Explicit Riemannian manifolds with unexpectedly behaving center of mass

The (relativistic) center of mass of an asymptotically flat Riemannian manifold is often defined by certain surface integral expressions evaluated along a foliation of the manifold near infinity, e. g. by Arnowitt, Deser, and Misner (ADM). There are also what we call 'abstract' definitions of the center of mass in terms of a foliation near infinity itself, going back to the constant mean curvature (CMC-) foliation studied by Huisken and Yau; these give rise to surface integral expressions when equipped with suitable systems of coordinates. We discuss subtle asymptotic convergence issues regarding the ADM- and the coordinate expressions related to the CMC-center of mass. In particular, we give explicit examples demonstrating that both can diverge -- in a setting where Einstein's equation is satisfied. We also give explicit examples of the same asymptotic order of decay with prescribed mass and center of mass. We illustrate both phenomena by providing analogous examples in Newtonian gravity. Our examples conflict with some results in the literature.Comment: examples with prescribed mass and center of mass included; asymptotic decay described in more detail; references update

Quantum dynamics in macrosystems with several coupled electronic states: hierarchy of effective Hamiltonians

We address the nonadiabatic quantum dynamics of macrosystems with several coupled electronic states, taking into account the possibility of multi-state conical intersections. The general situation of an arbitrary number of states and arbitrary number of nuclear degrees of freedom (modes) is considered. The macrosystem is decomposed into a system part carrying a few, strongly coupled modes, and an environment, comprising the vast number of remaining modes. By successively transforming the modes of the environment, a hierarchy of effective Hamiltonians for the environment is constructed. Each effective Hamiltonian depends on a reduced number of effective modes, which carry cumulative effects. By considering the system's Hamiltonian along with a few members of the hierarchy, it is shown mathematically by a moment analysis that the quantum dynamics of the entire macrosystem can be numerically exactly computed on a given time-scale. The time scale wanted defines the number of effective Hamiltonians to be included. The contribution of the environment to the quantum dynamics of the macrosystem translates into a sequential coupling of effective modes. The wavefunction of the macrosystem is known in the full space of modes, allowing for the evaluation of observables such as the time-dependent individual excitation along modes of interest, as well a spectra and electronic-population dynamics

Reachability in Cooperating Systems with Architectural Constraints is PSPACE-Complete

The reachability problem in cooperating systems is known to be PSPACE-complete. We show here that this problem remains PSPACE-complete when we restrict the communication structure between the subsystems in various ways. For this purpose we introduce two basic and incomparable subclasses of cooperating systems that occur often in practice and provide respective reductions. The subclasses we consider consist of cooperating systems the communication structure of which forms a line respectively a star.Comment: In Proceedings GRAPHITE 2013, arXiv:1312.706

Uniqueness of photon spheres in electro-vacuum spacetimes

In a recent paper, the authors established the uniqueness of photon spheres in static vacuum asymptotically flat spacetimes by adapting Bunting and Masood-ul-Alam's proof of static vacuum black hole uniqueness. Here, we establish uniqueness of suitably defined sub-extremal photon spheres in static electro-vacuum asymptotically flat spacetimes by adapting the argument of Masood-ul-Alam. As a consequence of our result, we can rule out the existence of electrostatic configurations involving multiple "very compact" electrically charged bodies and sub-extremal black holes.Comment: 16 pages. This paper extends the photon sphere uniqueness result obtained in arXiv:1504.05804 from the vacuum to the electro-vacuum setting. While the general proof method is similar, a number of new nontrivial issues aris