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Einstein tensor On a pseudo-Riemannian
1
manifold (M, g), the tensor G µν = R − Rg µν
µν
2
is the Ricci tensor and R is the Ricci
where R
E scalar of the Levi-Civita connection of the met-
µν
ric g.
By extension, on a manifold M with a con-
α
nection N , itisthe tensor with the same local
βµ
edge An unordered pair of nodes in a graph, expression where R µν is now the Ricci tensor
usually denoted as a tuple of arity two with the and R is the Ricci scalar of the connection fixed
twonodesasarguments. Thus(v ,v ) = (v ,v ) on M.
i
j
i
j
(the edge is symmetric). Because of Bianchi identities we have
Comment: Edges are usually represented as µ
∇ G ·µ = 0.
µ
lines or arcs in renderings of a graph or network.
Incidence relation and undirected edge are syn-
onyms. See directed edge. electric charge, Q Q Q Integral of the electric
current over time. The smallest electric charge
effective Lagrangian An effective Lagran- found on its own is the elementary charge, e, the
gian describes the behavior of a quantum field charge of a proton.
theory at large distances (low energy).
electrode potential, E E E Electromotive force
eigenspace See eigenvalue.
of a cell in which the electrode on the left is a
standard hydrogen electrode and the electrode
eigenvalue For a linear operator A : X →
on the right is the electrode in question.
X a scalar λ is called eigenvalue of A if there is
a nontrivial solution x to the equation Ax = λx.
Such an x is called eigenvector corresponding to electrodynamics Classical electrodynamics
λ. The subspace of all solutions of the equation is described by Maxwell’s equations of an elec-
(A − λI)x = 0 is called the eigenspace of A tromagnetic field. Quantum electrodynamics
corresponding to λ. (QED) is the theory of interaction of light with
matter and is described by the Dirac equations
eigenvector See eigenvalue. γ (i∇ − eA )ψ = mψ.
µ µ µ
eikonal equation In geometrical optics the
Hamilton-Jacobi equation H(x, dS(x)) = 0. electromagnetic field In modern geometric
terms a 2-form on space-time M given as fol-
lows. Let A be a connection 1-form (vector
Einstein Albert Einstein (1879–1955) Ger-
potential). Its curvature 2-form F , given by
man-American physicist. The inventor of special 1 A
F = dA + [A ∧ A], is the electromagnetic
and general relativity theory. Nobel prize in A 2 1 µ ν
physics 1922. Popularly regarded as a genius field. In local coordinates F = F dx dx .
µν
A
2
1
among geniuses, the greatest scientist in history. From the Lagrangian L =− (F ∧∗F ) (∗
A
A
2
Everything should be formulated as simple as the Hodge star operator) we obtain that classical
possible, but not simpler. equations of motion d ∗ F = 0. These together
A
with the Bianchi identity dF = 0 are Maxwell’s
A
Einstein equations Einstein’s field equa- equations (in empty space).
tion of general relativity (for the vacuum) is the
system of second-order partial differential equa-
electromagnetism One of the four elemen-
tions
tary forces in nature. Classical electrodynamics
R µν = 0
is governed by Maxwell’s equations for the elec-
where R is the Ricci curvature tensor. tromagnetic field.
µν
c
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC