6. Connections for Riemannian Manifolds and Gauge Theories 6.1 Introduction 6.2 Parallelism on Curved Surfaces 6.3 The Covariant Derivative 6.4 Components: 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
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6. Connections for Riemannian Manifolds and Gauge Theories 6.1Introduction 6.2Parallelism on Curved Surfaces 6.3The Covariant Derivative 6.4Components:
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6. Connections for Riemannian Manifolds and Gauge Theories
6.1 Introduction6.2 Parallelism on Curved Surfaces6.3 The Covariant Derivative6.4 Components: Covariant Derivatives of the Basis6.5 Torsion6.6 Geodesics6.7 Normal Coordinates6.8 Riemann Tensor6.9 Geometric Interpretation of the Riemann Tensor 6.10 Flat Spaces6.11 Compatibility of the Connection with Volume- Measure or the Metric6.12 Metric Connections6.13 The Affine Connection and the Equivalence Principle6.14 Connections & Gauge Theories: The Example of Electromagnetism6.15 Bibliography
6.1. Introduction
Affine connection → Shape & curvature.
Gauge connection : Gauge theory.
Amount of added structure:
Volume element < Connection < Metric
Connections are not part of the differential structure of the manifold.
6.2. Parallelism on Curved Surfaces
There is no intrinsic parallelism on a manifold.
Example: Parallelism on S2.
Direction of V at C depends on the route of parallel transport.
→ Absolute parallelism is meaningless.
Affine connection defines parallel transport.
Parallel transport = Moving a vector along a curve without changing its direction
6.3. The Covariant Derivative
Let C be a curve on M with tangent d
Ud
At point P, pick a vector PV T M
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.
0U
V V is parallel transported along C.
Let W be a vector defined everywhere on C. Then
0lim
U P C
W Q W P QW
where W (P →Q ) is W(P) parallel-transported to Q = C(λ+δλ ).
Reminder:
Lie dragging W along U requires the congruences of U & W around C.
→ UW 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.
Setting U
d ff
d we have
U U Uf W f W W f
Compatibility with the differential structure requires the covariant derivative to be a derivation (it satisfies the Leibniz rule) and additive in U.
U
d ff W W
d
U U U A B A B A BThus A, B = tensors
, , ,U U U
A A A
Under a change of parametrization λ → μ :
d d d dU
d d d d
d
gd
gU
d
d
With we have
0
limgU P C
W WW
0
limW Wd
d
U P C
g W
Combining with the additivity U V U VW W W
we have f U gV U VW f W g W where f, g are functions.
UW is a vector → the gradient W is a (11) tensor s.t.
; ,U
W U W (see Ex 6.1)
Caution: itself is not a tensor since its not linear: f W f W
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 i
j i jee e
Γkj i = Affine connection coefficients.
= Christoffel symbols for a metric connection
kk jie = vector
Thus, j
jU UV V e j j
j jU UV e V e
jj i
j i j
dVe V U e
d
j j j ki k ii
V V V , if
j
ij ji i
ji i i
dV de
V V d d
V e
jj i k
j k j i
dVe V U e
d
jk i j
k i j
dVV U e
d
→ where
,j j k
i k iV V ;j
iV
The parallel transport of V is then given by
k
i i i jj k
d xV P Q V P P V P
d
; ,
j j j ji i ii
kkV V V V
; ,k
j i i ii j k jj Ex. 6.6
6.5. Torsion
A connection is symmetric iff ,U V
V U U V
In a coordinate basis, a connection is symmetric iffk k
i j j i (Ex. 6.8 )
The torsion T is defined by , ki j j i i j k j ie e e e e T
→ T = 0 for symmetric connections
T is a (12) tensor (Ex.6.9)
The symmetric part of Γ is defined as 1
2k k k
i j i jS i j T
Torsion is usually neglected in most theories.
Ex.6.11 , ,j j
i j j iU iU U L ; ;
. .
j ji j j i
symm connU U
6.6 Geodesics
A geodesic parallel transport its own tangent U, i.e.,
0U
U ( Geodesic eq. )
0i
i j kj k
dUU U
d we get
and2
20
i j ki
j k
d x d x d x
d d d
dU
dSetting
i
i
d x
d [ Geodesic = x i (λ) ]
The geodesic eq. is invariant under the linear transform λ→ a λ + b.
λ is therefore an affine parameter. (see Ex.6.12)
Only symmetric part of Γ contributes to the geodesic eq.
→ Geodesics are independent of torsion.
Geometric effects of torsion :
Let U be the tangent at P of a geodesic C.
Let RP be the (n1)-D subspace of TP(M) consisting of vectors lin. indep. of U.
Construct a geodesic through P with tangent ξ RP .
Using Γ(S) , parallel transport U along ξ a small parameter distance εto point Q, i.e., (S) ξ 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.
• Parallel transport
• Lie dragging
ξ can now be transported along U in 2 ways:
By design: 0S
U
we have 1
2
ii j k
j kU T U Since 1
2k k k
i j i jS i j T
0U
By definition (§6.5), the torsion T is given by
i ij k k jT T
, ki j j i i j k j ie e e e e T
U i ξj both sides gives
, k i jk j iU
U U e T U
,U
U L k i jk j iU
U e T U →
If ξ is parallel transported along U,
→k i j
k j iUU e T U L
1
2k j i k i j
k j i k j ie T U e T U
1
2k j i
k i je T U since
i.e., the parallel transported ξ is ‘twisted’ by the torsion along the geodesics.
6.7. Normal Coordinates
Each vector UTP(M) defines a unique geodesic CU (λ) with tangent U at P.
A point Q near P can be associated with the unique vector UTP(M) that moves P to Q by a parallel-transport of distance Δλ = 1 along CU (λ) .
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 UTP(M) and all PM.
Useful property: Γijk |P = 0 in normal coordinates.
Proof:
0ij k P
Normal coordinates of Q a distance λ from P along geodesic CU(λ) are
i i
Qx U P so that 0i
Px
→
i
P ixU
ULet
in normal coordinates
2
20
id x
d Q on CU(λ)
Geodesic eq. for CU(λ) in arbitrary coordinates is 2
20
i j ki
j k
d y d y d y
d d d
wrt normal coordinates {xi} , 0j k
ij k
d x d x
d d
i.e.
on CU(λ).
0i j kj k Q U P U P Q on CU(λ)
Since this must be satisfied by arbitrary U(P), we must have 0ij k P
Reminder: 0ij k Q In
general,for Q P.
6.8. Riemann Tensor
The Riemann tensor R is defined by ,,,
U V U VU V
R
Its components are
,ij i jk l k lR e e e eR
, ,k l l k k li i i m i m i
j j j j ml k lj mkR
,
,k le ek l j je e
or
,i il k jk ljR e e e e R
R is a (13) tensor because it is a multiplicative operator
containing no differential operations on its arguments :
, ,U V f W f U V WR R
, ,f U V W f U V WR Rf = function
( Ex.6.13 )
In coordinate basis:
( Ex.6.14a )
In non-coordinate basis with , ij k j k ie e C e
, ,k l l k k l l ki i i m i
k l k lm i m i
j j j j m j m jmR C , i if e fwhere
, ,k l l k k li i i m i m i
j j j j ml k lj mkR ,
,k l
ij i j ek jk l elR e e e
→ Rijkl is anti-symmetric in k & l, i.e., 1
02
i ij k lj lk l
ikjR RR
Also 0ij k lR Ex.6.14(c)
Bianchi identities: ; 0k li
j mR
In coordinate basis:
, , , , , , 0i j k j k i k i j
Caution: Other definitions (with different signs & index orderings) of R exist.
The number of independent components of Rijkl in an n-D manifold is
4 2 2 21 1 11 1 2 1
2 3! 3n n n n n n n n n n Ex.6.14(d)
6.9. Geometric Interpretation of the Riemann Tensor
,,,
U V U VU V
R
The parallel transport of A along U = d/dλ from P (0) to Q (λ) is
UA Q P A P A P
0lim
U P C
W Q W P QW
0
limW Q P W P
for λ → 0
expU P
A for finite λ
Let V = d/dμ with [ U,V ] = 0 → λ & μ are good coordinates for a 2-D subspace.
expV Q
A R Q A exp expU V P
A R Q P A
exp expV U P
A R S P A
exp expU V P
A R Q P A
exp expV U P
A R S P A
A A R Q P A R S P exp , expU V
A
2 21 11 , 1 3
2 2U U U V V VA O
, 3U V
A O
→
, 3U V A O R since [ U,V ] = 0
3i i j k lj k lA R A U V O λμ = ‘area’ of loop
Geodesic Deviation
Consider the congruence of geodesics CU defined by 0U
U
Let ξ be a vector field obtained by Lie dragging ξ|P along U, i.e., 0U
L
, ,i i
i iUU U L
U U U UU L
; ;i i
i iU U
UU
(c.f. Ex.6.11)
UU since 0
U L
,U U
U U
,U
U since 0
UU
,U U U
U ,
,,U V U V
U V
Rwhere ,
U UU U R
, 0U
U L
i.e., i j ki j kU U
U U e ; ;
i j kj ki
U U e
,U UR k i j li j l kR U U e
or ; ;
i j k k i j lj i j li
U U R U U
; ; ;i j k i j k
i j j iU U U U
;i j k
j iU U since ; 0i j ji U
U U U
;i j k k i j l
j i i j lU U R U U Geodesic deviation equation
6.10. Flat Spaces
Definition: A manifold is flat if Euclid’s axiom of parallelism holds, i.e.,
The extensions of two parallel line segments never meet.
Hence , 0U U
U U R
where U is any geodesics & ξ 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 n with R 0
6.11. Compatibility of the Connection with Volume- Measure or the Metric
Compatibility issues arises when Γ & g or τ co-exist.
E.g., there are 2 ways to define the divergence of a vector field :
;i
iV V ,i i j
i j iV V
Vdiv V L
via covariant derivative
via volume n-form
Compatibility requires diV v V V
which is satisfied iff 0 Ex 6.17a
or ,
ln jj k
kg Ex 6.17b
E.g., inner product should be invariant under parallel transport :
g & Γ compatible iff 0 g
i.e., , , ,
1
2i im
jk m j k mk j j k mg g g g metric connection
Ex 6.18
Ex 6.20 :
i iV i jj jV V gL
0i j j iV V If V is a Killing vector,
Show that
6.12. Metric Connections
, , ,
1
2i i m
jk m j k mk j j k mg g g g 0 g
0ij k P
, 0l m n Pg
Ex 6.21-2 :
In normal coordinates
mi jk l im jk lR g R , , , ,
1
2 i l j k i k j l j k i l j l i kg g g g
→ i j k l k l i jR R
In which case, the number of independent components in R is
2 2 21 1 11 2 1 2 3 1
8 24 12n n n n n n n n n n
Ricci tensor :k
i j i k jR R
Ricci scalar :i i ki k jR R g R
j iR Ex 6.23
Bianchi’s identities;
10
2i j i j
j
R Rg ; 0i
j k l mR →
Weyl tensor :
1
23
i j i ji j i jk l k l k l k lC R R R
Every contraction between the indices of Cijkl vanishes.
Einstein tensor :
1
2i j i j i jG R Rg
Empty space : 0i jG 6 independent eqs
A geodesic is an extremum of arc length ,x
d dd
d d
g Ex 6.24
6.13. The Affine Connection and the Equivalence Principle
Γijk = 0 for flat space in Cartesian coordinates.
Γijk 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.
6.14. Connections & Gauge Theories: The Example of Electromagnetism
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
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)
Consider a neutral scalar particle with mass m governed by
2 0m Klein-Gordon eq.
3 * * 1d x with Conserved probability current density
If is a solution, so is ie , where is a constant.
i.e., the system is invariant under the gauge transformation ie
Restriction to = constant is equivalent to flat space + Cartesian coord.
Non-constant → EM forces.
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)
i xe General gauge transformation:
i xd d i d e
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 .
A gauge transformation is a cross-section of the U(1)-bundle.
i xe
i.e. d i xd e
is not invariant under the general gauge transformation. 2 0m
Remedy is to introduce a gauge-covariant derivative D s.t.
i xD D e
is invariant under the general gauge transformation.
2 0D D m &
i xe
This is accomplished by a 1-form connection A s.t.
i xe A A d and
D d i A
so that i x i xD d i d e i e A d
i xD e
Thus 2 0D D m 2i A i A m
D i A
A A
K.G. eq in an EM field with canonical momentum c
qp p A i
c
Affine connection: preserves parallelism.
Connection A : preserves phase of gradient under gauge transformation.
Curvature introduced by an affine connection: , V R V