boson stars in axisymmetry -...

Boson Stars in Axisymmetry

Kevin Lai

Department of Physics and AstronomyUniversity of British Columbia

Vancouver [email protected]

APS April Meeting 2004Denver, Colorado

March 3, 2004

COLLABORATOR: M. Choptuik, D. Choi

BOSON STARS IN AXISYMMETRY 1

Outline

• A Brief Introduction to Boson Stars

• The (2+1)+1 Formalism

• The GRAXI code

• Code Test

• Binary Collisions of Boson Stars

• Perturbation of Boson Stars by Real Scalar Field

• Summary


A Brief Introduction to Boson Stars

• Wheeler 1955:

? (EM) Geons: gravitating systems which are held together by gravitationalforces and are composed of fundamental, classical fields



• Wheeler 1955:

? (EM) Geons: gravitating systems which are held together by gravitationalforces and are composed of fundamental, classical fields

• Kaup 1968 & Ruffini 1969:

? Klein-Gordon geon: self-gravitating compact objects consists of scalarparticles which satisfy the Klein-Gordon equation

? Classical, massive complex scalar field

? Balance between attractive force of gravity and dispersive nature of wave

? Field solutions have different number of nodes, where the ground statehas no nodes



• Boson stars = Stationary solutions to Einstein Klein-Gordon system

• The action: ∫d4x√−g

[R

16π− 1

2(∇µφ∇µφ∗ + m2φ∗φ

)]• EOM:

Gµν = 8πTµν

∇µ∇µφ−m2φ = 0

where

Tµν =12

(∇µφ∗∇νφ +∇µφ∇νφ∗)−12

gµν(∇αφ∗∇αφ + m2φ∗φ

)



• Boson Stars Ansatz:

? 1D case

φ = φ0(r)e−iωt

? 2D case

φ = φ0(ρ, z)e−i(ωt−kϕ)



• A typical solution in spherical symmetry:


The (2+1)+1 Formalism

Z

• Project and describe the object as (2+1)+1

• Use the Killing vector ξ to define a project operator Hαβ ≡ δαβ −

ξαξβξγξγ

.


The GRAXI Code

• GRAXI (GR AXIsymmetric): developed by M. Choptuik, E. Hirschmann, S.Liebling, F. Pretorius

• Designed to solve the Einstein field equations for axisymmetric spacetimes

• Goal: study gravitational collapse, critical phenomena, head-on black holecollisions, head-on stars collisions

• 2+1+1 formalism

• Adaptive mesh refinement (AMR), multigrid, black hole excision


Code Tests

• Initial data: Interpolate spherically symmetric boson stars from 1D to 2D

Evolving (theoretically) static spacetime,

animation shows the modulus of the scalar field

|φ| for φ(0) = 0.02

Evolving (theoretically) static spacetime,

animation shows the conformal factor ψ for

φ(0) = 0.02


Code Tests

• Assume a harmonic time dependence of field perturbation ∼ eiσt:

Simulation: σ2 ≈ 0.00032 Perturbation Theory: σ2 ≈ 0.00035

maximum value of |φ| as a function of time. The

period of oscillation T ≈ 380

ADM mass as a function of time


Convergence Tests

maximum value of |φ| as a function of time

for 4 different resolutions. The value

converge to a constant

Convergence factor uhl+2−uhl+1

uhl+1−uhl

versus time. The average is close to 4 and

shows a second order convergence

ADM mass as a function of time for the 4

different resolutions. The value converge to

a constant


Binary Collisions of Boson Stars

• Initial data: Two identical boson stars boosted towards each other (nearcritical solution)

Supercritical evolution of two identical

boson stars with φ(0) = 0.02: the stars

merge on the first encounter

Subcritical evolution of two identical boson

stars with φ(0) = 0.02: the stars do not

merge on the first encounter, but merge on

the second


Critical Phenomena: Binary Collisions

Time of black hole formation tBH v.s.log(p∗z − pz)


Perturbation by Real Scalar Field: In Progress

Real massless perturbingscalar field φ

Complex massive scalarfield Φ (boson star)

max(Φ, φ)


Summary

• Boosted boson stars show solitonic behavior in relativistic regime

• Critical phenomena (scaling law) is observed in:

? axisymmetric binary boson stars collisions? perturbation by non-spherical real scalar field

• Momentum transfer?

boson stars in axisymmetry -...

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