Gravitational collapse of massless scalar field

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

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

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

4. Toy Model 2: one

5. Flat Spacetime Formalism

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

7. 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

8. 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

9. 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

10. Curved Spacetime Formalism Transformation variables:

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

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

13. 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…)

14. Discrete Self-Similarity

15. Self-Similarity: Discrete and Continuous

16. Critical Collapse in Spherical Symmetry Gundlach et al, 0711.4620

17. 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) Can the black hole be easily created in higher dimensions??? 4D can have naked singularity, while in higher dimensions, the cosmic censorship is protected

18. 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 Can the black hole formation be different for charged scalar in higher dimensions dS space??? (D = 11, ρ = 0.8) D = 7 (top, black), D = 8 (blue), D = 9 (green), D = 10 (red), D = 11(bottom, magenta). q=0.4 (brown) q=0.5 (blue) q=0.6 (green) q=0.7 (orange) q=0.8(red) q=0.9 (magenta). Konoplya, Zhidenko, PRL(09); Cardoso et al, PRD(09)

19. 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)

20. 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

21. 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

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

23. 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

24. 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

25. The Comparison of the Potentials same p Same electric field 4D dS 7D dS p* in 4D is bigger than p* in 7D 4DdS:   p*=0.215237 7DdS:   p*=0.17024757

26. 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

27. The Comparison of Different Spacetimes Q=0 4DdS:   p’*=0.215237 6DdS:   p’*=0.1715763 p’*<p*p*<p’* 4Dflat: p*=0.227824 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

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

29. 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 • Without electric field, the BH is more easily formed • With electric field, the BH is more difficult to be formed • How will the dimensional influence change with the increase of electric field? • Scaling law changes with dimensions and different kinds of spacetimes? • Generalize to the gravitational field perturbation More careful numerical computations are needed THANKS!

30. THANKS!