Planetoid String Solutions in 3 + 1 Axisymmetric Spacetimes

The string propagation equations in axisymmetric spacetimes are exactly solved by quadratures for a planetoid Ansatz. This is a straight non-oscillating string, radially disposed, which rotates uniformly around the symmetry axis of the spacetime. In Schwarzschild black holes, the string stays outside the horizon pointing towards the origin. In de Sitter spacetime the planetoid rotates around its center. We quantize semiclassically these solutions and analyze the spin/(mass$^2$) (Regge) relation for the planetoids, which turns out to be non-linear.Comment: Latex file, 14 pages, two figures in .ps files available from the author

Strings Next To and Inside Black Holes

The string equations of motion and constraints are solved near the horizon and near the singularity of a Schwarzschild black hole. In a conformal gauge such that $\tau = 0$ ($\tau$ = worldsheet time coordinate) corresponds to the horizon ($r=1$) or to the black hole singularity ($r=0$), the string coordinates express in power series in $\tau$ near the horizon and in power series in $\tau^{1/5}$ around $r=0$. We compute the string invariant size and the string energy-momentum tensor. Near the horizon both are finite and analytic. Near the black hole singularity, the string size, the string energy and the transverse pressures (in the angular directions) tend to infinity as $r^{-1}$. To leading order near $r=0$, the string behaves as two dimensional radiation. This two spatial dimensions are describing the $S^2$ sphere in the Schwarzschild manifold.Comment: RevTex, 19 pages without figure

String dynamics in cosmological and black hole backgrounds: The null string expansion

We study the classical dynamics of a bosonic string in the $D$--dimensional flat Friedmann--Robertson--Walker and Schwarzschild backgrounds. We make a perturbative development in the string coordinates around a {\it null} string configuration; the background geometry is taken into account exactly. In the cosmological case we uncouple and solve the first order fluctuations; the string time evolution with the conformal gauge world-sheet $\tau$--coordinate is given by $X^0(\sigma, \tau)=q(\sigma)\tau^{1\over1+2\beta}+c^2B^0(\sigma, \tau)+\cdots$, $B^0(\sigma,\tau)=\sum_k b_k(\sigma)\tau^k$ where $b_k(\sigma)$ are given by Eqs.\ (3.15), and $\beta$ is the exponent of the conformal factor in the Friedmann--Robertson--Walker metric, i.e. $R\sim\eta^\beta$. The string proper size, at first order in the fluctuations, grows like the conformal factor $R(\eta)$ and the string energy--momentum tensor corresponds to that of a null fluid. For a string in the black hole background, we study the planar case, but keep the dimensionality of the spacetime $D$ generic. In the null string expansion, the radial, azimuthal, and time coordinates $(r,\phi,t)$ are $r=\sum_n A^1_{n}(\sigma)(-\tau)^{2n/(D+1)}~,$ $\phi=\sum_n A^3_{n}(\sigma)(-\tau)^{(D-5+2n)/(D+1)}~,$ and $t=\sum_n A^0_{n} (\sigma)(-\tau)^{1+2n(D-3)/(D+1)}~.$ The first terms of the series represent a {\it generic} approach to the Schwarzschild singularity at $r=0$. First and higher order string perturbations contribute with higher powers of $\tau$. The integrated string energy-momentum tensor corresponds to that of a null fluid in $D-1$ dimensions. As the string approaches the $r=0$ singularity its proper size grows indefinitely like $\sim(-\tau)^{-(D-3)/(D+1)}$. We end the paper giving three particular exact string solutions inside the black hole.Comment: 17 pages, REVTEX, no figure

Effects of regulation on a self-organized market

Adapting a simple biological model, we study the effects of control on the market. Companies are depicted as sites on a lattice and labelled by a fitness parameter (some `company-size' indicator). The chance of survival of a company on the market at any given time is related to its fitness, its position on the lattice and on some particular external influence, which may be considered to represent regulation from governments or central banks. The latter is rendered as a penalty for companies which show a very fast betterment in fitness space. As a result, we find that the introduction of regulation on the market contributes to lower the average fitness of companies.Comment: 7 pages, 2 figure

An analytical proof of Hardy-like inequalities related to the Dirac operator

We prove some sharp Hardy type inequalities related to the Dirac operator by elementary, direct methods. Some of these inequalities have been obtained previously using spectral information about the Dirac-Coulomb operator. Our results are stated under optimal conditions on the asymptotics of the potentials near zero and near infinity.Comment: LaTex, 22 page

Strings Propagating in the 2+1 Dimensional Black Hole Anti de Sitter Spacetime

We study the string propagation in the 2+1 black hole anti de Sitter background (2+1 BH-ADS). We find the first and second order fluctuations around the string center of mass and obtain the expression for the string mass. The string motion is stable, all fluctuations oscillate with real frequencies and are bounded, even at $r=0.$ We compare with the string motion in the ordinary black hole anti de Sitter spacetime, and in the black string background, where string instabilities develop and the fluctuations blow up at $r=0.$ We find the exact general solution for the circular string motion in all these backgrounds, it is given closely and completely in terms of elliptic functions. For the non-rotating black hole backgrounds the circular strings have a maximal bounded size $r_m,$ they contract and collapse into $r=0.$ No indefinitely growing strings, neither multi-string solutions are present in these backgrounds. In rotating spacetimes, both the 2+1 BH-ADS and the ordinary Kerr-ADS, the presence of angular momentum prevents the string from collapsing into $r=0.$ The circular string motion is also completely solved in the black hole de Sitter spacetime and in the black string background (dual of the 2+1 BH-ADS spacetime), in which expanding unbounded strings and multi-string solutions appear.Comment: Latex, 54 pages + 2 tables and 4 figures (not included). PARIS-DEMIRM 94/01

Circular String-Instabilities in Curved Spacetime

We investigate the connection between curved spacetime and the emergence of string-instabilities, following the approach developed by Loust\'{o} and S\'{a}nchez for de Sitter and black hole spacetimes. We analyse the linearised equations determining the comoving physical (transverse) perturbations on circular strings embedded in Schwarzschild, Reissner-Nordstr\"{o}m and de Sitter backgrounds. In all 3 cases we find that the "radial" perturbations grow infinitely for $r\rightarrow 0$ (ring-collapse), while the "angular" perturbations are bounded in this limit. For $r\rightarrow\infty$ we find that the perturbations in both physical directions (perpendicular to the string world-sheet in 4 dimensions) blow up in the case of de Sitter space. This confirms results recently obtained by Loust\'{o} and S\'{a}nchez who considered perturbations around the string center of mass.Comment: 24 pages Latex + 2 figures (not included). Observatoire de Paris, Meudon No. 9305

Strings in Cosmological and Black Hole Backgrounds: Ring Solutions

The string equations of motion and constraints are solved for a ring shaped Ansatz in cosmological and black hole spacetimes. In FRW universes with arbitrary power behavior [R(X^0) = a\;|X^0|^{\a}\, ], the asymptotic form of the solution is found for both $X^0 \to 0$ and $X^0 \to \infty$ and we plot the numerical solution for all times. Right after the big bang ($X^0 = 0$), the string energy decreasess as $R(X^0)^{-1}$ and the string size grows as $R(X^0)$ for $0 1$. Very soon [ $X^0 \sim 1$] , the ring reaches its oscillatory regime with frequency equal to the winding and constant size and energy. This picture holds for all values of \a including string vacua (for which, asymptotically, \a = 1). In addition, an exact non-oscillatory ring solution is found. For black hole spacetimes (Schwarzschild, Reissner-Nordstr\oo m and stringy), we solve for ring strings moving towards the center. Depending on their initial conditions (essentially the oscillation phase), they are are absorbed or not by Schwarzschild black holes. The phenomenon of particle transmutation is explicitly observed (for rings not swallowed by the hole). An effective horizon is noticed for the rings. Exact and explicit ring solutions inside the horizon(s) are found. They may be interpreted as strings propagating between the different universes described by the full black hole manifold.Comment: Paris preprint PAR-LPTHE-93/43. Uses phyzzx. Includes figures. Text and figures compressed using uufile