The Geometry of Physics Frankel Solutions

The Geometry of Physics – Problem Solutions David Luposchainsky 30. 10. 2010 These are my solutions to problems given in Theodore Frankel’s book “The Geometry of Physics” (second edition). As I could not find any other sources, I do not know whether they are correct or not, so read with care (especially the index battles). If you have a solution that is not in here already, a better way of showing something, or just some useful comment, I’d like to hear about it1 .

Conventions If not mentioned differently, use the following conventions: P • Use Einstein summation. Sometimes, I’ll typeset a for clarification, though technically unnecessary. • The “+” used in the book will be used implicitly, i.e. multiindices are always assumed to be in ascending order. • Abbreviations concerning the metric tensor: g := | det ({gij }|), s := sign (det ({gij }))

1 e-mail:

stupid underscore name at gmx dot net

Contents

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Contents Manifolds, Tensors and Exterior Forms 2. Tensors and Exterior Forms . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4(2)(i) Contraction invariant under base transformation . . 2.4(2)(ii) Non-invariant “contraction” . . . . . . . . . . . . . . 2.4(3)(i) Transformation behavior of a contraction . . . . . . 2.4(3)(ii) Tensor? . . . . . . . . . . . . . . . . . . . . . . . . . 2.4(3)(iii) Tensor? – second attempt . . . . . . . . . . . . . . 2.5 The Graßmann or Exterior Algebra . . . . . . . . . . . . . . . . . 2.5(1) Basis expansion of a form . . . . . . . . . . . . . . . . 2.5(2) Components of α1 ∧ β 2 . . . . . . . . . . . . . . . . . . 2.6 Exterior Differentiation . . . . . . . . . . . . . . . . . . . . . . . . 2.6(1) Differential of a 3-Form in R4 . . . . . . . . . . . . . . 2.7 Pull-Backs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7(1) Proof of homomorphism . . . . . . . . . . . . . . . . . 2.7(2) Pull-back onto a surface . . . . . . . . . . . . . . . . . 2.10 Interior Products and Vector Analysis . . . . . . . . . . . . . . . 2.10(2) Components of the interior product . . . . . . . . . . 2.10(4) Vector analysis in R3 . . . . . . . . . . . . . . . . . . 2.10(5) Basis expansion of the cross product . . . . . . . . . . 3. Integration of Differential Forms . . . . . . . . . . . . . . . . . . . . . . . 3.1 Integration over a Parameterized Subset . . . . . . . . . . . . . . . 3.1(3)(i) Higher-dimensional cross product . . . . . . . . . . . 3.3 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3(1) ... in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3(2) ... in R4 . . . . . . . . . . . . . . . . . . . . . . . . . . 4. The Lie derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Lie Derivative of a Vector Field . . . . . . . . . . . . . . . . . 4.1(1) Coordinate expression for [X, Y] . . . . . . . . . . . . . 4.2 The Lie Derivative of a Form . . . . . . . . . . . . . . . . . . . . . 4.2(1) Coordinate expression for LX α1 . . . . . . . . . . . . . . 4.2(2) Compositions of derivations and antiderivations . . . . . 4.2(3) i[X,Y] = LX ◦ iY − iY ◦ LX . . . . . . . . . . . . . . . . . 4.2(3) Fugly proof of dα(X, Y) = X(α(Y))−Y(α(X))−α([X, Y]) 4.4 A problem set on Hamiltonian mechanics . . . . . . . . . . . . . . 4.4(1) Symplectic form . . . . . . . . . . . . . . . . . . . . . . . 4.4(1) Symplectic volume form . . . . . . . . . . . . . . . . . . P. 147: Derivation of Hamilton’s equations . . . . . . . . . . . 4.4(4) Hamilton in shrt . . . . . . . . . . . . . . . . . . . . . . 4.4(5) Lie derivative of the symplectic Poincaré 2-form . . . . . 4.4(8) Hmltn n shrtr . . . . . . . . . . . . . . . . . . . . . . . . 4.4(9) Lie derivative of the pre-symplectic Poincaré 2-form . . . 5. The Poincaré Lemma and Potentials . . . . . . . . . . . . . . . . . . . . 5.5 Finding potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5(1) Product of a closed and an exact form . . . . . . . . . .

4 4 4 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 7 8 8 8 8 8 8 9 10 10 10 10 10 10 11 11 12 12 12 12 13 13 13 14 14 14 14

Contents

3 7. R3 and Minkowski Space . . . . . . . . . . . . . . . . . . 7.2 Electromagnetism in Minkowski Space . . . . . . . 7.2(3) Field strength 2-Form . . . . . . . . . . 9. Covariant differentiation and Curvature . . . . . . . . . . 9.3 Cartan’s Exterior Covariant Differential . . . . . . 9.3(1) Basis expansion of the curvature form . 9.3(2) Covariant derivative of the identity form 9.4 Change of Basis and Gauge Transformations . . . 9.4(1) Transformation of the curvature form . . 15. Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 One-parameter subgroups . . . . . . . . . . . . . 15.2(1) Generator of rotations . . . . . . . . . . 15.2(2) Generator of A(1) . . . . . . . . . . . .

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Manifolds, Tensors and Exterior Forms

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Manifolds, Tensors and Exterior Forms 2. Tensors and Exterior Forms 2.4 Tensors 2.4(2)(i) A0ij

Contraction invariant under base transformation   0i k
∂x ∂x0i ∂xk 0 0i j ∂x A dxj , ∂ k = Aj j dx ∂ = = A(dx , ∂ i ) = A , k j ∂x0i ∂xj ∂x0i ∂x | {z } | {z } ∂xk ∂xj

=δjk

=Aj k

This is the transformation law of a scalar. 2.4(2)(ii)

Non-invariant “contraction”  X X X X  ∂xj ∂xk ∂xj ∂xk 0 0 0 Aii = A(∂ i , ∂ i ) = A ∂ , ∂ = A(∂ j , ∂ k ) j k ∂x0i ∂x0i ∂x0i ∂x0i | {z } i i i i =Ajk

X ∂xj ∂xk Ajk 6= Aii = ∂x0i ∂x0i i P Since the differential quotients do not cancel out, the value of i Aii is dependant on coordinates; a coordinate-dependant number is neither a scalar nor any other sort of tensor. 2.4(3)(i)

Transformation behavior of a contraction 0 0i gji v =

∂xk ∂xk ∂x` ∂x0i m ∂xk ∂x` ∂x0i m g v = gk` v ` g v = k` k` ∂x0j ∂x0i ∂xm ∂x0j |∂x0i{z∂xm} ∂x0j ` =δm

Thus, gji v i transforms like a vector. 2.4(3)(ii)

Tensor?  0i  ∂ ∂x k ∂ 2 x0i ∂x` k ∂x0i ∂v k ∂ 2 x0i ∂x` k ∂x0i ∂v k ∂x` ∂j0 v 0i = v = v + = v + 0j k ` k 0j k 0j ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x` ∂xk ∂x0j ∂xk |{z} ∂x` ∂x0j =∂` v k

2 0i

=

`

`

0i

∂ x ∂x k ∂x ∂x v + 0j ∂ vk ` ∂xk ∂x0j k ` ∂x ∂x ∂x | {z } 6=0

Although the second term is the correct tensor transformation law, the first term prevents ∂j v i from forming a tensor.

2. Tensors and Exterior Forms

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2.4(3)(iii) Tensor? – second attempt Using the result of (ii), one gets ∂x` ∂x0i ∂ 2 x0j ∂x` k ∂x` ∂x0j ∂ 2 x0i ∂x` k v + ∂` v k − v − ∂` v k ` k 0j 0j k ∂x ∂x ∂x ∂x ∂x ∂x` ∂xk ∂x0i ∂x0i ∂xk  2 0i   `  ∂ x ∂x` k ∂ 2 x0j ∂x` k ∂x ∂x0i ∂x` ∂x0j k k = v − v + ∂` v − ∂` v ∂x` ∂xk ∂x0j ∂x` ∂xk ∂x0i ∂x0j ∂xk ∂x0i ∂xk
∂x` ∂x0i ∂` v k − ∂k v ` 6= 0 + 0j k ∂x ∂x

∂j0 v 0i − ∂i0 v 0j =

2.5 The Graßmann or Exterior Algebra 2.5(1)

Basis expansion of a form
J aJ dxJ (∂ K ) = aJ dxJ (∂ K ) = aJ δK = aK = α (∂ K )

Since this is true for all ∂ K , α = aJ dxJ . 2.5(2)

Components of α1 ∧ β 2 (α1 ∧ β 2 )i