Approximate Killing Vectors for Computing Spin in ...
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Approximate Killing Vectors for Computing Spin in ...
Approximate Killing Vectors for Computing Spin in Black-Hole Initial Data and Evolutions Gregory B. Cook Wake Forest University Bernard F. Whiting University of Florida July 13, 2007
Measuring the Spin of a Black Hole • Spin is only rigorously defined at spatial/null infinity. • Must use quasi-local definition: e.g. Brown & York[2] or Ashtekar & Krishnan[1] I √ 2 1 i j Kij ξ s hd x S=− 8π BH ξi CK ξi = ξi
AKV
– Greg Cook – (WFU Physics)
˜ ij ⇒ conformal Killing vector of hij : Killing vector of h : Approximate Killing vector of hij
1
Approximate Killing Vectors Killing Transport on S 2[3] Diξj = ij L DiL = − 21 2Rij ξj • Equations assume KV exists! • Solution is “path dependent” • Solution violates equations • ξ i 6= ij Dj v
– Greg Cook – (WFU Physics)
2
Approximate Killing Vectors Killing Transport on S 2[3] Diξj = ij L DiL = − 21 2Rij ξj
New AKV Method ξ i = ij Dj v+Did Diξj = ij L + Sij +hij Λ
• Equations assume KV exists!
• Find solutions that minimize S ij !
• Solution is “path dependent”
• ξ i = ij Dj v by construction
• Solution violates equations
• L = 12 ij Diξ j by construction
• ξ i 6= ij Dj v
– Greg Cook – (WFU Physics)
2
New Approximate Killing Vectors • Minimize Sij S ij = (DiDj v)(DiDj v) − 12 (Dk Dk v)2 ~ 2 = (Div)(Div) = const. subject to constraint that |ξ| L ≡ Sij S ij + 12 2RΘ(Dk v)(Dk v)
•
δL = 0 δv
⇒
ξ i ≡ ij Dj υ
DiDiv + 2L = 0 1 i2 i 2 D DiL − (1 − Θ) 2 (D R)Div − RL = 0
– Greg Cook – (WFU Physics)
3
Corotating BBH “Spin AKVs” -6
-5⋅10
CKV KT AKV Θ
-5
1.5⋅10
-6
-4⋅10 -4
-5
10
-3⋅10
-5
Θ
ij
-6
10 10
-6
-6
10
-2⋅10
-7
10
-6
5⋅10
0
0 0
– Greg Cook – (WFU Physics)
0.025 0.05 0.075
0.025
0.1
0.05
-6
-10
mΩ0
0.075
0.1
0
4
Corotating BBH AKVs In addition to expected “spin AKV,” we find 2 more AKVs. No spin!
– Greg Cook – (WFU Physics)
φ1 Ω0
5
Corotating BBH AKVs In addition to expected “spin AKV,” we find 2 more AKVs. No spin!
φ1 Ω0
80 60
1
40
φ1 φ2
0.5
20
∆φ(radians)
0
degrees
radians
1.5
0
-5
10 -6 5⋅10 0 -6 -5⋅10 0
– Greg Cook – (WFU Physics)
0.025
0.05
mΩ0
0.075
0.1
5
Corotating BBH AKVs z 1 2
Corotating
-6
-5⋅10
-8
Θ
-4⋅10
-8
-2⋅10
-6
-2.5⋅10
0 0.025
0 0 – Greg Cook – (WFU Physics)
0.025
0.05
0.05
mΩ0
0.075
0.1 6
Corotating BBH AKVs z 1 2
Corotating
-6
-5⋅10
-8
Θ
-4⋅10
-8
-2⋅10
-6
-2.5⋅10
0 0.025
0 0 – Greg Cook – (WFU Physics)
0.025
0.05
0.05
mΩ0
0.075
0.1 6
radians
1.5 1
φ1 φ2- π/2
0.5
1.5
80
1
60
0.5
40
0
20 0.0172
0.01722
0
∆φ(radians)
degrees
Corotating BBH AKVs
0
-5
10 -6 5⋅10 0 -6 -5⋅10 0
Large sep. – Greg Cook – (WFU Physics)
0.025
0.05
mΩ0
Ω0
0.075
0.1
Dominant approx. φ1 symmetry axis in green
7
radians
1.5 1
φ1 φ2- π/2
0.5
1.5
80
1
60
0.5
40
0
20 0.0172
0.01722
0
∆φ(radians)
degrees
Corotating BBH AKVs
0
-5
10 -6 5⋅10 0 -6 -5⋅10 0
Small sep. – Greg Cook – (WFU Physics)
0.025
0.05
mΩ0
Ω0
0.075
0.1
Dominant approx. symmetry axis in green
φ1
7
Non-Spinning BBH AKVs 1.5
radians
1
60 40
0.5
degrees
φ1 φ2- π/2
80
20
z 1 2
0
Non-Spinning
0
-6
-5⋅10
0.025
0.05
MΩ0
0.075
0.1
-7
Θ
-4⋅10
0
-2⋅10
-6
-2.5⋅10
-7
0 0.025
0 0
0.025
– Greg Cook – (WFU Physics)
0.05
0.05
mΩ0
0.075
0.1
8
Summary • New method determines best AKV: smallest Sij S ij : ξ i = ij Dj v. • Computed spin essentially the same as from Killing Transport for corotation & non-spinning equal-mass cases. Differences may be more significant when higher spin rates or greater BH distortion are considered. • Only one AKV solution yields spin. Other AKV solutions provide new diagnostic tool for exploring horizon geometry of distorted BHs. • Algorithm for “conformal spheres” implemented and tested. Algorithm for general S 2 nearly finished. – Greg Cook – (WFU Physics)
9
References [1] A. Ashtekar and B. Krishnan. Dynamical horizons and their properties. Phys. Rev. D, 68:104030/1–25, 2003. 1 [2] J. D. Brown and J. W. York, Jr. Quasilocal energy and conserved charges derived from the gravitational action. Phys. Rev. D, 47:1407–1419, 1993. 1 [3] O. Dreyer, B. Krishnan, D. Shoemaker, and E. Schnetter. Introduction to isolated horizons in numerical relativity. Phys. Rev. D, 67:024018/1–14, Jan. 2003. 2
