Transcript Developing a Black Hole from Scalar Perturbations

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)
Toy Model 1:
If adding a damping term, ball loss energy
location (x(t), y(t)),
equation of motion
Toy Model 2:
one
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 infinity
Competing
The gravitational potential, if sufficiently dominant during the collapse,
will result in the trapping
Dynamical competition can be controlled by
tuning a parameter in the initial conditions
The Threshold of Black Hole Formation
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.
Gundlach, 0711.4620
The Threshold of Black Hole Formation
• Consider parametrized families of collapse solutions
• Parameter P to be either
P: (amplitude of the Gaussian, the width, center position)
• 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
Curved Spacetime Formalism
Transformation variables:
The Threshold of Black Hole Formation
t=0
r=0
t=0
r=0
The Black Hole Mass at The Critical Point
Type I
Type II
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, 0711.4620
Motivation to Generalize to High Dimensions
1503.06651
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
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)
Gravitational Collapse of Charged Scalar Field in de Sitter Space
Matter fields:
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,
Hod et al, (1996)
Gravitational Collapse of Charged Scalar Field in de Sitter Space
Matter fields:
The equation of motion of electromagnetic field
Expressed in canonical momentum,
Conserved current and charge
The energy-momentum tensor of 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
Repulsive force of the Electric field
wants to disperse the field to infinity
Competing
The gravitational potential, if sufficiently dominant during the collapse, will
result in the trapping
Dynamical competition can be controlled by
tuning a parameter in the initial conditions
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 dS
same p
Same electric field
7D dS
p* in 4D is bigger than p* in 7D
4DdS:
p*=0.215237
7DdS: p*=0.17024757
The Comparison of the Potentials
weak electric field
strong electric field
7D dS case
Same p
With stronger electric field, p* increases to form
a black hole
The Comparison of Different Spacetimes
Q=0
4DdS:
p’*=0.215237
p’*<p*
4Dflat:
p*=0.227824
6DdS: p’*=0.1715763
p*<p’*
6Dflat: p*= 0.167516
7DdS: p’*=0.170247
p*<p’*
7Dflat: p*=0.169756
Q not 0 ??
More exact signatures are waited to be disclosed
The Threshold of Black Hole Formation
r=CH
7D dS, p<p*, No BH
r=0
t=0
r=CH
r=0
t=0
r=CH
r=0
t=0
7D dS, p>p*,
with BH
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!
THANKS!