Black hole evolution with the BSSN system by pseudo-spectral methods

We present a new pseudo-spectral code for the simulation of evolution systems that are second order in space. We test this code by evolving a non-linear scalar wave equation. These non-linear waves can be stably evolved using very simple constant or radiative boundary conditions, which we show to be well-posed in the scalar wave case. The main motivation for this work, however, is to evolve black holes for the first time with the BSSN system by means of a spectral method. We use our new code to simulate the evolution of a single black hole using all applicable methods that are usually employed when the BSSN system is used together with finite differencing methods. In particular, we use black hole excision and test standard radiative and also constant outer boundary conditions. Furthermore, we study different gauge choices such as $1+\log$ and constant densitized lapse. We find that these methods in principle do work also with our spectral method. However, our simulations fail after about $100M$ due to unstable exponentially growing modes. The reason for this failure may be that we evolve the black hole on a full grid without imposing any symmetries. Such full grid instabilities have also been observed when finite differencing methods are used to evolve excised black holes with the BSSN system.Comment: 10 pages, 9 figure

On permutations of lacunary series

It is a well known fact that for periodic measurable $f$ and rapidly increasing $(n_k)_{k \geq 1}$ the sequence $(f(n_kx))_{k\ge 1}$ behaves like a sequence of independent, identically distributed random variables. For example, if $f$ is a periodic Lipschitz function, then $(f(2^kx))_{k\ge 1}$ satisfies the central limit theorem, the law of the iterated logarithm and several further limit theorems for i.i.d.\ random variables. Since an i.i.d.\ sequence remains i.i.d.\ after any permutation of its terms, it is natural to expect that the asymptotic properties of lacunary series are also permutation-invariant. Recently, however, Fukuyama (2009) showed that a rearrangement of the sequence $(f(2^kx))_{k\ge 1}$ can change substantially its asymptotic behavior, a very surprising result. The purpose of the present paper is to investigate this interesting phenomenon in detail and to give necessary and sufficient criteria for the permutation-invariance of the CLT and LIL for $f(n_kx)$

Geography lost and found in economics

The contribution will consist of two parts. The first part will demonstrate the contrast between the great and increasing importance of geographic factors for economic development and the small and decreasing attention mainstream economic theory paid to spatial aspects until recently. It thereby reveals how much mainstream economic theory cannot explain. The second part will elaborate the geographical innovations of modern economic theory, their under-exposed predecessors and it will try to investigate into the scope of a further approximation. The demonstration of the contrast between the importance of geography for economic development will start with the earliest history: Human settlements developed where the best geographical conditions for human survival existed. As civilisation spread out geographic dissimilarities, especially mineral wealth, induced trade, and for several thousand years the important trading routes dominated the geographical distribution of wealth around the world. The rise of (national) states since the 15th century partitioned the attention of geographic factors into domestic, i.e. regional aspects - central planning to imporve efficiency and strengthen the military potential - and foreign aspects - the importance of an export surplus for acquiring gold, synonymous with national wealth at that time. The dichotomy of geographical influences - regional as different from foreign - persisted until recently. Industrialisation and the steamship, subsequently the railway, enhanced the importance of geography for economics: For the first time a considerable geographical division of labor arose, benefiting from economies of scale, and based on locally available, region-specific factors of production and skill. Neo-classical economic theory nevertheless concentrated on the abstraction of the one-good/two-factor economy, modelled not as an area but as a point in space, and foreign trade theory modelled two such point economies trading with each other, the famous 2-by-2-by-2-case - two countries (points), two factors of production and two goods. The highly abstract neo-classical approach yielded important results, economics came out as a science, but geography was lost en route, for almost a century. Outsiders only dealt with spatial aspects of the economy. Integration in the Post-World War II world enforced a new interest in geography: The slowly increasing de-facto integration of the world economy in the last two decades demonstrated the unrealistic nature of the neo-classical convergence assumptions; why some countries or regions develop better than others got again the honour to be considered as a problem of economic theory. Even more important proved the rapidly increasing integration of Europe: Countries effectively lost their national demand-side policy instruments and a competitive race of regions started. All the old, outside-the-mainstream theories were unearthed, renovated, and marketed in a way, that disguises in many cases how little has actually been added. The second part of the paper therefore will try to collect all the new theoretical attempts to deal with geographic aspects, to investigate into their innovations and to evaluate their explanatory power. Has economics already found all its geographical connections?

Binary black hole mergers: large kicks for generic spin orientations

We present results from several simulations of equal mass black holes with spin. The spin magnitudes are $S/m^2=0.8$ in all cases, but we vary the spin orientations arbitrarily, in and outside the orbital plane. We find that in all but one case the final merged black hole acquires a kick of more than 1000 km/s, indicating that kicks of this magnitude are likely to be generic and should be expected for mergers with general spin orientations. The maximum kick velocity we find is 2500 km/s and occurs for initial spins which are anti-aligned in the initial orbital plane.Comment: 5 pages, 3 figures, to be submitted to PR

The final mass and spin of black hole mergers

We consider black holes resulting from binary black hole mergers. By fitting to numerical results we construct analytic formulas that predict the mass and spin of the final black hole. Our formulas are valid for arbitrary initial spins and mass ratios and agree well with available numerical simulations. We use our spin formula in the context of two common merger scenarios for supermassive galactic black holes. We consider the case of isotropically distributed initial spin orientations (when no surrounding matter is present) and also the case when matter closely aligns the spins with the orbital angular momentum. The spin magnitude of black holes resulting from successive generations of mergers (with symmetric mass ratio $\eta$) has a mean of $1.73\eta + 0.28$ in the isotropic case and 0.94 for the closely aligned case.Comment: Corrected typos, shortened introductio