1. Global Defects near Black Holes
Leandros Perivolaropoulos
Department of Physics, University of Ioannina, Greece
Talk based on:

2. Structure of talk
Previous Work - Questions to address Test particles in a Global Defect Background: Repulsion Global Fields/Defects in a Black Hole background: Repulsion? Is Derrick’s theorem applicable in curved backgrounds?
2. Generalized Derrick’s theorem in curved backgrounds
3. Spherical Domain Wall in a Black Hole background Gravitational Potential (Analytic Energetic Approximation)
Numerical Simulation
4. Global String Loop and Global Monopole in a Black Hole spacetime Gravitational Potential (Analytic Energetic Approximation)

3. Scattering of an infinitely thin (Nambu-Goto) string from a black hole
Nambu-Goto action:
Dynamical equations
Initial Conditions
Numerical simulation in isotropic coordinates
hAB is the internal metric on the worldsheet with determinant h, and GAB is the induced metric on the worldsheet,
ζ0 = τ , ζ3 = σ.
gμ is the spacetime metric.
A black hole deforms a cosmic string in an attractive manner

4. Test particles near global defects
Global monopole exterior metric:
Repulsive potential
Global string exterior metric:
Radial Geodesic:
Repulsive force
Domain wall weak field metric:
Geodesic:
Surface energy density
Repulsive acceleration:

5. Main Questions
1. What is the effect of a black hole background metric on the evolution of scalar fields? Is Derrick’s theorem still valid?
2. Do the repulsive gravitational effects of global defects on test particles persist when the test particle becomes a black hole and the global defect becomes a ‘test defect’?

6. Previous Work I: Static Wall with Black Hole
Static Numerical Solution of Thick Domain Wall Intersecting a Black Hole:

7. Previous Work II: Dynamic Wall with Black Hole
Schwarzschild space-time
Dynamical Simulation of Scattering of Thick Domain Wall on Black Hole:
Kerr spacetime

8. Global Field Dynamics in a Curved Isotropic Background
Dynamical equations from the action:
Isotropic metric:
Scalar field energy:

9. Evading Derrick’s theorem in a curved background
Rescale scalar field configuration:
Rescaled energy:
Causally Connected Range of integration:
For a static solution:
Thus it is possible to evade Derrick’s theorem and have a static solution provided that f’(r) <0 in some range (eg in SdS metric).

10. Spherical Domain Wall in an SdS background
Scalar field energy:

11. Spherical Domain Wall in an SdS background
expanding region
collapsing region
Force on the spherical wall:

12. Black Holes Repel Domain Walls
Force on the spherical wall:
Repulsive term in agreement with recent numerical simulations of planar domain walls close to black holes.

13. Numerical Evolution Λ=0
Solve dynamical equation:
Boundary conditions:
Initial condition (η=1):
Λ=0
Gravitating mass delays the collapse of the spherical wall.

14. Numerical Evolution Λ≠0
Solve dynamical equation:
Boundary conditions:
Initial condition (η=1):

15. Discretize Energy Density:
Energy Minimization
Discretize Energy Density:
N=200
Minimize total energy with fixed boundary conditions at the two horizons
m=0, Λ=0.2
For initial radius r0<r0crit the wall collapses. For r0>r0crit it expands to the outer horizon

16. Energy Minimization (boundary beyond the horizon)
Discretize Energy Density:
N=200
m=0, Λ=0.2
For initial radius r0<r0crit the wall collapses. For r0>r0crit it expands to the outer horizon
Minimize total energy with fixed boundary condition beyond the outer horizon.
Instabilities develop beyond the horizon
Boundary condition beyond the horizon

17. Global Monopole in a SdS spacetime
Global monopole field configuration:
Scalar field action:
Global monopole energy (coordinate center in monopole center):
Diverging angular terms:
Global monopole energy (coordinate center on black hole):

18. Global Monopole in a SdS spacetime
Scalar field energy (coordinate center on black hole):
Approximate form of non-diverging part of energy (r0>>Δr)
Approximations:
Simplified form of monopole energy:

19. Force acting on global monopole
Simplified form of monopole energy:
Force:
Global monopole behaves like a point particle with mass m = 2 π η .

20. Global String Loop in SdS spacetime
Approximate form of non-diverging part of energy (r0>>Δr)
Force:
Tension and cosmological constant determine the dynamics to lowest order (for large radius)

21. Conclusions
Simple energetic arguments tested by numerical simulations can determine the dynamics of global finite width topological defects in curved backgrounds
A black hole mass has a repulsive effect on a spherical domain wall surrounding it.
The repulsive force due to a cosmological constant can overcome the spherical domain wall tension and lead to expanding spherical domain walls if the initial loop radius is larger than a critical value.
The gravitational interaction between a SdS black hole and a global monopole is far from the black hole similar to that between a SdS black hole and a test particle.
A cosmological constant can lead to an expanding global string loop if the initial loop radius is larger than a critical value. In this case the loop tension is overcome by the vacuum energy repulsive force.

22. Scattering/capture of global defects by black holes
Outlook
Confirm numerically the analytic toy model results for the case of global monopole-black hole and global string loop-black hole interactions.
Strong gravity effects: gravitational waves from the collision of global defects with black holes (numerical simulation and analytical approximation)
Global defects in different curved backgrounds (RN or Kerr black holes etc)
Scattering/capture of global defects by black holes