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