Gravitational collapse of massless scalar fieldGravitational collapse of massless scalar field

Bin Wang

Shanghai Jiao Tong University

Outline:

• Classical toy models

• Gravitational collapse in the asymptotically flat space1. Spherical symmetric case2. Different dimensional influence massless scalar + electric field

• Gravitational collapse in de Sitter space1. Spherical symmetric case2. 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 for 0)

Competition in Dynamics

The kinetic energy of massless field wants to disperse the field to infinityThe 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 The gravitational potential, if sufficiently dominant during the collapse, will result in the trapping

Competin

gCompetin

g

Dynamical competition can be controlled by tuning a parameter in the initial conditions

The Threshold of Black Hole Formation

Gundlach, 0711.4620

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 formsHigh 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

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 curvaturePrecisely 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, 0711.4620

Motivation to Generalize to High Dimensions

Vaidya metric in N dimensions

The radial null geodesic The radial null geodesic

1503.06651

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 worldInstability 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: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 The energy-momentum tensor of matter fields

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 infinityThe 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

The gravitational potential, if sufficiently dominant during the collapse, will result in the trapping

Competin

gCompetin

g

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 infinityRepulsive 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 fieldp* is bigger than the neutral 4D dS case

More electric field make p* increase

The Comparison of the Potentials

4D dS 7D dSsame p

Same electric field

p* in 4D is bigger than p* in 7D

4DdS:   p*=0.215237 7DdS:   p*=0.17024757

The Comparison of the Potentials

7D dS caseSame p

weak electric field strong electric field

With stronger electric field, p* increases to form a black hole

The Comparison of Different Spacetimes

7DdS: p’*=0.170247

p*<p’*p*<p’*

7Dflat: p*=0.169756

Q=0

More exact signatures are waited to be disclosed

Q not 0 ??

4DdS:   p’*=0.215237 6DdS:   p’*=0.1715763 p’*<p*p’*<p* p*<p’* p*<p’*

4Dflat: p*=0.227824 6Dflat: p*= 0.167516

The Threshold of Black Hole Formation

7D dS, p<p*, No BH

7D dS, p>p*, with BH

r=0

r=0

r=CH

r=CH r=CH

r=0

t=0

t=0t=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 dynamics1. Without electric field, the BH is more easily formed2. 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!

THANKS!