## PHY 443: Black Hole Physics

3 credits | Prerequisites: PHY 430 for Major; PHY 230 for Minor

### Course rationale

Black holes are fascinating objects described by Einstein’s theory of General Relativity. One defining property of black holes is the existence of an event horizon, from where even light can not escape. Inside the event horizon, there is a special point, generally termed singularity where curvature diverges. Quantum mechanical analysis reveals that black holes behave like a thermal object, and the thermal properties are governed by Black Hole Thermodynamics. This course will describe the basics of black hole physics.

### Course content

Review of Differential Geometry: manifold, vectors, dual vectors, tensors, covariant differentiation, geodesic, Lie differentiation, Killing vector, curvature, geodesic deviation, Fermi normal coordinates, differential forms, tetrad formalism, Cartan’s structure equation; Geodesic Congruences: energy conditions, kinematics of deformable medium-expansion, shear, rotation, congruence of timelike geodesics-kinematics, Frobenius’ theorem, Raychaudhuri’s equation, focusing theorem, congruence of null geodesics-kinematics, Frobenius’ theorem, Raychaudhuri’s equation, focusing theorem; Geometry of Hypersurfaces: defining equations, normal vector, induced metric, integration on hypersurface, Gauss-Stokes theorem, tangent tensor fields, intrinsic covariant derivatives, extrinsic curvature, Gauss-Codazzi equations, initial value problem, junction conditions for thin shells, gravitational collapse; Lagrangian and Hamiltonian Formulation of General Relativity: EinsteinHilbert action, variation of the Hilbert term, variation of the boundary term, variation of the matter action, Einstein equation, Bianchi identities, 3+1 decomposition of the metric, field theory, foliation of the boundary, gravitational Hamiltonian, variation of the Hamiltonian, Hamilton’s equations, the Hamiltonian definition of mass and angular momentum, Komar formulae, Bondi-Sachs mass, transfer of mass and angular momentum; Reissner Nordstrom Black Hole: derivation of the metric-solution of Einstein and Maxwell equations, nature of spacetime, geodesic in Reissner-Nordstrom space-time-null and time like geodesics, the motion of charged particle; Rotating Black Holes: formation of rotating black hole, derivation of the Kerr metric, reference frame, event horizon, equation of motion, first integrals, motion in the equatorial plane-null and timelike geodesics, General Properties of Black Holes: asymptotic properties of Minkowski space-time, Penrose-Carter conformal diagram, event horizon, Penrose theorem, optical scalars, focusing theorem, Hawking area theorem, apparent horizon, trapped surface, singularities in general relativity, singularity theorems, cosmic censorship conjecture; Stationary Black Holes: no hair theorem, stationary space-time, static space-time, Penrose process, Killing horizon and properties, surface gravity, black hole uniqueness theorem; Black Hole Thermodynamics: black holes and thermodynamics, mass formula, four laws of black hole thermodynamics, generalized second law, entropy as Noether charge, black hole as a thermodynamic system, Euclidean approach, calculation of the Euclidean action, conical singularity method, statistical mechanics of black hole thermodynamics.

### Course objectives

- Deepen the understanding of differential geometry required for general relativity.
- Understand the Lagrangian and Hamiltonian formulation of general relativity.
- Analyze different kinds of black hole geometries.
- Understand important theorems related to black holes.
- Familiarize with black hole thermodynamics.

### References

- Black Hole Physics: Basic Concepts and New Developments by Valeri P. Frolov, and Igor D. Novikov
- The Mathematical Theory of Black Holes by Subrahmanyan Chandrasekhar
- The Large Scale Structure of Space-time by Stephen Hawking, and G. F. R. Ellis
- A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics (1st edition) by Eric Poisson
- General Relativity by Robert Wald
- Spacetime and Geometry: An Introduction to General Relativity (1st edition) by Sean M. Carroll
- Gravitation by Charles W. Misner, Kip S. Thorne, and John A. Wheeler
- Gravitation: Foundations and Frontiers (1st edition) by T. Padmanabhan
- Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity by Steven Weinberg