6. Connections for Riemannian Manifolds and Gauge Theories

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6. Connections for Riemannian Manifolds and Gauge
Theories
6.1 Introduction 6.2 Parallelism on Curved
Surfaces 6.3 The Covariant Derivative 6.4 Componen
ts Covariant Derivatives of the
Basis 6.5 Torsion 6.6 Geodesics 6.7 Normal
Coordinates 6.8 Riemann Tensor 6.9 Geometric
Interpretation of the Riemann Tensor 6.10 Flat
Spaces 6.11 Compatibility of the Connection with
Volume- Measure or the Metric 6.12 Metric
Connections 6.13 The Affine Connection and the
Equivalence Principle 6.14 Connections Gauge
Theories The Example of Electromagnetism 6.15 Bib
liography
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6.1. Introduction
Affine connection ? Shape curvature. Gauge
connection Gauge theory.
Connections are not part of the differential
structure of the manifold.
Amount of added structure Volume element lt
Connection lt Metric
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6.2. Parallelism on Curved Surfaces
There is no intrinsic parallelism on a manifold.
Example Parallelism on S2.
Parallel transport Moving a vector along a
curve without changing its direction
Direction of V at C depends on the route of
parallel transport. ? Absolute parallelism is
meaningless. Affine connection defines parallel
transport.
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6.3. The Covariant Derivative
Let C be a curve on M with tangent
At point P, pick a vector
An affine connection then allows us to define a
vector field V along C by parallel transport.
The covariant derivative ?U along U is defined
s.t.
? V is parallel transported along C.
Let W be a vector defined everywhere on C. Then
where W (P ?Q ) is W(P) parallel-transported to
Q C(?d? ).
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Reminder Lie dragging W along U requires the
congruences of U W around C. ? LUW requires U
W be defined in neighborhood of C. Parallel
transporting W along U requires only values of U
W on C. ? ?UW requires only U W be defined
on C.
Compatibility with the differential structure
requires the covariant derivative to be a
derivation (it satisfies the Leibniz rule) and
additive in U.
Thus
A, B tensors
Setting
we have
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Under a change of parametrization ? ? µ
With
we have
Combining with the additivity
we have
where f, g are functions.
?UW is a vector ? the gradient ?W is a (11)
tensor s.t.
(see Ex 6.1)
Caution ? itself is not a tensor since its not
linear
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6.4. Components Covariant Derivatives of the
Basis
Any tensor can be expressed as a linear
combination of basis tensors.
The basis tensors for ?V are
vector
Gkj i Affine connection coefficients.
Christoffel symbols for a metric connection
Thus,
?
where
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The parallel transport of V is then given by
Ex. 6.6
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6.5. Torsion
A connection is symmetric iff
(Ex. 6.8 )
In a coordinate basis, a connection is symmetric
iff
The torsion T is defined by
? T 0 for symmetric connections
T is a (12) tensor (Ex.6.9)
The symmetric part of G is defined as
Torsion is usually neglected in most theories.
Ex.6.11
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6.6 Geodesics
A geodesic parallel transport its own tangent U,
i.e.,
( Geodesic eq. )
Setting
Geodesic x i (?)
we get
and
The geodesic eq. is invariant under the linear
transform ?? a ? b. ? is therefore an affine
parameter. (see Ex.6.12)
Only symmetric part of G contributes to the
geodesic eq. ? Geodesics are independent of
torsion.
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Geometric effects of torsion
Let U be the tangent at P of a geodesic C. Let RP
be the (n?1)-D subspace of TP(M) consisting of
vectors lin. indep. of U. Construct a geodesic
through P with tangent ?? RP . Using G(S) ,
parallel transport U along ? a small parameter
distance eto point Q, i.e., ?(S) ? U
0. Construct another geodesic C ? with tangent U
through Q.
C ? will be roughly parallel to C.
A congruence of geodesics parallel to U can be
constructed around P in this manner.
? can now be transported along U in 2 ways

• Parallel transport
• Lie dragging

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By design
we have
Since
By definition (6.5), the torsion T is given by
U i ?j ? both sides gives
?
If ? is parallel transported along U,
?
since
i.e., the parallel transported ? is twisted by
the torsion along the geodesics.
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6.7. Normal Coordinates
Each vector U?TP(M) defines a unique geodesic CU
(?) with tangent U at P. A point Q near P can be
associated with the unique vector U?TP(M) that
moves P to Q by a parallel-transport of distance
?? 1 along CU (?) . The normal coordinates of
Q , with P as the origin, are defined as the
components U j of U wrt some fixed basis of
TP(M) . Thus, a normal coordinate system is a
1-1 map from M to TP(M) ? Rn. Since geodesics can
cross in a curved manifold, different normal
coordinate patches are required to cover it. The
map from TP(M) to M is called the exponential
map. It is well-defined even when the geodesics
cross. A manifold is geodesically complete if the
exponential map is defined for all U?TP(M) and
all P?M. Useful property Gijk P 0 in normal
coordinates.
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in normal coordinates
Proof
Let
Normal coordinates of Q a distance ? from P along
geodesic CU(?) are
so that
?
? Q on CU(?)
Geodesic eq. for CU(?) in arbitrary coordinates
is
? wrt normal coordinates xi ,
on CU(?).
i.e.
? Q on CU(?)
Since this must be satisfied by arbitrary U(P),
we must have
Reminder
In general,
for Q ? P.
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6.8. Riemann Tensor
The Riemann tensor R is defined by
Its components are
or
R is a (13) tensor because it is a
multiplicative operator containing no
differential operations on its arguments
f function
( Ex.6.13 )
In coordinate basis
( Ex.6.14a )
In non-coordinate basis with
where
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? Rijkl is anti-symmetric in k l, i.e.,
Also
Ex.6.14(c)
Bianchi identities
In coordinate basis
The number of independent components of Rijkl in
an n-D manifold is
Ex.6.14(d)
Caution Other definitions (with different signs
index orderings) of R exist.
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6.9. Geometric Interpretation of the Riemann
Tensor
The parallel transport of A along U d/d? from P
(0) to Q (?) is
for ? ? 0
for finite ?
Let V d/dµ with U,V 0 ? ? µ are good
coordinates for a 2-D subspace.
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since U,V 0
?
?µ area of loop
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Geodesic Deviation
Consider the congruence of geodesics CU defined by
Let ? be a vector field obtained by Lie dragging
?P along U, i.e.,
(c.f. Ex.6.11)
?
since
since
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where
i.e.,
or
since
Geodesic deviation equation
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6.10. Flat Spaces
Definition A manifold is flat if Euclids axiom
of parallelism holds, i.e., The extensions of two
parallel line segments never meet.
Hence
where U is any geodesics ? is Lie dragged by U.
The sufficient condition for this to hold is R
0, i.e., R is a measure of the curvature of the
manifold.

• Properties of a flat space
• Parallel transport is path-independent so that
  there is a global parallelism.
• All TP(M) can be made identical (not merely
  isomorphic).
• M can be identified with any TP(M).
• Exponentiation can be extended throughout any
  simply-connected regions.

Ex.6.16 Polar coordinates in En with R ? 0
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6.11. Compatibility of the Connection with
Volume- Measure or the Metric
Compatibility issues arises when G g or t
co-exist.
E.g., there are 2 ways to define the divergence
of a vector field
via covariant derivative
via volume n-form
Compatibility requires
? V
Ex 6.17a
which is satisfied iff
or
Ex 6.17b
E.g., inner product should be invariant under
parallel transport
g G compatible iff
Ex 6.18
i.e.,
metric connection
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Ex 6.20
Show that
? If V is a Killing vector,
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6.12. Metric Connections
?
?
Ex 6.21-2
In normal coordinates
?
In which case, the number of independent
components in R is
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Ricci tensor
Ex 6.23
Ricci scalar
Bianchis identities
?
Weyl tensor
Every contraction between the indices of Cijkl
vanishes.
Einstein tensor
Empty space
6 independent eqs
A geodesic is an extremum of arc length
Ex 6.24
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6.13. The Affine Connection and the Equivalence
Principle
Gijk 0 for flat space in Cartesian
coordinates. Gijk ? 0 for flat space in
curvilinear coordinates.
Principle of minimal coupling ( between physical
fields curvature of spacetime) Strong
principle of equivalence Laws of physics take
the same form in curved spacetime as in flat
spacetime with curvilinear coordinates.
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6.14. Connections Gauge Theories The Example
of Electromagnetism
For an introduction to gauge theories, see Chaps
8 12 of I.D.Lawrie, A unified grand tour of
theoretical physics, 2nd ed., IoP (2002)
Basic feature of gauge theories Invariance
under a group of gauge transformations. E.g.,
electromagnetism Variables 1-form A Gauge
transformations A ? A d f
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Consider a neutral scalar particle with mass m
governed by
Klein-Gordon eq.
with
Conserved probability current density
If ? is a solution, so is
, where ? is a constant.
i.e., the system is invariant under the gauge
transformation
Special relativity Lorentz transformations
(flat spacetime Cartesian coord). Generalization
to curvilinear coord introduces an affine
connection. Relaxation to non-flat connections ?
gravitational effects (general relativity)

Restriction to ? constant is equivalent to flat
space Cartesian coord. Non-constant ? ? EM
forces.

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General gauge transformation
Since e i ? is a point on the unit circle in the
complex plane, the gauge transformation is a
representation of the group U(1) on ?. The
geometric structure is a fibre bundle ( called
U(1)-bundle ) with base manifold M Minkowski
spacetime, and typical fibre U(1) unit
circle in C. A gauge transformation is a
cross-section of the U(1)-bundle.
?
i.e.
is not invariant under the general gauge
transformation.
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Remedy is to introduce a gauge-covariant
derivative D s.t.
?
is invariant under the general gauge
transformation.

This is accomplished by a 1-form connection A s.t.
?
and
so that
Thus
K.G. eq in an EM field with canonical momentum
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Affine connection preserves parallelism. Connect
ion A preserves phase of gradient under gauge
transformation.
Curvature introduced by an affine connection
Curvature introduced by A
Faraday tensor
?
or
Gauge transformation
?