Semi-Riemann Geometry and General Relativity

Another free online geometry textbook written by Shlomo Sternberg from Harvard University. This text, written in 2003, stems from a one semester course, taught by the author, covering an introduction to Riemannian geometry and its principle physical application – Einstein’s theory of general relativity.

Students should have a background in linear algebra and advanced calculus with a concentration on differential forms.

Sternberg has also written the advanced mathematics textbooks – Theory of Functions of a Real Variable and Lie Algebras.

Table of Contents

0.1 Introduction

1 The principal curvatures

1.1 Volume of a thickened hypersurface

1.2 The Gauss map and the Weingarten map

1.3 Proof of the volume formula

1.4 Gauss’s theorema egregium

1.4.1 First proof, using inertial coordinates

1.4.2 Second proof. The Brioschi formula

1.5 Problem set – Surfaces of revolution

2 Rules of calculus

2.1 Superalgebras

2.2 Differential forms

2.3 The d operator

2.4 Derivations

2.5 Pullback

2.6 Chainrule

2.7 Lie derivative

2.8 Weil’s formula

2.9 Integration

2.10 Stokes theorem

2.11 Lie derivatives of vector fields

2.12 Jacobi’s identity

2.13 Left invariant forms

2.14 The Maurer Cartan equations

2.15 Restriction to a subgroup

2.16 Frames

2.17 Euclidean frames

2.18 Frames adapted to a submanifold

2.19 Curves and surfaces – their structure equations

2.20 The sphere as an example

2.21 Ribbons

2.22 Developing a ribbon

2.23 Parallel transport along a ribbon

2.24 Surfaces in R3

3. Levi-Civita Connections

3.1 Definition of a linear connection on the tangent bundle

3.2 Christoffel symbols

3.3 Parallel transport

3.4 Geodesics

3.5 Covariant differential

3.6 Torsion

3.7 Curvature

3.8 Isometric connections

3.9 Levi-Civita’s theorem

3.10 Geodesics in orthogonal coordinates

3.11 Curvature identities

3.12 Sectional curvature

3.13 Ricci curvature

3.14 Bi-invariant metrics on a Lie group

3.14.1 The Lie algebra of a Lie group

3.14.2 The general Maurer-Cartan form

3.14.3 Left invariant and bi-invariant metrics

3.14.4 Geodesics are cosets of one parameter subgroups

3.14.5 The Riemann curvature of a bi-invariant metric

3.14.6 Sectional curvatures

3.14.7 The Ricci curvature and the Killing form

3.14.8 Bi-invariant forms from representations

3.14.9 The Weinberg angle

3.15 Frame fields

3.16 Curvature tensors in a frame field

3.17 Frame fields and curvature forms

3.18 Cartan’s lemma

3.19 Orthogonal coordinates on a surface

3.20 The curvature of the Schwartzschild metric

3.21 Geodesics of the Schwartzschild metric

3.21.1 Massive particles

3.21.2 Massless particles

4. The bundle of frames

4.1 Connection and curvature forms in a frame field

4.2 Change of frame field

4.3 The bundle of frames

4.3.1 The form θ

4.3.2 The form θ in terms of a frame field

4.3.3 The definition of ω

4.4 The connection form in a frame field as a pull-back

4.5 Gauss’ theorems

4.5.1 Equations of structure of Euclidean space

4.5.2 Equations of structure of a surface in R3

4.5.3 Theorema egregium

4.5.4 Holonomy

4.5.5 Gauss-Bonnet

5 Connections on principal bundles

5.1 Submersions, fibrations, and connections

5.2 Principal bundles and invariant connections

5.2.1 Principal bundles

5.2.2 Connections on principal bundles

5.2.3 Associated bundles

5.2.4 Sections of associated bundles

5.2.5 Associated vector bundles

5.2.6 Exterior products of vector valued forms

5.3 Covariant differentials and covariant derivatives

5.3.1 The horizontal projection of forms

5.3.2 The covariant differential of forms on P

5.3.3 A formula for the covariant differential of basic forms

5.3.4 The curvature is dω

5.3.5 Bianchi’s identity

5.3.6 The curvature and d2

6 Gauss’s lemma.

6.1 The exponential map

6.2 Normal coordinates

6.3 The Euler field E and its image P

6.4 The normal frame field

6.5 Gauss’ lemma

6.6 Minimization of arc length

7 Special relativity

7.1 Two dimensional Lorentz transformations

7.1.1 Addition law for velocities

7.1.2 Hyperbolic angle

7.1.3 Proper time

7.1.4 Time dilatation

7.1.5 Lorentz-Fitzgerald contraction

7.1.6 The reverse triangle inequality

7.1.7 Physical significance of the Minkowski distance

7.1.8 Energy-momentum

7.1.9 Psychological units

7.1.10 The Galilean limit

7.2 Minkowski space

7.2.1 The Compton effect

7.2.2 Natural Units

7.2.3 Two-particle invariants

7.2.4 Mandlestam variables

7.3 Scattering cross-section and mutual flux

8. Die Grundlagen der Physik

8.1 Preliminaries

8.1.1 Densities and divergences

8.1.2 Divergence of a vector field on a semi-Riemannian manifold

8.1.3 The Lie derivative of of a semi-Riemann metric

8.1.4 The covariant divergence of a symmetric tensor field

8.2 Varying the metric and the connection

8.3 The structure of physical laws

8.3.1 The Legendre transformation

8.3.2 The passive equations

8.4 The Hilbert “function”

8.5 Schrodinger’s equation as a passive equation

8.6 Harmonic maps

9 Submersions

9.1 Submersions

9.2 The fundamental tensors of a submersion

9.2.1 The tensor T

9.2.2 The tensor A

9.2.3 Covariant derivatives of T and A

9.2.4 The fundamental tensors for a warped product

9.3 Curvature

9.3.1 Curvature for warped products

9.3.2 Sectional curvature

9.4 Reductive homogeneous spaces

9.4.1 Bi-invariant metrics on a Lie group

9.4.2 Homogeneous spaces

9.4.3 Normal symmetric spaces

9.4.4 Orthogonal groups

9.4.5 Dual Grassmannians

9.5 Schwarzschild as a warped product

9.5.1 Surfaces with orthogonal coordinates

9.5.2 The Schwarzschild plane

9.5.3 Covariant derivatives

9.5.4 Schwarzschild curvature

9.5.5 Cartan computation

9.5.6 Petrov type

9.5.7 Kerr-Schild form

9.5.8 Isometries

9.6 Robertson Walker metrics

9.6.1 Cosmogeny and eschatology

10 Petrov types

10.1 Algebraic properties of the curvature tensor

10.2 Linear and antilinear maps

10.3 Complex conjugation and real forms

10.4 Structures on tensor products

10.5 Spinors and Minkowski space

10.6 Traceless curvatures

10.7 The polynomial algebra

10.8 Petrov types

10.9 Principal null directions

10.10 Kerr-Schild metrics

11 Star

11.1 Definition of the star operator

11.2 Does⋆:∧kV →∧n−kV determine the metric?

11.3 The star operator on forms

11.3.1 For R2

11.3.2 For R3

11.3.3 ForR1,3

11.4 Electromagnetism

11.4.1 Electrostatics

11.4.2 Magnetoquasistatics

11.4.3 The London equations

11.4.4 The London equations in relativistic form

11.4.5 Maxwell’s equations

11.4.6 Comparing Maxwell and London

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Semi-Riemann Geometry and General Relativity