Fall 2018 M637: Theory of Gravitation

Shouhong Wang (5-8350, showang@indiana.edu)

Office hours

Class Meeting Time and Location

Description

This is the first of a sequence of three courses, taught in three consecutive fall semesters:

• M637: Theory of gravitation 
• M655: Mathematical Foundations of Quantum Mechanics 
• M656: Kinetic Theory and Statistical Mechanics 

These courses will be taught independently to each other, and the students can take any of them in any order.  My intention is to offer interested students a first-principle approach and a global mathematical view to  theoretical physics, as envisioned by Galileo Galilei, James Clerk Maxwell, Albert Einstein, and Paul Dirac. Also, in these courses, we emphasize in particular the symbiotic interplay between modern physics and advanced mathematics. 
Topics to be covered in this course include 

• Einstein's Theory of General Relativity (GR): The focus is on the Einstein Two principles: the principle of equivalence (PE) and the principle of general relativity (PGR). PGR is a symmetry principle, which, together with the simplicity of law of nature, {\bf uniquely determines} the Einstein-Hilbert action. The law of gravity is given by the  Einstein equations, which is the Euler-Lagrange equation of the Einstein-Hilbert action. The recent discovery of gravitational waves offers another strong experimental support of the Einstein theory of GR.
• Theory of black holes: We derive the Schwarzschild solution and the Tolman-Oppenheimer-Volkoff solution of the Einstein equations, and study the implications of these solutions to the structure and formation of black holes. In particular, we shall show that assuming the validity of Einstein's GR, a black hole is closed with nothing gets inside  and nothing comes out of a black hole.
• New Law of Gravity, dark energy and dark matter: The presence of dark matter and dark energy phenomena suggests the need for modification of the Einstein GR. We will show that  to preserve  the conservation of energy-momentum and the Einsteins two principles: PE and PGR, and to incorporate the dark matter and dark energy effect, the unique route for altering the Einstein GR is through  the principle of interaction dynamics, which takes variation of the Einstein-Hilbert action subject to energy momentum conservation constraint. This leads to unique new gravitational field equations, and demonstrates that {\it dark matter and dark energy are intrinsic properties of gravity.
• Geometry and structure of the universe:  On the very large scale, our Universe obeys the cosmological principle (CP): the universe is homogeneous and isotropic. Under GR and  CP, we show that the universe is a 3-dimensional sphere $S^3$, the dark matter is due to the curvature effect and the dark energy plays the role of negative pressure preventing the universe from collapsing.  We shall also examine the stability of the Friedmann-Lema\^itre-Robertson-Walker metric, as well as  the formation mechanism of stars and galaxies. 

Exams and Prereq

References

I will distribute some lecture notes, and the following will be consulted

• L. D. Landau & E. M. Lifshitz, Classical Theory of Fields, Vol 2 of the Courses on Theoretical Physics

• Tian Ma & Shouhong Wang, Mathematical Principles of Theoretical Physics, Science Press, 2015