arXiv subjects

Christian Röken

Publications and source records attributed to Christian Röken.

At least 19 recordsLinked to original sources

arXiv2026

The Construction and Application of Penrose Diagrams, with a Focus on the Maximally Analytically Extended Schwarzschild Spacetime

We present a detailed, mathematically rigorous description of the construction procedure of Penrose diagrams for the example of the maximal analytic extension of the exterior Schwarzschild spacetime. To this end, we first outline the central idea underlying Penrose diagrams, state the general requirements on the spacetimes to be visualized, and give a definition of Penrose diagrams. We then construct the Penrose diagram of the maximally analytically extended Schwarzschild spacetime and discuss its components and characteristics. As an application, we work out the differences between the Eddington-Finkelstein and Penrose coordinate representations of the Schwarzschild spacetime by visually analyzing-and comparing-the Penrose diagram of the maximally analytically extended Schwarzschild spacetime equipped with, on the one hand, a foliation by the level sets of the Eddington-Finkelstein time coordinate and, on the other hand, a foliation by the level sets of the Penrose null coordinate. Through the whole of the paper, we provide explanatory accounts of the relevant parts of the seminal research papers on the exterior Schwarzschild solution pertaining to coordinatizations and possible extensions by Schwarzschild himself, Kruskal, Eddington, Finkelstein, and Penrose. This paper is primarily of pedagogical nature aimed at graduate students in physics and applied mathematics, serving mainly as an introduction to Penrose diagrams along with descriptions and analyses of coordinate representations and extensions of the exterior Schwarzschild spacetime.

gr-qc↗

arXiv2025

A Family of Horizon-penetrating Coordinate Systems for the Schwarzschild Black Hole Geometry with Cauchy Temporal Functions

We introduce a new family of horizon-penetrating coordinate systems for the Schwarzschild black hole geometry featuring time coordinates that are Cauchy temporal functions for which the level sets are smooth, asymptotically flat, spacelike Cauchy hypersurfaces. Coordinate systems of this kind are well suited for the study of the temporal evolution of matter and radiation fields in the joined exterior and interior regions of the Schwarzschild black hole geometry, whereas the associated foliations can be employed as initial data sets for the globally hyperbolic development under the Einstein flow. For their construction, we formulate an explicit method that utilizes the geometry of - and structures inherent in - the Penrose diagram of the Schwarzschild black hole geometry, thus relying on the corresponding metrical product structure. As an example, we consider an integrated algebraic sigmoid function as the basis for the determination of such a coordinate system. Finally, we generalize our results to the Reissner-Nordström black hole geometry up to the Cauchy horizon. The geometric construction procedure presented here can be adapted to yield similar coordinate systems for various other spacetimes with the same metrical product structure.

gr-qc↗

arXiv2025

A Quasi-local, Functional Analytic Detection Method for Stationary Limit Surfaces of Black Hole Spacetimes

We present a quasi-local, functional analytic method to locate and invariantly characterize the stationary limit surfaces of black hole spacetimes with stationary regions. The method is based on ellipticity-hyperbolicity transitions of the Dirac, Klein-Gordon, Maxwell, and Fierz-Pauli Hamiltonians defined on spacelike hypersurfaces of such black hole spacetimes, which occur only at the locations of stationary limit surfaces and can be ascertained from the behaviors of the principal symbols of the Hamiltonians. Therefore, since it relates solely to the effects that stationary limit surfaces have on the time evolutions of the corresponding elementary fermions and bosons, this method is profoundly different from the usual detection procedures that employ either scalar polynomial curvature invariants or Cartan invariants, which, in contrast, make use of the local geometries of the underlying black hole spacetimes. As an application, we determine the locations of the stationary limit surfaces of the Kerr-Newman, Schwarzschild-de Sitter, and Taub-NUT black hole spacetimes. Finally, we show that for black hole spacetimes with static regions, our functional analytic method serves as a quasi-local event horizon detector and gives rise to relational concepts of event horizons and black hole entropy.

gr-qc↗

arXiv2024

An exact analytical solution for the weakly magnetized flow around an axially symmetric paraboloid, with application to magnetosphere models

Rotationally symmetric bodies with longitudinal cross sections of parabolic shape are frequently used to model astrophysical objects, such as magnetospheres and other blunt objects, immersed in interplanetary or interstellar gas or plasma flows. We discuss a simple formula for the potential flow of an incompressible fluid around an elliptic paraboloid whose axis of symmetry coincides with the direction of incoming flow. Prescribing this flow, we derive an exact analytical solution to the induction equation of ideal magnetohydrodynamics for the case of an initially homogeneous magnetic field of arbitrary orientation being passively advected in this flow. Our solution procedure employs Euler potentials and Cauchy's integral formalism based on the flow's stream function and isochrones. Furthermore, we use a particular renormalization procedure that allows us to generate more general analytical expressions modeling the deformations experienced by arbitrary scalar or vector-valued fields embedded in the flow as they are advected first toward and then past the parabolic obstacle. Finally, both the velocity field and the magnetic field embedded therein are generalized from incompressible to mildly compressible flow, where the associated density distribution is found from Bernoulli's principle.

astro-ph.SR↗

arXiv2024

On Penrose's Analogy between Curved Spacetime Regions and Optical Lenses

We present a detailed analysis of Penrose's gravito-optical analogy between the focusing effects of particular families of Ricci- and Weyl-curved spacetime regions on the one hand, and anastigmatic and astigmatic optical lenses on the other. We put the analogy in its historical context, investigate its underlying assumptions, its range of validity, its proof of concept, and its application in Penrose's study of the notion of energy flux in general relativity. Finally, we examine the analogy within the framework of Norton's material theory of induction.

physics.hist-ph↗

arXiv2022

An Exact, Time-dependent Analytical Solution for the Magnetic Field in the Inner Heliosheath

We derive an exact, time-dependent analytical magnetic field solution for the inner heliosheath, which satisfies both the induction equation of ideal magnetohydrodynamics in the limit of infinite electric conductivity and the magnetic divergence constraint. To this end, we assume that the magnetic field is frozen into a plasma flow resembling the characteristic interaction of the solar wind with the local interstellar medium. Furthermore, we make use of the ideal Ohm's law for the magnetic vector potential and the electric scalar potential. By employing a suitable gauge condition that relates the potentials and working with a characteristic coordinate representation, we thus obtain an inhomogeneous first-order system of ordinary differential equations for the magnetic vector potential. Then, using the general solution of this system, we compute the magnetic field via the magnetic curl relation. Finally, we analyze the well-posedness of the corresponding Dirichlet-type initial-boundary value problem, specify compatibility conditions for the initial-boundary values, and outline the implementation of initial-boundary conditions.

astro-ph.SR↗

arXiv2019

The Fermionic Signature Operator in the Exterior Schwarzschild Geometry

The structure of the solution space of the Dirac equation in the exterior Schwarzschild geometry is analyzed. Representing the space-time inner product for families of solutions with variable mass parameter in terms of the respective scalar products, a so-called mass decomposition is derived. This mass decomposition consists of a single mass integral involving the fermionic signature operator as well as a double integral which takes into account the flux of Dirac currents across the event horizon. The spectrum of the fermionic signature operator is computed. The corresponding generalized fermionic projector states are analyzed.

math-ph↗

arXiv2019

The Massive Dirac Equation in the Kerr Geometry: Separability in Eddington-Finkelstein-type Coordinates and Asymptotics

The separability of the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating advanced Eddington-Finkelstein-type coordinates is shown. To this end, the Kerr geometry is described in the Newman-Penrose formalism by a regular Carter tetrad and the Dirac spinors and matrices are defined in a chiral Newman-Penrose dyad representation. Applying Chandrasekhar's mode ansatz, the Dirac equation is separated into radial and angular systems of ordinary differential equations. Asymptotic radial solutions at infinity, the event horizon, and the Cauchy horizon are derived, and the decay of the associated errors is analyzed. Moreover, specific aspects of the angular eigenfunctions and eigenvalues are discussed. Finally, as an application, the scattering of massive Dirac particles by the gravitational field of a rotating Kerr black hole is studied. This work provides the basis for a Hamiltonian formulation of the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating coordinates and for the construction of an integral spectral representation of the Dirac propagator that yields the dynamics of Dirac particles outside, across, and inside the event horizon, up to the Cauchy horizon.

gr-qc↗

arXiv2018

An Integral Spectral Representation of the Massive Dirac Propagator in the Kerr Geometry in Eddington-Finkelstein-type Coordinates

We consider the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating advanced Eddington-Finkelstein-type coordinates and derive a functional analytic integral representation of the associated propagator using the spectral theorem for unbounded self-adjoint operators, Stone's formula, and quantities arising in the analysis of Chandrasekhar's separation of variables. This integral representation describes the dynamics of Dirac particles outside and across the event horizon, up to the Cauchy horizon. In the derivation, we first write the Dirac equation in Hamiltonian form and show the essential self-adjointness of the Hamiltonian. For the latter purpose, as the Dirac Hamiltonian fails to be elliptic at the event and the Cauchy horizon, we cannot use standard elliptic methods of proof. Instead, we employ a new, general method for mixed initial-boundary value problems that combines results from the theory of symmetric hyperbolic systems with near-boundary elliptic methods. In this regard and since the time evolution may not be unitary because of Dirac particles impinging on the ring singularity, we also impose a suitable Dirichlet-type boundary condition on a time-like inner hypersurface placed inside the Cauchy horizon, which has no effect on the dynamics outside the Cauchy horizon. We then compute the resolvent of the Dirac Hamiltonian via the projector onto a finite-dimensional, invariant spectral eigenspace of the angular operator and the radial Green's matrix stemming from Chandrasekhar's separation of variables. Applying Stone's formula to the spectral measure of the Hamiltonian in the spectral decomposition of the Dirac propagator, that is, by expressing the spectral measure in terms of this resolvent, we obtain an explicit integral representation of the propagator.

gr-qc↗

arXiv2017

Flaring of Blazars from an Analytical, Time-dependent Model for Combined Synchrotron and Synchrotron Self-Compton Radiative Losses of Multiple Ultrarelativistic Electron Populations

We present a fully analytical, time-dependent leptonic one-zone model that describes a simplified radiation process of multiple interacting ultrarelativistic electron populations, accounting for the flaring of GeV blazars. In this model, several mono-energetic, ultrarelativistic electron populations are successively and instantaneously injected into the emission region, i.e., a magnetized plasmoid propagating along the blazar jet, and subjected to linear, time-independent synchrotron radiative losses, which are caused by a constant magnetic field, and nonlinear, time-dependent synchrotron self-Compton radiative losses in the Thomson limit. Considering a general multiple-injection scenario is, from a physical point of view, more realistic than the usual single-injection scenario invoked in common blazar models, as blazar jets may extend over tens of kiloparsecs and, thus, most likely pick up several particle populations from intermediate clouds. We analytically compute the electron number density by solving a kinetic equation using Laplace transformations and the method of matched asymptotic expansions. Moreover, we explicitly calculate the optically thin synchrotron intensity, the synchrotron self-Compton intensity in the Thomson limit, as well as the associated total fluences. In order to mimic injections of finite duration times and radiative transport, we model flares by sequences of these instantaneous injections, suitably distributed over the entire emission region. Finally, we present a parameter study for the total synchrotron and synchrotron self-Compton fluence spectral energy distributions for a generic three-injection scenario, varying the magnetic field strength, the Doppler factor, and the initial electron energy of the first injection in realistic parameter domains, demonstrating that our model can reproduce the typical broad-band behavior seen in observational data.

astro-ph.HE↗

arXiv2017

Kerr Isolated Horizons in Ashtekar and Ashtekar-Barbero Connection Variables

The Ashtekar and Ashtekar-Barbero connection variable formulations of Kerr isolated horizons are derived. Using a regular Kinnersley tetrad in horizon-penetrating Kruskal-Szekeres-like coordinates, the spin coefficients of Kerr geometry are determined by solving the first Maurer-Cartan equation of structure. Isolated horizon conditions are imposed on the tetrad and the spin coefficients. A transformation into an orthonormal tetrad frame that is fixed in the time gauge is applied and explicit calculations of the spin connection, the Ashtekar and Ashtekar-Barbero connections, and the corresponding curvatures on the horizon 2-spheres are performed. Since the resulting Ashtekar-Barbero curvature does not comply with the simple form of the horizon boundary condition of Schwarzschild isolated horizons, i.e., on the horizon 2-spheres, the Ashtekar-Barbero curvature is not proportional to the Plebanski 2-form, which is required for an SU(2) Chern-Simons treatment of the gauge degrees of freedom in the horizon boundary in the context of loop quantum gravity, a general method to construct a new connection whose curvature satisfies such a relation for Kerr isolated horizons is introduced. For the purpose of illustration, this method is employed in the framework of slowly rotating Kerr isolated horizons.

gr-qc↗

arXiv2017

The Fermionic Signature Operator and Quantum States in Rindler Space-Time

The fermionic signature operator is constructed in Rindler space-time. It is shown to be an unbounded self-adjoint operator on the Hilbert space of solutions of the massive Dirac equation. In two-dimensional Rindler space-time, we prove that the resulting fermionic projector state coincides with the Fulling-Rindler vacuum. Moreover, the fermionic signature operator gives a covariant construction of general thermal states, in particular of the Unruh state. The fermionic signature operator is shown to be well-defined in asymptotically Rindler space-times. In four-dimensional Rindler space-time, our construction gives rise to new quantum states.

math-ph↗

arXiv2016

Generalized multi-polytropic Rankine-Hugoniot relations and the entropy condition

The study aims at a derivation of generalized \RH relations, especially that for the entropy, for the case of different upstream/downstream polytropic indices and their implications. We discuss the solar/stellar wind interaction with the interstellar medium for different polytropic indices and concentrate on the case when the polytropic index changes across hydrodynamical shocks. We use first a numerical mono-fluid approach with constant polytropic index in the entire integration region to show the influence of the polytropic index on the thickness of the helio-/astrosheath and on the compression ratio. Second, the Rankine-Hugonoit relations for a polytropic index changing across a shock are derived analytically, particularly including a new form of the entropy condition. In application to the/an helio-/astrosphere, we find that the size of the helio-/astrosheath as function of the polytropic index decreases in a mono-fluid model for indices less than $γ=5/3$ and increases for higher ones and vice versa for the compression ratio. Furthermore, we demonstrate that changing polytropic indices across a shock are physically allowed only for sufficiently high Mach numbers and that in the hypersonic limit the compression ratio depends only on the downstream polytropic index, while the ratios of the temperature and pressure as well as the entropy difference depend on both, the upstream and downstream polytropic indices.

astro-ph.SR↗

arXiv2016

Self-Adjointness of the Dirac Hamiltonian for a Class of Non-Uniformly Elliptic Boundary Value Problems

We consider a boundary value problem for the Dirac equation in a smooth, asymptotically flat Lorentzian manifold admitting a Killing field which is timelike near and tangential to the boundary. A self-adjoint extension of the Dirac Hamiltonian is constructed. Our results also apply to the situation that the space-time includes horizons, where the Hamiltonian fails to be elliptic.

math-ph↗

arXiv2016

The Fermionic Projector in a Time-Dependent External Potential: Mass Oscillation Property and Hadamard States

We give a non-perturbative construction of the fermionic projector in Minkowski space coupled to a time-dependent external potential which is smooth and decays faster than quadratically for large times. The weak and strong mass oscillation properties are proven. We show that the integral kernel of the fermionic projector is of Hadamard form, provided that the time integral of the spatial sup-norm of the potential satisfies a suitable bound. This gives rise to an algebraic quantum field theory of Dirac fields in an external potential with a distinguished pure quasi-free Hadamard state.

math-ph↗

arXiv2016

Dynamical Gravitational Coupling as a Modified Theory of General Relativity

A modified theory of general relativity is proposed, where the gravitational constant is replaced by a dynamical variable in space-time. The dynamics of the gravitational coupling is described by a family of parametrized null geodesics, implying that the gravitational coupling at a space-time point is determined by solving transport equations along all null geodesics through this point. General relativity with dynamical gravitational coupling (DGC) is introduced. We motivate DGC from general considerations and explain how it arises in the context of causal fermion systems. The underlying physical idea is that the gravitational coupling is determined by microscopic structures on the Planck scale which propagate with the speed of light. In order to clarify the mathematical structure, we analyze the conformal behavior and prove local existence and uniqueness of the time evolution. The differences to Einstein's theory are worked out in the examples of the Friedmann-Robertson-Walker model and the spherically symmetric collapse of a shell of matter. Potential implications for the problem of dark matter and for inflation are discussed. It is shown that the effects in the solar system are too small for being observable in present-day experiments.

gr-qc↗

arXiv2016

Toward more realistic analytic models of the heliotail: Incorporating magnetic flattening via distortion flows

Both physical arguments and simulations of the global heliosphere indicate that the tailward heliopause is flattened considerably in the direction perpendicular to both the incoming flow and the large-scale interstellar magnetic field. Despite this fact, all of the existing global analytical models of the outer heliosheath's magnetic field assume a circular cross section of the heliotail. To eliminate this inconsistency, we introduce a mathematical procedure by which any analytically or numerically given magnetic field can be deformed in such a way that the cross sections along the heliotail axis attain freely prescribed, spatially dependent values for their total area and aspect ratio. The distorting transformation of this method honors both the solenoidality condition and the stationary induction equation with respect to an accompanying flow field, provided that both constraints were already satisfied for the original magnetic and flow fields prior to the transformation. In order to obtain realistic values for the above parameters, we present the first quantitative analysis of the heliotail's overall distortion as seen in state-of-the-art three-dimensional hybrid MHD-kinetic simulations.

astro-ph.SR↗

arXiv2015

First-order quantum-gravitational correction to Friedmannian cosmology from covariant, holomorphic spinfoam cosmology

The first-order loop quantum gravity correction of the simplest, classical general-relativistic Friedmann Hamiltonian constraint, emerging from a holomorphic spinfoam cosmological model peaked on homogeneous, isotropic geometries, is studied. The quantum Hamiltonian constraint, satisfied by the EPRL transition amplitude between the boundary cosmological coherent states, includes a contribution of the order of the Planck constant $\hbar$ that also appears in the corresponding semiclassical symplectic model. The analysis of this term gives a quantum-gravitational correction to the classical Friedmann dynamics of the scale factor yielding a small decelerating expansion (small accelerating contraction) of the universe. The robustness of the physical interpretation is established for arbitrary refinements of the boundary graphs. Also, mathematical equivalences between the semiclassical cosmological model and certain classical fluid and scalar field theories are explored.

gr-qc↗