Christian Röken | Rheinische Friedrich-Wilhelms-Universität Bonn (original) (raw)

Papers by Christian Röken

We present a detailed analysis of Penrose's gravito-optical analogy between the focusing effects ... more 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.

We present a quasi-local, functional analytic method to locate and invariantly characterize the s... more 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 a relational concept of black hole entropy.

Rotationally symmetric shapes with parabolic cross sections are frequently used to model astrophy... more Rotationally symmetric shapes with parabolic cross sections are frequently used to model astrophysical objects such as magnetospheres and other blunt objects immersed in interplanetary or interstellar gas or plasma flows. We present 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. We then derive an exact analytical solution to the induction equation of ideal magnetohydrodynamics, thereby obtaining explicit expressions for an initially homogeneous magnetic field of arbitrary orientation being passively advected in this flow. The solution procedure employs Euler potentials and the method of Cauchy's Integral based on the flow's stream function and its isochrones. Furthermore, a novel renormalization procedure allows us to generate more general analytic expressions modeling the deformation experienced by arbitrary scalar or vector-valued fields embedded into the flow as they are advected first towards and then past the parabolic obstacle. Finally, the flow field is generalized from incompressible to mildly compressible velocities, where the associated density distribution is found from Bernoulli's principle. Note: Some parts of this manuscript are not completely finalized, and its contents still has to be checked prior to submission to the journal. Please exercise caution when using it for derivative works, or else wait for the updated version or the official publication.

We derive an exact, time-dependent analytical magnetic field solution for the inner heliosheath, ... more 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 boundary value problem, specify compatibility conditions for the boundary values, and outline the implementation of boundary conditions.

We introduce a new family of horizon-penetrating coordinate systems for the Schwarzschild black h... more We introduce a new family of horizon-penetrating coordinate systems for the Schwarzschild black hole geometry that feature time coordinates, which are specific Cauchy temporal functions, i.e., the level sets of these time coordinates 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.

The structure of the solution space of the Dirac equation in the exterior Schwarzschild geometry ... more 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.

To explain the flux variabilities of active galactic nuclei, especially blazars, we assume a scen... more To explain the flux variabilities of active galactic nuclei, especially blazars, we assume a scenario of multiple injections of ultrahigh energy radiating electrons in powerful cosmic nonthermal radiation sources with dominant magnetic field self-generation leading to a series of bursts. Therefore, we examine analytically the cases of electron energy losses in the form of synchrotron cooling with a constant magnetic field and with a partition condition between the energy densities of the magnetic field and the injected relativistic electrons. Thus, assuming partition conditions, the magnetic field strength becomes time dependent changing both the synchrotron emissivity and the intrinsic temporal evolution of the relativistic particle energy spectrum after injection. In this paper, the linear and nonlinear kinetic equations for the intrinsic temporal evolution of relativistic electrons are solved for the case of multiple instantaneous monoenergetic injections of relativistic electrons. The solutions are applied and compared in the calculations of the optically thin synchrotron radiation intensities and the synchrotron fluences. They show significant differences in the optically thin synchrotron spectral distributions at different times and in the synchrotron light curves at different frequencies.

The imaginary unit on the right-hand side of Eq. (15) should not be there, namely the product αβ ... more The imaginary unit on the right-hand side of Eq. (15) should not be there, namely the product αβ has to be real-valued,

Using analytical and numerical methods, estimates are given of future predictions in astrophysics... more Using analytical and numerical methods, estimates are given of future predictions in astrophysics that can be gathered from a sequence of observed events, for example for γ-ray bursts. Some general probability considerations are provided and then a maximum likelihood estimation , together with an approximation for the large number limit of possible events. Illustrations are given of the numerical maximum likelihood estimation programs in the situations of both a large number and a finite number of events. The effects of data uncertainty are also considered. Numerical calculations and comparisons with theoretical expectations are presented too.

A theoretical radiation model for the flaring of TeV blazars is discussed here for the case of a ... more A theoretical radiation model for the flaring of TeV blazars is discussed here for the case of a nonlinear electron synchrotron cooling in these sources. We compute analytically the optically thick and thin synchrotron radiation intensities and photon density distributions in the emission knot as functions of frequency and time followed by the synchrotron self-Compton intensity and fluence in the optically thin frequency range using the Thomson approximation of the inverse Compton cross section. At all times and frequencies, the optically thin part of the synchrotron radiation process is shown to provide the dominant contribution to the synchrotron self-Compton quantities, while the optically thick part is always negligible. Afterwards, we compare the linear to the nonlinear synchrotron radiation cooling model using the data record of PKS 2155-304 on MJD 53944 favouring a linear cooling of the injected monoenergetic electrons. The good agreement of both the linear and the nonlinear cooling model with the data supports the relativistic pickup process operating in this source. Additionally, we discuss the synchrotron self-Compton scattering, applying the full Klein-Nishina cross section to achieve the most accurate results for the synchrotron self-Compton intensity and fluence distributions.

The vast improvement of the sensitivity of modern ground-based air Cherenkov telescopes, together... more The vast improvement of the sensitivity of modern ground-based air Cherenkov telescopes, together with the sensitive flux measurements at lower frequencies, requires accurate elaborations of the theoretical radiation models for flaring blazars. Here the flaring of TeV blazars due to the synchrotron-self Compton (SSC) process is considered. We assume that, at the moment t = t 0 , a flare in the emission knot occurs due to the instantaneous injection of monoenergetic (E 0) ultrarelativistic electrons. The ultrarelativistic electrons are injected uniformly over the knot volume and at later times are subject to linear synchrotron radiation cooling in a magnetic field whose strength remains constant during the time evolution of the relativistic electrons.The generated synchrotron photons are subject to multiple Thomson-scattering off the cold electrons in the source giving rise to spatial photon diffusion. Optically thick and thin synchrotron radiation intensities and photon density distributions in the emission knot as functions of frequency and time are analytically determined. The synchrotron photons serve as target photons for the SSC process, which is calculated in the optically thin frequency range using the Thomson approximation of the inverse Compton cross section. It is shown that the optically thick part of the synchrotron radiation process provides a negligible contribution to the resulting SSC intensity at all frequencies and times.Because the high-energy TeV photons undergo no elastic multiple Compton scatterings, we neglect the influence of photon diffusion in the calculation of the SSC intensity and fluence distribution with energy. The SSC fluence exhibits a break at E f = 15.8b −1/3 GeV from a ∝E −1/4 s-power law spectrum at lower photon energies E t ≤ E s ≤ E f to a ∝E −2 s [1 − (E s /E 0) 7/3 ]-distribution at high energies E f ≤ E s ≤ E 0. The application to the observed TeV fluence spectrum of the flare of PKS 2155-304 on July 28, 2006 yields δb −1/3 = 27.1 ± 6.5. The emergent SSC light curve is independent of spatial photon diffusion and determined by the temporal variations on the relativistic electron density distribution and the synchrotron photon density. The comparison of the observed with the theoretical monochromatic synchrotron light curve determines the photon escape distribution.

A previously published analytical magnetohydrodynamic model for the local interstellar magnetic f... more A previously published analytical magnetohydrodynamic model for the local interstellar magnetic field in the vicinity of the heliopause (Röken et al. 2015) is extended from incompressible to compressible, yet predominantly subsonic flow, considering both isothermal and adiabatic equations of state. Exact expressions and suitable approximations for the density and the flow velocity are derived and discussed. In addition to the stationary induction equation, these expressions also satisfy the momentum balance equation along stream lines. The practical usefulness of the corresponding, still exact analytical magnetic field solution is assessed by comparing it quantitatively to results from a fully self-consistent magnetohydrodynamic simulation of the interstellar magnetic field draping around the heliopause.

The study aims at a derivation of generalized Rankine-Hugoniot relations, especially that for the... more The study aims at a derivation of generalized Rankine-Hugoniot 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-Hugoniot 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.

A fully analytical, time-dependent leptonic one-zone model that describes a simplified radiation ... more A fully analytical, time-dependent leptonic one-zone model that describes a simplified radiation process of multiple interacting relativistic electron populations and accounts for the flaring of blazars is presented. In this model, several mono-energetic, relativistic electron populations are successively and instantaneously injected into the emission region and subjected to linear, time-independent synchrotron and nonlinear, time-dependent synchrotron self-Compton radiative losses. The corresponding electron number density is computed analytically by solving a transport equation using an approximation scheme that employs specific asymptotics. Moreover, the optically thin synchrotron intensity, the synchrotron self-Compton intensity in the Thomson limit, as well as the associated total fluences are explicitly calculated. In order to mimic injections of finite duration times and radiative transport, flares are modeled by sequences of these instantaneous injections, suitably distributed over the entire emission region. The total synchrotron and synchrotron self-Compton fluence spectral energy distributions are plotted for a generic three-flare scenario with a set of realistic parameter values, reproducing the typical broad-band behavior seen in observational data.

The fermionic signature operator is constructed in Rindler space-time. It is shown to be an unbou... more 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.

A modified theory of general relativity is proposed, where the gravita-tional constant is replace... more A modified theory of general relativity is proposed, where the gravita-tional 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 gravi-tational 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.

We consider a boundary value problem for the Dirac equation in a smooth, asymptotically flat Lore... more 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.

We consider the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating ad... more 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.

The separability of the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetr... more 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.

Both physical arguments and simulations of the global heliosphere indicate that the tailward heli... more 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.

We present a detailed analysis of Penrose's gravito-optical analogy between the focusing effects ... more 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.

We present a quasi-local, functional analytic method to locate and invariantly characterize the s... more 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 a relational concept of black hole entropy.

Rotationally symmetric shapes with parabolic cross sections are frequently used to model astrophy... more Rotationally symmetric shapes with parabolic cross sections are frequently used to model astrophysical objects such as magnetospheres and other blunt objects immersed in interplanetary or interstellar gas or plasma flows. We present 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. We then derive an exact analytical solution to the induction equation of ideal magnetohydrodynamics, thereby obtaining explicit expressions for an initially homogeneous magnetic field of arbitrary orientation being passively advected in this flow. The solution procedure employs Euler potentials and the method of Cauchy's Integral based on the flow's stream function and its isochrones. Furthermore, a novel renormalization procedure allows us to generate more general analytic expressions modeling the deformation experienced by arbitrary scalar or vector-valued fields embedded into the flow as they are advected first towards and then past the parabolic obstacle. Finally, the flow field is generalized from incompressible to mildly compressible velocities, where the associated density distribution is found from Bernoulli's principle. Note: Some parts of this manuscript are not completely finalized, and its contents still has to be checked prior to submission to the journal. Please exercise caution when using it for derivative works, or else wait for the updated version or the official publication.

We derive an exact, time-dependent analytical magnetic field solution for the inner heliosheath, ... more 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 boundary value problem, specify compatibility conditions for the boundary values, and outline the implementation of boundary conditions.

We introduce a new family of horizon-penetrating coordinate systems for the Schwarzschild black h... more We introduce a new family of horizon-penetrating coordinate systems for the Schwarzschild black hole geometry that feature time coordinates, which are specific Cauchy temporal functions, i.e., the level sets of these time coordinates 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.

The structure of the solution space of the Dirac equation in the exterior Schwarzschild geometry ... more 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.

To explain the flux variabilities of active galactic nuclei, especially blazars, we assume a scen... more To explain the flux variabilities of active galactic nuclei, especially blazars, we assume a scenario of multiple injections of ultrahigh energy radiating electrons in powerful cosmic nonthermal radiation sources with dominant magnetic field self-generation leading to a series of bursts. Therefore, we examine analytically the cases of electron energy losses in the form of synchrotron cooling with a constant magnetic field and with a partition condition between the energy densities of the magnetic field and the injected relativistic electrons. Thus, assuming partition conditions, the magnetic field strength becomes time dependent changing both the synchrotron emissivity and the intrinsic temporal evolution of the relativistic particle energy spectrum after injection. In this paper, the linear and nonlinear kinetic equations for the intrinsic temporal evolution of relativistic electrons are solved for the case of multiple instantaneous monoenergetic injections of relativistic electrons. The solutions are applied and compared in the calculations of the optically thin synchrotron radiation intensities and the synchrotron fluences. They show significant differences in the optically thin synchrotron spectral distributions at different times and in the synchrotron light curves at different frequencies.

The imaginary unit on the right-hand side of Eq. (15) should not be there, namely the product αβ ... more The imaginary unit on the right-hand side of Eq. (15) should not be there, namely the product αβ has to be real-valued,

Using analytical and numerical methods, estimates are given of future predictions in astrophysics... more Using analytical and numerical methods, estimates are given of future predictions in astrophysics that can be gathered from a sequence of observed events, for example for γ-ray bursts. Some general probability considerations are provided and then a maximum likelihood estimation , together with an approximation for the large number limit of possible events. Illustrations are given of the numerical maximum likelihood estimation programs in the situations of both a large number and a finite number of events. The effects of data uncertainty are also considered. Numerical calculations and comparisons with theoretical expectations are presented too.

A theoretical radiation model for the flaring of TeV blazars is discussed here for the case of a ... more A theoretical radiation model for the flaring of TeV blazars is discussed here for the case of a nonlinear electron synchrotron cooling in these sources. We compute analytically the optically thick and thin synchrotron radiation intensities and photon density distributions in the emission knot as functions of frequency and time followed by the synchrotron self-Compton intensity and fluence in the optically thin frequency range using the Thomson approximation of the inverse Compton cross section. At all times and frequencies, the optically thin part of the synchrotron radiation process is shown to provide the dominant contribution to the synchrotron self-Compton quantities, while the optically thick part is always negligible. Afterwards, we compare the linear to the nonlinear synchrotron radiation cooling model using the data record of PKS 2155-304 on MJD 53944 favouring a linear cooling of the injected monoenergetic electrons. The good agreement of both the linear and the nonlinear cooling model with the data supports the relativistic pickup process operating in this source. Additionally, we discuss the synchrotron self-Compton scattering, applying the full Klein-Nishina cross section to achieve the most accurate results for the synchrotron self-Compton intensity and fluence distributions.

The vast improvement of the sensitivity of modern ground-based air Cherenkov telescopes, together... more The vast improvement of the sensitivity of modern ground-based air Cherenkov telescopes, together with the sensitive flux measurements at lower frequencies, requires accurate elaborations of the theoretical radiation models for flaring blazars. Here the flaring of TeV blazars due to the synchrotron-self Compton (SSC) process is considered. We assume that, at the moment t = t 0 , a flare in the emission knot occurs due to the instantaneous injection of monoenergetic (E 0) ultrarelativistic electrons. The ultrarelativistic electrons are injected uniformly over the knot volume and at later times are subject to linear synchrotron radiation cooling in a magnetic field whose strength remains constant during the time evolution of the relativistic electrons.The generated synchrotron photons are subject to multiple Thomson-scattering off the cold electrons in the source giving rise to spatial photon diffusion. Optically thick and thin synchrotron radiation intensities and photon density distributions in the emission knot as functions of frequency and time are analytically determined. The synchrotron photons serve as target photons for the SSC process, which is calculated in the optically thin frequency range using the Thomson approximation of the inverse Compton cross section. It is shown that the optically thick part of the synchrotron radiation process provides a negligible contribution to the resulting SSC intensity at all frequencies and times.Because the high-energy TeV photons undergo no elastic multiple Compton scatterings, we neglect the influence of photon diffusion in the calculation of the SSC intensity and fluence distribution with energy. The SSC fluence exhibits a break at E f = 15.8b −1/3 GeV from a ∝E −1/4 s-power law spectrum at lower photon energies E t ≤ E s ≤ E f to a ∝E −2 s [1 − (E s /E 0) 7/3 ]-distribution at high energies E f ≤ E s ≤ E 0. The application to the observed TeV fluence spectrum of the flare of PKS 2155-304 on July 28, 2006 yields δb −1/3 = 27.1 ± 6.5. The emergent SSC light curve is independent of spatial photon diffusion and determined by the temporal variations on the relativistic electron density distribution and the synchrotron photon density. The comparison of the observed with the theoretical monochromatic synchrotron light curve determines the photon escape distribution.

A previously published analytical magnetohydrodynamic model for the local interstellar magnetic f... more A previously published analytical magnetohydrodynamic model for the local interstellar magnetic field in the vicinity of the heliopause (Röken et al. 2015) is extended from incompressible to compressible, yet predominantly subsonic flow, considering both isothermal and adiabatic equations of state. Exact expressions and suitable approximations for the density and the flow velocity are derived and discussed. In addition to the stationary induction equation, these expressions also satisfy the momentum balance equation along stream lines. The practical usefulness of the corresponding, still exact analytical magnetic field solution is assessed by comparing it quantitatively to results from a fully self-consistent magnetohydrodynamic simulation of the interstellar magnetic field draping around the heliopause.

The study aims at a derivation of generalized Rankine-Hugoniot relations, especially that for the... more The study aims at a derivation of generalized Rankine-Hugoniot 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-Hugoniot 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.

A fully analytical, time-dependent leptonic one-zone model that describes a simplified radiation ... more A fully analytical, time-dependent leptonic one-zone model that describes a simplified radiation process of multiple interacting relativistic electron populations and accounts for the flaring of blazars is presented. In this model, several mono-energetic, relativistic electron populations are successively and instantaneously injected into the emission region and subjected to linear, time-independent synchrotron and nonlinear, time-dependent synchrotron self-Compton radiative losses. The corresponding electron number density is computed analytically by solving a transport equation using an approximation scheme that employs specific asymptotics. Moreover, the optically thin synchrotron intensity, the synchrotron self-Compton intensity in the Thomson limit, as well as the associated total fluences are explicitly calculated. In order to mimic injections of finite duration times and radiative transport, flares are modeled by sequences of these instantaneous injections, suitably distributed over the entire emission region. The total synchrotron and synchrotron self-Compton fluence spectral energy distributions are plotted for a generic three-flare scenario with a set of realistic parameter values, reproducing the typical broad-band behavior seen in observational data.

The fermionic signature operator is constructed in Rindler space-time. It is shown to be an unbou... more 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.

A modified theory of general relativity is proposed, where the gravita-tional constant is replace... more A modified theory of general relativity is proposed, where the gravita-tional 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 gravi-tational 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.

We consider a boundary value problem for the Dirac equation in a smooth, asymptotically flat Lore... more 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.

We consider the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating ad... more 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.

The separability of the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetr... more 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.

Both physical arguments and simulations of the global heliosphere indicate that the tailward heli... more 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.

Doctoral Thesis, 2009

We investigate the double-peak profile of the emission of powerful cosmic non-thermal radiation s... more We investigate the double-peak profile of the emission of powerful cosmic non-thermal radiation sources with dominant magnetic field self-generation like TeV blazars. Therefore, we assume a flare to occur in the emission knot due to the uniform instantaneous injection of monoenergetic ultra-relativistic electrons via a relativistic pick-up process. The electrons are subjected to a linear or nonlinear synchrotron radiation cooling and the synchrotron photons are multiple Thomson scattered off their generating electrons (SSC process). We work out the differences between single and multiple instantaneous injections of monoenergetic relativistic electrons. This is of great interest, because it is very likely that injections into the plasmoids occur repeatedly, so that this would explain the short−time energy variability of blazars. We also compute for the first time the nonlinear SSC radiation quantities using a Thomson limit approximation and the full Klein-Nishina cross section.

Doctoral Thesis, 2020

The main objective of this doctoral thesis is the derivation of an integral spectral representati... more The main objective of this doctoral thesis is the derivation of an integral spectral representation of the massive Dirac propagator in the nonextreme Kerr geometry in horizon-penetrating advanced Eddington-Finkelstein-type coordinates. To this end, we divide the doctoral thesis into the following three parts. In the first part, we describe the nonextreme Kerr geometry in the Newman-Penrose formalism by means of a Carter tetrad in advanced Eddington-Finkelstein-type coordinates, which are regular across the event and the Cauchy horizon, respectively, and feature a temporal function for which the level sets are partial Cauchy surfaces. On this background geometry, we define the massive Dirac equation in the Weyl representation in 2-spinor form with a Newman-Penrose dyad basis for the spinor space. We perform Chandrasekhar’s mode analysis and thus show the separability of the massive Dirac equation expressed in such horizon-penetrating coordinates into systems of radial and angular ordinary differential equations. We compute asymptotic radial solutions at infinity, the event horizon, and the Cauchy horizon, and demonstrate that the corresponding errors have suitable decay. Furthermore, we study specific aspects of the set of eigenfunctions and the eigenvalue spectrum of the angular system. In the second part, we introduce a new method of proof for the essential self-adjointness of the Dirac Hamiltonian for a particular class of nonuniformly elliptic mixed initial-boundary value problems on smooth asymptotically flat Lorentzian manifolds, combining results from the theory of symmetric hyperbolic systems with near-boundary elliptic methods. Finally, in the third part, we present the Hamiltonian formulation of the massive Dirac equation in the nonextreme Kerr geometry in advanced Eddington-Finkelstein-type coordinates and, within this framework, derive an explicit integral spectral representation of the massive Dirac propagator, which yields the full time-dependent dynamics of massive spin-1/2 fermions outside, across, and inside the event horizon, up to the Cauchy horizon. For the construction of this propagator, we first prove that the Dirac Hamiltonian in the extended Kerr geometry is essentially self-adjoint by employing the method introduced in the second part, and then use the spectral theorem for unbounded self-adjoint operators as well as Stone’s formula, which links the spectral measure of the Dirac Hamiltonian to the associated resolvent. We determine the resolvent in a separated form in terms of the projector onto a finite-dimensional invariant spectral eigenspace of the angular operator and the radial Green’s matrix both obtained within the mode analysis of the Dirac equation presented in the first part. This propagator may be applied to study the long-time dynamics and the decay rates of massive Dirac fields in a rotating Kerr black hole spacetime. It can furthermore be used in the formulation of an algebraic quantum field theory.