The course is taught in English by
Gabriel Martinez-Pinedo, Tel. 06159 71 2750, g.martinez at gsi.de, Schlossgartenstraße 2, room 304
Exercises:
Ninoy Rahman, Tel. 6159 71 3558, n.rahman at gsi.de, Schlossgartenstraße 2, room 04

It provides an introduction of the General Theory of Relativity and discusses its main results

Main topics include:

0. Course description
1. Preliminaries: Newtonian Gravity, Geometry as Physics, Special Relativity
2. Gravity as Geometry, Newtonian Gravity in Spacetime Terms: Static Weak Field Metric
   • Notes on the meaning of proper time
   • Gravity as geometry
3. Curved spacetimes: Tangent space, vectors and dual vectors (one-forms), Tensors
   • General Relativity basic ideas
   • Description curved spacetime (Note on the use of differential as Coordinate basis state)
4. Motion test particle in curved spacetime: Geodesics, Covariant derivative, Parallel transport
   • Covariant derivative
   • Invariance under transformations (Lie derivative)
5. Spacetimes around Spherical Objects and Black Holes: Equations of Motion in Schwarzschild geometry.
   • Orbits of test masses and photons
6. Observational tests of General Relativity
   • Slides
7. Sources of curvature
   • Number density and stress-energy tensor
8. Curvature and Einstein Equation
   • Riemann curvature tensor, Ricci Curvature tensor, Einstein Tensor
9. Spherically symmetric solutions
   • Schwarzschild geometry and Relativistic stars
10. Cosmological models (slides)
11. Linearized theory of Gravity: Gravitational Waves
    • Introduction to Gravitational Waves
    • Linearized theory of Gravity
    • Gravitational Wave Emission from a Binary Star System (Mathematica Notebook, pdf)

Lecture notes (covering item 1 above)

Exercises

• Exercise Sheet 1 (solution)
• Exercise Sheet 2 (solution)
• Exercise Sheet 3 (solution)
• Exercise Sheet 4 (solution)
• Exercise Sheet 5 (solution)
• Exercise Sheet 6 (solution, Reissner-Nordstrøm metric: Mathematica Notebook, pdf, Jupyter notebook using einsteinpy: notebook)

Additional material

• Equations of motion in General Relativity (Mathematica Notebook, pdf,
  see also Janna Levin and Gabe Perez-Giz, A periodic table for black hole orbits, Phys. Rev. D 77, 103005 (2008))
• Calculation of Riemann, Ricci and Einstein Tensors for several metrics (Mathematica Notebook, pdf)
• B. P. Abbott, The Basic Physics of the Binary Black Hole Merger GW150914, Ann. Phys. 529, 1600209 (2017): Basic discussion of the first gravitational event detected by the LIGO and VIRGO collaborations.

Bibliography

The lectures are mainly based on the book (several printings available):

• James B. Hartle, Gravity: An Introduction to Einstein's General Relativity, Addison Wesley, 1st ed. 2002 (Hardcover)
• James B. Hartle, Gravity: An Introduction to Einstein's General Relativity, Pearson, 1st ed. 2013 (Paperback)
• James B. Hartle, Gravity: An Introduction to Einstein's General Relativity, Pearson India, 1st ed. 2014 (Cheap Indian edition, preface and some appendices missing)

Supplemental material including Errata, Mathematica Programs and Web Supplements is available at authors website

Some particular deductions are taken from:

• Bernard Schutz, A First Course in General Relativity, Cambridge University Press, 2nd edition, 2009

Additional bibliography:

• Sean M. Carroll, Spacetime and Geometry, Cambridge University Press, Cambridge, 2019
• Robert M. Wald, General Relativity, University of Chicago Press, Chicago, 1984
• Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, Princeton University Press, 2017
• Steven Weinberg, Gravitation And Cosmology: Principles And Applications Of The General Theory Of Relativity, Wiley Student Edition, 2008
• Ray D'Inverno, Introducing Einstein's Relativity, Oxford University Press, 1990