Gravitational Collapse and Holographic Thermalization
- Autores: Javier Abajo Arrastia
- Directores de la Tesis: Esperanza Lopéz Manzanares (dir. tes.)
- Lectura: En la Universidad Autónoma de Madrid ( España ) en 2016
- Idioma: español
- Tribunal Calificador de la Tesis: Germán Sierra Rodero (presid.), Enrique Álvarez Vázquez (secret.), Óscar Dias (voc.), Erik Tonni (voc.), Joan Simon Soler (voc.)
- Materias:
- Enlaces
- Tesis en acceso abierto en: Biblos-e Archivo
- Resumen
The entanglement entropy, computed according to Ryu-Takayanagi's holographic proposal, has been the main character along this work. It has been calculated for two different gravitational collapse scenarios, and it has allowed confirming and predicting results regarding to the equilibration dynamics of the dual quantum systems. Gravitational collapse in AdS has thus confirmed as an appropriate holographic model of thermalization of a quantum system.
The difficulties overcome to get the present results had their origin mainly in the numerical computations required to solve the involved equations. Shooting and relaxation methods have been used to solve the extremization equations with prescribed boundary conditions that define the surface related to the holographic entanglement entropy. In addition, solving the scalar-gravity system has required a lot of effort to attain the neces- sary stability and convergence. At each slice of constant time, equations have been solved using four points stencils integrators. A fourth order Runge-Kutta method has been used to advance in time. Boundary points have demanded a special care, and ghost points have extended the interval according to the power series form of the corresponding solution, as well as dissipation routines for the propagation of numerical errors have been implemented.
In Chapter 4 the Vaidya metric in Poincaré coordinates in AdS_3 was used as a first holographic model of thermalization. The behaviour of the holographic entanglement entropy reproduced the one found for the quantum quench dynamics of two dimensional CFTs by Cardy and Calabrese, essentially characterized by a causal propagation of the quasiparticles sourced by the perturbation at a finite velocity and a local thermalization as results of it. The horizon effect has appeared with the associated expected velocity v_E = 1. Also the effective thermalization of subsystems smaller than 2v_Et has been described holographically in terms of the particular form of the geodesics in this regime, which reach very close to the horizon but do not traverse it, and their length is actually the length of the horizon (confirming at the local level of a subsystem the correspondence between its thermal entropy and the Bekenstein-Hawking entropy of the dual AdS black hole).
Moreover, the analysis of the infinite size intervals has allowed obtaining more details and has determined that after a characteristic time this causal behavior is the only effect that drives the evolution. The early time dynamics has also been analyzed, finding that the apparent horizon does not play a relevant role at this stage. On the other side, for the later causal dynamics the spacelike geodesics relevant for the entanglement entropy contain a piece inside the horizon. The contribution to the total length due to this part grows linearly with time and accounts for the increase of the entanglement entropy in the dual subsystem as time evolves. In addition, it has been remarked how the holographic entanglement entropy proposal implies a manifest unitary evolution in collapse scenarios dual to thermalization processes, as long as there is no singularity in the asymptotic past of the spacetime manifold.
In Chapter 5 the objective was to study holographically some aspects of the equi- libration dynamics of closed quantum systems. Following the gravity works regarding instability of AdS under scalar perturbations, the aim was to further analyze the gravitational collapse process and to use holography to describe the relaxation dynamics of the dual finite size quantum systems. For small enough amplitudes, this setup contains solutions corresponding to gravitational collapse processes requiring several bounces off the AdS boundary to generate a horizon. These spacetimes have been shown able to model holographically field theory relaxation processes with large dephasing times. Contrarily, they have not proven suitable to realize evolutions with a prethermalization stage. This is consistent with the fact that this gravity system only has the mass as conserved quantity, while prethermalization is typically associated to the closeness of a system to an integrable theory.
It has been proposed in Chapter 5 that the radial position of the collapsing scalar shell encodes the typical separation of entangled components in the dual out of equilibrium quantum system. When the pulse is close to the boundary, entanglement is concentrated among nearby excitations. Its fall towards the interior geometry corresponds to entangled excitations moving apart, as the holographic model for a local quench of Takayanagi et al. The travel back of the shell to the boundary is then associated to the entangled components approaching again on the boundary sphere. This interpretation is aligned with works that directly link the entanglement pattern of a field theory with a higher dimensional geometry.
As in the Vaidya model, the entanglement entropy evolution follows the one of the metric, and in this case it is shown by the obtained oscillatory pattern. For narrow scalar profiles, the periodicity of the entanglement entropy oscillations is always close to π. This is indeed the time needed by the quasiparticle excitations of the dual field theory process to travel along the equator of the boundary sphere and meet again , and provides the main support for the advocated picture. Consistently, the minima of the entanglement entropy happen when the scalar shell bounces at the boundary and the maxima are reached when the shell is close to the origin.
The entanglement entropy is not only able to detect propagation, but also interaction effects related to the internal dynamics of the shell. A main characteristic of this dynamics is weak turbulence, that governs the appearance of a horizon for profiles of small mass. It has been shown how, besides the generation of a sufficiently sharp front for collapse, there is an accompanying effect of radial dispersion in the scalar profile leftover by the emerging horizon. Interestingly, this flow of energy towards opposite scales in the bulk, is reflected in a decrease of the entanglement entropy maxima along the pre-horizon evolution.
In order to get a complete picture of the field theory relaxation process, the gravitational evolution was followed until a static black hole was almost established. For narrow initial pulses, the fraction of the scalar profile escaping from the emergent horizon exhibits radial localization along several further cycles of partial absorptions and bounces against the boundary. According to the proposed interpretation, this suggests that while part of the system has dephased, a subset of degrees of freedom maintains quantum coherence. Moreover, a typical separation can be associated to the remaining entangled components, which still evolves with a period close to π. This stepwise pattern of relaxation is an important result of the analysis.
The amplitude of the entanglement entropy oscillations neatly reflects the formation of a horizon, decreasing as the trapped mass increases. In spite of this, the emergence of an apparent horizon does not leave a sharp imprint on the entanglement entropy. Neither do the regions of its stepwise growth. Along the post-horizon evolution the entanglement entropy exhibits damped but smooth oscillations.
Finally, in Chapter 6 some details of the gravitational collapse solutions found in Chapter 5 have been given, by defining and analyzing the evolution of spectral decompositions of the scalar field and the energy. This point of view has complemented the description made in the previous chapter which considers the radial distribution of the energy pulse along the evolution. Also, the mode analysis has stressed that the non linear self interaction of the scalar field, mediated by the coupling to gravity, is responsible for the concentration of energy in smaller scales. This has been manifest for narrow pulses, which travel almost freely through AdS and the interaction takes place at the shocks in the origin. Correspondingly, the envelope of the spectral distributions is approximately constant except at the instants of the shocks, when a steep increase is observed that im- plies the excitation of higher modes. Broad pulses have not shown this behavior, as the transfer of energy to higher modes is disordered and continuous, which is consistent with the substructure of such pulses as described in Chapter 5, since they contain subpulses going forth and forward and thus interactions take place continuously.
