On the stability of a superspinar Ken-ichi Nakao (OCU, Japan) In - - PowerPoint PPT Presentation
On the stability of a superspinar Ken-ichi Nakao (OCU, Japan) In collaboration with Hideyuki Tagoshi (ICCR, Japan) Tomohiro Harada (Rikkyo University, Japan) Pankaj S. Joshi (TIFR, India) Prashant Kocharakota (TIFR, India) Jun-qi Guo
arXiv:1707.07242 Based on
ds2 = − ΣΔ A dt2 + A Σ sin2θ dϕ − 2aMr A dt $ % & ' ( )
2
+ Σ Δ dr2 + Σdθ 2
Σ r,θ
Δ r
( ) = r2 − 2Mr + a2
A r,θ
( ) = r2 + a2
2 − Δ r
( )a2 sin2θ
a2 > M2 : always D(r) > 0 holds. No event horizon. where M : ADM mass a : Kerr parameter (angular momentum L=aM) In Boyer-Lindquist coordinates Ring singularity at r = 0, q = p/2 is naked. a2 ≤ M2 : larger root of D(r) = 0 𝑠 = 𝑁 + 𝑁% − 𝑏%
−∞ < t < +∞, −∞ < r < +∞, 0 ≤θ ≤ π, 0 ≤ϕ < 2π
Maxwell field, static, spherically symmetric, asymptotically flat Q2 > M2 : always f(r) > 0 holds. No event horizon. where M : ADM mass Q : Charge parameter Singularity at r = 0 is naked. Q2 ≤ M2 : larger root of f(r) = 0 𝑠 = 𝑁 + 𝑁% − 𝑅%
𝑒𝑡% = −𝑔 𝑠 𝑒𝑢% + 𝑒𝑠% 𝑔 𝑠 + 𝑠%𝑒𝜄% + 𝑠%sin%𝜄𝑒𝜒% 𝑔 𝑠 = 1 − 2𝑁 𝑠 + 𝑅% 𝑠%
Metric
Gauge one-form 𝐵9 = − 𝑅 𝑠 , 0,0,0
..
𝜖 𝜖𝑢 , 𝜖 𝜖𝑢 = − 1 − 2𝑁𝑠 Σ > 0 𝑁 − 𝑁% − 𝑏%cos%𝜄
𝑁% − 𝑏%cos%𝜄
𝜖𝑢 : time coordinate basis=Killing vector field
Ergo-region (ergo-sphere): is spacelike Killing “energy” of a particle: 𝐹 = −𝒗 F 𝜖
𝜖𝑢 can be negative in the ergo-region, 𝜖 𝜖𝑢 ds2 = − ΣΔ A dt2 + A Σ sin2θ dϕ − 2aMr A dt $ % & ' ( )
2
+ Σ Δ dr2 + Σdθ 2
Σ r,θ
Δ r
( ) = r2 − 2Mr + a2
A r,θ
( ) = r2 + a2
2 − Δ r
( )a2 sin2θ
where In Boyer-Lindquist coordinates even if 𝒗 is future directed timelike vector.
𝐹I = 𝑛 2𝑁 − 𝑀L 2𝑁 − 𝑀%
𝐹L = 𝑛, 𝑀L 𝐹% = 𝑛, 𝑀% 𝐹N = 2𝑛 − 𝐹I
𝜔 = 𝑓WXYZ[X\]𝑆 𝑠 𝑇 𝜄 Master variable of the perturbations: Teukolsky equations determine perturbations in Kerr spacetime ∆Wa 𝑒 𝑒𝑠 ∆a[L 𝑒𝑆 𝑒𝑠 + 𝐿% − 2𝑗𝑡 𝑠 − 𝑁 𝐿 ∆ + 4𝑗𝑡𝜕𝑠 − 𝜇 𝑆 = 0 1 sin 𝜄 𝑒 𝑒𝜄 sin 𝜄 𝑒𝑇 𝑒𝜄 + 𝑏𝜕 cos 𝜄 + 𝑡 % − 𝑛 + 𝑡 cos 𝜄 sin 𝜄
%
− 𝑡 𝑡 − 1 + 𝐵 𝑇 = 0 where 𝐿 ≔ 𝑠% + 𝑏% 𝜕 − 𝑏𝑛 𝜇 ≔ 𝐵 + 𝑏%𝜕% − 2𝑏𝑛𝜕 𝐵 = 𝐵Yh\a 𝑚 ≥ 𝑛𝑏𝑦 𝑛 , |𝑡| : Eigen value of the equation for S 𝑡 = 0 𝑡 = 1 𝑡 = 2 : scalar : EM : GW
𝜔 = 𝑓WXYZ[X\]𝑆 𝑠 𝑇 𝜄 Master variable of the perturbations: Teukolsky equations determine perturbations in Kerr spacetime ∆Wa 𝑒 𝑒𝑠 ∆a[L 𝑒𝑆 𝑒𝑠 + 𝐿% − 2𝑗𝑡 𝑠 − 𝑁 𝐿 ∆ + 4𝑗𝑡𝜕𝑠 − 𝜇 𝑆 = 0 1 sin 𝜄 𝑒 𝑒𝜄 sin 𝜄 𝑒𝑇 𝑒𝜄 + 𝑏𝜕 cos 𝜄 + 𝑡 % − 𝑛 + 𝑡 cos 𝜄 sin 𝜄
%
− 𝑡 𝑡 − 1 + 𝐵 𝑇 = 0 Outgoing GW ΨN = 𝑠 − 𝑗𝑏 cos 𝜄 WN𝜔 with 𝑡 = −2
𝜔 = 𝑓WXYZ[X\]𝑆 𝑠 𝑇 𝜄 Master variable of the perturbations: ∆Wa 𝑒 𝑒𝑠 ∆a[L 𝑒𝑆 𝑒𝑠 + 𝐿% − 2𝑗𝑡 𝑠 − 𝑁 𝐿 ∆ + 4𝑗𝑡𝜕𝑠 − 𝜇 𝑆 = 0 1 sin 𝜄 𝑒 𝑒𝜄 sin 𝜄 𝑒𝑇 𝑒𝜄 + 𝑏𝜕 cos 𝜄 + 𝑡 % − 𝑛 + 𝑡 cos 𝜄 sin 𝜄
%
− 𝑡 𝑡 − 1 + 𝐵 𝑇 = 0 Outgoing GW ΨN = 𝑠 − 𝑗𝑏 cos 𝜄 WN𝜔 with 𝑡 = −2 𝑆 → for 𝑠 → 𝑠
[
𝐷 𝑠 𝑓XYn for 𝑠 → ∞
for Quasi-normal mode (QNM)
∆ 𝑠 = 0: singular point of the radial Teukolsky equation Horizon Regularity at horizon Unique boundary condition 𝑆 →
In black hole case
𝜔 = 𝑓WXYZ[X\]𝑆 𝑠 𝑇 𝜄 Master variable of the perturbations: ∆Wa 𝑒 𝑒𝑠 ∆a[L 𝑒𝑆 𝑒𝑠 + 𝐿% − 2𝑗𝑡 𝑠 − 𝑁 𝐿 ∆ + 4𝑗𝑡𝜕𝑠 − 𝜇 𝑆 = 0 1 sin 𝜄 𝑒 𝑒𝜄 sin 𝜄 𝑒𝑇 𝑒𝜄 + 𝑏𝜕 cos 𝜄 + 𝑡 % − 𝑛 + 𝑡 cos 𝜄 sin 𝜄
%
− 𝑡 𝑡 − 1 + 𝐵 𝑇 = 0 Outgoing GW ΨN = 𝑠 − 𝑗𝑏 cos 𝜄 WN𝜔 with 𝑡 = −2 𝑆 → 𝐷 𝑠 𝑓XYn for 𝑠 → ∞
for Quasi-normal mode (QNM)
∆ 𝑠 > 0 : no singular point of the radial Teukolsky equation on the real axis of 𝑠. No horizon No unique boundary condition unless we know what is a superspinar.
In superspinar (over-spinning Kerr) case
Singularity Cardoso, Pani, Cadoni, Cavaglia (2008); Pani, Barausse, Berti and Cardoso (2010)
Reflecting BC Absorbing BC
Imaginary part of 𝜕 is positive →unstable 𝜔 = 𝑓WXYZ[X\]𝑆 𝑠 𝑇 𝜄 Quasi-normal modes of perturbations (no incoming wave at infinity) 𝑠 = 0
Boundary condition at 𝑠 = 𝑠
p =constant
𝑍r = −𝑗𝑙𝑍 𝑠 = 𝑠
p
𝑍 = ∆a %
⁄
𝑠% + 𝑏% L %
⁄ 𝑆
where 𝑍” + 𝑊 𝑠 𝑍 = 0 𝑙 = 𝑊 𝑠
p
𝜕w > 0: unstable
Reflection Boundary Condition
(angular momentum) (angular momentum) (Radius of Boundary) (Radius of Boundary)
(Imaginary Part of Angular Frequency) (Real Part of Angular Frequency) (Real Part of Angular Frequency) (Imaginary Part of Angular Frequency)
𝜕w > 0: unstable
Absorbing Boundary Condition
Reflecting Boundary Condition
In superspinar case, replace the the stability problem with “Does there exist the boundary condition under which the superspinar is stable?”
“Does there exist the boundary condition under which the superspinar is stable?”
In superspinar case, replace the the stability problem with We assume stable angular frequency, i.e., 𝜕 = 𝜕• + 𝑗𝜕w 𝜕w with negative Input Parameter
Solve the angular Teukolsky equation by imposing regularities at
1 sin 𝜄 𝑒 𝑒𝜄 sin 𝜄 𝑒𝑇 𝑒𝜄 + 𝑏𝜕 cos 𝜄 + 𝑡 % − 𝑛 + 𝑡 cos 𝜄 sin 𝜄
%
− 𝑡 𝑡 − 1 + 𝐵 𝑇 = 0
𝜇 ≔ 𝐵 + 𝑏%𝜕% − 2𝑏𝑛𝜕
𝜄 = 0, 𝜌
𝐵 is determined.
Solve the angular Teukolsky equation by imposing regularities at
1 sin 𝜄 𝑒 𝑒𝜄 sin 𝜄 𝑒𝑇 𝑒𝜄 + 𝑏𝜕 cos 𝜄 + 𝑡 % − 𝑛 + 𝑡 cos 𝜄 sin 𝜄
%
− 𝑡 𝑡 − 1 + 𝐵 𝑇 = 0
Solve the radial Teukolsky equation with QNM boundary condition
∆Wa 𝑒 𝑒𝑠 ∆a[L 𝑒𝑆 𝑒𝑠 + 𝐿% − 2𝑗𝑡 𝑠 − 𝑁 𝐿 ∆ + 4𝑗𝑡𝜕𝑠 − 𝜇 𝑆 = 0
𝜇 ≔ 𝐵 + 𝑏%𝜕% − 2𝑏𝑛𝜕
𝜄 = 0, 𝜌
𝐵 is determined. No singular point on real axis of 𝑠
The boundary condition at, for example, 𝑠 = 0 is found. Solve the angular Teukolsky equation by imposing regularities at
1 sin 𝜄 𝑒 𝑒𝜄 sin 𝜄 𝑒𝑇 𝑒𝜄 + 𝑏𝜕 cos 𝜄 + 𝑡 % − 𝑛 + 𝑡 cos 𝜄 sin 𝜄
%
− 𝑡 𝑡 − 1 + 𝐵 𝑇 = 0
Solve the radial Teukolsky equation with QNM boundary condition
∆Wa 𝑒 𝑒𝑠 ∆a[L 𝑒𝑆 𝑒𝑠 + 𝐿% − 2𝑗𝑡 𝑠 − 𝑁 𝐿 ∆ + 4𝑗𝑡𝜕𝑠 − 𝜇 𝑆 = 0
𝜇 ≔ 𝐵 + 𝑏%𝜕% − 2𝑏𝑛𝜕
𝜄 = 0, 𝜌
𝐵 is determined. No singular point on real axis of 𝑠
Infinite number of boundary conditions under which the superspinar is stable.
Reflecting Boundary Condition