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Lecture 6
Monday, September 6
Contents
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Differential Geometry
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Differential Forms
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Differential Geometry
Vectors in real space have magnitude and direction. They transform under rotations in R3 according to an
irreducible 3-dimensional representation of SO(3). Gauss’ Law is the scalar product of gradient and electric
field vectors
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X
∂
Ei = ρ ,
∂r
i
i=1
in a Cartesian coordinate system. Vector calculus gives a convenient and compact notation for vectors that
does not depend on any coordinate system
∇·E=ρ.
Differential geometry generalizes vector calculus to irreducible representations of Lie groups on Rn in a
coordinate independent way. Altland’s ebook on Geometry in Physics[1] summarizes three major advantages:
(1) the underlying structure of the theory becomes maximally visible because formulas are not cluttered
with indices, (2) transformations between different coordinate systems are exposed in transparent and easily
recognizable terms, and (3) formulas in specific coordinate systems are written to expose the underlying
conceptual structure of objects.
Differential geometry is based on the concept of differential forms. The textbook on Gravitation by Misner,
Thorne and Wheeler was one of the earliest to use differential forms in physics in a systematic way: the
equations of General Relativity, which are usually a mess of indicies, are expressed in compact and completely
index-free notation.
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Maxwell’s Equations
The Maxwell equations were originally published as partial differential equations in vector component form,
for example in Maxwell’s 1860 article On Physical Lines of Force in the Philosophical Magazine.
The vector calculus notation used in most physics textbooks was introduced by J.W. Gibbs and by O.
Heaviside towards the end of the nineteenth century. This greatly simplified writing Maxwell’s equations:
compare Maxwell’s component notation (columns 1,3) with modern vector notation (columns 2,4)
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Equations F,G,H,I and L in column 3 in Gothic letters look like vector notation, but actually represent W.R.
Hamilton’s Quaternions, which were used by Maxwell in his 1873 textbook A Treatise on Electricity and
Magnetism.
Covariant Relativistic Form
The electromagnetic field in relativistic notation is a second rank tensor with two spacetime indices. Its
covariant components are given by
Fµν = ∂µAν − ∂ν Aµ ,
Aµ = (Φ, −A)
The dual electromagnetic tensor interchanges electric and magnetic fields:
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Feµν = µνκλF κλ
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The two Maxwell equations with sources are given by
∂µF µν = ∂µ (∂ µAν − ∂ ν Aµ) = Aν − ∂ ν (∂µAµ) = J ν
The divergence of the dual tensor gives the two source free Maxwell equations:
∂µFeµν = ∂µµνκλ (∂κAλ − ∂λAκ) = 0
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Differential Forms
An Integral is the weighted sum of the values of a function f over a range R of values x:
Z
dµ f (x)
R
The measure µ(x) specifies the weight to be assigned to the value of of f at x.
A Differential form ω on a submanifold S of a differentiable manifold M is everything under the integral
sign in an integral of a function over the submanifold:
Z
ω
S
Differential forms on p-dimensional submanifolds of an n-dimensional manifold are called p-forms on M .
The degree p of the differential form (also called grade) can take values p = 0, 1, 2, 3, . . . , n, corresponding
to points, curves, area, volumes, hypervolumes, . . ..
Real space is an Euclidean manifold M 3 with n = 3. Objects in real space can be defined in terms of
p-forms with p = 0, 1, 2, 3 corresponding to scalars, vectors, pseudo-vectors, and the volume measure d3r.
Electrodynamics is defined on Minkowski spacetime M 1,3 with n = 4. Electrodynamics objects in spacetime
can defined in terms of p-forms with p = 0, 1, 2, 3, 4. There are 24 = 16 independent differential forms.
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Zero-Forms and Dual Basis
A 0-dimensional submanifold of M is a set of one or more points in M . A differential 0-form is defined to
be a smooth (differentiable) function on M .
The gauge function Λ(x), which transforms the gauge of the 4-vector potential according to
Aµ(x) −→ Aµ(x) + ∂ µΛ(x)
is an example of a 0-form on spacetime.
Each component xµ of points x in spacetime is a 0-form. These component 0-forms are associated with a
special set of 1-forms. Consider, for example, a segment along the y axis with length
Z b
Z
b−a=
dy = e2
a
S
This defines a 1-form e2, where 2 denotes second or y component, because its integral over a 1-dimensional
subspace is the line integral along the 2-axis with integrand f (y) = 1.
The coordinate 1-forms
eµ ,
µ = 0, 1, 2, 3
are defined to give the corresponding line integrals
e0 , e1 , e2 , e3
⇔
dt, dx, dy, dz
and provide a natural basis for constructing p-forms on M 1,3.
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Note that the superscript µ is not a relativistic 4-vector index. It simply labels the direction in spacetime of
the integration axis.
The Potential 1-Form
The electromagnetic 4-vector potential is associated with the potential 1-form
A = Φe0 − Axe1 − Ay e2 − Az e3
⇔
Φdt − A · dr = Aµdxµ
The potential 1-form has a simple geometrical interpretation: it represents the 3-dimensional hypersurface
that is perpendicular to the direction of the contravariant 4-vector Aµ(x).
In flat Euclidean space perpendicular directions differ by 90◦. Perpendicular directions in spacetime are
defined by zero inner product using the metric gµν .
The potential 1-form A can be visualized as a family of 3- dimensional hypersurfaces in M 3,1. It provides
a line-integral measure at each spacetime point along the normal direction to this family of hypersurfaces,
as illustrated in the Figure from Misner, Thorne and Wheeler.
Wedge Product and Exterior Algebra
Differential forms on a manifold represent volume elements of submanifolds.
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h
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Areas and volumes in 3-dimensional Euclidean space can be constructed using the vector Cross product
introduced by Gibbs. Vector algebra based on the cross product works only in flat 3-dimensional space.
The generalization to manifolds of any dimension with non-Euclidean geometry is the Exterior algebra based
on the wedge product.
A simple example of a wedge product is the 2-form
eµ ∧ eν = −eν ∧ eµ
The wedge product of two 1-forms is antisymmetric. There six different 2-forms constructed from the four
basis 1-forms provide a basis for 2-forms in Minkowski space.
Electromagnetic Field 2-Form
In 3-dimensional vector notation, the electric field is a vector and the magnetic field is a pseudo-vector. In
4-vector notation they are combined into a single second rank antisymmetric tensor.
In differential geometry, the electromagnetic field tensor is represented by the field 2-form
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Fµν eµ ∧ eν
2
= Exe0 ∧ e1 + Ey e0 ∧ e2 + Ez e0 ∧ e3 + Bxe2 ∧ e3 + By e3 ∧ e1 + Bz e1 ∧ e2
F =
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Exterior Derivative
The Exterior derivative d is a differential operator on a p-form that produces a p+1 form.
A 0-form is just a continuous function f (x) on space time. The exterior derivative of a 0-form is the 1-form
(covariant derivative)
∂f
∂f
eµ
df = µ eµ ⇔ ∂f =
∂x
∂xµ
corresponding to the 4-gradient of f (x).
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The exterior derivative of a p-form is defined to satisfy two properties
d dα = 0 ,
d(α ∧ β) = ( dα) ∧ β + (−1)pα α ∧ ( dβ)
These requirements determine the exterior derivative of the potential 1-form:
1
dA = Fµν eµ ∧ eν = F
2
and imply that the exterior derivative of the electromagnetic field 2-form
dF = d dA = 0
The Hodge Dual (Star Operator)
The Hodge dual or Hodge star operator converts a p-form into an (n−p)-form, where n is the dimension
of the manifold.
Its action on a basis p-form is defined to be
? ei1 ∧ ei2 ∧ . . . eip = eip+1 ∧ eip+2 ∧ . . . ein
where i1, i2, . . . , in is an even permutation of 1, 2, . . . , n.
The Hodge dual of a function f (x) or 0-form is a 4 − 0 = 4 form
?f = f (x)e0 ∧ e1 ∧ e2 ∧ e3
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which represents the function as integrand and the volume measure so
Z
Z
?f = d4x f (x)
The electromagnetic current 3-form
?J = Jµ ?eµ = 1 µνκλ j µ ∧ eν ∧ eκ ∧ eλ
3!
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2
= ρ e ∧ e ∧ e 3 − Jx e 2 ∧ e 3 ∧ e 0 − Jy e 3 ∧ e 0 ∧ e 1 − Jz e 0 ∧ e 1 ∧ e 2
The Hodge dual of the current 3-form is the current density 1-form
? ?J = J = ρe0 − jxe1 − jy e2 − jz e3
⇔
ρdt − j · dr = jµdxµ
The Hodge dual of the electromagnetic field 2-form is also an n − 2 = 2 form
?F = Bxe0 ∧ e1 + By e0 ∧ e2 + Bz e0 ∧ e3 + Exe2 ∧ e3 + Ey e3 ∧ e1 + Ez e1 ∧ e2
1
= Feµν eµ ∧ eν
2
and represents the dual field tensor Feµν .
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The Maxwell Equations
The Maxwell equations can be generated as a system of differential forms starting from the potential and
current 1-forms and applying the exterior derivative operator d and the Hodge dual operator ? repeatedly.
The derivative of the potential 1-form gives the electromagnetic field 2-form
F = dA
Applying derivation and Hodge star to the field 1-form
?F dual field 2-form
dF = d dA = 0 ,
gives the source-free Maxwell equations and the dual field.
Applying derivation and Hodge star to the dual electromagnetic field 2-form
d ?F = ?J ,
? ?F = −F
gives the Maxwell equations with sources and recovers the field 2-form.
Finally, applying derivation to the current 3-form
d ?J = d d ?F = 0
gives the current conservation equation.
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References
[1] A. Altland’s teaching webpage http://www.thp.uni-koeln.de/alexal/teaching.html, Geometry in Physics,
http://www.thp.uni-koeln.de/alexal/pdf/geometry.pdf.
[2] G.W. Gibbons’s webpage http://www.damtp.cam.ac.uk/people/g.w.gibbons/ Applications of Differential Geometry to Physics, http://www.damtp.cam.ac.uk/research/gr/members/gibbons/gwgPartIII˙DGeometr
1.pdf.
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