# General Relativity

The aim of this course is to present a selection of advanced topics in classical gravitational dynamics.

After reminder of the basics of General Relativity (GR) and Riemannian geometry, we initially focus on black-holes. We analyse different asymptotically flat solutions (including Schwarzschild, Kerr) and their properties, introducing Penrose diagrams and discussing event horizons and Killing horizons.

Then we focus on different observational signatures of black holes, including photon geodesics, lensing and light rings, and black-hole shadows. We also discuss test fields (scalar, vector and tensor) in the background of a Schwarzschild black-hole, leading on to the study of quasi-normal modes relevant for black-hole perturbation theory and gravitational wave detection. In the case of a Kerr background we discuss superradiace and the Penrose process. This leads on to black-hole thermodynamics. We approach Hawking radiation by analysing the Unruh effect.

After a brief discussion of cosmological solutions, including black-holes in Friedmann-Robertson Walker Lemaitre spacetimes, we focus on gravitational waves (GWs). This is a very topical subject as GWs have recently detected from individual sources such as binary black holes (LIGO-Virgo-Kagra), and there are very strong hints of the detection of stochastic gravitational wave background using Pulsar timing experiments (EPTA, Nanograv… 2023)

Thus we discuss GWs and their sources (for instance bound as well as unbound binary systems), going from the linearised Einstein equations (including the quadrupole formula and the energy in GWs) to determining the GW signal from black-hole binaries to first order in the post-Newtonian expansion. We also discuss the latest results from the LIGO-Virgo GW detectors, and how they can be used to measure for example the Hubble constant, as well as determine the properties of the populations of black-holes observed, and also test general relativity. If there is time, we will discuss the stochastic gravitational wave background and its properties.

The last part of the course touches on more advanced topics : this may include theories of gravity beyond GR; the 3+1 decomposition of GR; and possibly a quantum field theoretic description of general relativity, namely a modern approach to gravity in which the gravitational force is carried by spin-2 gravitons.

A first basic course in GR is assumed, and hence some familiarity with

– tensors (including Riemann, Ricci etc);

– covariant and Lie derivatives

– Einstein equations

– Schwarzschild Black Hole solution

– geodesic equation

– tests of GR in the solar system.

*• Overview*

• *Brief reminders *(including Einstein equations+EH action, symmetries, Killing vectors, Penrose diagrams etc.)*• Some useful geometrical aspects (*including hypersurfaces, normals, extrinsic curvature; n-forms, exterior derivs..)

*• Black holes*

– Schwarzschild BH: Finkelstein diagram, white holes, Kruskal extension, Einstein-Rosen bridge. Geodesics, light rings and blackhole shadows. Instability of circular null geodesics at the light ring, and its time scale. Damping time of BH perturbations induced by a massless scalar field: quasinormal modes.

– Kerr BH (event horizons, ergoregion, Killing horizons, mass/charge/spin,); geodesics (Carter constant etc). Superradiance and BH thermodynamics.

– QFT in curved spacetime: the Unruh effect and BH evaporation.

*• Cosmological solutions*

– FRLW, and also black hole solutions in FRLW (e.g. Kerr-deS; Sch-deS, Sch-FRLW...)

*• Gravitational waves:**– *Linearized Einstein equations, quadrupole formula, energy in GWs

– GW signal from a binary pair of (bound and un-bound) BHs Post-Newtonian expansion

– BH perturbation theory

– From GW data to properties of the source + measurement of cosmological parameters – Hellings & Downs curve (observed for first time in June 2023 by pulsar timing experiments, *nHz *)

*• 3+1 decomposition of General Relativity **• Beyond general relativity: Modified gravity*

• Lecture Notes on General Relativity, M.Blau,

http://www.blau.itp.unibe.ch/GRLecturenotes.html

• Spacetime and geometry, S.Carroll, (2014, Pearson publishers).

• General Relativity, R.Wald, (1984, The University of Chicago Press)

• Gravitation and Cosmology : Principles and applications of the general theory of relativity, S.Weinberg (1972, Wiley)

• Cosmology, S.Weinberg, (2008, Oxford University Press)

• Gravitational Waves : Volume 1 : Theory and Experiments, M.Maggiore, (Oxford University Press, 2008).

• Gravitational Waves : Volume 2 : Astrophysics and cosmology, M.Maggiore, (Oxford University Press, 2018).

• Gravity. Newtonian, post-newtonian, relativistic, E.Poisson and C.M.Will, (Cambridge University Press, 2014.)