Gravitational collapse of massless scalar field Bin Wang Shanghai Jiao Tong University

Outline: Classical toy models Gravitational collapse in the asymptotically flat space 1.Spherical symmetric case 2.Different dimensional influence massless scalar + electric field Gravitational collapse in de Sitter space 1.Spherical symmetric case 2.Different dimensional influence massless scalar + electric field

Classical Toy Models A small ball on a plane (x, y); Potential V(x, y)equation of motion location (x(t), y(t)), Toy Model 1: If adding a damping term, ball loss energy

one Toy Model 2:

Flat Spacetime Formalism

Curved Spacetime Formalism measure proper time of a central observer Auxiliary scalar field variables Equations of motion Initial conditions: 0)=0 Gaussian for0)

Competition in Dynamics The kinetic energy of massless field wants to disperse the field to infinity The gravitational potential, if sufficiently dominant during the collapse, will result in the trapping Competing Dynamical competition can be controlled by tuning a parameter in the initial conditions

The Threshold of Black Hole Formation Gundlach, Any trajectory beginning near the critical surface, moves almost parallel to the critical surface towards the critical point. Near the critical point the evolution slows down, and eventually moves away from the critical point in the direction of the growing mode.

The Threshold of Black Hole Formation Parameter P to be either Consider parametrized families of collapse solutions Demand that family “interpolates” between flat spacetime and black hole Black hole formation at some threshold value P Low setting P: no black hole forms High setting: black hole forms P: (amplitude of the Gaussian, the width, center position)

Transformation variables: Curved Spacetime Formalism

The Threshold of Black Hole Formation r=0 t=0 r=0

Type I Type II The Black Hole Mass at The Critical Point Depends on the perturbation fields

Critical Phenomena Interpolating families have critical points where black hole formation just occurs  sufficiently fine-tuning of initial data can result in regions of spacetime with arbitrary high curvature  Precisely critical solutions contain nakes singularities Phenomenology in critical regime analogous to statistical mechanical critical phenomena Mass of the black hole plays the role of order parameter Power-law scaling of black hole mass Scaling behavior of critical solution  Discrete self-similarity (scalar, gravitational, Yang-Mills waves..)  Continued self-similarity (perfect fluid, multiple-scalar systems…)

Discrete Self-Similarity

Self-Similarity: Discrete and Continuous

Critical Collapse in Spherical Symmetry Gundlach et al,

Motivation to Generalize to High Dimensions Vaidya metric in N dimensions Vaidya metric in N dimensions The radial null geodesic Comparing the slope of radial null geodesic and the slope of the apparent horizon near the singular point (v=0,t=0) 4D can have naked singularity, while in higher dimensions, the cosmic censorship is protected Can the black hole be easily created in higher dimensions???

Motivation to Generalize to de Sitter Space Instability of higher dimensional charged black holes in the de Sitter world unstable for large values of the electric charge and cosmological constant in D>=7 (D = 11, ρ = 0.8) q=0.4 (brown) q=0.5 (blue) q=0.6 (green) q=0.7 (orange) q=0.8(red) q=0.9 (magenta). D = 7 (top, black), D = 8 (blue), D = 9 (green), D = 10 (red), D = 11(bottom, magenta). Konoplya, Zhidenko, PRL(09); Cardoso et al, PRD(09) Can the black hole formation be different for charged scalar in higher dimensions dS space???

Gravitational Collapse of Charged Scalar Field in de Sitter Space The total Lagrangian of the scalar field and the electromagnetic field Consider the complex scalar field and the canonical momentum The Lagrange becomes The equation of motion of scalar Expressed in canonical momentum, Matter fields: Hod et al, (1996)

The equation of motion of electromagnetic field Gravitational Collapse of Charged Scalar Field in de Sitter Space Expressed in canonical momentum, Conserved current and charge The energy-momentum tensor of matter fields Matter fields:

Gravitational Collapse of Charged Scalar Field in de Sitter Space Spherical metric Electromagnetic field with scalar field: The equation of motion of scalar EM field: The equation of motion of electric field

Gravitational Collapse of Charged Scalar Field in de Sitter Space Metric constraints: Initial conditions

Competition in Dynamics The kinetic energy of massless field wants to disperse the field to infinity The gravitational potential, if sufficiently dominant during the collapse, will result in the trapping Competing Dynamical competition can be controlled by tuning a parameter in the initial conditions Repulsive force of the Electric field wants to disperse the field to infinity

The Comparison of the Potentials p<p*p>p* 4D dS case Same electric field p* is bigger than the neutral 4D dS case More electric field make p* increase

The Comparison of the Potentials 4D dS7D dS same p Same electric field p* in 4D is bigger than p* in 7D 4DdS: p*= DdS: p*=

The Comparison of the Potentials 7D dS case Same p weak electric fieldstrong electric field With stronger electric field, p* increases to form a black hole

The Comparison of Different Spacetimes 7DdS: p’*= p*<p’* 7Dflat: p*= Q=0 More exact signatures are waited to be disclosed Q not 0 ?? 4DdS: p’*= DdS: p’*= p’*<p* p*<p’* 4Dflat: p*= Dflat: p*=

The Threshold of Black Hole Formation 7D dS, p<p*, No BH 7D dS, p>p*, with BH r=0 r=CH r=0 t=0 How will the scaling law change in diffreent spacetimes with different dimensions???

Outlooks Try to understand dynamics in different spacetimes and dimensions With the increase of dimensions, the formation of BH can be easier Q=0: In low d case, BH can be formed more easily in the dS than in the asymptotically flat space, but the result is contrary in high d Q non zero?? Try to understand the electric field influence on the dynamics 1.Without electric field, the BH is more easily formed 2.With electric field, the BH is more difficult to be formed 3.How will the dimensional influence change with the increase of electric field? 4.Scaling law changes with dimensions and different kinds of spacetimes? Generalize to the gravitational field perturbation More careful numerical computations are needed THANKS!