Research

Here you can find scientific topics which I am interested in, including results of my own research.

Meissner effect for axially symmetric charged black holes

Norman Gürlebeck, Martin Scholtz     |     Physical Review D 97 084042, 2018

In our previous work [N. Gürlebeck and M. Scholtz, Phys. Rev. D 95, 064010 (2017)], we have shown that electric and magnetic fields are expelled from the horizons of extremal, stationary and axially symmetric uncharged black holes; this is called the Meissner effect for black holes. Here, we generalize this result in several directions. First, we allow that the black hole carries charge, which requires a generalization of the definition of the Meissner effect. Next, we introduce the notion of almost isolated horizons, which is weaker than the usual notion of isolated horizons, since the geometry of the former is not necessarily completely time independent. Moreover, we allow the horizon to be pierced by strings, thereby violating the usual assumption on the spherical topology made in the definition of the weakly isolated horizon. Finally, we spell out in detail all assumptions entering the proof and show that the Meissner effect is an inherent property of black holes even in full nonlinear theory.

On the implications of the Bekenstein bound for black hole evaporation

Giovanni Acquaviva, Alfredo Iorio, Martin Scholtz     |     Annals of Physics 387 317-333, 2017

We elaborate on the possible impact of the Bekenstein bound on the unitarity of black hole evaporation. As such maximal bound on the entropy of any system may be regarded as due to the existence of entities more elementary than the ordinary ones, and since at our energy scales such fundamental degrees of freedom must organize themselves into quantum fields acting on classical space- times, we then propose that both, quantum fields and geometries, are emergent phenomena stemming from the same underlying dynamics. We investigate the kinematical and model independent effects of this ‘‘quasi-particle picture’’ on black hole evaporation within a simple toy model, that we construct. We conclude that the information associated to the quantum fields in the ‘‘phase’’ before the formation of the black hole is, in general, only par- tially recovered in the ‘‘phase’’ after the black hole has evaporated. This information loss is shown to be due to the entanglement between fields and geometry. Such modifications of the Page curve should be regarded as common features of any theory of quantum gravity.

Kerr-Newman black hole in the formalism of isolated horizons

Martin Scholtz, Aleš Flandera, Norman Gürlebeck     |     Phys. Rev. D 96 064024, 2017

The near horizon geometry of general black holes in equilibrium can be conveniently characterized in the formalism of weakly isolated horizons in the form of the Bondi-like expansions (Krishnan B, Class.\ Quantum Grav.\ 29, 205006, 2012). While the intrinsic geometry of the Kerr-Newman black hole has been extensively investigated in the weakly isolated horizon framework, the off-horizon description in the Bondi-like system employed by Krishnan has not been studied. We extend Krishnan's work by explicit, non-perturbative construction of the Bondi-like tetrad in the full Kerr-Newman spacetime. Namely, we construct the Bondi-like tetrad which is parallelly propagated along a nontwisting null geodesic congruence transversal to the horizon and provide all Newman-Penrose scalars associated with this tetrad. This work completes the description of the Kerr-Newman spacetime in the formalism of weakly isolated horizons and is a starting point for the investigation of deformed black holes.

Meissner effect for weakly isolated horizons

Norman Guerlebeck, Martin Scholtz     |     Physical Review D 95,2017

Black holes are important astrophysical objects describing an end state of stellar evolution, which are observed frequently. There are theoretical predictions that Kerr black holes with high spins expel magnetic fields. However, Kerr black holes are pure vacuum solutions, which do not include accretion disks, and additionally previous investigations are mainly limited to weak magnetic fields. We prove for the first time in full general relativity that generic rapidly spinning black holes including those deformed by accretion disks still expel even strong magnetic fields. Analogously to a similar property of superconductors, this is called the Meissner effect.

On the Bondi mass of Einstein-Klein-Gordon spacetimes

Martin Scholtz, Lukas Holka     |     General Relativity and Gravitation 46,2014

In this paper we calculate the Bondi mass of asymptotically flat spacetimes with interacting electromagnetic and scalar fields. The system of coupled Einstein–Maxwell–Klein–Gordon equations is investigated and corresponding field equations are written in the spinor form and in the Newman–Penrose formalism. Asymptotically flat solution of the resulting system is found near null infinity. Finally we use the asymptotic twistor equation to find the Bondi mass of the spacetime and derive the Bondi mass-loss formula. We compare the results with our previous work (Bičák et al. in Class Quantum Gravity 27(17):175011, 2010) and show that, unlike the conformal scalar field, the (Maxwell–)Klein–Gordon field has negatively semi-definite mass-loss formula.

On the Existence and Properties of Helically Symmetric Systems

Jiri Bičák, Martin Scholtz, Paul Tod     |     In proceedings Relativity and gravitation, Springer 2014

By an argument similar to that of Gibbons and Stewart, but in a different coordinate system and less restrictive gauge, we show that any weakly asymptotically simple, analytic vacuum or electro-vacuum spacetime which is periodic in time is necessarily stationary. We generalized this theorem to the presence of scalar fields and, among other results, derived new expressions for the Bondi mass in this case. Here we summarize these results and also briefly discuss some new considerations concerning the periodic solutions within linearized theory of gravity.

Helical symmetry, spinors and periodic solutions in general relativity

Martin Scholtz     |     Lambert Academic Publishing, 2012

This book is an extended version of the thesis defended by the author in 2011. It contains an introduction to the formalism of two-component spinors in general relativity and conformal techniques related to asymptotically flat spacetimes. These techniques are then applied to the problem of the non-existence of asympotically flat periodic solutions of Einstein's equations. It is shown that such solutions in fact do not exist which, roughly speaking, means that gravitational field of isolated gravitating systems cannot be periodic. Proof of this statement arose with the collaboration of the author, his former supervisor prof. Jiri Bičák from Charles University in Prague and prof. Paul Tod from University of Oxford. In the book, the notion of helical symmetry is introduced as a special kind of periodicity and new helically symmetric solutions in electrodynamics and linearized gravity are presented. In the second part, the formalism of 2-spinors is introduced together with the Newman-Penrose formalism. Finally, in the third part, we present the proof of the non-existence of periodic solutions of Einstein's equations which was published in Classical and Quantum Gravity journal in 2010.

ISBN: 978-3-659-28580-6     |

Periodic solutions of Einstein's equations (Czech language)

Martin Scholtz     |     Czechoslovak journal for physics3 (62), 2012

In the Newtonian theory of gravity, exact solutionsof the two-body problem are well known, while in general relativity we expect that periodic solutions cannot describe isolated systems. In the presented paper we give a rigorous proof of the non-existence of asymptotically flat periodic solutions to Einstein’s equations. Moreover, we introduce several mathematical methods used in the proof. The paper is written in the Czech language.

ISSN:0009-0700     |     Manuscript

On asymptotically flat solutions of Einstein's equations periodic in time: II. Spacetimes with scalar-field sources

Jiri Bičák, Martin Scholtz, Paul Tod     |     Classical and quantum gravity 27 (5), 2010

We extend the work in our earlier paper (Bičák J et al 2010 Class. Quantum Grav. 27 055007) to show that time-periodic, asymptotically flat solutions of the Einstein equations analytic at ${\cal I}$, whose source is one of a range of scalar-field models, are necessarily stationary. We also show that, for some of these scalar-field sources, in stationary, asymptotically flat solutions analytic at $\mathcal{I}$, the scalar field necessarily inherits the symmetry. To prove these results we investigate miscellaneous properties of massless and conformal scalar fields coupled to gravity, in particular Bondi mass and its loss.

On asymptotically flat solutions of Einstein's equations periodic in time: I. Vacuum and electrovacuum solutions

Jiri Bičák, Martin Scholtz, Paul Tod     |     Classical and quantum gravity 27 (5), 2010

By an argument similar to that of Gibbons and Stewart (1984 Absence of asymptotically flat solutions of Einstein's equations which are periodic and empty near infinity Classical General Relativity (London, 1983) ed W Bonnor, J N Islam and M A H Callum (Cambridge: Cambridge University Press) pp 77-94), but in a different coordinate system and less restrictive gauge, we show that any weakly asymptotically simple, analytic vacuum or electrovacuum solutions of the Einstein equations which are periodic in time are necessarily stationary.

Turbulence and quantum field theory

In my diploma thesis, I was studying problems related to fully developed turbulence under supervision of RNDr. Marián Jurčišin Ph.D. In particular, we were examining the influence of anisotropy on stability of the scaling regimes.

What is interesting about these problems is that they can be analysed employing the methods of quantum field theory and the technique of renormalization group. Any problem of stochastic dynamics can be reformulated as the problem of quantum field theory with the number of fields doubled (new fields are considered non-physical). Then one can construct correlation functions (Green functions) using the standard technique of Feynman diagrams and remove divergencies using the renormalization.

$G_0[J] = \frac{1}{N} \int e^{i\,S_0[\phi] + i\int J\,\phi\,\mathrm{d}^4 x}\,{\mathcal D}\phi$

Numerical investigation of scaling regimes in a model of an anisotropically advected vector field

E. Jurčišinová, M. Jurčišin, R. Remecký, M. Scholtz     |     Physics of Particle and Nuclei Let. 3 (5) 2008

Influence of strong uniaxial small-scale anisotropy on the stability of inertial-range scaling regimes in a model of a passive transverse vector field advected by an incompressible turbulent flow is investigated by means of the field theoretic renormalization group. Turbulent fluctuations of the velocity field are taken to have the Gaussian statistics with zero mean and defined noise with finite correlations in time. It is shown that stability of the inertial-range scaling regimes in the three-dimensional case is not destroyed by anisotropy, but the corresponding stability of the two-dimensional system can be corrupted by the presence of anisotropy. A borderline dimension $d_c$ below which the stability of the scaling regime is not present is calculated as a function of anisotropy parameters.

Combined Effects of Small Scale Anisotropy and Compressibility on Anomalous Scaling of a Passive Scalar

E. Jurčišinová, M. Jurčišin, R. Remecký, M. Scholtz     |     Int. Journal of Modern Physics B 21 (22) 2008

The model of a passive scalar field advected by the compressible Gaussian strongly anisotropic velocity field with the covariance $\propto \delta(t-t')|{\bf x}-{\bf x}'|^{2\epsilon}$ studied using the field theoretic renormalization group and the operator product expansion. The inertial-range stability of the corresponding scaling regime is established. The anomalous scaling of the single-time structure functions is studied and the corresponding anomalous exponents are calculated. Their dependence on the compressibility parameter and anisotropy parameters is analyzed. It is shown that, as in the isotropic case, the presence of compressibility leads to the decrease of the critical dimensions of the important composite operators, i.e., the anomalous scaling is more pronounced in the compressible systems. All calculations are done to the first order in $\epsilon$.

Influence of anisotropy and compressibility on anomalous scaling of a passive scalar field

Eva Jurčišinová, Marián Jurčišin, Richard Remecký, Martin Scholtz     |     talk presented by R. Remecký

Influence of uniaxial small-scale anisotropy and compressibility on the stability of scaling regime and on the anomalous scaling of structure functions of a scalar field is investigated in the model of a passive scalar field advected by the compressible Gaussian strongly anisotropic velocity field with the covariance $\propto \delta(t-t^{\prime})|{\bf x}-{\bf x^{\prime}}|^{2\epsilon}$ by using the field theoretic renormalization group and the operator product expansion. The inertial-range stability of the corresponding scaling regime is established. The anomalous scaling of the single-time structure functions is studied and the corresponding anomalous exponents are calculated. Their dependence on the compressibility parameter and anisotropy parameters is analyzed. It is shown that the presence of compressibility leads to the decreasing of the critical dimensions of the important composite operators, i.e., the anomalous scaling is more pronounced in the compressible systems. This result is demonstrated for the structure function of the third order. All calculations are done to the first order in $\epsilon$.

Influence of weak anisotropy on scaling regimes in a model of advected vector field

Eva Jurčišinová, Marián Jurčišin, Richard Remecký, Martin Scholtz     |     talk presented by M. Scholtz

Influence of weak uniaxial small-scale anisotropy on the stability of inertial-range scaling regimes in a model of a passive transverse vector field advected by an incompressible turbulent flow is investigated by means of the field theoretic renormalization group. Weak anisotropy means that parameters which describe anisotropy are chosen to be close to zero, therefore in all expressions it is enough to leave only linear terms in anisotropy parameters. Turbulent fluctuations of the velocity field are taken to have the Gaussian statistics with zero mean and defined noise with finite correlations in time. It is shown that stability of the inertial-range scaling regimes in the three-dimensional case is not destroyed by anisotropy but the corresponding stability of the two-dimensional system can be destroyed even by the presence of weak anisotropy. A borderline dimension $d_c$ below which the stability of the scaling regime is not present is calculated as a function of anisotropy parameters.

Complex variable in theory of 2D inviscid flow

Martin Scholtz, Petr Vesely     |     talk presented by M. Scholtz, May 2014

This is a short talk intended as an introduction to fluid dynamics for students. We introduce the notion of the potential $\phi$ defined by $v_x = \frac{\partial \phi}{\partial x}, \qquad v_y =\frac{\partial \phi}{\partial y}$ and the notion of the stream function $\psi$ defined by $v_x = \frac{\partial \psi}{\partial y}, \qquad v_y =- \frac{\partial \psi}{\partial x}$ Then, the potential and the stream function are combined into a single complex potential $F = \phi + i\,\psi,$ such that the complex velocity becomes $v = \overline{\frac{\partial F}{\partial z}}.$ We show that global characteristics of the flow can be conveniently represented by the complex integral $\oint \bar{v}\,\mathrm{d}z = \Gamma + i\,Q,$ where the real part $\Gamma$ is circulation and imaginary part $Q$ is the flux of the flow.

Then, the elementary flows are introduced: uniform flow, source/sink, dipole, vortex line. Without proof we present the complex potential for the flow past the cylinder, which acquires the form $F(z) = \bar{v}_\infty \, z + (v_\infty-u)\,\frac{R^2}{z} + \frac{\Gamma}{2\,\pi\,i}\,\log z.$

Next we explain the method of vortex panels and apply it to a problem of flow past arbitrary airfoil. We explain the "no-penetration condition", the Joukowski condition and present our code in Mathematica which solves the problem.

This talk has been given in Ondřejov near Prague, Czech Republic for students and researchers of Charles University in Prague. Petr Vesely is my former bachelor student who wrote the thesis about these methods under my supervision.

Quantum field theoretical methods in the theory of turbulence

Martin Scholtz     |     talk presented by M. Scholtz, May 2014

This is a short talk given for students and members of Institute of theoretical physics, Charles University in Prague. Presentation starts with brief explanation of Kolmogorov-Obuchov phenomenological theory of fully developed turbulence. The transition from laminar to turbulent flow is explained following book by U. Frish closely. Next we introduce the correlation (structure) functions and present Kolmogorov hypotheses about energy spectrum of turbulent flow.

The main part of the talk is devoted to the formalism of the path integrals and Feynman diagrams in ordinary quantum mechanics and in quantum field theory. We proceed very intuitively and non-rigorously. However, we derive the generating functional for the free scalar field and for self-interacting scalar field. We construct free propagators and first-order corrections represented by 1-loop Feynman diagram.

Next we show how the problem of stochastic dynamics can be reformulated as the problem of quantum field theory and apply this theorem to the Navier-Stokes equation of fluid dynamics. Again we construct free propagators (for Navier-Stokes eqution without non-linear term) and diagramatically represent first-order corrections.

Quasi-local quantities in General relativity. Bondi mass.

Martin Scholtz

In the last year or two I gave several talks devoted to construction of quasi-local quantities in general relativity. The content of the presentations is essentialy the same: definition of energy-momentum (or angular momentum) in the flat spacetime of special relativity, difficulties with the construction in curved spacetimes, Penrose's suggestion how to avoid these difficulties (the Penrose mass) and the Bondi mass as the limit of the Penrose mass.

Our exposition follows an excellent review paper by Laszlo Szabados: Quasi-Local Energy-Momentum and Angular Momentum in General Relativity. The construction of the Bondi mass for the spacetimes with electro-scalar sources (Einstein-Maxwell-Klein-Gordon spacetimes) is new and details can be found in my paper On the Bondi mass of Einstein-Klein-Gordon spacetimes. Many technical details on spinors in General relativity and spinorial techniques employed in the construction of quasi-local quantities can be found in my publication Helical symmetry, spinors and periodic solutions in general relativity and mainly in the thesis of my diploma student Lukas Holka.