A COMPANION TO THE TRACK 2 MATHEMATICS OF MTW

(Modern Mathematical Methods

In Relativity Theory II)

Spring 2023

ULRICH H. GERLACH

VOLUME II

Tentative Table of Contents and Reading List

1. Rapid course in special relativity: Key ideas: Principle of relativity [T-W 3.1-3.3 (1.3)]; inertial (free float) frames [MTW Fig 1.7; T-W 2.2, 2.3, 2.4 (1.2, 1.4)]; Isotropy of space [T-W (1.3)].

2. Key ideas: Invariance of the interval [T-W 3.6, 3.7, 3.8 (1.5)].

3. Euclidean geometry vs. Lorentz geometry [T-W 5.2, 5.3 (1.1, 1.6)]; Lorentz transformations [T-W L.3, L.4, L.5, L.6] 4. Important mathematical properties of a Lorentz transformation. Causal

classification of events [T-W 6.1, 6.2, 6.3 (1.7)].

5. Spacetime as a vector space; four velocity; four acceleration; wave propagation four vector. [Box 1.3, 2.4; Sect. 2.1-2.3 in MTW].

6. Significance and utility of the projection invariant; worldline of an accelerated observer [MTW Ch 6].

7. One parameter family; Instantaneous Lorentz frames along a given worldline. Fermi-Walker transport [MTW Ch. 6].

8. Fermi-Walker transport and curvilinear spacetime coordinates [MTW Ch. 6].

9. Covectors and Vectors: Definition of a linear function; the dual vector space; bracket notation; construction of linear functions; basis for dual space; no natural isomorphism between vector space and its dual; basis covectors = coordinate functions..[MTW 2.5 - 2.7, 9.1 - 9.5]

10. Bilinear functional; metric on a vector space; natural isomorphism between vectors and duals; basis representation of a metric; lowering the indices on the components of a vector; reciprocal vector basis; correspondence between reciprocal basis vectors and basis covectors; covariant vs. contravariant components of a vector.

11. Metric as an isomorphism between the vector space and its dual space; representation of the metric relative to a given basis; vector components related to those of its image in V^{∗}; reciprocal basis. [ MTW §2.4, Box 8.4, §13.2.]

12. Tensors as multilinear maps; components of a tensor relative to a chosen basis; tensor product; tensor product basis; space of tensors; constructing new tensors by “lowering” indeces, by contraction.

13. Examples of tensors of various rank; the Levi-Civita tensor; exterior product [MTW 3.5, 4.1-4.2]. Flux tensor in Euclidean space; Flux tube structure; particle (or charge) worldline density tensor. [Read MTW Box 4.2, Fig. 4.2-4.5, Box 4.4].

14. Particle (or charge) worldline density tensor; Symplectic inner product [MTW, Box 4.5]; Euclidean basis vs. symplectic basis [V. Arnold §41].

15. Euclidean rotation vs. Lorentz rotation vs. symplectic transformation [V. Arnold §42]; “Raising” and “lowering” of indices, contraction. Coordinate charts, atlas, manifold; coordinate representative of a function [MTW Ch. 9; Singer & Thorpe 97-99]; Example: Minkowski spacetime with a Rindler atlas.

16. Example of coordinate charts: Minkowski spacetime with Rindler charts, coordinate functions. Tangent vector [Singer & Thorpe §5.1; Hicks Ch. 1; MTW Ch. 9].

17. Tangent vector; transforming its components; vector as a derivation [Singer & Thorpe §5.1; Hicks Ch. 1].

18. Vector field, differential 1-form [MTW 2.5, 9.1] [MTW 2.5, 9.1-9.5]; tangent to a curve; integral curves; [Singer & Thorpe §5.1; Hicks Ch. 1].

19. The Rotation Group SO(3) as a manifold with three independent nowhere zero vector fields [MTW Problem 9.13].

20. Integral curve of a vector field [Singer & Thorpe pp. 125-126]. Commutator of two vector fields [MTW Box 8.4 E, Box 9.2; §9.6; Singer & Thorpe p. 127]. The differential (one form) of a function [MTW §9.4; T. Apostol: Math.’l Analysis pp. 103-107].

21. Differential 1-form as the linear approximation of a function [MTW §9.4]; Differential p-forms: antisymmetric tensor field of rank \binom{0}{p}; exterior product of two forms; exterior derivative of a p-form [MTW §4.1, §14.5, Ex. 14.5].

22. Parallel transport between tangent spaces [MTW Box 10.2, 10.3]; covariant differential of a vector [MTW §14.5]; covariant derivative [MTW §10.3].

23. Parallel transport in the Euclidean plane.

24. Commutator of O.N. polar basis vectors; covariant derivative of a general vector and a general covector [MTW §10.4]. Commutator vs. covariant derivative; pointwise linearity of a tensor map [MTW Box 10.3B]; parallel vector field [MTW Box 10.2], the torsion tensor [MTW §14.5 pp. 353-354]. [Note MTW always assume zero torsion.]

25. Cartan’s 1st Structural equation; Stoke’s Theorem: its infinitesimal version [MTW §14.5]; Riemann Curvature: Parallel transport around a closed loop [MTW §11.14]. Cartan’s 2nd Structural Equation [MTW §14.5]; components of Riemann relative to a coordinate basis [MTW §11.3].

26. Jacobi’s equation of geodesic deviation [MTW §11.3, Box 11.4].

27. Summary: Cartan’s derivation of his two structure equations [MTW §14.5]; compatibility between mietric and parallel transport [MTW §13.3, §14.5]; Christoffel symbols; geodesic on a two sphere; coordinate components vs. orthonormal components of a vector and a covector.

28. Metric induced symmetries of the curvature tensor [MTW §11.6, 14.5].

29. Curvature from a metric: the Cartan-Misner procedure [MTW Box 14.5]. Example: the 2-sphere. Ricci tensor; Curvature Invariant.

30. Metric on a three-sphere; the metric for a Robertson-Walker universe; curvature inside spherical star in free-fall collapse; its Einstein tensor [MTW Box 14.5].

Resource Texts

1. C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation, (Freeman, San Francisco, 1973).

2. U.H. Gerlach, Notes on ModernMathematical Methods in Relativity Theory I, (Unpublished, Department of Mathematics, OSU, 2012)

3. V.I. Arnold, Mathematical Methods of Classical Mechanics, (Springer, 1978).

4. I.M. Singer and Thorpe, Lecture Notes on Elementary Topology and Geometry.

5. N.J. Hicks, Notes on Differential Geometry, (Van Norstround, Princeton, N.J., 1964.)

6. E.F. Taylor and J.A. Wheeler, Spacetime Physics., SECOND EDITION (1st Edition)

7. T. Apostol, Mathematical Analysis.