A COMPANION TO THE TRACK II MATHEMATICS

OF MTW

(Modern Mathematical

Methods In Relativity Theory II)

Spring 2023

ULRICH H. GERLACH

PREFACE

1. The theme of Math 5757 is the mathematization of of the relation between matter and gravitation. The second manifests itself by the state of motion of the first, which in turn controls and the first the second

gravitation in geometrical terms.

The subject is motion of matter in response to geometry, and geometry’s reponse to matter.

4. One of the most glaring elements missing from Einstein’s original 1916 line of reasoning is the physical and geometrical meaning of the tensor on the l.h.s. of his field equations

R_{μν} − \frac{1}{2}g_{μν}R = \frac{8πG}{c^{2}}T_{μν} .

That element was supplied by Cartan and Wheeler who identified the l.h.s. as

R_{μν} − \frac{1}{2}g_{μν}R = “moment of rotation per spacetime 3-volume” .

The underlying concept which made this possible is the “moment” concept, which is familiar from physics in other contexts. In order to mathematize this concept geometrically and make it generalizable to 4-d spacetime, these lectures chewed and rechewed it in order to integrate it with the already familiar physics knowledge about it. This fact is already evident from the lecture content as spelled out in the ensuing table of contents.

Contents

Lecture 0: How Newton was led to his law of universal gravitation Lecture 1: The concept “Gravitation”: Where does mit come from?

(a) Gravitation’s observational basis

(b) Mathematization of gravitation according to i. Galilileo ii. Kepler iii. Newton iv. Euler, Lagrange & Hamilton v. Lagrangian Mechanics and the Three-body Problem vi. Einstein

Lecture 2: (a)
\begin{matrix}0. & The Twin Paradox \\ I. & The WHY of Extremal Length \\ II & Generalization \\ III & The Variational Principle \end{matrix} \} Lecture 2

\begin{matrix}IV. & Parametrization Invariance \\ 1. & Noether’s Theorem \\ 2. & Underdetermined System \\ V. & Torsionless Metric-compatible Parallel Transport \\ VI. & Constant of Motion \end{matrix} \} consigned to Lecture 3

Lecture 3: Continuation from Lecture 2 (a) Reparametrization

(b) Torsionless metric-compatible parallel transport

(c) Constant of motion

(d) Q: Why a geometrical interpretation of free particle mechanics?

(e) A: Look at free particle motion relative to a rotating frame

Lecture 4: Geometrization of Newtonian mechanics:

(a) Free particle in a rotating frame

(b) Free body motion vs geodesic motion

(c) Free body motion in an accelerated frame Lecture 5: (a) Geometrization of free body motion in an accelerated frame

(b) Geometrization of gravity i. The Equivalence Principle ii. Geometrized gravity via geometrized inertial motion and the

Equivalence Principle iii. “Physical Foundations of a theory of Gravitation” by A. Ein-

stein (1914):

Lecture 6: (a) Electromagnetism vs. gravitation: Equations of motion and field

equations

(b) Matter and Motion: Its thinkers in history

(c) Momentum: i. Newtonian ii. Relativistic

Lecture 7: (a) Momentum conservation plus the Principle of Relativity implies

enrgy consrvation

(b) The momenergy (=”energy-momentum) four-vector

Lecture 8: The particle density-flux 3-form (a) Derivation of ^{∗}S

(b) The particle current 4-vector

LEcture 9: The particle density-flux 3-form: Its physical, geometrical and algebraic

properties

Lecture 10: The density-flux 3-form (a) Algebraic properties

(b) Four geometrical properties ***************************************

Lecture 11: Momenergy (a) World line vs. world tube (b) Density-flux of charge, momenergy. Their conservation

(c) Physical meaning of the momenergy tensor components

(d) Momenergy for a mixture of particles

Appendix: Q: Why is the stress tensor symmetric?

A: No intrinsic spin

Lecture12: The stress tensor (a) Stress tensor as the area-force relation

(b) The “area vector”

i. the area vector as a vectorial 2-form in 3-d ii. Surface integral in 3-d: How to construct it relative to generic

curvilinear coordinates iii. Example for a spherical surface

Lecture 13: Momenergy (continued)

(a) Momenergy for a mixture of particles

(b) ^{∗}S vs S

(c) Conservation of momenergy

Lecture 14: (a) Orientation

(b) Momenergy conservation mathematized by exterior differentiation

Lecture 15: (a) Maxwell’s vs Einstein’s equations compared and contrasted

(b) The 2-3 version of Stokes’ theorem i. Vectorial version ii. Scalar version iii. Tensor version

Lecture 16: (a) Matter in motion: Kinematics i. Volume as a geometrical attribute ii. Matter in motion iii. Change in comoving volume

(b) Matter in motion: Dynamics i. Momenergy for a perfect fluid ii. Momenergy for e.&m. iii. Conservation of momenergy

A. Electro-magnetism (“Laser”) driven fluid dynamics

Lecture 17: Relativistic hydrodynamics (a) Generalized Newtonian perspective

(b) Energy conservation

(c) Chemical potential

(d) Euler’s 3-d equations

Lecture 18: History of Gravitation: Newton → Einstein → Cartan-Wheeler Lecture 19: The Einstein Field Equations (E.F.E.)

(a) Einstein’s tensorial line of reasoning

(b) Cartan, Misner, and Wheeler’s geometrization of the E.F.E.

(c) “Rotation” as a tensor

(d) Curvature as rotation

Lecture 20: (a) Rotation as a tensor (b) The Hodge dual

(c) Curvature-induced rotation

Lecture 21: (a) Curvature-induced rotation applied to a 3-cube (b) Reminder about curvature (Can be skipped on the 1^{st} reading)

(c) Bianchi’s identity

(d) ∂∂C = 0 and the fundametal Theorem of Calculus

Appendix to Lec 21: Translation of Chapter 8: The Bianchi Identities in Cartan’s “Geome-

trie des Espaces de Riemann”.

Lecture 22: Force, rotation, and their moments mathematized as geometrical objects:

(a) Mathematization of rotational changes induced by curvature in-

side and on the boundary of a 3-cube

Lecture 23: (a) Einstein’s mathematization of gravitation (b) How to form a concept

(c) Solid cube in translational equilibrium: its mathematization

(d) Rotational equilibrium:

(e) Its mathematization via the moment of force

(f) Non-equilibrium

Lecture 24-25: (a) The moment of force i. 3-cube with an electric dipole moment ii. Translational response to a uniform electric field iii. Moment of force method vs the cross product method iv. Arbitrarily located fulcrum v. Cartan’s “diisplacement” (′′dP′′) conceptualized vi. Total moment of force as a generator of rotation

(b) Comparison with Einstein’s Field Equations

(c) The ⋆ isomorphism

(d) Conclusion: torque generates rotation

Appendix M: Moment of the e.m. (Faraday) field

Lecture 26: (a) Surface force density

(b) Translational equilibrium

Lecture 27: (a) Rotational kinematics (b) Lever arms

(c) Moment of force

(d) Torque: Its two mathematizations

Lecture 28: The equations of geometrodynamics

(a) Einstein’s incomplete line of reasoning; Cartan and Wheeler’s elucidation of the Einstein curvature tensor via the “moment” concept (b) Moment of force‘

(c) Translational equilibrium

(d) Moment of force as torque

(e) Equivalence via Hodge dual

Lecture 29: (a) The method of moments to arrive at bulk properties from environmentally induced surface properties i. Moment of force vs ii. moment of rotation

(b) Einstein’s field equations

(c) Two equivalent momenergy representations

Appendix: The Hodge Dual Isomorphism for E^{3} and for M^{4} Lecture 30: (a) Moment of rotation

(b) Einstein’s Field Equations (E.F.E.) stated geometrically

(c) Scholium on the linear algebra of two mappings

(d) Transition from Cartan-Wheeler’s to Einstein’s representation of

his field equations

Lecture 31: E.F.E. for Spherically symmetric Systems (a) The 2+2 decomposition of M^{4} into M^{4} = M^{2} × S^{2}

(b) The decomposed E.F.E.

Appendix to Lec. 31: Computation of Einstein’s tensor field equations for spherical systems

Lecture 32: (a) Integration of the E.F.E. using a conservation law

(b) Mass distribution determines spatial geometry

Lecture 33: Non-Euclidean geometry in the equatorial plane Lecture 34: Relativistic stars

(a) Their equations for hydrostatic equilibrium

(b) Their geometrical environment

(c) Equilibrium: Stable or unstable

(d) Superdense stars

Lecture 35: Ultimate purpose: Integrate wave dynamics with particle dynamics

(a) Wave equation (Klein-Gordon, Schrödinger) and its solutions in

the the short wave, high frequency asymptotic approximation.

Appendix: Schrödinger wave equation as the non-relativistic limit of the Klein-Gordon wave equation; the Kogelnik equation as the paraxial limit of the Helmholtz equation

Lecture 36: Reconstructing the classical trajectoriesusing the Principle of Construc-

tive interference: the basic idea

Lecture 37: Reconstruction of classical worldlines from the principle of constructive inteference in 4-d spacetime via Hamilton’s equations of motion: Plane waves ⇒ straight worldlines

Lecture 38: Hamilton-Jacobi analysis of particle orbits in a spherical static space-

time geometry

Lecture 39: (a) Orbital motion and light deflection in a static spherical gemetry (b) The static Schwarzschild solution to the E.F.E.

(c) Geometry and topology of Shwarzschild spacetime

Lecture 40: Ingoing and outgoing Eddington-Finkelstein (E-F) coordinates for the

Schwarzschild gemetry

Lecture 41: Black Hole: Its Topology and Global Structure; The global Kruskal cooerdinate system for the Schwarzschild geometry;

(a) The tortoise coordinate

(b) Ingoing and outgoing E-F coordinates

(c) Kruskal null coordinates via rescaling

(d) Global Kruskal atlas from double images

(e) Kruskal spacetime diagram

Appendix to Lec. 41: Global Kruskal Schwarzschild coordinates atlas via the Rindler cordinate atlas of Minkowski spacetime

The ensuing notes comprise the second half of OSU’s Mathematics 5756-5757 sequence “Modern Mathematical Methods in Relativity Theory”.

The first half developed inductively the mathematical methods and principles, observations of the physical world as conceptualized by one’s mind. Mathematically they constitute a non-trivial marriage between linear algebra and differential calculus.

Its off-springs are the applications that in Math 5757 mathematize the fundamental classical laws of the world.