PH-721 General Relativity and Cosmology

1. Principles of special relativity and relativistic mechanics: the addition of velocities and Michelson-Morley experiment, Einstein’s resolution and its consequences, space-time, time dilation and twin paradox, Lorentz boosts, four vectors, special relativistic kinematics and dynamics, variational principle for free particle motion, light rays, observers and observations.
2. The curved space time of general relativity: testing of equality of gravitational and inertial mass, equivalence principle, geodesics, metric coordinate transformations, Christoffel symbols, geodesics and coordinate transformations.
3. The physics and geometry of geodesics: geodesic equation from variational principle, the Newtonian limit, the gravitational red-shift, locally inertial and Riemann normal coordinates, affine and non-affine parameterization.
4. Tensor algebra and tensor analysis: from Einstein equivalence principle to the principle of general covariance, tensor algebra, tensor density, the covariant derivative of vector fields, extension of covariant derivative and other tensor fields, main properties of covariant derivatives, the principle of minimal coupling, covariant differentiation along a curve, parallel transport and geodesics, generalizations.
5. Physics in a gravitational field: particle mechanics and electrodynamics in a gravitational field, conserved quantities from covariantly conserved currents and tensors.
6. Lie derivatives, symmetries and Killing vectors: symmetries of a metric, the Lie derivative for scalars, vector fields, tensor fields, metric and Killing vectors.
7. Curvature: the Riemann curvature tensor, intrinsic geometry, parallel transport, geodesic derivative equations, Einstein equations, weak field limit, Bianchi identities, cosmological constant, Weyl tensor, Einstein-Hilbert action, the matter Lagrangian and consequence of the variational principle.
8. Tests of general relativity: black holes, relativistic star models, cosmological models, early stages of evolution of the universe, gravitational waves.

Recommended texts:

• Gravity: an Introduction to Einstein’s General Relativity, J.B.Hartle (Addison-Wesley 2003).
• Space-Time and Geometry: An Introduction to General Relativity, Sean Caroll (Addison-Wesley 2004).
• Gravitation and Cosmology, S. Weinberg (Wiley, 1972).