Università Roma Tre

§I. Introduction: inertia and covariance

Galilean Relativity and Special Relativity. The Equivalence Principle. Motivations for general covariance. Local inertial frames.

§II. Dynamical spacetime: basics of General Relativity

Curvilinear coordinates. Vectors and tensors under general coordinate transformations. Parallel transport and Christoffel symbols. Covariant derivatives and metric compatibility. Covariantly conserved vectors and conserved vector densities. Transformation of the Christoffel symbols. Ricci's theorem. Geodesics and their Newtonian limit. Time-like, null and space-like curves. Normal coordinates and local inertial frames. Covariant derivation along curves. Riemann curvature tensor. Algebraic properties. Bianchi identities. Local inertial frames. Characterisation of flat spaces. Geodesic deviation. Einstein-Hilbert action and equations of motion. The Palatini identity. Non-dynamical sources: contracted Bianchi identity, matter energy-momentum tensor and covariant conservation. Minimal coupling to scalar fields and to Maxwell fields.

§III. Linear approximation and gravitational waves

Motivations. Weak fluctuations over flat space-time: linearised Riemann tensor and its abelian gauge invariance. Equations of motion: de Donder gauge and gravitational waves. Comparing to Maxwell's theory.

§IV. Isometries and maximally symmetric spaces

Symmetries of tensors: form-invariance. Killing equations. Integrability condition and maximal number of isometries. The Lie derivative. Killing vectors and conservation laws. The example of Minkowski space. Translational isometries.

§V. Basics of the Schwarschild solution

Schwarzschild metric. Spherical symmetry and Birkhoff's theorem (without proof). Killing vectors of Schwarzschild's metric. Gravitational redshift.

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-Dirac P A M General Theory of Relativity (Princeton University Press, 1996)
-Hartle S Gravity: An Introduction to Einstein's General Relativity (Cambridge University Press, 2021)
-Rovelli C, General Relativity - the Essentials, (Cambridge University Press, 2021).
-Weinberg S, Gravitation and Cosmology - principles and applications of the general theory of relativity, (John Wiley \& Sons, 1972).