● Introduction
●GR equations for simulation
●Prior work of critical phenomena in general relativity
● Axisymmetrical Code
●Idea, implement, advantage
●Convergence tests: resolution, boundary position
● Critical Phenomena in axisymmetric Collapse of Neutron Stars
●Universality, convergence, index
●Possibility of existence in the universe
● Conclusion
Can Critical Collapse Occur in Nature ?Ke Jian Jin
Sec I-2 Prior work on critical phenomena in GR
• Begin with Matthew W. Choptuik • Most studies spherical (with few exceptions)
– Scalar field, YM, MP, Wave;– Perfect fluid: Wave pocket
• Limitations:– System not realistic– Do not know much beyond spherical
Sec II Axisymmetrical Code
For a system of axisymmetry, only need to develop one radial slice Based on 3D cartesian coordinate code
Very stable Evolution On A Quasi-2D Grid Lots of utilities available
Implement as Boundary Condition after Evolution Advantage
323^3 (3D) = 2560x5x2560 (2D)
3 R K 2 K ijKij 16 ADM 0,
j Kij ij
j K 8 j i 0.Hamiltonian constraint equation
Momentum constraint equations
Practically they should be convergent to zero in certain order with raising resolution:
For IVP(t=0), there should be 2nd order convergent (~h2);For evolution(t>0), it should be 1st order convergent (~h1) in the peak due to TVD, and so for the long time run.
Read the order from Fig.:Raise resolution: h2 = (1/2)*h1, error e2 = (1/2)*e1;Scaling: e2 -> 2*e2 , => e2 = e1, the two curves will overlap.
Sec II-2 Convergence Test of the Axisymmetric Code
Convergence of headon process (4) Boundary Effect Sec II-2
The momentum constraints is convergent over step length, but not over boundary effect. The boundary effect will be bigger than the interior value for higher resolution run. For small sized grid, the boundary effect will propagate in and ruin the convergence over long time.
Convergence of headon process (5) Boundary (Grid Size) Effect Sec II-2
Short time, nearly same Momentum Constraints Longer time, small grid size one worse
Boundary (Grid Size) Effect Momentum vs. hamiltonian at t=324 Sec II-2
MomentumObvious, Convergent
HamiltonianNearly Independent
Sec III Critical Phenomena in Axisymmetric Collapse of Neutron Stars
Movie: density oscillates with time in one collision which is near critical point
Density as a heightmovie on wugrav
Critical Phenomena Universality (1) Sec III-1Minimum lapse vs. time (varying density, falling from infinity)
Two heavy stars collide into a black hole; the lapse in collision center dips into zero. On the other hand, two light stars collide, the lapse dips, rebounds up. When two stars with the critical density collide, the lapse will dip, rebound, dip, rebound,...
Critical Phenomena Universality (2) Sec III-1
Minimum lapse vs. time (varying velocity, fixed density & separation)
Critical Phenomena Universality (3) Sec III-1Minimum lapse vs. time (varying density, fixed velocity & separation)
Critical Phenomena Get the departure (from the critical curve) time Sec III-1
vs. time vs. time
When α(t,ρ) or α(t,v) departures from the critical one α*? We made several criterions, that are: 5%, 10%, 15%, 20%. The following figure is for (α-α*)/α* = 5%.
Convergent Critical Index & Universality (1) Section III-1
varying density, and showing the convergence
t const
1 0
0
lo g .
Convergent Critical Index & Universality (2) Section III-1
varying velocity, and showing the convergence
tv v
vconst
1 0
0
lo g .
Convergent Critical Index & Universality (3) Section III-1
Evidence showing universality (from dx=0.12):
vary density falling from infinity: 10.87+-0.04 vary velocity with fixed d & s: 10.78+-0.05 vary density with fixed v & s: 10.67+-0.06
Sec III-2 Possibility of Existence in the Universe
● Fine tuning of parameter hard to realize● EOS could vary continuously● The duration of EOS varying longer than collapse
– T of EOS varying: 10 sec.– T of critical collapse: 0.05 millisec.
● Is there Critical Phenomena for EOS varying ?