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!