### Julian Adamek : Relativistic Simulations for Cosmology

Cosmological N-body simulations are one of the most versatile tools for studying the evolution of large-scale structure in the Universe. While the Newtonian limit of general relativity can be used for most purposes within the basic LCDM model, the true nature of the dark components (dark matter and dark energy) is unknown and may ultimately require a relativistic description. Also the neutrinos from the standard model are relativistic for most of the cosmic history if they have a mass within the range allowed by cosmological and laboratory constraints. In this course I will introduce a framework for relativistic N-body simulations that can treat any relativistic degrees of freedom self-consistently. Furthermore, I will discuss the important aspect of how the simulation data are mapped to observables by constructing the past light cone of an observation event. A metric-based approach is presented that is also suitable for treating a wide range of models beyond LCDM.

Suggested reading:

* Chapters 1 and 2 of Baumgarte & Shapiro, "NUMERICAL RELATIVITY - Solving Einstein's Equations on the Computer"

* Sections 2 and 3 of "Relativistic N-body simulations with massive neutrinos" (Adamek, Durrer & Kunz, arXiv:1707.06938)

### Vladimir Karas : Relativistic effects in spectra and polarization from black hole accretion disks

Astrophysical black holes are often characterized by only two parameters, namely, mass and angular momentum. However, cosmic black holes are not completely isolated. Instead, they interact with their gaseous and stellar environment, and so the astrophysically realistic models may require additional information to describe the spacetime metric and to determine the state of the surrounding gaseous environment. We will review a fruitful approach to study variety of electromagnetic signatures from accretion disks in strong gravity regime. Transfer functions can be introduced, pre-computed, and then employed to generate model spectra and to fit them to the electromagnetic signal in X-rays. We have been developing this approach to analyse spectra and light curves and to predict the polarimetric properties. Other groups adopt different schemes which we will briefly outline, too. Models can be finally tested with the current and upcoming observations. Some of recent results challenge the expectations based on standard accretion scenarios.

### Mikołaj Korzyński : Redshift drift, position drift and parallax in general relativity

I will discuss the redshift and the position drifts in general relativity, i.e. the temporal variations of the redshift and the position on the sky of a light source, as registered by an arbitrary observer. With the recent advancements in astrometry, the drifts of distant sources are likely to become important observables in cosmology in the near future. In my lecture I will present the derivation of exact relativistic formulas for the drifts. I will show how the drifts may be expressed in terms of the kinematical variables characterizing the motions of the source and the observer, i.e. their momentary 4-velocities and 4-accelerations, as well as the spacetime curvature along the line of sight. The formulas we derive are completly general and involve automatically all possible GR effects. They may be regarded as the counterpart of the Sachs optical equations for temporal variations of the standard observables. I will discuss their physical consequences and their possible applications to the gravitational lensing theory, cosmology and pulsar timing. Building on the same formalism I will also consider the trigonometric parallax effect in general relativity, and show how we can measure the mass density along the line of sight by comparing the parallax distance and the angular diameter distance to a single source.

Literature:

"Optical drift effects in general relativity", M. Korzyński, J. Kopiński, Journal of Cosmology and Astroparticle Physics 03 (2018) 012

"Geometric optics in general relativity using bilocal operators", M. Grasso, M. Korzyński, J. Serbenta, Phys. Rev. D 99 (2019) no.6, 064038

"Geometric optics in relativistic cosmology: new formulation and a new observable", M. Korzyński, E. Villa, Phys. Rev. D 101 (2020) no.6, 063506

Prerequisites:

Basic general relativity: worldlines, null and timelike geodesics, parallel transport, curvature tensor, Einstein equations

Basic cosmology: Friedmann equations, cosmological distances, redshift

Somewhat more advanced topics in general relativity: geodesic deviation equation, optical Sachs equations (recommended but not necessary, I will introduce this material during the course)

### Patryk Mach : Matter around black holes - self-gravitating systems

Matter around black holes is usually modeled neglecting its self-gravity, i.e., assuming a fixed background metric. In these lectures I will focus on the opposite case, in which the self-gravity of matter is taken into account. Simple general-relativistic systems in which the effects of self-gravity can be studied include axially symmetric configurations - stationary disks (or tori) around black holes - or spherically symmetric steady accretion flows. I will consider mostly hydrodynamical or magentohydrodynamical models. The interplay between the structure of spacetime affected by the self-gravity of matter and the motion of matter around black holes leads to several interesting phenomena, which I will shortly discuss: bifurcations of solutions, occurrence of various ergoregions, changes in the phase-space of geodesic orbits.

**Olivier Sarbach :** Particle motion and dynamics of a Vlasov gas in the exterior of a Kerr black hole

These lectures start with a discussion of some of the properties of the most important black hole solution in general relativity and relativistic astrophysics: the Kerr black hole. In particular, the notions of static and stationary observers, ergospheres, horizons, causal structure and the motion of free-falling massive and massless particles will be reviewed. Next, some tools are introduced to understand the geometry of the cotangent bundle associated with a (generic) curved spacetime (M,g). Based on these tools, a manifestly covariant theory is derived describing a relativistic Vlasov gas, that is, a gas consisting of collisionless particles propagating in (M,g). In the final part of the lectures, this theory is applied to the study of the dynamics of a Vlasov gas consisting of particles which follow spatially bound timelike geodesics in the exterior of a Kerr black hole. To this purpose, generalized action-angle variables are introduced in which the geodesic flow simplify considerably and the relativistic Vlasov equation can be solved analytically. Based on this representation, it is shown that - even though it is collisionless - the gas undergoes a relaxation process and settles down to a stationary, axisymmetric configuration. The underlying mechanism for this effect, which is due to phase mixing, will be explained.

### Marek Szczepańczyk : The quest to detect gravitational waves

Gravitational-Wave Astrophysics, a newly established field, is an exciting frontier for scientific discovery. It opens up possibilities to investigate phenomena that were previously inaccessible through time-domain astronomy. Notably, among the nearly hundred gravitational-wave detections, the first binary black hole merger and an intermediate-mass black hole have been remarkable observations that challenged our understanding of the Universe. The fourth observing run of LIGO, Virgo, and KAGRA presents great opportunities for discoveries. During the course, I will review a range of topics about the Gravitational-Wave Astrophysics. I will outline the gravitational-wave detectors, their operation, noise sources, and future designs. During the lectures, I will give a brief overview of the gravitational-wave data analysis and focus on the model-independent methods. I will discuss the compact binary sources detected centering around the exceptional events. Finally, I will talk about the current observing run of the gravitational-wave detectors.