Riccati equations for holographic 2-point functions

Any second order homogeneous linear ordinary differential equation can be transformed into a first order non-linear Riccati equation. We argue that the Riccati form of the linearized fluctuation equations that determine the holographic 2-point functions simplifies considerably the numerical computation of such 2-point functions and of the corresponding transport coefficients by computing directly the response functions, eliminating the arbitrary source from the start. Moreover, it provides a neat criterion for the infrared regularity of the fluctuations. In particular, it is shown that the infrared regularity conditions for scalar and tensor fluctuations coincide, and hence they are either both regular or both singular. We demonstrate our numerical recipe based on the Riccati equations by computing the holographic 2-point functions for the stress tensor and a scalar operator in a number of asymptotically anti de Sitter backgrounds of bottom up scalar-gravity models. Analytical results are obtained for the 2-point function of the transverse traceless part of the stress tensor in two confining geometries, including a geometry that belongs to the class of IHQCD. We find that in this background the spin-2 spectrum is linear and, as expected, the position space 2-point function decays exponentially at large distances at a rate proportional to the confinement scale.Comment: 33 pages, 5 figures, 2 appendices. Changes with respect to V1: major extension of the numerical and analytical analysis. Added lemma 5.1, appendices A and B and references. Corrected typo

Mini-Black-Hole production at RHIC and LHC

We argue that heavy-ion collisions provide the best testing ground for mini-black hole physics as $M_P\simeq 4 GeV$ for the gravity dual of YM and give concrete evidence for a new extra dimension, that is visible only to the strong interactions. We analyse the process of production evolution and decay of the mini-black-holes by using recent results on gravity duals of YM. There are several novelties compared with the traditional story of black hole evaporation, including Bjorken scaling instead of sphericity, evaporation via bubble nucleation instead of the Hawking mechanism and lepton-poor final states. Multiplicities are estimated using shock-wave scattering techniques. It is argued that high-multiplicity/high energy pp collisions will also show similar characteristics of mini-black-hole production and decay.Comment: Latex, 10 pages, 2 figure

Evolving Geometries in General Relativity

The problem of collisions of shockwaves in gravity is well known and has been studied extensively in the literature. Recently, the interest in this area has been revived trough the anti-de-Sitter space/Conformal Field Theory correspondence (AdS/CFT) with the difference that in this case the background geometry is Anti de Sitter in five dimensions. In a recent project that we have completed in the context of AdS/CFT, we have gained insight in the problem of shockwaves and our goal in this work is to apply the technique we have developed there in the case of ordinary gravity. In the current project, each of the shockwaves correspond to a point-like Stress-Energy tensor that moves with the speed of light while the collision is asymmetric and involves an impact parameter (b). Our method is to expand the metric $(g_{\mu \nu})$ in the background of flat space-time in the presence of the two shockwaves and compute corrections that satisfy causal boundary conditions taking into account back-reactions of the Stress-Energy tensor of the two point-like particles. Our solution respects causality as expected but this casual dependence takes place in an intuitive way. In particular, $g_{\mu \nu}$ at any given point $\vec{r}$ on the transverse plane at fixed $\tau$ evolves according from whether the propagation from the center of each of the shockwaves or from both shockwaves has enough proper time ($\tau$) to reach the point under consideration or not. Simultaneously around the center of each shockwave, the future metric develops a $\delta$-function profile with radius $\tau$; therefore this profile expands outwards from the centers (of the shockwaves) with the speed of light. Finally, we discuss the case of the zero impact parameter collision which results to the violation of conservation and we argue that this might be a signal for the formation of a black hole.Comment: 60 pages, 7 figures; Master Thesis: Department of Mathematics in The Ohio State Universit