Sub-Kerr Axisymmetric Collapse

Introduction
Evolution of Density Profile
Evolution of Lagrangian Matter Tracers
Evolution of the Lapse Function
Analytic Tests for Kerr Black Hole Parameters


Introduction


Fig. 1-1 Initial Shape of the Rotating Star

Evolution is performed on a 600 x 600 grid, with outer boundaries at 14M. The equatorial radius of the initial star is Req = 4.2M. In this simulation, the spin parameter is J/M2 = 0.92 and 0 < t/M < 74. The star begins collapsing immediately at t/M = 0, due to pressure depletion. Shortly after the black hole appears, the singularity is excised at t/M = 31. At this point, the excision radius is Rexc/M = 0.41, while the polar radius of the apparent horizon is Rpol/M = 1.4, and its equatorial radius is Req/M = 1.8. All of the matter falls into the black hole within 20M after excision. Stable evolution of space surrounding the black hole continues until t/M = 74, at which point the simulation is terminated.


Evolution of the Density Profile

In the meridional and equatorial clips, the density is plotted on a logarithmic scale normalized to the central density of the star at the start of evolution (Fig 2-1). The apparent horizon is denoted by a black area and the region excised is white. As can be seen from figures 2-3, 2-4, and 2-5, a significant portion of the matter is still outside the black hole at excision. By t/M = 54 (Fig. 2-5, 2-8), all of the mass is inside the black hole. The ergoregion (hatched region) is displayed at this time. Stable evolution of the space surrounding the black hole continues until t/M = 74.



Meridional Plane


Fig. 2-1 Color code for density profile

Fig. 2-2 Density profile at t/M = 0

Fig. 2-3 Excision (white region) at t/M = 31

Fig. 2-4 Apparent horizon measurements at t/M = 31 (black region)

Fig. 2-5 Ergoregion at t/M = 54 (hatched region)

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**A zoom occurs just prior to excision so visual inspection should not be used to compare sizes.


Equatorial Plane


Fig. 2-6 Density profile at t/M = 0

Fig. 2-7 Black hole excision at t/M = 31

Fig. 2-8 Ergoregion at t/M = 54

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**A zoom occurs just prior to excision so visual inspection should not be used to compare sizes.


Evolution of Lagrangian Matter Tracers

In these clips, we track 100,000 Lagrangian matter tracers (particles) that represent fluid elements. The initial distribution of Lagrangian tracers is proportional to the initial rest mass density. We then calculate the trajectories of the tracers by integrating fluid velocities. The particles are also color-coded according to the density at their current location. The color coding is the same as the density profile clips (Fig. 2-1). The black region is the apparent horizon and the innermost white sphere is the excised region.



Meridional View


Fig. 3-1 Lagrangian matter tracers at t/M = 0

Fig. 3-2 Black hole excision at t/M = 31

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**A zoom occurs just prior to excision so visual inspection should not be used to compare sizes.


Equatorial View


Fig. 3-3 Lagrangian matter tracers at t/M = 0

Fig. 3-4 Black hole excision at t/M = 31

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**A zoom occurs just prior to excision so visual inspection should not be used to compare sizes.


Evolution of the Lapse Function

This clip shows the evolution of log(α) in the equatorial plane of the star. Excision is denoted by a white stripe on the surface (Fig. 4-3). Anything inside of the white stripe is excised. Note that log(α) stays finite on the excision boundary (Fig. 4-4) at late time. Hence our time slicing condition for α is "horizon penetrating".


Fig. 4-1 Initial star with equatorial plane

Fig. 4-2 Lapse function at t/M = 0

Fig. 4-3 Black hole excision at t/M = 31

Fig. 4-4 Value of lapse on excision boundary at t/M = 31

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Analytic Tests for Kerr Black Hole Parameters

In the Sub-Kerr rotation case, a Kerr black hole forms containing all the matter. We therefore check the validity of our evolution code by computing several invariant parameters characterizing the final black hole formed in the simulation and comparing them with analytic values for a Kerr black hole. We compare both the polar and equatorial coordinate circumferences of the event horizon, and the areas of both the event horizon and the ergoregion. In addition, we give the angular momentum J, mass M, and J/M2 at the end of the simulation to check how well they are conserved (some small mass-energy is lost due to gravitational waves, while J should be strictly conserved).


Fig. 5-1 Ergoregion and event horizon at t/M = 54

Fig. 5-2 Measurement of event horizon equatorial circumference at t/M = 54

Fig. 5-3 Measurement of event horizon polar circumference at t/M = 54

The results are summarized in the table below.

AEH/AEH(Kerr)1.01
Ceq/Ceq(Kerr)1.01
Cpol/Cpol(Kerr)1.00
Aergo/Aergo(Kerr)1.10
M/M(0)1.00
J/J(0)0.94
J/M20.87

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last updated 4 December 2014 by aakhan3

Center for Theoretical Astrophysics---University of Illinois at Urbana-Champaign

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