The Curvature, the Einstein Equations, and the Black Hole

This lecture note is an extended version of the series of lectures I have given in the physics seminar at the University of Southern Mississippi. The majority of the audience were graduate students who have never had any prior encounter with differential geometry. Therefore, I tried to maintain mathematical rigor and technicalities at a minimum when discussed differential geometric concepts, instead mostly used hand-waving and rudimentary arguments with emphases on physical ideas and intuition.

  • Lecture Notes

    • Click on linked topics to view lecture notes.

  • Further Reading

    • Not in particular order. There are so many wonderful books on general relativity and cosmology. I have listed below only some of those books on general relativity and cosmology that I am familiar with and also that I want to suggest you for further reading. In addition, I also included a couple of books on differential geometry which can be helpful for studying general relativity and cosmology (the last two books on the list).

    • Spacetime and Geometry: An Introduction to General Relativity, Sean Carroll, Pearson, 2016. This book grew out of Sean Carroll's much earlier Lecture notes on General Relativity.
    • Lecture Notes on General Relavitiy, Matthias Blau, 950+ pages as of October 2019!
    • A Relativist's Toolkit, The Mathematics of Black-Hole Mechanics, Eric Poisson, Cambridge University Press, 2004
    • An Introduction to General Relativity, L. P. Hughston and K. P. Tod, Cambridge University Press, 1990
    • Relativity, An Introduction to Special and General Relativity, 3rd Edition, Hans Stephani, Cambridge University Press, 2004
    • Black Holes and Time Warps, Einstein's Outrageous Legacy, Kip Thorne and Stephen Hawking (Foreword), W. W. Norton & Company, 1995
    • Lorentzian Wormholes, From Einstein to Hawking, Matt Visser, AIP Series in Computational and Applied Mathematical Physics, 2008
    • Advanced General Relativity, John Stewart, Cambridge University Press, 1991
    • General Relativity and Relativistic Astrophysics, Norbert Straumann, Springer-Verlag, 1984
    • Relativity, Thermodynamics and Cosmology, Richard C. Tolman, Oxford at the Clarendon Press, 1934

      • This is a wonderful classical book on the subject and is still well-worth reading.

    • Relativity on Curved Manifolds, F. De Felice and C. J. S. Clark, Cambridge University Press, 1990

      • This book contains a good bit of materials on differential geometry.

    • Lectures on General Relativity, A. Papapetrou, D. Reidel Publishing Company, 1974

      • This book also contains a good bit of materials on differential geometry.

    • Principles of Cosmology and Gravitation, Michael V. Berry, Cambridge University Press, 1976
    • Gravitation, Charles W. Misner, Kip S. Thorne and John Archibald Wheeler, W. H. Freeman and Company, 1973

      • A pretty big book with more than 1300 pages. This book is dubbed the bible of general relativity.

    • The Large Scale Structure of Space-Time, S. W. Hawking and G. F. R. Ellis, Cambridge University Press, 1973

      • Not for the faint of heart.

    • General Relativity, Robert M. Wald, The University of Chicago Press, 1984
    • An Introduction to Differentiable Manifolds and Riemannian Geometry, William M. Boothby, Academic Press, 1986
    • Semi-Riemannian Geometry with Applications to Relativity, Barrett O'Neill, Academic Press, 1983