Lecture series

• Martin Krššák
  Introduction to General Relativity
  This course is an introduction to general relativity where we start with a concept of spacetime and its description using the Riemannian and pseudo-Riemannian geometry. We then discuss physical motivation for introducing pseudo-Riemannian geometry and introduce Einstein field equations. We derive the solution of Einstein field equations in the spherically symmetric case known as the Schwarzschild solution and explore the motion of observers in this spacetime. We discuss the concept of curvature and coordinate singularities and show how the latter can be removed using suitable coordinates and obtain the Kruskal extension.  We conclude with discussing the action principle for general relativity where we show derivation of the field equations using Einstein-Hilbert action and consequences of diffeomorphism invariance. We also discuss boundary terms in the gravitational action and introduce the Gibbons-Hawking-York boundary term and show some applications in physically interesting situations.

Lectures

• Jerzy Kijowski
  Variational Principles (On the dangers and ambushes awaiting a theoretical physicist as soon as he starts to apply them)
  Standard prejudices and misconceptions about variational principles in physics will be discussed in detail and ways to avoid them will be shown. Application of these ideas in the theory of gravitation will be presented.
• Salvatore Capozziello
  Non-local gravity cosmology
  Recently the so-called non-Local Gravity acquired a lot of interest as an effective field theory towards the full Quantum Gravity. In this talk, we sketch its main features, discussing, in particular, possible infrared effects at astrophysical and cosmological scales. In particular, we focus on general non-local actions including curvature invariants like the Ricci scalar and the Gauss-Bonnet topological invariant, in metric formalism, or the torsion scalar, in teleparallel formalism. In both cases, characteristic lengths emerge at cosmological and astrophysical scales. Furthermore, it is possible to fix the form of the Lagrangian and to study the cosmological evolution considering the existence of Noether symmetries.
• Jacek Jezierski
  On conformal Yano-Killing tensors and its applications in GR
• Ján Brajerčík
  Spherical symmetry: discussion
  We consider the action of the rotation group on the set R³ \ {(0,0,0)} and the induced actions on R x R³ \{(0,0,0)} and on S¹ x R³ \ {(0,0,0)}. The corresponding coordinate formulas are recalled (local and global cases) and the corresponding (global) invariant metric fields are derived. These expressions determine in a standard way the corresponding solution of the Einstein vacuum equations. The non-standard method of solving the Einstein equations does not include an assumption on the metric signature. These solutions can be directly interpreted on two topologically non-equivalent space-times R x R³\ {(0,0,0)} and on S¹ x R³ \ {(0,0,0)}. Possible innovations are discussed.
• Demeter Krupka
  The Hilbert variational principle
  Mathematical theory of the Hilbert variational principle on smooth manifolds is presented. As preliminaries we need the concepts like differential group, differential invariants of the metric field, and their classification. Also, basic variational notions for fibred manifolds are introduced, and specified for the bundles of metrics. Then the Hilbert Lagrangian form is considered, and the corresponding first variational formula is derived. The discussion includes the concept of a conservation law for Lagrangians of this type.