6 Connections for Riemannian Manifolds and Gauge Theories

6. Connections for Riemannian Manifolds and Gauge Theories 6. 1 6. 2 6. 3 6. 4 6. 5 6. 6 6. 7 6. 8 6. 9 6. 10 6. 11 6. 12 6. 13 6. 14 6. 15 Introduction Parallelism on Curved Surfaces The Covariant Derivative Components: Covariant Derivatives of the Basis Torsion Geodesics Normal Coordinates Riemann Tensor Geometric Interpretation of the Riemann Tensor Flat Spaces Compatibility of the Connection with Volume- Measure or the Metric Connections The Affine Connection and the Equivalence Principle Connections & Gauge Theories: The Example of Electromagnetism Bibliography

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 < Connection < Metric

6. 2. Parallelism on Curved Surfaces There is no intrinsic parallelism on a manifold. Example: Parallelism on S 2. 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.

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(λ+δλ ).

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 Setting A, B = tensors we have

Under a change of parametrization λ → 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:

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 Γkj i = Affine connection coefficients. = Christoffel symbols for a metric connection Thus, → where

The parallel transport of V is then given by Ex. 6. 6

6. 5. Torsion A connection is symmetric iff 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 Γ is defined as Torsion is usually neglected in most theories. Ex. 6. 11 (Ex. 6. 8 )

6. 6 Geodesics A geodesic parallel transport its own tangent U, i. e. , ( Geodesic eq. ) Setting [ Geodesic = x i (λ) ] we get and 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 (n 1)-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. ξ can now be transported along U in 2 ways: • Parallel transport • Lie dragging

By design: Since we have By definition (§ 6. 5), the torsion T is given by U i ξj both sides gives → If ξ is parallel transported along U, → since i. e. , the parallel transported ξ is ‘twisted’ by the torsion along the geodesics.

6. 7. Normal Coordinates Each vector U TP(M) defines a unique geodesic CU (λ) 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 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 U TP(M) and all P M. Useful property: Γijk |P = 0 in normal coordinates.

in normal coordinates Proof: Let Normal coordinates of Q a distance λ from P along geodesic CU(λ) are so that Q on CU(λ) → Geodesic eq. for CU(λ) in arbitrary coordinates is wrt normal coordinates {xi} , on CU(λ). Q on CU(λ) i. e. Since this must be satisfied by arbitrary U(P), we must have Reminder: In for Q P.

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. 14 a ) In non-coordinate basis with where

→ 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.

6. 9. Geometric Interpretation of the Riemann Tensor The parallel transport of A along U = d/dλ from P (0) to Q (λ) is for λ → 0 for finite λ Let V = d/dμ with [ U, V ] = 0 → λ & μ are good coordinates for a 2 -D subspace.

since [ U, V ] = 0 → λμ = ‘area’ of loop

Geodesic Deviation Consider the congruence of geodesics CU defined by Let ξ be a vector field obtained by Lie dragging ξ|P along U, i. e. , (c. f. Ex. 6. 11) since

where i. e. , or since 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 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 En 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 : via covariant derivative via volume n-form Compatibility requires V which is satisfied iff Ex 6. 17 a or Ex 6. 17 b E. g. , inner product should be invariant under parallel transport : g & Γ compatible iff i. e. , Ex 6. 18 metric connection

Ex 6. 20 : Show that If V is a Killing vector,

6. 12. Metric Connections Ex 6. 21 -2 : In normal coordinates → In which case, the number of independent components in R is

Ricci tensor : Ex 6. 23 Ricci scalar : Bianchi’s identities → Weyl tensor : Every contraction between the indices of Cijkl vanishes. Einstein tensor : Empty space : A geodesic is an extremum of arc length 6 independent eqs 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 For an introduction to gauge theories, see Chaps 8 & 12 of I. D. Lawrie, “A unified grand tour of theoretical physics”, 2 nd ed. , Io. P (2002) Basic feature of gauge theories : Invariance under a group of gauge transformations. E. g. , electromagnetism: Variables: 1 -form A Gauge transformations: A→A+df

Consider a neutral scalar particle with mass m governed by Klein-Gordon eq. with If is a solution, so is Conserved probability current density , 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.

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. is not invariant under the general gauge transformation.

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

Affine connection: preserves parallelism. Connection A : preserves phase of gradient under gauge transformation. Curvature introduced by an affine connection: Curvature introduced by A: → Gauge transformation: Faraday tensor or