/04;
Zlosnik et al PRD(07)ap/06
[as dark matter alternative]; Afshordi a0807 [and thermodynamic solution to
cosmological constant problem].
> Online resources:
Arminjon's page [ether-based
gravity].
Euclidean Group, Metric, Theories
Euler Angles > see lie groups; rotation.
Euler Equation > see fluid.
Euler Function > see Wikipedia page.
Euler's Totient Function > see Wikipedia page.
Euler-Calogero-Sutherland Model > see bianchi I.
Euler-Lagrange Equations > s.a.
lagrangian dynamics.
@ References: Gamboa Saraví & Solomin JPA(03) [global version].
Euler-Mascheroni Constant
* Value: The number
:= –0infty
e–u (ln u) du.
@ References: Derbyshire 03 [I].
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: Giannitrapani JMP(98)
[quantum coordinates in local algebras];
Zafiris FPL(01)
[categorical scheme], FPL(01)
[interpretation], FPL(04)
[object of truth values]; > 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.
Exotic Differentiable / Smoothness Structure > see differentiable manifolds.
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
ab := q(am qb)n m
un , := aa = 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 ab := 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 > s.a.
Occam's Razor; philosophy
of science; Understanding.
* 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
ones.
@ References: Glennan PhSc(02)sep
[mechanistic explanations]; McGrew BJPS(03);
Kelly PhSc(07)dec
[truth and simplicity, puzzle of simplicity]; Bokulich BJPS(08)
[of quantum phenomena
in terms of classical structures]; Grimm BJPS(08)
[understanding the need for explanation].
Exponential Function
* Stretched exponential:
A function of the form f(t) = exp{–(t/
)b},
where the stretching parameter b is between 0 and 1; Applied to
the description of relaxation phenomena.
@ References: Cardona et al a0710,
Berberan-Santos et al a0804 [stretched
exponential, history]; > s.a. Wikipedia page.
Exponential Hilbert Space / Representation > see fock space.
Exponential Mapping
$ In a group: The map
exp: Te G → G that
takes 
exp :=
ggamma(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.
Extended Objects > s.a. fluid; gravitating
matter.
@ Dynamics: Collet & Eckmann 90 [instabilities and fronts]; Capovilla
et al CQG(04)ht [Hamiltonian,
in Minkowski]; Avron & Kenneth NJP(06)mp [swimming
in curved spacetime].
@ Order: Mazenko 02 [fluctuations and defects]; Olemskoi PhyA(05) [long-range,
theory].
Extension of a Group > see group theory.
Extensors > see tensors.
Exterior Algebra > s.a. grassmann.
Exterior Calculus > s.a. forms.
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
14 oct 2009