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.Comment: 31 pages, 1 figure, details added, references added, minor correction

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.Comment: 26 pages, LaTeX, 1 figure, minor improvements, references added (published version

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.Comment: 43 pages, LaTeX, 9 figures, 6 ancillary file

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

We derive an exact, time-dependent analytical magnetic eld solution for the inner heliosheath, which satis es both the induction equation of ideal magnetohydrodynamics in the limit of in nite electric conductivity and the magnetic divergence constraint. To this end, we assume that the magnetic eld is frozen into a plasma ow 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 rst-order system of ordinary di erential equations for the magnetic vector potential. Then, using the general solution of this system, we compute the magnetic eld via the magnetic curl relation. Finally, we analyze the well-posedness of the corresponding Dirichlettype initial-boundary value problem, specify compatibility conditions for the initial-boundary values, and outline the implementation of initial-boundary conditions

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.Comment: 19 pages, 1 figur

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.Comment: 27 pages, LaTeX, more details on self-adjoint extension (published version

A family of horizon-penetrating coordinate systems for the Schwarzschild black hole geometry with Cauchy temporal functions

The author is grateful to Miguel Sanchez for useful discussions. Furthermore, the author thanks the anonymous referees for helpful and constructive comments. This work was partially supported by the research project MTM2016-78807-C2-1-P funded by MINECO and ERDF.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.Spanish Government MTM2016-78807-C2-1-PEuropean Commissio