Syllabus

Module 1: (6 hours)
Principle of equivalence and the geometrical description of gravity, uniformly accelerated observers, and the Rindler spacetime. Diffeomorphism, Covariant and Contravariant tensors under general coordinate transformations, tensor algebra in curved spacetime, metric tensor and its connection with gravity.

Module 2: (7 hours)
Parallel transport, Christoffel symbols, geodesic equation from action principle, Lie transport, Killing vectors and their association with symmetries of the metric tensor. Parallel transport around a closed curve and the Riemann curvature tensor, connection of the Riemann tensor with spacetime curvature, algebraic and differential properties of the Riemann tensor, tidal acceleration produced by gravity and geodesic deviation equation. Ricci tensor, Weyl tensor, and conformal transformations, Einstein tensor from Bianchi identities.

Module 3: (7 hours)
Geodesic congruence and Raychaudhuri equation: time-like and null congruence. Einstein equation from the action principle, deriving Poisson’s equations from Einstein’s equations. Variation of the matter action and stress-energy tensors, dust, and ideal fluid in curved spacetime, classical field theory in curved spacetime.

Module 4: (5 hours)
Vacuum solution of Einstein’s equations, Birkhoff’s theorem and Schwarzchild solution, geodesics, and particle trajectories in Schwarzschild spacetime, experimental tests of the general theory of relativity, e.g. perihelion precession, bending of light and gravitational redshift.

Module 5: (5 hours)
Reissner Nordstrom geometry, Einstein Equation with matter, Tolman-Oppenheimer-Volkoff Equation, Maximal extension of the Schwarzschild spacetime

Module 6: (6 hours)
Cosmology: Cosmological principle, maximally symmetric spaces and Friedman-Robertson-Walker
metric, distance measurement, luminosity distance, Hubble’s law, Friedman’s equations, Cosmological constant and
dark energy. Thermal history of the Universe, Cosmic Microwave Background Radiation.

To understand gravity as the manifestation of curvature of spacetime, the importance of tensors, and appreciating why the laws of physics should be diffeomorphism invariant.

Understanding the necessity to introduce the covariant derivative in curved spacetime, the concept of parallel transport, the symmetries of the metric tensor, and the Lie transport.

Understanding the importance and properties of the Riemann curvature tensor and the tensors constructed from it.

To understand Einstein’s equations, its solutions and its applications in predicting the planetary motions.

CO1: On successful completion of this course, the students should be able to appreciate the inescapable connection between gravity and the curvature of spacetime.

CO2: They should be able to explain the motion of particles in curved spacetime and how the presence of matter-energy causes curvature of space-time.

CO3: They should be able to appreciate how singularities are inescapable in general relativity through Raychaudhuri’s equations and how Einstein’s equations predict mysterious objects like black holes.

CO4: They should be able to explain how light bends while moving in curved spacetime and how planetary motions can be accurately predicted by incorporating general relativistic corrections.

CO5: They should be able to appreciate the applications of general relativity in understanding cosmology and the large scale structure of the Universe.

S. Carroll, Space time and geometry: an introduction to general relativity, CAMBRIDGE UNIVERSITY PRESS

T. Padmanabhan, Gravitation, CAMBRIDGE UNIVERSITY PRESS

Bernard Schutz, A First Course in General Relativity, CAMBRIDGE UNIVERSITY PRESS

J. Hartel, Gravity: An introduction to Einstein’s general relativity, CAMBRIDGE UNIVERSITY PRESS