Main Program (August 1418, 2023)
Monday 

09:0009:15  Summer school opening  
09:1510:00  P. Nurowski, Andrzej Trautman (90): contribution to mathematical physics  
10:0010:45  J. Jezierski, Jerzy Kijowski (80): contribution to mathematical physics  
11:0012:15  M. Krššák, Introduction to General Relativity  
12:3013:00  Welcome meeting  
15:0017:00  Guided tour of Prešov  
Tuesday 

09:0010:30  M. Krššák, Introduction to General Relativity  
11:0012:30  J. Kijowski, Variational Principles (On the dangers and ambushes awaiting a theoretical physicist as soon as he starts to apply them)  
15:0015:45  M. Fecko, Hodge star operator in Galilean and Carrollian spacetimes  
15:4516:30  C. Ferrara, Equivalent Gravities: common features and differences  
16:4517:30  D. Saunders, Some remarks on the Fundamental Lepage Equivalent  
17:3018:00  Discussions  
Wednesday 

09:0010:30  M. Krššák, Introduction to General Relativity  
11:0012:30  S. Capozziello, Nonlocal gravity cosmology  
14:30  Field trip at Šariš Castle  


09:0010:30  M. Krššák, Introduction to General Relativity  
11:0012:30  J. Jezierski, On conformal YanoKilling tensors and its applications in General Relativity  
15:0015:45  D. Krupka, The Hilbert variational principle  
15:4516:30  J. Brajerčík, Spherical symmetry: discussion  
18:30  Conference dinner  
Friday 

09:0010:30  M. Krššák, Introduction to General Relativity  
11:0012:30  Discussions, Poster session  
12:30  Summer school conclusion 
 Pawel Nurowski
Andrzej Trautman (90): contribution to mathematical physics  Jacek Jezierski
Jerzy Kijowski (80): contribution to mathematical physics
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 pseudoRiemannian geometry. We then discuss physical motivation for introducing pseudoRiemannian 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 EinsteinHilbert action and consequences of diffeomorphism invariance. We also discuss boundary terms in the gravitational action and introduce the GibbonsHawkingYork 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
Nonlocal gravity cosmology
Recently the socalled nonLocal 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 nonlocal actions including curvature invariants like the Ricci scalar and the GaussBonnet 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 YanoKilling tensors and its applications in GR  Ján Brajerčík
Spherical symmetry: discussion
We consider the action of the rotation group on the set R^{3 }\ {(0,0,0)} and the induced actions on R x R^{3} \{(0,0,0)} and on S^{1} x R^{3} \ {(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 nonstandard method of solving the Einstein equations does not include an assumption on the metric signature. These solutions can be directly interpreted on two topologically nonequivalent spacetimes R x R^{3}\ {(0,0,0)} and on S^{1} x R^{3} \ {(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.
Workshop lectures
 D. Saunders
Some remarks on the Fundamental Lepage Equivalent
In this talk I shall describe how to construct, for any given fibred manifold, a procedure which will process a Lagrangian (of arbitrary order) and produce a global Lepage equivalent satisfying the closure condition: that the Lepage equivalent is closed precisely when the Lagrangian is null. The procedure will depend on the choice of a linear connection on the base of the fibred manifold, and uses a theorem of Ian Anderson.  M. Fecko
Hodge star operator in Galilean and Carrollian spacetimes
Differential forms on Lorentzian spacetimes are a wellestablished topic. On Galilean and Carrollian spacetimes it does not seem to be quite so. The reason may be the absence of the Hodge star operator. However, there are potentially useful analogs of the Hodge star operator in the latter two spacetimes as well. We present two ways to find them. We also show what their application to Galilean and Carrollian electrodynamics yields.  C. Ferrara
Equivalent Gravities: common features and differences
In this lecture, I discuss equivalent representations of gravity in the framework of metricaffine geometries (MAGs). Starting from the formalism that characterizes MAGs, which is the tetrad formalism, I focus my attention to describe the socalled Geometric Trinity of Gravity. Specifically, I consider General Relativity, constructed upon the metric tensor and based on the curvature R; Teleparallel Equivalent of General Relativity, formulated in terms of torsion T and relying on tetrads and spin connection; Symmetric Teleparallel Equivalent of General Relativity, built up on nonmetricity Q, constructed from metric tensor and affine connection. I analyze their dynamical equivalence at three levels: (1) the variational method; (2) the field equations; (3) the solutions. Regarding the second point, I provide a procedure starting from the (generalized) second Bianchi identity and then deriving the field equations. Referring to the third point, I compare spherically symmetric solutions in vacuum recovering the Schwarzschild metric and the Birkhoff theorem in all the approaches. Finally, I consider the meaning that Equivalence Principle acquires in these two specific classes of teleparallel geometries.