6. connections for riemannian manifolds and gauge theories 6.1introduction 6.2parallelism on curved...
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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
Curvature introduced by A: , FD D
, i A i A i A i AD D
i A A A A
i A A
→ F i A A F i d A or
Gauge transformation: A A d F F d A d A
Faraday tensor