Kerr-Schild Method and Geodesic Structure in Codimension-2 BBH

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1 1 Kerr-Schild Method and Geodesic Structure in Codimension-2 Brane Black Holes B. Cuadros-Melgar Departamento de Ciencias Físicas, Universidad Nacional Andrés Bello, Santiago, Chile Abstract We consider the black hole solutions of five-dimensional gravity with a Gauss-Bonnet term in the bulk and an induced gravity term on a 2-brane of codimension-2. Applying the Kerr-Schild method we derive additional solutions which include charge and angular momentum. Moreover, we study the geodesic structure of such spacetimes. * In collaboration with S. Aguilar and N. Zamorano (Universidad de Chile).

2 2 BTZ on codimension-2, E.Papantonopoulos, M.Tsoukalas, V.Zamarias, Phys.Rev.Lett. 100, Action S grav = M (5) 3 { 2 ( + α + rc (2008) d 5 x g (5) [ R (5) R (5)2 4R (5) MN R(5)MN + R (5) d 3 x g (3) R (3)δ(ρ) 2πb } + MNKL R(5)MNKL)] d 5 xl bulk + d 3 xl brane δ(ρ) 2πb.

3 3 Metric ds 2 5 = f 2 (ρ) ) ( n(r) 2 dt 2 + n(r) 2 dr 2 + r2 l 2 dφ2 + dρ 2 + b 2 (ρ)dθ 2. (1)

4 3 Metric ds 2 5 = f 2 (ρ) ) ( n(r) 2 dt 2 + n(r) 2 dr 2 + r2 l 2 dφ2 + dρ 2 + b 2 (ρ)dθ 2. (1) Einstein Equations G (5)N M + r2 cg (3)ν µ g µ M gn ν δ(ρ) 2πb αhn M = 1 M 3 (5) [ T (B)N M + T (br)ν µ g µ M gn ν ] δ(ρ) 2πb,

5 4 Junction Conditions: Einstein Equations on the brane G (3) µν = 1 M 3 (5) (r2 c + 8π(1 β)α) T (br) µν + 2π(1 β) r 2 c + 8π(1 β)α g µν. (2)

6 5 Solutions Table 1: Bulk Solutions n(r) f(ρ) b(ρ) Λ 5 Notes ( ) cosh ρ 2 3 b(ρ) α 4α l 2 = 4α ( ) BTZ cosh ρ 2 β ) α sinh l 2 = 4α 2 α ( ρ 2 α ±1 γ 1/2 sinh ( γ 1/2 ρ ) 3 γ = 2Λ l 2 5 ( ) n(r) cosh ρ 2 2 β ( ) a sinh ρ α 2 3 α 4α T br ( ) corrected cosh ρ 2 2 β ( ) α sinh ρ α 2 3 α 4α l 2 = 4α BTZ ±1 2 β ( ) α sinh ρ l 2 = 12α 2 α 3 4α 1 4α 3+4αΛ 5 BTZ: BTZ-corrected: n 2 (r) = M + r2 l 2, n 2 (r) = M + r2 l 2 ζ r.

7 6 Brane Equations n(r) T br Brane BTZ ( Λ 3 = 1/l 2 ) Vacuum BTZ-corrected ζ ζ,, ζ Scalar field 2r 3 2r 3 r 3 Table 2: Brane Solutions

8 7 Applying the Kerr-Schild Method Background Metric New Solution g MN = g MN + 2H(r, ρ)l M l N. (3)

9 7 Applying the Kerr-Schild Method Background Metric New Solution g MN = g MN + 2H(r, ρ)l M l N. (3) Null geodesic vector l M = C 1, l M;N l N = 0, (4) l M l M = 0. (5) ( ) 1 C n 2 C r 2 + ξ2 n 2 ξ, C 2, 2 f 2 C2 3 b 2, C 3. (6)

10 8 Main solutions Charged BTZ string H(r, ρ) = f 2 (ρ) Q2 2 ln r ( ) ds 2 = f 2 ˆn 2 dˆt 2 + dr2 ˆn 2 + r2 dφ 2 with ˆn 2 = M + r 2 /l 2 Q 2 ln r. + dρ 2 + b 2 dθ 2, (7) Brane energy-momentum tensor, T ν µ = diag ) ( Q2 2r 2, Q2 2r 2, Q2 2r 2. (8)

11 9 BTZ string + brane scalar field H(r, ρ) = f 2 (ρ) ζ 2r ˆn 2 = M + r 2 /l 2 ζ/r

12 9 BTZ string + brane scalar field H(r, ρ) = f 2 (ρ) ζ 2r ˆn 2 = M + r 2 /l 2 ζ/r Charged BTZ string + brane scalar field Background metric: Charged BTZ string Brane energy-momentum tensor, Tµ ν = diag ( Q2 2r 2 + ζ 2r 3, Q2 2r 2 + ζ 2r 3, Q2 2r 2 ζ ) r 3, (9)

13 10 BTZ string with angular momentum H(r, ρ) = f 2 (ρ)c ds 2 = f 2 [ ˆn 2 dˆt 2 + dr2 ˆn 2 + R2 ( ) ] 2 J 2R 2dˆt + d ˆφ + dρ 2 + b 2 dθ 2, (10) where ˆn 2 (r) = M + R 2 /l 2 + J 2 /4R 2. Analogous solutions when we use f(ρ) = 1 and b(ρ) = γ sinh(ρ/γ).

14 11 Geodesic Structure Lagrangian L = f 2 (ρ) ( n(r) 2 ṫ 2 + ) ṙ2 n(r) 2 + r2 φ2 + ρ 2 + b 2 (ρ) θ 2. (11) L ṫ = 2E, (12) L φ = 2L, (13) L θ = 2K, (14)

15 12 Then, ( 2L = cosh 2 ρ ) 2 α E 2 cosh 4 ( ρ 2 α ) n(r) 2 + ṙ2 n(r) 2 + L 2 ( r 2 cosh 4 ρ 2 α ) + ρ 2 + K 2 ( ) = h, (15) 4β 2 α sinh 2 ρ 2 α h = 0, 1 lightlike and timelike geodesics.

16 13 Geodesics on the Brane Effective Potential Orbits dr dλ = ( L Veff 2 = n 2 2 ) (r) r 2 h. (16) ( L E 2 n 2 2 ) (r) r 2 h. (17)

17 14 BTZ Case Radial geodesics (L = 0) Veff 2 0 K2 K4 K6 K r E = 15 E = 0 E = 5 r l K4 K2 2 4 K10 l K20 K30 K10 Figure 1: Effective potential and orbits for timelike radial particles on the brane.

18 15 L 0 1, r l 0,5 K0,4 K0,2 0 0,2 0,4 l K0,5 E=5 E=0 E=1 r l K4 K2 2 4 K10 l K20 E = 15 E = 5 E = 0 K1,0 K30 Figure 2: Orbits for lightlike (left) and timelike (right) geodesics with L.

19 16 Charged BTZ and BTZ + Scalar field Case 2 15 Q=2 Q=1 Q=3 Veff 0 K2 K4 K6 K8 K r z=5 z=1 z=10 Veff K r Figure 3: Effective potential for timelike geodesic on the brane.

20 17 Geodesics in the Bulk: f(ρ) = cosh L = K = 0 ( ) ρ 2, b(ρ) = 2β ( ) α sinh ρ α 2 α 10 8 r l l E = 0 E = 5 E = 20 Figure 4: Timelike orbits for geodesics in the bulk.

21 18 L 0, K r 2 r l E=5 E=15 E= l E=5 E=15 E=8 Figure 5: Orbits for lightlike (left) and timelike (right) particles with L.

22 19 f(ρ) = 1, b(ρ) = γ sinh(γ 1 ρ) 100 Veff 50 0 K r K100 Figure 6: Effective potential for f = 1.

23 20 BTZ with Angular Momentum h = f 2 [ ( 4M + 4r2 l 2 + J 2 ) l 4 ( 2r 2 E + LJ) 2 r 2 f 4 (4Ml 2 r 2 4r 4 J 2 l 2 ) 2 4ṙ r2 ( Jl2( 2r2 E + LJ)) 4M + 4r 2 /l 2 + J2 r 2 f 2 (4Ml 2 r 2 4r 4 J 2 l 2 ) r 2 ] + 2( JEl2 + 2Ml 2 L 2Lr 2 ) f 2 (4Ml 2 r 2 4r 4 J 2 l 2 ) 2 + ρ 2 + K2 b 2, (18)

24 21 Geodesics on the Brane 40 1,5 1,0 r l 20 r l 0,5 K6 K4 K l K20 K40 E=-0.1 E=2 K1,5 K1,0 K0,5 0 0,5 1,0 1,5 l K0,5 K1,0 K1,5 E=2 E=0 E=5 E=0.7 Figure 7: Timelike (left) and likelike (right) brane geodesics for BTZ with J.

25 22 Geodesics in the Bulk R(t) r(t) R(t) r(t) K1 K2 K4 K2 Figure 8: Lightlike (left) and timelike (right) phaseportrait for geodesics in the bulk for BTZ with J.

26 23 Conclusions By applying the Kerr-Schild method it was possible to find more solutions using as background the BTZ string-like solutions in codimension-2 branes.

27 23 Conclusions By applying the Kerr-Schild method it was possible to find more solutions using as background the BTZ string-like solutions in codimension-2 branes. These additional solutions include charge, angular momentum, and brane scalar fields coupled to the BBH.

28 23 Conclusions By applying the Kerr-Schild method it was possible to find more solutions using as background the BTZ string-like solutions in codimension-2 branes. These additional solutions include charge, angular momentum, and brane scalar fields coupled to the BBH. The geodesic behaviour on the brane and in the bulk for each of the main solutions was studied.

29 23 Conclusions By applying the Kerr-Schild method it was possible to find more solutions using as background the BTZ string-like solutions in codimension-2 branes. These additional solutions include charge, angular momentum, and brane scalar fields coupled to the BBH. The geodesic behaviour on the brane and in the bulk for each of the main solutions was studied. The orbits present several features, some of them can be inferred from an effective potential, others can be integrated directly, and a phaseportrait graph was needed in some cases.