Black Hole solutions in Einstein-Maxwell-Yang-Mills-Born-Infeld Theory

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2 Black Hole solutions in Einstein-Maxwell-Yang-Mills-Born-Infeld Theory S Habib Mazharimousavi Eastern Mediterranean University, north Cyprus S. Habib Mazharimousavi and M. Halilsoy, Phys. Rev. D 76 (2007) ; Phys. Rev. D 78 (2008) ; Phys. Lett. B 665 (2008) 125; Phys. Lett. B 659 (2008) 471; J. of Cosmology and Astroparticle Phys. (JCAP) 12 (2008) 005.

3 Black Hole solutions in Einstein-Maxwell-Yang-Mills-Born-Infeld Theory 3 1 Outline and some initial comments 1) We introduce Maxwell eld alongside with Yang-Mills (YM) eld in general relativity. 2) We present static spherically symmetric EMYM black hole solutions in any higher dimensions. 3) These two gauge elds, one Abelian (Maxwell) the other non-abelian(ym), are coupled through gravity. 2) Our treatment of YM eld like Maxwell eld is completely classical (i.e. non-quantum). 3) From physics standpoint, electromagnetism has long range effects and dominates outside the nuclei of natural matter. 4) YM eld on the other hand is conned to act inside nuclei, however, the existence of exotic and highly dense matter encourages us to use YM eld in a broader sense. 4) To obtain exact solutions the Maxwell eld is chosen pure electric while the YM eld is pure magnetic. 2 Action, Field Equations and our ansaetze The action which describes Einstein-Maxwell-Yang-Mills gravity with a cosmological constant in N dimensions: I G = 1 Z dx Np (N 1) (N 2) g R F F Tr F (a) F (a) (1) 2 3 M

4 Black Hole solutions in Einstein-Maxwell-Yang-Mills-Born-Infeld Theory 4 Tr (:) = (N 1)(N 2)=2 X a=1 (:) : (2) F (a) A (a) F A A (a) C(a) (b)(c) A(b) A (c) ; (3),! C (a) (N 1)(N 2) (b)(c) stands for the structure constants of 2 parameter Lie group G and is a coupling constant. A (a) are the SO(N 1)gauge group YM potentials while A represents the usual Maxwell potential. (fa; b; c; :::g do not differ whether in covariant or contravariant form). Field equations: Stress-energy tensor: T = G + (N 1) (N 2) g = T ; (4) 6 2F 1 F 2 F F g + Tr 2F (a) F (a) 1 2 F (a) F (a) g : (5)

5 Black Hole solutions in Einstein-Maxwell-Yang-Mills-Born-Infeld Theory 5 YM and Maxwell equations: F (a) ; + 1 C(a) (b)(c) A(b) F (c) = 0; F ; = 0; (6) N-dimensional static spherically symmetric line element: ds 2 = Higher dimensional Wu-Yang ansatz: f(r) dt 2 + dr2 f(r) + r2 d 2 N 2; (7) In 4D with x 1 = x; x 2 = y and x 3 = z : A (a) = Q r 2 (x idx j x j dx i ) ; Q = charge, r 2 = 2 j + 1 i N 1; and 1 a (N 1) (N 2)=2; NX 1 x 2 i ; (8) i=1 A (1) = Q (ydx xdy) ; (9) r2 A (2) = Q (zdx xdz) ; (10) r2 A (3) = Q (zdy ydz) ; (11) r2

6 Black Hole solutions in Einstein-Maxwell-Yang-Mills-Born-Infeld Theory 6 in which x = r sin cos ; y = r sin sin and z = r cos : In 5D with x 1 = x; x 2 = y; x 3 = z and x 4 = w : A (a) = A (a) (; ) =) F (a) = F (a) (; ) : (12) A (1) = Q (ydx xdy) ; (13) r2 A (2) = Q (zdx xdz) ; (14) r2 A (3) = Q (zdy ydz) ; (15) r2 A (4) = Q (wdx xdw) ; (16) r2 A (5) = Q (wdy ydw) ; (17) r2 A (6) = Q (wdz zdw) ; (18) r2 where x = r cos sin sin ; y = r sin sin sin ; z = r cos sin ; w = r cos and therefore A (a) = A (a) (; ; ) =) F (a) = F (a) (; ; ) : (19)

7 Black Hole solutions in Einstein-Maxwell-Yang-Mills-Born-Infeld Theory 7 The Maxwell potential 1-form: A = q r dt; N 3 N 4 q ln (r) dt; N = 3 The energy-momentum tensor for the Maxwell and YM elds for N 4 (3-dimensional case will be studied separately): 3 EMYM solution for N 6 Metric function: (20) T a Max b = (N 3)2 q 2 r 2(N 2) diag [1; 1; 1; 1; ::; 1] ; (21) TYM a (N 3) (N 2) Q2 b = 2r 4 diag [1; 1; ; ; ::; ] ; (22) = N 6 N 2 : f (r) = 1 2M r N 3 3 r2 + (N 3) 2q2 (N 2) r 2(N 3) (N 3) Q 2 (N 5) r 2 ; (23)

8 Black Hole solutions in Einstein-Maxwell-Yang-Mills-Born-Infeld Theory 8 Newtonian-like potential, (r)(for = 0): f (r) = (r) (24) (r) = M (N 3) q2 (N 3) Q 2 + rn 3 (N 2) r 2(N 3) 2 (N 5) r2: (25) The active force through F = r : F (r) = M (N 3) r (N 2) + 2q2 (N 3) 2 (N 2) r 2N 5 (N 3) Q 2 (N 5) r 3 : (26) #) The signs of the Maxwell and YM terms reveal that while the Maxwell is repulsive the YM becomes attractive. #) One can easily show that for r! 1 the YM term dominates (let = 0), namely: lim f (r)! 1 (N 3) Q 2 r!1 (N 5) r : 2 (27)

9 Black Hole solutions in Einstein-Maxwell-Yang-Mills-Born-Infeld Theory 9 #) For r! 0 + we have the opposite case: lim r!0 (N 3) 2q2 +f (r)! 1 + (N 2) r 2(N 3); (28) which may be interpreted as an "asymptotic independence " from one type of charge (or the other )in different limits. #) For the mini black holes this has the striking effect that the Hawking temperature depends only on the electric charge. 3.1 The case N=5 Metric function: f (r) = 1 2M r 2 3 r2 + 4q2 3r 4 2Q 2 ln (r=r 0 ) r 2 ; (29) The Hawking temperature T H : T H = 2 = 1 4 jf 0 (r h )j = r h 3 r h where r h is the radius of the event horizon and stands for the surface gravity. 4 3 q 2 rh 5 Q 2 r 3 h (30)

10 Black Hole solutions in Einstein-Maxwell-Yang-Mills-Born-Infeld Theory 10 #) The corresponding Newtonian-like YM force term: which implies that it is attractive (repulsive) for r > p e (r < p e). #) The Maxwell term remains always repulsive. #) For r! 1 (for = 0) the YM term dominates over the Maxwell force Q2 (1 2 ln r) ; (31) r3 #) In the limit r! 0 +, on the other hand we obtain lim f (r)! 1 2Q 2 ln (r) : r!1 r 2 (32) lim r!0 which is in conrm with the behavior for N 6: 4q2 +f (r)! 1 + 3r4; (33) 3.2 The case N=4 Metric function: f (r) = 1 2M r 3 r2 + q2 + Q 2 ; r 2 (34)

11 Black Hole solutions in Einstein-Maxwell-Yang-Mills-Born-Infeld Theory The case N=3 The Maxwell potential 1-form: YM gauge potential 1-forms: A = q ln (r) dt; (q = electric charge), (35) A (1) = Q cos () ln (r) dt; (36) A (2) = Q sin () ln (r) dt; A (3) = Qd; (Q = YM charge), which satisfy the Maxwell and YM equations, respectively. Metric function: Line element: f (r) = M 1 3 r2 2 q 2 + Q 2 ln (r) ; (37) ds 2 = f (r) dt 2 + dr2 f (r) + r2 d 2 : (38)

12 Black Hole solutions in Einstein-Maxwell-Yang-Mills-Born-Infeld Theory 12 A negative cosmological constant leads to a black hole solution: f (r) = M jj r2 2 q 2 + Q 2 ln (r) (39) The horizons: " r + = exp " r = exp! M 1 2 LamberW 1; jj e Q 2 +q 2 3 (Q 2 + q 2 )! M 1 2 LamberW 0; jj e Q 2 +q 2 3 (Q 2 + q 2 ) # M 2 (Q 2 + q 2 ) # M 2 (Q 2 + q 2 ) (40) (41) The energy density: Hawking temperature: T H = 2 = 1 2 = T tt = Q2 + q 2 jj 3 r + r 2 (42) Q 2 + q 2 : (43) r +

13 Black Hole solutions in Einstein-Maxwell-Yang-Mills-Born-Infeld Theory 13 4 The Geometry Outside of the 5-dimensional EMYM black hole The Killing vectors in 5D: Conserved quantities: = (1; 0; 0; 0; 0)! symmetry under displacements in the direction t; (44) = (0; 0; 0; 0; 1)! symmetry under displacements in the direction ; (45) e = u = g u = f (r) u t! the energy density (46) ` = u =g u = r 2 sin 2 () sin 2 () u! the angular momentum per unit mass (47) u= u t ; u r ; u ; u ; u! ve-velocity (48) We restrict the particle to stay on the plane = 2 ; = 2 with u = u = 0: Geodesics equation: g u u = f (r) u t f (r) (ur ) 2 + r 2 u 2 = 1 (49),! u t = dt d ; ur = dr d ; u = d d ; u = d d ; u,! is the proper time measured by the observer moving with the particle. = d d (50)

14 Black Hole solutions in Einstein-Maxwell-Yang-Mills-Born-Infeld Theory 14 =) =) dr + 1 `2 d 2 r + 1 f (r) 1 2 r dr 2r d = 4 (E V eff (r)); `2 = e2 1 2 (51) V eff (r) = 1 2 `2 r f (r) 1 : (52)

15 Black Hole solutions in Einstein-Maxwell-Yang-Mills-Born-Infeld Theory 15 (53)

16 Black Hole solutions in Einstein-Maxwell-Yang-Mills-Born-Infeld Theory Newtonian motion In the weak eld limit, (and setting to be zero): Radial force: The radial equation of motion: (r) = M r 2 + 2q2 3r 4 Q 2 ln (r) r 2 ; (54) F r = d 2M dr (r) = 8q 2 Q 2 r 3 3r 5 r + 2Q2 ln (r) : 3 r 3 (55) d 2 r dt 2 = `2 r 3 As a particular case we set 1 + Q2 2M 8q 2 Q 2 r 3 3r 5 r + 2Q2 ln (r) ; u = 1 3 r 3 r =) (56) d 2 u d Q2 2M u + 2Q2 `2 `2 8q 2 3`2 u3 = 0: (57) q 2M `2 = 0; ~ Q = p 2Q ` and ~q = 8 3 q ` : d 2 u d 2 + ~ Q 2 u ln u ~q 2 u 3 = 0; (58)

17 Black Hole solutions in Einstein-Maxwell-Yang-Mills-Born-Infeld Theory 17 such that the inverse of the solution of this equation is plotted (i.e., r = 1 u versus ):: =) The roles of Maxwell and YM charges outside the black hole are in contrast with each other. (59)

18 Black Hole solutions in Einstein-Maxwell-Yang-Mills-Born-Infeld Theory 18 5 What may be done next?...non-linear electrodynamic! 5.1 Field Equations and the metric ansatz for EMYMBI gravity Einstein Maxwell-Yang-Mills-Born-Infeld (EMYMBI) action: S = 1 Z d n+1 x p n (n 1) g R + L M (F ) + L Y M F ; (60) 16 3 is given by M =) Maxwell-Born-Infeld (MBI) Lagrangian L M (F) = 4 2 M 1 F = F F =) Yang-Mills-Born-Infeld (YMBI) Lagrangian L Y M F = 4 2 Y M 1 F = Tr(F (a) F (a) ) s 1 + F 2 2 M s! 1 + F 2 2 Y M! (61) (62) (63) (64)

19 Black Hole solutions in Einstein-Maxwell-Yang-Mills-Born-Infeld Theory 19 lim L M (F ) = F; M!1 lim M!0 L M (F ) = 0; (65) lim L Y M F = F ; lim L Y M F = 0: (66) Y M!1 Y M!0 T = T + T M Y M (67) lim M!1 lim Y M!1 T M = 1 2 L M 4F F (L M ) F ; (LM ) F M (F ) ; T = 1 Y M 2 L Y M 4Tr F (a) F (a) (L Y M ) F ; (L Y M ) F Y M F ; (69) T = 1 M 2 F 4F F! Enistein-Maxwell-EMT; lim F 4F (a) F (a) T = 1 Y M 2 M!1 T M! Enistein-Yang Mills-EMT; lim Y M!1 6 EMYMBI Black hole solution in ve dimensions The metric function = 0; (70) T = 0;(71) Y M

20 Black Hole solutions in Einstein-Maxwell-Yang-Mills-Born-Infeld Theory 20 where f (r) = 1 (r) = 2M 2 M 2 Y M r 2 4 r 2 3 3r ( 2 M (r) + Y M (r)) ; Z q q Mr 6 dr; (r) = Z q 3Q Y Mr 4 dr: (72) This is a black hole solution and M is an integration constant to be identied as the mass of the black hole. One can show that in the limit of M;Y M! 1; L (F ) and f (r) reduce to the case of EMYM as we mentioned above, i.e., lim f (r) = 1 2M!1 3 r2 r 2 + q2 3r 4 2Q 2 ln (r) r 2 ; (73) while in the limit of! 0 they reduce to the pure gravity with the cosmological constant lim f (r) = 1 2M!0 r 2 3 r2 : (74) This black hole solution asymptotically behaves like a de-sitter spacetime (Anti de-sitter) such that lim f (r) = 1 r!1 3 r2 and for = 0; it is asymptotically at. The Born-Infeld parameter modies the radius of the horizon,

21 Black Hole solutions in Einstein-Maxwell-Yang-Mills-Born-Infeld Theory 21 as we plot in gure. In fact, for = 0 the solution matches with the pure gravity while for = 1 it gives the horizon of the EYM black hole. We notice that, BI parameter interpolates the horizon of the corresponding black hole, between the two extremal values of the radii of the horizons for = 0 and = 1:

22 Black Hole solutions in Einstein-Maxwell-Yang-Mills-Born-Infeld Theory 22 7 Summery and conclusion ##) We have introduced higher dimensional magnetically Yang-Mills one-form potentials which is N-dimensional version of Wu-Yang ansatz. ##) We have found EMYM black hole solutions in any dimensions. ##) Our exact solution suggests that for N 5, the Maxwell and YM elds compete for dominance in the asymptotic regions. That is, for r! 0 (r! 1) the Maxwell charge q (the YM charge Q) dominates. This may shed light on the problem of gravitational connement (i.e., accretion, collapse) versus the Maxwell and YM charges. ##) The Newtonian approximation in the polar plane consisting of the coordinates (r; ) reveals that the YM charge deepen the potential well to form bound states. ##) In lower dimensions (i.e., N = 3; N = 4), however the roles of q and Q remain indistinguishable. The presence of a logarithmic term for N = 3 and N = 5 is a distinctive property compared to other dimensions. ##) A black hole solution in EYMBI theory was introduced.

23 Black Hole solutions in Einstein-Maxwell-Yang-Mills-Born-Infeld Theory 23 THANK YOU