Book reviews

Teaching/Communication/Extension/Profession,

The Newtonian potential of thin disks

The one-dimensional, ordinary differential equation (ODE) by Hur\'e & Hersant (2007) that satisfies the midplane gravitational potential of truncated, flat power-law disks is extended to the whole physical space. It is shown that thickness effects (i.e. non-flatness) can be easily accounted for by implementing an appropriate "softening length" $\lambda$. The solution of this "softened ODE" has the following properties: i) it is regular at the edges (finite radial accelerations), ii) it possesses the correct long-range properties, iii) it matches the Newtonian potential of a geometrically thin disk very well, and iv) it tends continuously to the flat disk solution in the limit $\lambda \rightarrow 0$. As illustrated by many examples, the ODE, subject to exact Dirichlet conditions, can be solved numerically with efficiency for any given colatitude at second-order from center to infinity using radial mapping. This approach is therefore particularly well-suited to generating grids of gravitational forces in order to study particles moving under the field of a gravitating disk as found in various contexts (active nuclei, stellar systems, young stellar objects). Extension to non-power-law surface density profiles is straightforward through superposition. Grids can be produced upon request.Comment: Accepted for publication in A&

A substitute for the singular Green kernel in the Newtonian potential of celestial bodies

The "point mass singularity" inherent in Newton's law for gravitation represents a major difficulty in accurately determining the potential and forces inside continuous bodies. Here we report a simple and efficient analytical method to bypass the singular Green kernel 1/|r-r'| inside the source without altering the nature of the interaction. We build an equivalent kernel made up of a "cool kernel", which is fully regular (and contains the long-range -GM/r asymptotic behavior), and the gradient of a "hyperkernel", which is also regular. Compared to the initial kernel, these two components are easily integrated over the source volume using standard numerical techniques. The demonstration is presented for three-dimensional distributions in cylindrical coordinates, which are well-suited to describing rotating bodies (stars, discs, asteroids, etc.) as commonly found in the Universe. An example of implementation is given. The case of axial symmetry is treated in detail, and the accuracy is checked by considering an exact potential/surface density pair corresponding to a flat circular disc. This framework provides new tools to keep or even improve the physical realism of models and simulations of self-gravitating systems, and represents, for some of them, a conclusive alternative to softened gravity.Comment: Accepted for publication in A&A; 7 pages, color figure

A Lifshitz Black Hole in Four Dimensional R^2 Gravity

We consider a higher derivative gravity theory in four dimensions with a negative cosmological constant and show that vacuum solutions of both Lifshitz type and Schr\"{o}dinger type with arbitrary dynamical exponent z exist in this system. Then we find an analytic black hole solution which asymptotes to the vacuum Lifshitz solution with z=3/2 at a specific value of the coupling constant. We analyze the thermodynamic behavior of this black hole and find that the black hole has zero entropy while non-zero temperature, which is very similar to the case of BTZ black holes in new massive gravity at a specific coupling. In addition, we find that the three dimensional Lifshitz black hole recently found by E. Ayon-Beato et al. has a negative entropy and mass when the Newton constant is taken to be positive.Comment: 11 pages, no figure; v2, a minor error correcte

Non-relativistic metrics from back-reacting fermions

It has recently been pointed out that under certain circumstances the back-reaction of charged, massive Dirac fermions causes important modifications to AdS_2 spacetimes arising as the near horizon geometry of extremal black holes. In a WKB approximation, the modified geometry becomes a non-relativistic Lifshitz spacetime. In three dimensions, it is known that integrating out charged, massive fermions gives rise to gravitational and Maxwell Chern-Simons terms. We show that Schrodinger (warped AdS_3) spacetimes exist as solutions to a gravitational and Maxwell Chern-Simons theory with a cosmological constant. Motivated by this, we look for warped AdS_3 or Schrodinger metrics as exact solutions to a fully back-reacted theory containing Dirac fermions in three and four dimensions. We work out the dynamical exponent in terms of the fermion mass and generalize this result to arbitrary dimensions.Comment: 26 pages, v2: typos corrected, references added, minor change

Asymptotic behaviour of a semilinear elliptic system with a large exponent

Consider the problem \begin{eqnarray*} -\Delta u &=& v^{\frac 2{N-2}},\quad v>0\quad {in}\quad \Omega, -\Delta v &=& u^{p},\:\:\:\quad u>0\quad {in}\quad \Omega, u&=&v\:\:=\:\:0 \quad {on}\quad \partial \Omega, \end{eqnarray*} where $\Omega$ is a bounded convex domain in $\R^N,$ $N>2,$ with smooth boundary $\partial \Omega.$ We study the asymptotic behaviour of the least energy solutions of this system as $p\to \infty.$ We show that the solution remain bounded for $p$ large and have one or two peaks away form the boundary. When one peak occurs we characterize its location.Comment: 16 pages, submmited for publicatio

Topologically Massive Gravity and Ricci-Cotton Flow

We consider Topologically Massive Gravity (TMG), which is three dimensional general relativity with a cosmological constant and a gravitational Chern-Simons term. When the cosmological constant is negative the theory has two potential vacuum solutions: Anti-de Sitter space and Warped Anti-de Sitter space. The theory also contains a massive graviton state which renders these solutions unstable for certain values of the parameters and boundary conditions. We study the decay of these solutions due to the condensation of the massive graviton mode using Ricci-Cotton flow, which is the appropriate generalization of Ricci flow to TMG. When the Chern-Simons coupling is small the AdS solution flows to warped AdS by the condensation of the massive graviton mode. When the coupling is large the situation is reversed, and warped AdS flows to AdS. Minisuperspace models are constructed where these flows are studied explicitly

Wide Field Photometry of the Galactic Globular Cluster M22

We present wide field photometry of the Galactic Globular Cluster M~22 in the B, V and I passbands for more than 186,000 stars. The study is complemented by the photometry in two narrowband filters centered on H$_{\alpha}$ and the adjacent continuum, and by infrared J, H and K magnitudes derived from the 2 MASS survey for $\sim$2000 stars. Profiting from this huge database, we completely characterized the evolved stellar sequences of the cluster by determining a variety of photometric parameters, including new photometric estimates of the mean metallicity, reddening and distance to the cluster. In particular, from our multi-wavelength analysis, we re-examined the long-standing metallicity spread problem in M~22. According to our dataset, we conclude that most of the observed width of the red giant branch must be due to differential reddening, which amounts to a maximum of $\Delta E(B-V)\simeq0.06$, although the presence of a small metallicity spread cannot be completely ruled out. More specifically, the maximum metallicity spread allowed by our data is of the order of $\Delta$[Fe/H]$\simeq 0.1\div 0.2$ dex, i.e., not much more than what allowed by the photometric errors. Finally, we identified most of the known variable stars and peculiar objects in our field of view. In particular, we find additional evidence supporting previous optical identifications of the central star of the Planetary Nebula IRAS 18333-2357, which is associated with M~22.Comment: 15 pages, 16 figures, accepted for publication in MNRA

Maximal parabolic regularity for divergence operators including mixed boundary conditions

We show that elliptic second order operators $A$ of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth, the coefficients of $A$ are discontinuous and $A$ is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented.Comment: 39 pages, 4 postscript figure

A boundary stress tensor for higher-derivative gravity in AdS and Lifshitz backgrounds

We investigate the Brown-York stress tensor for curvature-squared theories. This requires a generalized Gibbons-Hawking term in order to establish a well-posed variational principle, which is achieved in a universal way by reducing the number of derivatives through the introduction of an auxiliary tensor field. We examine the boundary stress tensor thus defined for the special case of `massive gravity' in three dimensions, which augments the Einstein-Hilbert term by a particular curvature-squared term. It is shown that one obtains finite results for physical parameters on AdS upon adding a `boundary cosmological constant' as a counterterm, which vanishes at the so-called chiral point. We derive known and new results, like the value of the central charges or the mass of black hole solutions, thereby confirming our prescription for the computation of the stress tensor. Finally, we inspect recently constructed Lifshitz vacua and a new black hole solution that is asymptotically Lifshitz, and we propose a novel and covariant counterterm for this case.Comment: 25 pages, 1 figure; v2: minor corrections, references added, to appear in JHE