Black Holes in High Energy Physics

Numerical Relativity in D-dimensional Spacetimes

Collisions of Black Holes and Wave Extraction

Helvi Witek,

V. Cardoso, L. Gualtieri, C. Herdeiro, A. Nerozzi, U. Sperhake, M. Zilh˜ao

CENTRA / IST, Lisbon, Portugal

September 10, 2010 Phys. Rev. D 81, 084052 (2010)

arXiv:1006.3081 work in progress

ERE 2010, Granada, 6 - 10 September, 2010

Black Holes in High Energy Physics

Hoop - Conjecture (Thorne ’72):

black hole formation if:

circumference of particle<2πrS

in high energy collisions: E = 2γm0c²>EPlanck

gravity is dominant

particular nature of particlenotimportant for understanding of process

Sperhake et al. ’09

⇒high energy collisions of particles are well described by BH collisions

ultra relativistic collision of solitons: black hole formation if boostγc ≥2.9 (Choptuik & Pretorius ’10)

head-on collisions of highly boosted BHs: E_∞/M ≈14±3%

(Sperhake et al. ’08)

grazing collisions and scattering processes of highly boosted BHs:

E_∞/M≤35±5%

(Shibata et al. ’08, Sperhake et al. ’09)

⇒compute cross-section and energy loss in high energy scattering process

Black Holes in High Energy Physics

above the Planck scale:

gravity is dominant interaction inD= 4:

mEW ∼10³GeV, MPl∼10¹⁸GeV

⇒“hierarchy problem”

consider theories of gravity in higher dimensions

flat, compact

(Arkani-Hamed, Dimopoulos & Dvali ’98) warped (Randall & Sundrum ’99) flat, non-compact

(Dvali, Gabadadze & Porrati ’00) inD>4: lowering of Planck scale

MPl,D∼mEW ⇒M_Pl,4² ∼M_Pl,D^D−2R^D−4 MPl,6∼TeV

Black Holes in High Energy Physics

TeV gravity scenarios

⇒black hole production in high energy collision of particles at the Large Hadron Collider

http://lhc.web.cern.ch/lhc/

in Cosmic Rays interactions

http://www.phy.olemiss.edu/GR/

⇒compute cross-section of BH production and

energy emitted in gravitational radiation for BH event generators

Numerical Relativity in D Dimensions

Numerical Relativity in D Dimensions

consider highly symmetric problems

dimensional reduction by isometry on a (D-4)-sphere

D dimensional vacuum Einstein Eqs. ⇒4D Einstein Eqs. plus scalar field different higher dimensions manifest in scalar field

Wave Extraction in D > 4

Generalization of Regge-Wheeler-Zerilli formalism by Kodama & Ishibashi ’03 Master function

Φ_,t= (D−2)r^(D−4)/2 2rF_,t−F_t^r k²−D+ 2 + ^{(D−2)(D−1)}₂ ^r

D−3 S

r^D−3

, k =l(l+D−3)

Energy flux & radiated energy

dEl

dt = (D−3)k²(k²−D+ 2)

32π(D−2) (Φ^l_,t)², E =

∞

X

l=2

Z ∞

−∞

dtdEl

dt

Momentum flux & recoil velocity inD= 5

dP dt = 1

4πΦ^l=3_,t 5Φ^l=2_,t + 21Φ^l=4_,t

, v_recoil=

Z ∞

−∞

dtdP dt

Head-on collisions in D = 5

Numerical Setup

use Sperhake’s extendedLeancode (Zilh˜ao et al. ’10):

3+1 Einstein equations with scalar field BSSN system with moving puncture approach dynamical variables: χ, ˜γij,K, ˜Aij, ˜Γⁱ,ζ,Kζ

modified puncture gauge

∂tα = −2α(ηKK+ηK_ζKζ) +β^k∂kα

∂tβⁱ = 3

4˜Γⁱ−ηββⁱ+β^k∂kβⁱ Brill Lindquist type initial data

ψ= 1 +r_S,1^D−3/4r₁^D−3+r_S,2^D−3/4r₂^D−3 inital positions: z1=−z2= 3.185rS

unequal mass head-on with mass ratiosq=r_S,1^D−3/r_S,2^D−3= 1,¹₂,¹₃,¹₄ measure lengths in terms ofrS with

r_S^D−3= 16π (D−2)A^SD−2M

Head-on in D = 5 - Gravitational Waves

Modes of Φ

0 0.04

,tl=2

q = 1

0 0.005

,tl=3

q = 1:2

0 0.0009

,tl=4

q = 1:3

0 0.004

,tl=5

q = 1:4

-20 0 20 40 60 80

t / r_S

0 6e-05

,tl=6

∆

ΦΦ ΦΦΦ

characteristic ringdown frequency forl= 2 mode ofq= 1:

rSω= 0.955±0.005−ı(0.255±0.005) (rSω= 0.9477 +ı0.2561, e.g., Berti et al. ’09)

Head-on in D = 5 - Convergence Test

∆Φ

for q = 1 : 4

0 50 100 150 200 250 300

t / r

S

-0.0003 -0.0002 -0.0001 0 0.0001 0.0002 0.0003

t

h_c - h_m 1.47 ( h_m - h_f )

∆ Φ,

simulations at resolutionsh_c= 1/72,h_m= 1/78,h_f = 1/84 obtain 4^th order convergence⇒∆Φ,t/Φ,t = 1.5%

Head-on Collision in D = 5 - Radiated Energy

0 0.2 0.4 0.6 0.8 1 1.2

q 2e-04

4e-04 6e-04 8e-04 1e-03

E / M

Fit D = 4 Fit D = 5 D = 4 D = 5

maximum energy atq= 1

D = 4:

Emax/M= 0.055%

D = 5:

Emax/M= 0.089%

Fitting function

(see M.Lemos ’10, MSc thesis,http://blackholes.ist.utl.pt/) E/M=AE

q²

(1 +q)⁴, AE = 0.009 (D= 4) and AE = 0.014 (D= 5)

Head-on Collision in D = 5 - Recoil Velocity

0 0.2 0.4 0.6 0.8 1 1.2

q 0

5 10 15

vrecoil [km/s]

Fit D = 4 Fit D = 5 D = 4 D = 5

maximum kick velocity atq= 0.38

D = 4:

vrecoil = 4.4km/s D = 5:

v_recoil = 12.8km/s

Fitting function

(see Fitchett ’83; M.Lemos ’10, MSc thesis,http://blackholes.ist.utl.pt/ ) v=Avq² 1−q

(1 +q)⁵, Av = 248km/s(D= 4) and Av = 716km/s(D= 5)

Conclusions and Outlook

evolution of equal mass head-on inD= 5

quasinormal ringdown with characteristic frequency rSω= 0.955±0.005−ı(0.255±0.005)

radiated energyE^rad/M= 0.089%

evolution of unequal mass head-on collisions inD= 5 withq= 1,¹₂,¹₃,¹₄ maximum kick velocity atq= 0.38 with

vrecoil^max = 4.4km/s (D= 4) v_recoil^max = 12.8km/s (D= 5) ToDo:

dependence on initial separation (work in progress) higher mass ratios (work in progress)

numerical simulations of black hole collisions inD≥6

study head-ons of BHs with non-zero initial velocity and spinning BHs

http://blackholes.ist.utl.pt