Spinning a Charged Dilaton Black Hole

Home , Gyromagnetic ratio

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S = d4x[R − 2(∇φ)2 − exp(2aφ)F 2] , (1) Z where a is the parameter which determines the strength of the coupling between the Maxwell field F and the dilaton field φ. For a = 0, the action reduces to the one which governs the usual Einstein-Maxwell system with a free scalar field. A charged spherical black hole is described by the Reissner-Nordström solution in the system. For a = 1, the action coincides with the one which is derived from low-energy string theory. The solution for a charged spherical black hole in the system governed by the general action (1) can be written in the form [1, 2]

− ds2 = −f dt2 + f 1dr2 + R2(r)(dθ2 + sin2 θdϕ2) , (2)

where 2 2 (1−a )/(1+a ) f = (1 − r+/r)(1 − r−/r) , (3) and 2 2 2 2 2a /(1+a ) R = r (1 − r−/r) ,. (4)

In this solution r+ and r− are integration constants and they are connected to the physical quantities of the black hole through the relations

1 − a2 2M = r + r− , (5) + 1+ a2 r−r Q2 = + , (6) 1+ a2 where M and Q are the mass and the electric charge of the black hole, respec- tively. The horizon is located at r = r+, and the maximal charge is attained 2 1/2 when r+ = r−, i.e., Qmax = (1+ a ) M. The dilation and the electric ﬁeld behave as

2 2 2a /(1+a ) exp(2aφ)=(1 − r−/r) , (7) and Q exp(−2aφ)F = dt ∧ dr (8) R2

(or, equivalently, At = Q/r). If an infinitesimal angular momentum j (parallel to the z-axis) is added, we need only to know a few components of the gauge field and the metric. They 1 are Aφ and gtφ, which are of the order of j . The corrections to the other components of the field and the metric are O(j2), and ignored in the present analysis. Note that the equation of motion for the dilaton gets no correction

2 at the first order and thus the behavior of the dilaton is unchanged from (7). Therefore we find that j2/M 2 cannot be as much as Q2. The asymptotic form of the field components is given by

sin2 θ A = µ + O(1/r2) , (9) φ r sin2 θ g = 2j + O(1/r2) , (10) tφ r where the constant µ is interpreted as the magnetic moment. We are looking for the solution which behaves as (9) and (10) at inﬁnity. Solving the linearized equation of motion derived from the action (1), we obtain

sin2 θ A = Qα , (11) φ r and

2 2 2 2 2 2 r 1+ a 1+ a (3a −1)/(1+a ) − − − gtφ = α sin θ 2 2 2 [(1 r /r) 1] r− 1 − a 1 − 3a 2 2 r− (1−a )/(1+a ) 2 2 − − (1 − r+/r) (1 − r−/r) , for a =16 ,a =16 /3 , (12) r i 2 2 2r 2 − − − gtφ = α sin θ 2 [r /r + (1 − r /r) ln(1 − r /r)] − (1 − r+/r) , for a =1 , (13) r− 2 2 2r 1/2 2 − − − gtφ = α sin θ − 2 [r /r + ln(1 − r /r)] − (1 − r+/r) (1 − r /r) , for a =1(14)/3 , r− where α is a constant. Using the asymptotic behaviors, we find that the gyromagnetic ratio is given by µ Q (1 + a2)r + (1 − a2)r− Q + ≡ = 2 2 g . (15) j M (1 + a )r+ + (1 − a /3)r− 2M For a = 0, (15) reduces to the well-known result for a Kerr-Newman black hole [11]: g = 2 (like an electron!). For finite a, g is expressed as a function of Q/M. g is smaller than 2 for finite Q/M in general, whereas the relation

lim g(Q/M)=2 (16) Q/M→0 is satisﬁed for any value of a. The dependence on Q/M for some cases is shown in ﬁg. 1. Before discussing the physical implication, we will investigate another physical quantity. The angular velocity at the horizon r = r+ is given in the leading order by

g (r = r ,θ = π/2) ≡ tφ + Ω+ 2 . (17) R (r+)

3 Figure 1: Fig. 1. The g factor of the charged dilaton black hole with small angular momentum versus the charge-to-mass ratio of the black hole. (—–) a2 =0; (- - - -) a2 = 1/3; (– – –) a2 = 1/2; (– · –) a2 = 1; (– ·· –) a2 = 3; (······) a2 = 5.

The explicit expression for Ω+ can be obtained by using (12)–(14). The moment of inertia for the charged dilaton hole with inﬁnitesimal angular momentum, I = j/Ω+,is exhibited in ﬁg. 2. We see, for any value of a, that I lim 2 =1 . (18) Q/M→0 MR (r+) 2 2 2 The critical value for a is 1. For a < 1, I/MR (r+) diverges at Q = Qmax; for 2 2 2 2 a > 1, I/MR (r+) approaches zero when Q → Qmax. For a = 1, I/MR (r+) diverges logarithmically near Q = Qmax.

Figure 2: Fig. 2. The moment of inertia, I, for the charged dilaton black hole with small angular momentumversus the charge-to-mass ratio ofthe black hole. 2 The moment of inertia I is normalized by MR (r+). Designations as in ﬁg 1.

Now we are working with the assumption of inﬁnitesimal rotation; we must notice that the present result is not very trustworthy in the extremal limit, unfortunately, since even small rotation may change the extremity condition.

4 2 Except for the region where Q ∼ Qmax, I/MR (r+) is smaller than 1. This result, when combined with the previous result g < 2, indicates that the ratio of the magnetic moment to the angular velocity decreases when a2 =6 0 and Q2 is finite. Therefore one can say that the “charge distribution” must be changed by the configuration of the dilaton. (Here we wish to say that if the charge distribution of an object remains unchanged, the reduction of the moment of inertia leads to an increase of the gyromagnetic ratio of the object according to classical electromagnetism.) Actually an interpretation is possible in the case with small Q (i.e., r−/r+ ≪ 1). The charge “observed” at the horizon is deduced from the Gauss law and (7), (8): 2 2 2 2a /(1+a ) 2a r− − − ∼ − Qobs/Q = (1 r /r+) 1 2 . (19) 1+ a r+ If the “observed” charge at the horizon induces the magnetic moment of the hole, the g factor must take the above correction. Compared with (15), unfortunately, the explanation is only a qualitative one (in spite of all the elaboration of the other factors). In summary, we have managed to give a charged dilaton hole a small angular momentum. We have found that the g factor of the black hole depends not only on a but also on Q/M. The moment of inertia for the black hole has also been obtained in the limit of infinitesimal angular momentum. We have found that both the g factor and the (normalized) moment of inertia of the black hole decrease as the parameter a increases. Both quantitites decrease also as the charge of the hole increases in most of the parameter region except for Q ∼ Qmax. Quantum processes associated with the curved space near the dilaton black hole may well be worth studying. Incidentally, the angular velocity ofthe dilaton black hole obtained here is relevant to the phenomenon of super radiance [12]. The future study of quantum processes will clarify various aspects of the thermodynamics of black holes.

Note added. After the submission of the present paper, the author has been informed of refs. [13] and [14]. Ref. [13] treats a similar study while ref. [14] claims that the effective action (1) with a = 1 does not exactly correspond to that of string theory because of the excitation ofan antisymmetric field strength by the non-zero electric and magnetic field via dH = F ∧ F . I should like to thank the referee for this important information.

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