Università Roma Tre

§I.Relativistic Field Theory

The Poincaré Group. Symmetries: global vs local. Noether's first and second theorems and conservation laws. The canonical stress-energy and angular momentum tensors. Improvements. Belinfante's argument and symmetric energy-momentum tensor. Local symmetries and conserved quantities.

§II.Gravity as a relativistic field theory

Particles and fields in Special Relativity. Irreps of the Poincaré group: Wigner's induced representation method. Massless particles: ISO(D-2) little group and gauge invariance. From relativistic massless spin-2 particles to full GR. Fierz-Pauli quadratic Lagrangian. Nöther method and non-linear completions. Nöther's construction of Yang-Mills Lagrangian. The transverse-traceless gravitational cubic vertex. Weinberg's Equivalence Principle from relativistic invariance of the S matrix. Spin and the sign of static forces.

§III.Elements of differential geometry

Topological spaces. Manifolds. Diffeomorphisms. Tangent spaces and vectors. Coordinate basis. Derivative operators on manifolds. Levi-Civita connection. Torsion. Differential forms: definition, wedge product, interior and exterior derivatives, Hodge dual. Lie derivative of forms and Cartan's formula. Yang-Mills theory in the language of forms. Weyl tensor. Riemann and Weyl tensors in various dimensions: counting components for irreps of GL(D). Conformal transformations of the metric tensor. Conformally flat spaces. Conformally coupled scalar fields.

§IV. The Cartan-Weyl formulation of GR and Fermionic couplings

Local inertial frames. The frame field and its relation to the metric field. Local Lorentz transformations. The spin connection. The vielbein postulate. Torsion constraint and second-order formulation. The contorsion tensor. Local Lorentz curvature. Gravity as a gauge theory of the Poincaré algebra. Connection one-forms on the Poincar\'e algebra. Local Poincar\'e transformations. Torsion and curvature over the Poincar\'e algebra. First-order formulation and Cartan-Weyl's action. Relation between gauge transformations and diffeomorphisms. Spinors on curved manifolds. Minimally coupled Fermionic matter. Dirac Lagrangian.

§V. Maximally symmetric spaces

Homogeneous and isotropic spaces. Characterisation of maximally symmetric spaces: curvature constant and signature. MSS as vacuum solutions to the EH equations with cosmological constant. Construction from embedding in (D+1) pseudo-Lorentzian spaces: metric and Christoffel coefficients.

§VI. The Schwarzschild black hole

Spherically symmetric spaces. The Schwarzschild solution. Birkhoff's theorem. Singularities, definitions and criteria: curvature singularites and geodesic incompleteness. Free-fall towards the horizon. The tortoise coordinate. Extension of a space-time. Eddington-Finkelstein coordinates. Event horizons, black holes and white holes. Kruskal-Szekeres coordinates. Maximal extension of the Schwarzschild solution. Kruskal's diagram and eternal black holes. (A)dS-Schwarzschild space-time.

§VII. More general black holes

Conformal diagrams. Event horizons. Reissner-Nordström and Kerr black holes. Black hole thermodynamics.

§VII. Gravitational energy

Conserved quantities in gauge theories: the example of Yang-Mills theory. Covariant conservation and ordinary conservation. Einstein-Hilbert equations for asymptotically flat metrics. Candidate for gravitational energy-momentum tensor. The superpotential. ADM energy and momentum. Example: ADM energy of the Schwarzschild solution. The positive-energy theorem (without proof). Generic background with Killing vectors. Quadrupole radiation.

§VIII. Asymptotic symmetries

General notion of asymptotic symmetry group. The example of Maxwell's theory in flat space. Covariant phase space formalism. Asymptotically flat spacetime and Bondi-van der Burg-Metzner-Sachs supertranslations. Applications: soft theorems and memory effects.

Note: some topics may be assigned as homework problems, as an alternative to the oral exam

-Carroll S, Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley 2014/Cambridge University Press, 2019)
-Weinberg S, Gravitation and Cosmology - principles and applications of the general theory of relativity, (John Wiley \& Sons, 1972)
-Wald R, General Relativity (The University of Chicago Press, 1984)