We present the separation of the Teukolsky master equation for the test field of arbitrary spin on the background of the rotating C-metric. We also summarize and simplify some known results about Debye potentials of these fields on type D background. The equation for the Debye potential is also separated.
Solving for the Debye potential of the electromagnetic field we show that on the extremely rotating C-metric no magnetic field can penetrate through the outer black hole horizon -- we thus recover the Meissner effect for the C-metric.
The optical matrix of type III in 6D has been derived using Bianchi identities. It allows non-zero shear in an arbitrary dimension greater than four -- contrary to the famous Goldberg-Sachs theorem valid in 4D.
We derive a generalization of the Bonnor-Melvin solution where the cylindrically symmetric magnetic field is homogeneous. This requires a specific, non-vanishing cosmological constant. We investigate the properties of the spacetime and discover who discovered it.
We study a class of higher dimensional spacetimes that lead to a separable Klein-Gordon equation. Motivated by Carter's work in four dimensions, we introduce an ansatz for the separable metric in higher dimensions and find solutions of the Klein-Gordon equation. For such a metric we solve the Einstein equations and regain the Kerr–NUT–(A)dS spacetime and the Einstein-Kahler metric of a Euclidean signature. We construct a warped geometry of two Klein-Gordon separable spaces with a properly chosen warped factor. We show that the corresponding Klein-Gordon equation can also be solved by separation of variables. By solving the Einstein equations for the warped geometry we find new solutions.
It is well-known that the black hole horizon obeys the equation of motion for a viscous fluid. A statistical model of this fluid can help us obtain some insight into the thorny problems of the microscopic black hole degrees of freedom and black hole entropy. In this talk, we shall take this fluid to be of a physical origin and construct a theory of fluctuations for it. In particular, I shall show that the Langevin equation governing the energy transported from outside into the horizon-fluid corresponds to the Raychaudhuri equation for the null congruences on the black hole horizon. We shall also briefly outline a method, that uses the Kubo formula to compute the coefficient of bulk viscosity of the horizon fluid. We shall show that this comes out to be negative due to the teleological nature of the black hole event horizon. Finally, I shall briefly outline how it is possible to have a statistical mechanical understanding of the negative specific heat of black holes in a similar manner.
Partial differential equations are formulated and some of them quite well understood for single-valued (i.e. scalar) or vector-valued functions. The heat equation or the Laplace equation are examples of equations for scalar and the Maxwell equations are examples of those for vector fields. Their formulations on manifolds or on bundles over manifolds are known as well. PDEs for maps with values in specific Banach bundles were studied by Mishchenko and Fomenko (MSU) and their colleagues already in the seventies. They tried to generalize the Atiyah-Singer index theorem. In mathematics, their results became a part of so called analytic and geometric K-theory.
In Hodge theory one focuses on the systems of PDE's, more precisely, on the so called complexes of PDE's that are "elliptic". One of the important results of Hodge theory is the existence of an isomorphism of the cohomology of a complex with the kernel of generalized Laplacian associated uniquely to the complex. In the case of Banach bundles, the Hodge theory encounters crucial problems already at a topological level. The cohomology of such complexes may be an infinite dimensional TVS (which seems reasonable), but the induced topology on the cohomology does not separate points. Otherwise said, there exist convergent sequences whose limits can be any point of the vector space. Especially, it seems that no reasonable classical analysis on them is possible. In the current Physics, there is no known canonical way for choosing such a topology. There are no equations of physics which would determine it. Quite worrying is this situation in the case of the Becchi-Rouet-Stora-Tyutin (BRST) theory - a quantum field theory for constrained systems which gained a prominent role among theories on quantizing the gauges of potential type.
We explain basic notions briefly, give examples of the above-mentioned difficulties and show for which complexes one can get around them. Namely, the Hodge theory can be established for elliptic complexes on finitely generated projective K-bundles. One may imagine these bundles as bundles of state spaces whose fibers are acted upon by an algebra of observables in a way that they are finite with respect to this action. The projectivity means that the fibers are not too warped with respect to the observable algebra.
Cosmic microwave background anisotropies are given by radiation energy density perturbations at the time of recombination and effect influencing photons during their way to the observer. Both can be explained by perturbation theory in general relativity applied to case of early universe. Basics of this theory will be summarized. Equations governing evolution of cosmological perturbations generated during inflation, effects influencing photons after last scattering and computation of CMB angular power spectrum will be explained. Angular power spectrum of CMB is nonzero only if perturbations are non-Gaussian. This non-Gaussianity may originate in some inflationary models. In the last part I will talk about effect of presence of solid matter on CMB anisotropies.
Since its formulation in the mid seventies, the standard model of particle physics has been very successful in describing the known interactions between fundamental particles and predicting the outcome of many high-energy physics experiments. However, the standard model is widely believed to be an incomplete theory. For example, it provides no explanation for the origin and value of the fermion masses, the existence of three particle generations, and the observed strengths of the fundamental interactions. The standard model does not include gravity and fails to explain why the latter is much weaker than all other forces, what is become to be known as the "hierarchy problem".
If large extra dimensions exist, the hierarchy problem may be solved by lowering the Planck scale to the electroweak scale. If that is the case, microscopic black holes may be produced in particle collisions at this energy scale. Searches for black hole production at the Large Hadron Collider have so far produced null results. The absence of observed black hole events in experimental data can be used to constrain the value of the Planck scale and other parameters related to microscopic black hole formation in models with large extra dimensions. In this talk we present the results of a recent analysis based on data from the Compact Muon Solenoid detector at the LHC. We derive new lower bounds on the Planck scale and the minimum allowed mass for black hole formation in this class of models.