Curvature without metric: the Penrose construction for half-flat pp-waves
We derive the Penrose data for half-flat pp-waves and extend his original construction for the Weyl spinor of plane waves in terms of this data.
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arXiv subjects
Peter C. Aichelburg
Publications and source records attributed to Peter C. Aichelburg.
Is the equivalence principle useful for understanding general relativity?
The Equivalence Principle (EP) is at the heart of General Relativity (GR), tested in many aspects. It is often used to discuss qualitatively the influence of gravity on physical phenomena. But can this be made more precise? We compare clock rates, frequency shifts, light deflection and time delay in simple static spacetimes to the analogous phenomena seen by accelerated observers in Minkowski space. In contrast to previous studies, we do not assume that the gravitational field is weak and see, as we proceed, how the field is constrained by the EP. Special care is taken that results are only observer-, but not coordinate-dependent. By this we clarify some of the issues raised in the literature and show which gravitational effects can and which cannot be simulated by acceleration. The paper may also serve as a contribution for critical discussions on the implications of the EP.
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Scattering of Classical and Quantum Particles by Impulsive Fields
We investigate the scattering of classical and quantum particles in impulsive backgrounds fields. These fields model short outbursts of radiation propagating with the speed of light. The singular nature of the problem will be accounted for by the use of Colombeau's generalized function which however give rise to ambiguities. It is the aim of the paper to show that these ambiguities can be overcome by implementing additional physical conditions, which in the non-singular case would be satisfied automatically. As example we discuss the scattering of classical, Klein-Gordon and Dirac particles in impulsive electromagnetic fields.
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On stable self-similar blow up for equivariant wave maps: The linearized problem
We consider co-rotational wave maps from (3+1) Minkowski space into the three-sphere. This is an energy supercritical model which is known to exhibit finite time blow up via self-similar solutions. The ground state self-similar solution $f_0$ is known in closed form and based on numerics, it is supposed to describe the generic blow up behavior of the system. In this paper we develop a rigorous linear perturbation theory around $f_0$. This is an indispensable prerequisite for the study of nonlinear stability of the self-similar blow up which is conducted in a companion paper. In particular, we prove that $f_0$ is linearly stable if it is mode stable. Furthermore, concerning the mode stability problem, we prove new results that exclude the existence of unstable eigenvalues with large imaginary parts and also, with real parts larger than 1/2. The remaining compact region is well-studied numerically and all available results strongly suggest the nonexistence of unstable modes.
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Spectral Properties and Linear Stability of Self-Similar Wave Maps
We study co--rotational wave maps from $(3+1)$--Minkowski space to the three--sphere $S^3$. It is known that there exists a countable family $\{f_n\}$ of self--similar solutions. We investigate their stability under linear perturbations by operator theoretic methods. To this end we study the spectra of the perturbation operators, prove well--posedness of the corresponding linear Cauchy problem and deduce a growth estimate for solutions. Finally, we study perturbations of the limiting solution which is obtained from $f_n$ by letting $n \to \infty$.
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On the mode stability of a self-similar wave map
We study linear perturbations of a self-similar wave map from Minkowski space to the three-sphere which is conjectured to be linearly stable. Considering analytic mode solutions of the evolution equation for the perturbations we prove that there are no real unstable eigenvalues apart from the well-known gauge instability.
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A note on the eigenvalues for equivariant maps of the SU(2) sigma-model
We numerically calculate the first few eigenvalues of the perturbations of self-similar solutions of the spherically symmetric co-rotational SU(2) sigma-model on Minkowski space.
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Bifurcation and fine structure phenomena in critical collapse of a self-gravitating $σ$-field
Building on previous work on the critical behavior in gravitational collapse of the self-gravitating SU(2) $σ$-field and using high precision numerical methods we uncover a fine structure hidden in a narrow window of parameter space. We argue that this numerical finding has a natural explanation within a dynamical system framework of critical collapse.
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News from Critical Collapse: Bondi Mass, Tails and Quasinormal Modes
We discuss critical gravitational collapse on the threshold of apparent horizon formation as a model both for the discussion of global aspects of critical collapse and for numerical studies in a compactified context. For our matter model we choose a self-gravitating massless scalar field in spherical symmetry, which has been studied extensively in the critical collapse literature. Our evolution system is based on Bondi coordinates, the mass function is used as an evolution variable to ensure regularity at null infinity. We compute radiation quantities like the Bondi mass and news function and find that they reflect the DSS behavior. Surprisingly, the period of radiation at null infinity is related to the formal result for the leading quasi-normal mode of a black hole with rapidly decreasing mass. Furthermore, our investigations shed some light on global versus local issues in critical collapse, and the validity and usefulness of the concept of null infinity when predicting detector signals.
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A New Transition between Discrete and Continuous Self-Similarity in Critical Gravitational Collapse
We analyze a bifurcation phenomenon associated with critical gravitational collapse in a family of self-gravitating SU(2) $σ$-models. As the dimensionless coupling constant decreases, the critical solution changes from discretely self-similar (DSS) to continuously self-similar (CSS). Numerical results provide evidence for a bifurcation which is analogous to a heteroclinic loop bifurcation in dynamical systems, where two fixed points (CSS) collide with a limit cycle (DSS) in phase space as the coupling constant tends to a critical value.
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Type II Critical Collapse of a Self-Gravitating Nonlinear $σ$-Model
We report on the existence and phenomenology of type II critical collapse within the one-parameter family of SU(2) $σ$-models coupled to gravity. Numerical investigations in spherical symmetry show discretely self-similar (DSS) behavior at the threshold of black hole formation for values of the dimensionless coupling constant $\ccbeta$ ranging from 0.2 to 100; at 0.18 we see small deviations from DSS. While the echoing period $Δ$ of the critical solution rises sharply towards the lower limit of this range, the characteristic mass scaling has a critical exponent $γ$ which is almost independent of $\ccbeta$, asymptoting to $0.1185 \pm 0.0005$ at large $\ccbeta$. We also find critical scaling of the scalar curvature for near-critical initial data. Our numerical results are based on an outgoing-null-cone formulation of the Einstein-matter equations, specialized to spherical symmetry. Our numerically computed initial-data critical parameters $p^*$ show 2nd order convergence with the grid resolution, and after compensating for this variation in $p^*$, our individual evolutions are uniformly 2nd order convergent even very close to criticality.
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SU(2) Cosmological Solitons
We present a class of numerical solutions to the SU(2) nonlinear $σ$-model coupled to the Einstein equations with cosmological constant $Λ\geq 0$ in spherical symmetry. These solutions are characterized by the presence of a regular static region which includes a center of symmetry. They are parameterized by a dimensionless ``coupling constant'' $β$, the sign of the cosmological constant, and an integer ``excitation number'' $n$. The phenomenology we find is compared to the corresponding solutions found for the Einstein-Yang-Mills (EYM) equations with positive $Λ$ (EYM$Λ$). If we choose $Λ$ positive and fix $n$, we find a family of static spacetimes with a Killing horizon for $0 \leq β< β_{max}$. As a limiting solution for $β= β_{max}$ we find a {\em globally} static spacetime with $Λ=0$, the lowest excitation being the Einstein static universe. To interpret the physical significance of the Killing horizon in the cosmological context, we apply the concept of a trapping horizon as formulated by Hayward. For small values of $β$ an asymptotically de Sitter dynamic region contains the static region within a Killing horizon of cosmological type. For strong coupling the static region contains an ``eternal cosmological black hole''.
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Sigma Model on de Sitter Space
We discuss spherically symmetric, static solutions to the SU(2) sigma model on a de Sitter background. Despite of its simplicity this model reflects many of the features exhibited by systems of non-linear matter coupled to gravity e.g. there exists a countable set of regular solutions with finite energy; all of the solutions show linear instability with the number of unstable modes increasing with energy.
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Magnetically Charged Black Holes and their Stability
We study magnetically charged black holes in the Einstein-Yang-Mills-Higgs theory in the limit of infinitely strong coupling of the Higgs field. Using mixed analytical and numerical methods we give a complete description of static spherically symmetric black hole solutions, both abelian and nonabelian. In particular, we find a new class of extremal nonabelian solutions. We show that all nonabelian solutions are stable against linear radial perturbations. The implications of our results for the semiclassical evolution of magnetically charged black holes are discussed.
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