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Einstein-Hilbert the by governed is gravitation S µν hsi h lc ormr ntecoesmlrt (but similarity close the on remark to place the is This esaldsrb h rmwr ntelnug fa of language the in framework the describe shall We = r oendb h obndaction combined the by governed are eial)cagdbakhlshv simple have holes black charged netically) e,tennxsec fpeieyextremal precisely of nonexistence the les, − lc oefml.Aogtenotable the Among family. hole black c L eodlaw ouinfracagdshrclblack spherical charged a for solution 16 P 1 s h tews icl calculation difficult otherwise the es; π lcrc ed rnprn.W re- We transparent. fields electric) ∗ ( = F nltcetnino h relevant the of extension analytic Z aiblt of variability t µν ~  G/c = e − 2 κ 2 1 3 ψ ǫ ) F µναβ ≡ 1 ∗ / µν F 2 l (4 F µν ≈ F µν π αβ ; e α ν 1 easneo the of absence he ~ − . + c 0 = h yaisof dynamics The . 616 ntewk of wake the in 2 ) ψ − κ 2 1 where , 2 / × 2 ψ ,µ 10 with ψ − , ψ F µ 33 ψ  µν sara scalar real a is l ( m n that and cm, → − eghpa- length a ψ ssupposed is g ψ → ) 1 const. + / 2 const., ψ d 4 S x, and it ; (1) (2) 2 equivalent of the framework’s length scale l must be LP, black hole are found ab initio in Sec. VIII; we give a trick but in the framework l is a free parameter to be de- that considerably simplifies the calculation of the electric termined experimentally. The framework was thought potential. We remark, in agreement with earlier opinions, to predict [9] that with l LP departures from the that cosmological α growth does not compromise the sec- weak equivalence principle occur≥ which would have been ond law; likewise, it cannot drive a charged black hole to ruled out by the classic experiment of Dicke et al [16]. become a naked singularity. In the appendix we derive Olive and Pospelov [17] investigated a modification of directly the solution and properties of the magnetically the framework designed to eliminate this problem (ψ charged black hole in the framework; they coincide with field couples more strongly to dark matter than to bary- those one would expect from duality considerations. onic). However, more detailed analysis [10, 18] has sug- In what follows we take the metric signature as gested that equivalence principle violations in the original 1, 0, 0, 0 and denote the temporal coordinate by t and framework may actually be suppressed (see Ref. 19 for a {−the others} by xi with i =1, 2, 3. Greek indices run from differing opinion). 0 to 3. The fact that α variability goes hand in hand with a modification of Maxwellian electrodynamics means that the charged black holes in the framework must be distinct II. EQUATIONS AND BOUNDARY from the Reissner-Nordstr¨om (RN) and Kerr-Newman CONDITIONS families of solutions of the Einstein-Maxwell theory. A family of spherically symmetric charged black holes of the From the action follow the equations relevant for a bare dilaton theory which supplant the RN black holes has black hole (it is convenient to subtract the trace part of long been known [12, 13, 14, 15] (these dilatonic black Einstein’s equations): holes have recently been extended to take into account −2ψ µν phantom matter sources [20]). But it was not clear that 0 = (e F );ν (3) the dilatonic black holes are the unique spherical black 2 µ κ −2ψ µν hole family in face of α variability. Neither were the co- ψ,µ; = e Fµν F (4) − 2 ordinates in which the metric for a dilatonic black hole is 2G −2ψ α 1 αβ usually couched particularly conducive to visualization Rµν = 4 e Fµ Fνα gµν F Fαβ of its geometry. We have thus here derived ab initio, c − 4 1 h   and in standard isotropic coordinates, the unique geom- + 2 ψ,µψ,ν . (5) etry and electric field of a spherical static charged black κ i hole within the variable α framework. We show them Eq. (3) shows that whereas e−2ψ is the vacuum electri- to be those of a dilatonic black hole (by transforming to cal permittivity, e2ψ plays the role of vacuum magnetic the coordinates used by Garfinkle, Horowitz and Stro- permeability. For in 3-d dimensional language appropri- minger (GHS) [14]). We elucidate the geometric, elec- ate to an inertial frame with 4-velocity ∂/∂t and Carte- trodynamic and thermodynamic properties of the modi- sian spatial coordinates, it reads ~ (e−2ψE~ ) = 0 and fied black holes, including some that were previously un- ~ −2ψ ~ −1 −2ψ ~ ∇ · known. (e B)= c ∂(e E)/∂t. Accordingly, the speed of∇× light c is constant in the framework; we shall hence- In Sec. II we collect the equations to be solved for forth set c = 1. a static spherical system with electromagnetism as the Eqs. (3)-(5) must be supplemented by suitable bound- only matter source. In Sec. III we delineate the expected ary conditions. Asymptotically we must require that the features expected of a black hole solution, and in light geometry approach Minkowski’s, and that F µν 0 while of them we solve the aforementioned equations to obtain ψ ψ = const., physically the coeval value of→ψ in the the generic black hole solution. Sec. IV provides a metric, c cosmological→ model in which our solution is embedded. alternative to the isotropic one, which allows exploration At the putative black hole horizon we must require of the black hole interior as well as two other sectors that all physical quantities, such as theH curvature (Ricci) which may be interpreted, respectively, as a spacelike scalar R = gµν R as well as F F µν be bounded (oth- naked singularity in asymptotically flat spacetime, and a µν µν erwise would be a singularity). Now from the trace of world lying between two timelike singularities. In Sec. V Einstein’sH equation (5) we get we show that our solution corresponds to the extended dilatonic black hole family. 2G R = (ψ,µψ ). (6) Sec. VI connects the mass and charge of a varying α κ2 ,µ black hole with the formal parameters of the solution. ,µ Sec. VII shows that the framework provides no exact Regularity of R at thus requires that ψ ψ,µ be counterpart to extremal RN black holes, but that nearly bounded there. H extremal black holes have very special properties. We dis- It is easy to check that the Schwarzschild and Kerr cuss the geometry of externally imposed electromagnetic black holes are exact solutions of Eqs. (3)-(5) with ψ = fields in the background geometry of a nearly extremal const. and F µν = 0. On the other hand, the RN black black hole. The thermodynamic functions of a varying α hole with ψ = ψc is not a solution because the r.h.s. 3 term of Eq. (4) prevents derivatives of ψ from vanishing Integrating once more and demanding that C(r) 0 as µν → (Fµν F = 0 for RN). r (asymptotic flatness) leads to We now6 assume a static spherically symmetric situa- → ∞ tion. In isotropic coordinates ( x1, x2, x3 = r, θ, ϕ ) εb2 { } { } eC =1 , (17) the metric is − r2 ds2 = e2A(r)dt2 + e2B(r)(dr2 + r2dΩ2), (7) where εb2 stands for an integration constant. We should − 2 2 2 2 thus consider the choices ε =0or 1. The b will be taken where dΩ dθ + sin θ dϕ . The boundary conditions as positive (no loss of generality± as will become clear in require that≡A(r) 0 and B(r) 0 as r . As- → → → ∞ III B); its dimension is that of length. suming there is only means that the only 2B 2 µν 01 tr Now (r)=4πe r is the area of a 2D-section of component of F is F = F , which will be a function space concentricA with the origin of the coordinates. With of r only. Also ψ = ψ(r). Hence Eq. (3) gives help of Eq. (17) this may be rewritten as 2ψ−A−3B tr e 2 F = Q , (8) εb 2 − r2 (r)=4πr2 1 e 2A. (18) A − r2 with Q an integration constant. Asymptotically we ex-  The horizon must be a null and spherically symmetric pect to recover a radial field. We thus identify H Qe2ψc as the electric charge measured a la Gauss from hypersurface, so it corresponds to one of those sections infinity. and is represented by the equation r rH = 0 with rH a constant. Further, the Killing vector−ξµ corresponding From the last result we may compute 2A to stationarity is 1, 0, 0, 0 with norm gtt = e . This − { } − e4ψ 4B vector must become null on the horizon, for otherwise ’s F αβF = 2Q2 . (9) H αβ − r4 generator would not be in a symmetry direction. Hence e2A 0 on . Thus the last factor in (r) diverges at Substituting this in Eq. (4) we have the scalar equation . → H A H However, (rH) cannot be infinite, for if it were so, e2ψ+A−B A+B 2 ′ ′ 2 2 (r) wouldA be decreasing with r just outside . One (e r ψ ) = κ Q 2 . (10) r Aconsequence would be the existence of a sphericallyH sym- where here and henceforth “ ′ ” denotes the derivative metric outgoing congruence of null rays, launched from just outside , with initially decreasing area, that is with with respect to r. H We now make Eq. (5) explicit: initial positive convergence. But according to the focus- ing theorem, in the presence of fields obeying the null ′ ′′ 2A ′ ′ ′ positive energy condition [which holds for action (2)], tt : A + + A B + A 2 = H (11) r such congruence must reach a caustic (singularity) in a 2B′ 2G finite stretch of affine parameter. Of course a spherically rr : A′′ +2B′′ + A′B′ + A′2 = H ψ′2 (12) r − − κ2 symmetric singularity outside would negate its horizon ′ ′ character. H ′′ ′ ′ ′2 A 3B θθ : B + A B + B + + = H (13) Thus we must require (rH) < . We conclude that r r − A ∞ 2ψ−2B the product of the first factors in Eq. (18) must cancel e − 2 the divergence of e 2A. Now for ε = 1 such product H GQ 4 (14) ≡ r can vanish only as r , but the horizon− cannot lie at infinity. Thus there→ can ∞ be no black hole solution for ε = 1. We proceed to show that assuming ε = 0 also III. SPHERICALLY SYMMETRIC STATIC − SOLUTIONS fails to produce black holes. For ε = 0 we must have eC = 1 and so B = A. − A. Excluding non black holes The divergence of is defused if rH = 0 and if, for r 0, e2A vanishesA as r2 or slower. In the second case we→ would have 0 as r 0. This is unacceptable There are actually several families of solutions of since a zero areaA surface → cannot→ be traversed by a finite Eqs. (10)-(13). We proceed to exclude the irrelevant ones. sized object, so that were the horizon to have zero area, We define the variables C = A + B and D = A B. − the black hole’s interior could not be accessed by real Adding Eqs. (11) and (13) gives particles. This does not correspond to the usual notion ′ of black hole. Thus for ε = 0 only the behavior e2A r2 ′ ′′ C C 2 + C +3 =0, (15) as r 0 is acceptable. ∝ r Let→ us subtract Eq. (13) from Eq. (11): which can be integrated to ′ 2ψ+2A ′′ 2A e 3 C ′ ′ A + = GQ2 . (19) r (e ) =0. (16) r r4  4

Now since as r 0, A ln r, we deduce from this equa- so that the only unknown now is ψ. tion that eψ must→ remain∼ bounded and nonvanishing at We now focus on the scalar equation (10) where we ′ H so that ψ cannot blow up there. It follows that rψ 0 substitute e2A from the above results. In terms of the as r |0.| This would imply that the l.h.s. of Eq. (19)→ new field variable with →A = B must vanish at the horizon. However, with 2A 2 − 2ψ 2 e r and e nonvanishing the r.h.s. is nonvanishing u 2(1 + G/κ )(ψ ψc)+4bξ (26) there.∝ The contradiction can be traced to the assump- ≡ − tion that ε = 0, which we must thus reject. The only ε (whose range is exactly that of ξ) it becomes which is consistent with black holes is thus ε = 1. d2u = q2eu; q2 2(G + κ2)Q2e2ψc . (27) dξ2 ≡ B. Black hole solutions This is the equation of motion in time ξ of a particle For ε = 1 the divergence of is prevented if rH = moving in an exponential potential. Its first integral is b = 0 and e2A vanishes for r Ab as (r b)2 or slower. 6 → − 2 Again exclusion of zero horizon area leaves us only the 1 du 2 u − = E + q e , (28) first option. In this case by Eq. (17) e 2B is bounded at 2 dξ αβ   . Thus the requirement of bounded F Fαβ tells us, again,H that ψ must not tend to + as r b. Further- where E is the analog of the particle’s conserved energy. ′ ∞ → ′ more, according to Eq. (6) ψ must remain bounded as Since ψ remains bounded as r b, dψ/dξ 0 and r b. This rules out behavior like ψ at . du/dξ 4b at . In addition u→ , so→ the q2eu → → −∞ H1 → H → −∞ Subtracting Eqs. (11) and (13), rewriting A = 2 (C + term vanishes. Thus the above equation requires that 1 2 D) and B = 2 (C D) and replacing the source term by E =8b . − means of Eq. (10) gives Since u 0 as r + , and the above equation has → → ∞ ′ ′ no turning point, it is clear thats u increases with ξ. Thus C D 2G ′ D′C′ + D′′ +2 = r2eC ψ′ . (20) the integral of the above equation is − r r κ2r2eC  du Using Eq. (17) we get after some algebra = √2 dξ (29) 8b2 + q2eu ′ 2b2 2G ′ Z Z D′(r2 b2) = (r2 b2)ψ′ , (21) − − r2 κ2 − with positive root.p which equation integrates to  The integration is done as follows. In terms of the variable v u ln(8b2/q2) the integral of Eq. (29) with ≡ − 2b2 2G the correct boundary conditions included is D′(r2 b2)+ = (r2 b2)ψ′ + ab, (22) − r κ2 − 0 dv′ where a is a new (dimensionless) integration constant. 4bξ = ′ . (30) 2 2 √1+ ev Let us now introduce the new radial coordinate Zln(q /8b ) dr 1 r b v 2 ξ = = ln − . (23) Now substitute e = sinh x to obtain the integral of r2 b2 2b r + b 2csch x over x which is 2 ln[csch x(cosh x 1)]. Simplify- Z   − ing with help of identities for hyperbolic− functions, and We note that ξ at and ξ 0 at spatial infinity. reverting to variable u one obtains Dividing Eq. (22)→ by −∞r2 bH2, going→ over to variable ξ, and − performing an additional integration we obtain 4bξ = ln Σ (q2/8b2)eu ln σ; (31) − 2 −1 −1 b 2G Σ(z) 1+2z 2z √1+ z , (32) D + ln 1 abξ = (ψ ψc), (24) ≡ − − r2 − κ2 − σ Σ(q2/8b2), (33)   ≡ where the new integration constant has been chosen so as where the square root is expressly the positive one. to enforce the asymptotic conditions ψ ψc and D 0 as r + . → → It follows from Eq. (33) that Now→ the∞ first two terms in Eq. (24) add up to 2A which, (32b2/q2)σ = (1 σ)2, (34) we have seen, must behave as 2 ln(r b) as r b. Since − ψ is bounded there, the term abξ−must diverge→ there It is clear from this that σ cannot be negative. Further, as 2 ln(r b), which is possible− only if a = 4. Thus Eq. (33) shows that 0 <σ< 1. Eq.− (24) can− be written as To recover ψ(ξ) first solve Eq. (32) for z: r b 2G 2A = 2 ln − + (ψ ψ ), (25) −2 r + b κ2 − c z = 4Σ(z) 1 Σ(z) . (35)   −  5

In this last substitute on the l.h.s. z = (q2/8b2)eu This is precisely the RN metric in isotropic coor- and on the r.h.s. Σ(z) = exp(4bξ + ln σ⇒), and invoke dinates (see for example Ref. 21). Anticipating Eq. (23) to get ⇒ Eqs. (57) and (58) we find the parameter corre- 1 spondence to be b = (R+ R−) and σ = R−/R+ − 4 2 2 2 − 32b2 r b r b with R± = GM G2M 2 GQ2e2ψc and M the eu = σ − 1 σ − . (36) ± − q2 r + b − r + b mass.       ! p Now take the logarithm and substitute u here from IV. REGIONS AND SECTORS OF THE Eq. (26); after canceling a term and taking Eq. (34) into SOLUTION account we get

2 It is worthwhile noting that under the change of vari- κ 1 σ 2 ψ = ψc + 2 ln − − 2 . (37) able r ֒ ρ b /r, the metric (7) with Eqs. (38) and G + κ 1 σ r b ! → ≡ − r+b (39) goes over into itself with ρ everywhere replacing r.  Thus the interval r (0,b) is mapped onto the black At this point recall Eq. (25); Exponentiating it and hole exterior r (b, ∈ ) with the horizon being a fixed substituting from our last result gives point. Just as in∈ the∞ case of the RN solution in isotropic

2G coordinates, the the metric (7) with Eqs. (38) and (39) 2 G+κ2 covers the black hole exterior twice, but does not cover 2A r b 1 σ e = − − − 2 (38) any part of the black hole interior. r + b 1 σ r b !   − r+b In the RN case this sort of problem is resolved by pass- ing to Schwarzschild style coordinates (squared radial co- Further, from B = C A and Eq. (17) follows − ordinate gives area). Such a transformation is intractable 2G here as it entails solution of a higher order algebraic equa- − 2 G+κ2 b 4 1 σ r b tion. Because of this we opt for a slightly different radial e2B = 1+ − r+b (39) r 1 σ coordinate   −  ! r b 2 We have so far supposed that b> 0. Are values b 0 ̟ − . (42) ≤ ≡ r + b permitted too? Note that b = 0 amounts to taking ε =0   in Eq. (17) and that case was already ruled out there. This can be inverted to get As for b < 0, suppose we formally switch the sign of b 2B in our solution (37)-(39). It is then clear that e = 0 1+ √̟ 2A r = b . (43) and e = on the surface r = b , meaning that surface 1 √̟ has zero area∞ and is an infinite blueshift| | surface. Thus no − physical body could penetrate from r> b while photons | | Actually a second solution may be obtained by switching launched outwardly from near the surface would reach the signs of the radicals. This doubling corresponds to distant observers with arbitrary large energy. The object the two radial coordinates r and b2/r which cover the in question would obviously be pathological. This means same (exterior) region, as discussed earlier in this section. b must be positive, as assumed. Transforming metric (7) with Eqs. (38) and (39) from Eqs. (37)-(39) specify the unique exact solution for a r to ̟ we have nonrotating charged black hole in general relativity with 2G varying α electrodynamics. Several special cases are of 1 σ G+κ2 ds2 = ̟ − dt2 (44) interest. − 1 σ̟  −  2G For Q 0, σ 0, and the solution takes the 16b2 1 σ̟ G+κ2 d̟2 • → → + − + dΩ2 . form of a Schwarzschild metric in isotropic coor- (1 ̟)2 1 σ ̟(1 ̟)2     dinates with ψ = ψc. As mentioned in Sec. II, the − − − Schwarzschild solution should be a solution also in Likewise writing down the field ψ from Eq. (37) we have the varying α theory. κ2 1 σ ψ = ψ + ln − . (45) For κ 0, ψ ψc and both exponents in the last c G + κ2 1 σ̟ • factors→ in Eqs.→ (38) and (39) become 2. Thus  − 

2 It should be clear that the new coordinates are suitable r2 b2 so long as σ < 1; the case σ 1 is discussed in Sec. VII. e2A = − (40) → 2 2 1+σ The black hole exterior is covered by the coordinate r + b +2b − r ! 1 σ domain ̟ (0, 1) with ̟ = 1 being spatial infinity and 2 1 1+ σ ̟ = 0 being∈ the event horizon (r = b). The change of e2B = r2 + b2 +2b r (41) r4 1 σ coordinate has now put the horizon’s interior in view:   −   6 it is the domain ̟ ( , 0) wherein the t coordinate also a singularity which is timelike in character because ∈ −∞ becomes spacelike and ̟ timelike. g̟̟ > 0 as one approaches it from lower ̟. It can be We identify ̟ = as the central singularity. The shown from Eq. (47) that d /d̟ < 0 throughout the reasoning is as follows.−∞ From Eqs. (44) and (45) and domain under discussion: theA area of 2-surfaces increases comparing with Eq. (6) we have monotonically with ̟ as we go from singularity to spatial infinity. Thus ̟ (1, 1/σ) spans the static asymptoti- 2 − 4G+2κ ∈ ,µ 4 2 cally flat spacetime around a naked timelike singularity. R ψ ψ = ̟(1 ̟ ) 1 σ̟ G+κ . (46) ∝ ,µ − − This spacetime is obviously distinct from the black hole Thus the scalar curvature diverges for̟ signify- one. ing that ̟ = is a true singularity. Since→ −∞ it borders a Finally we study the domain ̟ (1/σ, ). Metric −∞ (44) is real there only when 2G/(G∈+ κ2) is∞ a rational region where g̟̟ < 0 this singularity is spacelike (nor- mal with negative norm). Further, from the area of a number which, when maximally reduced, is either of form ̟ = const. surface, (2n1 + 1)/(2n2 +1) or 2n1/(2n2 + 1) with n1,n2 two integers. However, in the first case t becomes spacelike 2G 2 −2 in the said domain while ̟, θ and ϕ all become timelike. (̟) (1 σ̟ G+κ (1 ̟) , (47) A ∝ − − The metric’s signature is thus unphysical and we must we observe that the singularity has vanishing area. Thus reject any physical interpretation of this case. 2 ̟ = is a central singularity, in all respects like the By contrast, for 2G/(G+κ )=2n1/(2n2 +1) the met- one in−∞ the Schwarzschild solution, and in contrast to the ric’s signature is the usual one, with t a timelike coordi- nate in the whole domain ̟ (1/σ, ). According to timelike singularity of the RN solution. ∈ ∞ For κ = 0 there is no second (inner or Cauchy) horizon, Eq. (46) the scalar curvature diverges both as ̟ 1/σ 6 and ̟ from within the domain, so that the→ said such as we have in the RN solution. It is true that gtt can → ∞ vanish not only at ̟ = 0 but, provided κ2 < G, also at boundaries are timelike singularities. The spacelike dis- ̟ = . However, as clear from Eq. (46), ̟ = is, like tance between them, ∞ ∞ ̟ = , a point of unbounded curvature, and cannot G ∞ 2 be a−∞ horizon. Thus we are left with a single horizon, 4b (1 σ̟) G+κ d̟ ℓ = G − 2 , (50) ̟ = 0. G+κ2 √̟ (1 ̟) (1 σ) Z1/σ − But then how does the RN solution (for κ = 0) man- − age to have two horizons? For this Maxwellian electro- converges at both limits of the integral, so we may speak dynamics case the scalar R vanishes identically since ψ of a finite static spacetime lying between two spherically must be constant (see Eqs. (37) and (46)). There is then symmetric timelike singularities of vanishing area. This no longer any reason for ̟ = to be a singularity, and spacetime is evidently not asymptotically flat. thus it might be a horizon. To∞ see that it is introduce The two singularities are point charges of clearly oppo- the area radial coordinate site signs and the same magnitude, since by Gauss’ law the electric flux lines issuing from one must, by dint of 4b 1 σ̟ ̺ = − , (48) the symmetry, end up in the other. How come the two 1 σ 1 ̟ charges do not pull each other together? A simple cal- − − culation shows that a freely falling particle will oscillate for which (̺)=4π̺2 just as with the radial coordinate A between the two singularities, and find it impossible to in the usual form of Schwarzschild’s metric. In terms of approach either, no matter how large its conserved energy ̺ we have is. Gravity is thus repulsive in nature in the said region, and this must be the agent that balances the charges’ (1 σ)̺ 4b (1 σ)̺ 4bσ g = − − − − . (49) attraction. tt − (1 σ)2̺2 −  It is now clear that there two horizons, at ̺ = R±, with V. IDENTITY WITH DILATONIC BLACK 1 R−/R+ = σ and 4 (R+ R−) = b. These last are pre- HOLES cisely the relations quoted− for RN at the end of Sec. III B. In agreement with our previous remarks we note that, We now demonstrate the identity of our solution with according to Eq. (48), the inner horizon R = R− cor- responds to ̟ = . Thus RN is the only charged the spherical dilatonic black holes by comparing with spherical black hole−∞ sporting an inner horizon. More on GHS’s formulation [14]. GHS use a radial coordinate such that g = 1/gRR. To convert metric (44) to this, in light of GHS, in Sec. V. R tt − Metric (44) describes not only black holes, but also this form we obviously have to require other denizens of the gravitational world. 4b d̟ = d . (51) In the domain ̟ (1, 1/σ), t is a timelike coordinate. (1 ̟)2 ± R The boundary (̟ =∈ 1) is spatial infinity as can be seen − because as ̟ 1+ with R vanishing there. And Taking the positive sign so that increases with ̟, and because AR →diverges ∞ as→̟ 1/σ, this second boundary is choosing the integration constantR (the zero of ) with → R 7 hindsight we get To obtain the observable mass M we expand Eq. (38) in powers of 1/r: 4b 4bσ = + . (52) R 1 ̟ 1 σ 4b 2Gσ − − − e2A =1 1+ + (r 2) (56) − r (G + κ2)(1 σ) O Inverting this we can put metric (44) in the form  −  2 2G Because asymptotically r is the radius, one can identify 2 2 2 d 2 4bσ G+κ2 2 ds = λ dt + R2 + 1 dΩ , (53) the coefficient of 1/r here as 2GM. Replacing the σ in − λ R − (1 σ) 2 − − R terms of (1 σ) by means of Eq. (34), and substituting   − with σ’s value from Eq. (33) we finally get the observable mass,

− 2 G κ − 2 4b 4bσ G+κ2 1 1+ 1 (G + κ2)b 2Q2e2ψc 1 λ 1 1 . (54) M =2b + 4 − (57) ≡ − (1 σ) − (1 σ) G (G + κ2)  − R  − R  p ! We may also translate ψ from Eq. (45) to the form where, again, the positive square root is chosen. κ2 4bσ Solving this expression for b in terms of M and Q by ψ = ψ + ln 1 . (55) c G + κ2 − (1 σ) first manipulating it into the form of a quadratic in b we  − R  get, for κ2 = G, Strictly speaking dilaton theory is based on equations 6 (3)-(5) but with the choice κ2 = G [14]. To investi- Gκ2M + G4M 2 (G κ2)G2Q2e2ψc b = − − − (58) gate stability of the dilatonic black holes with respect to 2 p 2(G κ ) changes of the parameters of the theory, GHS also con- − sidered the case of generic κ (which they denote a) but The second solution to the quadratic—corresponding to a negative signed radical—turns out to be extraneous. It setting ψc = 0. Our Eqs. (53) and (54) agree in form with theirs; the difference between their expression for would be the solution to Eq. (57) if the radical in the lat- the dilaton field and our Eq. (55) is immediately under- ter were negative: the solution procedure described above stood if we recall that our ψ/κ corresponds to GHS’s entails squaring that radical, and so adds an unphysical dilaton, and that GHS deal with the magnetic charge solution which must be rejected by hand. 2 case, c.f. Eq. (A.8) of the appendix below. By contrast, if κ = G we naturally get just one solu- GHS remark that the zero of λ2 at = 4bσ/(1 σ) tion: is really a singularity, except in the caseR κ = 0 in which− b = 1 GM 1 Q2e2ψc /M. (59) it marks the inner horizon of the RN geometry. This 2 − 4 tallies with the point made in Sec. IV that ̟ = is To avoid negative b, which would be meaningless for an −∞ − the central singularity in the generic case. Together with horizon, one demands Q √2GMe ψc . Gibbons and Maeda [13] and GHS [14] we conclude that Thus for any κ2, the| “no|≤ hair” principle is satisfied: the inner horizon becomes unstable and metamorphoses for any pair M,Q there is just one black hole, spec- into a singularity as κ departs from zero. ified by Eqs.{ (38) and} (39) with the parameters given We shall connect the two black hole parameters to ob- by Eq. (58) or Eq. (59) together with Eq. (33). We do servables in a different way than did GHS. For this pur- not count90 eψc as a black hole parameter since it is set pose we revert to metric (7) with Eqs. (38) and (39). by the cosmological model in which our solution is to be embedded. Whether for κ2 > G (strongly coupled α variability) or VI. PHYSICAL OBSERVABLES OF BLACK κ2 < G (weakly coupled varying α theory) Eq. (58) gives HOLES WITH VARYING α − nonnegative b only for Q √G + κ2 Me ψc . Thus for all values of κ2, charged| black| ≤ holes can be had only for How do the black hole parameters b and Q relate to − the observable properties of the black hole for generic κ? Q G + κ2 Me ψc . (60) In Eq. (8) the radial electric field as r is Qe2ψc /r2. | |≤ By asymptotic flatness the 4πr2 is, in→ this ∞ limit, a good For fixed M the blackp hole family is a one-parameter measure of the area of a sphere concentric with the black sequence; the parameter can be either b or Qeψc or hole, so that we must conclude that the charge, as in- σ. Along the sequence b decreases monotonically with ferred from Gauss’ law, is Qe2ψc . This is the observable Q eψc from b = GM/2 at Q = 0 towards its zero point | | − charge (as seen from infinity). We refer the reader to at Q = √G + κ2 Me ψc , in the vicinity of which b | | ∝ the discussion in Ref. 9 to the effect that Q, the strength (√G + κ2 Q eψc ). According to Eq. (34) σ Q2e2ψc parameter in the electric current, is rather the conserved as Q 0− while | | σ 1 for b 0. ∝ charge. In agreement with this, the observed charge at From| | → Eq. (39) with→ r = b we→ have for the horizon area infinity would undergo evolution in an expanding uni- verse (where ψ would be given by the time dependent 64πb2 c (b)= . (61) solution to the cosmological problem). A G~(1 σ)2G/(G+κ2) − 8

Numerically it is found that (b) decreases monotoni- the effective permeability would be below unity through- cally with Q eψc at fixed M.A As evident from Eq. (34), out space. That would make the black hole exterior like | | − (b) tends to zero as Q √G + κ2Me ψc and σ 1. a diamagnetic medium which tends to expel magnetic AAs Q 0, (b)| tends| → to its Schwarzschild value→ fields. Therefore, for such strong coupling the black hole 16πG| 2|M →2. A would tend to bend external magnetic lines away from itself, much as a superconductor will push out magnetic flux. And a sufficiently strong magnetic field would be VII. NEARLY EXTREMAL BLACK HOLES able to bodily push the black hole away. It is easy to see what changes would take place were the black hole charged magnetically. The basis is explained In Einstein-Maxwell theory, for which the charged in the appendix. black holes are RN, the extremal black holes are those for which Q attains its least upper bound √GM. In our framework extremality would correspond to the sat- VIII. BLACK HOLE THERMODYNAMICS uration of inequality (60). We have just seen that this corresponds to b 0 and σ 1 in which limit the → → The easiest way to calculate the black hole temper- horizon area vanishes. But as argued in Sec. IIIA, an µ object with zero horizon area cannot be regarded as a ature is via the surface gravity. At fixed point x = t, r, 0, 0 has an acceleration vector aµ = 0,ar, 0, 0 black hole. Hence, in the framework there is no exactly { r } t 2 { } extremal black hole (for κ = 0). This agrees with our with a = Γrr(dt/dτ) . For a diagonal metric like that 6 in Eq. (7)− this gives ar = e−2BA′. Consequently, the in- rejection of solutions with ε = 0 in Sec. II; ε = 0 is µ −B ′ equivalent to b = 0. variant acceleration is √a aµ = e A . To these corre- sponds the local Unruh temperature T = (~/2π)e−BA′. In what follows we shall be concerned with the nearly U Redshifting this to infinity we get the global temperature extremal black holes in the framework. Many of their A T = e T . It is reasonable to take T = lim → T . properties can be ascertained most easily by developing g U BH r rH g Using the specific metric (7) we get in the limit r b the generic formulae in Taylor series in b while setting → (1 σ) =4√2 b/q in accordance with Eq. (34). For ~ − 2G/(G+κ2) example, in the extremal limit TBH = (1 σ) . (65) − 16πb − 2 (G + κ2)M 2G/(G+κ ) An independent approach is afforded by the the Eu- e2A = 1+ + (b), (62) r O clidean framework. According to Eqs. (7) and (39) the   2B −2A increment of radial proper length ℓ from the horizon to e = e , (63) point r is given by dℓ = eBdr so that 2 2 2 −κ /(G+κ ) ψ ψc (G + κ )M G e = e 1+ + (b).(64) − 2 G+κ2 r O r b 2 1 σ r b   ℓ = 1+ − r+b (66) b r 1 σ  ! It is interesting that to leading order eψ is just some Z   − − 2 power of e2A. = 4(1 σ) G/(G+κ )(r b)+ (r b)2 . (67) − − O − One interesting application of the above is as follows. Solving r b in terms of ℓ in Eq. (38) gives  It is known [22] that Maxwellian electrodynamics in a − static background geometry can be regarded as electro- 2 2 ℓ dynamics in flat spacetime filled with a medium with elec- e2A = (1 σ)4G/(G+κ ) + (ℓ3), (68) tric permittivity and magnetic permeability both equal to − 64b2 O 1/√ g . But we have remarked that varying α electro- tt so that the Euclidean metric of the t-r plane takes the dynamics− is related to the Maxwellian one by a vacuum form permitivitty e−2ψ and a e2ψ. By 2 compounding the two cases we see that the new electro- 2 ℓ ds2 = (1 σ)4G/(G+κ ) dτ 2 + dℓ2 (69) dynamics in the curved spacetime represented by metric Euc − 64b2 (7) functions like the Maxwellian brand in flat spacetime filled with a medium with permittivity e−A−2ψ and per- where τ is Euclidean “time”. If we wish to interpret this meability e−A+2ψ. τ as an angle, a conical singularity will occur unless we For the special case of coupling κ2 = G/2, we find from require that its period be Eqs. (62) and (64) that the exterior of a nearly extremal 2 −1/2 electrically charged black hole has unit effective perme- 2 ℓ Π=2π (1 σ)4G/(G+κ ) . (70) ability throughout! This would mean that magnetic lines − 64b2 produced by distant currents would not be bent by the   hole’s presence (with bending judged with respect to the This periodicity is equivalent to a thermal ensemble with 2 flat asymptotic space). And for larger coupling κ > G/2 temperature ~/Π. This last is precisely TBH of Eq. (65). 9

Yet a third alternative approach is to start with the since, as discussed in Sec. II, Qe2ψc is the charge observ- black hole entropy as a quarter of the horizon area in able from infinity. Since this leads to a very complicated units of Planck’s length squared: computation, we here determine ΦBH as the value of the 2 At in the limit r b (provided At (b) 16πb vanishes asymptotically).− The logic for→ this prescription SBH = A = 2 . (71) 4G~ G~(1 σ)2G/(G+κ ) is as follows. The Lagrangian for a charged particle with − mass µ and (conserved) charge e is Then the black hole temperature is

−1 α β α ∂SBH dx dx dx T = . (72) = µ gαβ + eAα , (78) BH ∂M L − − dτ dτ dτ  Q r where A is the electromagnetic vector potential: F = To evaluate this we should regard σ a function of b and α αβ A A . In a gaugefor which A is time independent, b a function of M. Then β,α α,β α the Lagrangian− is also t independent, and we have the 2 G~ (1 σ)2G/(G+κ ) conserved canonical momentum TBH = − (73) 16πb ∂b Gb ∂σ/∂b ∂ dt 2 ∂M q 1+ G+κ2 1−σ Pt = dtL = µgtt + eAt. (79)   ∂ dτ dτ The easiest way to obtain  ∂σ/∂b is to take the loga- rithm of Eq. (34) and differentiate the result with respect Since the first term in Pt is minus the particle’s rest plus to b at fixed q: kinetic energy, Pt is minus the total energy and eAt must be the particle’s electric energy at the correspond-− ∂σ 2σ(1 σ) = − (74) ing point. The electric potential of the black hole as mea- ∂b q − b(1 + σ) sured from infinity can thus be deduced from the value   Therefore, Eq. (73) takes the new form this energy takes on as the particle nears the horizon:

−2ψc 2G/(G+κ2) ΦBH = lim e At. (80) G~ (1 σ) − r→b TBH = − (75) 16πb ∂b G 2σ 2 1 2 −2ψc ∂M q − G+κ 1+σ Here the factor e accounts for the fact that the ob- 2ψc    servable charge of the particle is e e. Harmony of this with Eq. (65) would entail the identity Turning to Eq. (8) we have −1 ∂b G 2σ 2ψ+A−B 1 G 1 . (76) 2A+2B tr e 2 2 A = e F = Q . (81) ∂M q ≡ − G + κ 1+ σ t,r r2     Although we have been unable to establish this analyt- Substitution from Eqs. (37)-(39) gives ically, we have checked it numerically for a large set of 2 values of Q/M and various κ /G. Thus the Euclidean Qe2ψc (r2 b2) A = − . (82) and thermodynamic calculations of TBH agree. This, by t,r 2 2 2 1+σ the way, demonstrates that the area formula for SBH r + b +2 1−σ br does not get corrections from α-variability.   As already mentioned in Sec. VI, SBH as well as (b) This integrates to vanish in the limit b 0 in which inequality (60) wouldA → Qe2ψc r be saturated and the black hole would become extremal. A = , (83) t 2 2 1+σ This is in contrast to the situation of the extremal RN −r + b +2 1−σ br black hole which has nonvanishing area and entropy. The TBH has an even more curious behavior. As mentioned which appropriately vanishes as r . Now taking the limit r b in accordance with Eq.→ (80) ∞ gives in Sec. VII the would be extremal black hole is reached → in the limit b 0 with (1 σ) b. It may be seen from → 2 − ∼ Q(1 σ) Eq. (65) that for κ < G the temperature vanishes in ΦBH = − . (84) that limit (just as it does for the RN black hole), while 4b 2 for κ > G it diverges. As already noticed by GHS, for We have checked numerically (for a variety of values 2 the true dilatonic black holes (κ = G) TBH remains of Q/M and κ2/G) that the Maxwell thermodynamic finite [14]. relation In the thermodynamic approach the electric potential ∂(1/T ) ∂(Φ /T ) of the black hole, ΦBH is BH = BH BH (85) 2ψc ∂(Qe ) M − ∂M Qe2ψc ∂SBH ∂M     ΦBH = TBH = . − ∂(Qe2ψc ) ∂(Qe2ψc ) is indeed obeyed. Further, we recall that as κ 0 we  M  SBH (77) should recover the RN result. According to Sec. III→ B, for 10 the RN black hole 4b = R R− and σ = R−/R , where The above ignores the effects of radiation. Suppose + − + R± are the radii of the two horizons in Schwarzschild-like the hole is so small (and hot) that Hawking emission coordinates. With these our potential ΦBH reduces to dominates both radiation accretion and M’s growth due Q/R+, which is the correct RN result. Both above checks to α evolution, but not hot enough to emit charges, so support the correctness of our result for ΦBH . Note that that Q remains fixed. Then it seems that bound (60) ψc −ψc ΦBH depends on e through b and σ. will eventually be surpassed since e decreases. Let An application of the above is to the issue of whether us check if the hole can reach the limiting point of the black hole thermodynamics can usefully constrain the inequality in a finite time. cosmological rate of change of α? Davies, Davis and The energy emission rate M˙ should, on physical | | 4 Lineware (DDL) [23] argued from the form of the black grounds, be proportional to (b)TBH . We see from hole entropy of a RN black hole that it could not help but Eqs. (61) and (65) that near theA limiting point (b = 0) decrease if α is increasing as claimed by Webb’s group [7]. 2 6G − 4G−κ They propose to rule out a cosmologically growing α on M˙ = const. (1 σ) G+κ2 b 2 = const. b G+κ2 , (87) the ground it would violate the generalized second law (if − × − − × one ignores the entropy produced by Hawking emission). where the second equality follows from the fact (Sec. VII) DDL discount the possibility that variability of α could that near b = 0, b 1 σ. Using Eq. (76) we convert modify charged black hole properties enough to overturn this into ∼ − their conclusion. db G(G + κ2) 4G−2κ2 But DDL’s implicit assumption that the mass of a = const. b G+κ2 . (88) black hole in the expanding universe is constant is in- dt − × κ2 correct. Expansion makes the black hole environment For κ2 G the integral time dependent, and there is no reason for mass (Hamil- ≤ tonian) to be conserved (even if no radiation flows in or b 2− ′ 2κ 4G ′ out). The expansion timescale is generally long compared b G+κ2 db (89) to the hole’s dynamical timescale, so the process is adia- Z0 batic. And as pointed out by Fairbairn and Tytgat [24] diverges at the lower limit, which shows that Hawking and Flambaum [25], it is rather the black hole entropy radiation cannot bring the hole to the limiting point (or horizon area) which is unchanged under these circum- in a finite time. But for κ2 > G the integral con- stances (adiabatic invariance of the black hole area was verges. But before concluding that it takes but a fi- established earlier by Mayo [26] and by one of us [27]). nite time for b to shrink to zero and for the hole to Consulting Eq. (61) we see that the black hole parameters 2 −ψc 2 achieve Q = √G + κ Me and become a naked sin- should evolve with b (1 σ)G/G+κ . Then Eqs. (33) gularity| even| without help from varying α, we should ∼ − and (57) determine the dependence M(eψc ), and hence recall our assumption that no charged particles are emit- M’s temporal variation. Similar remarks are made by ted. Sufficiently near b = 0 the temperature diverges as 2 2 2 Fairbairn and Titgart on the basis of the κ = G case b(G−κ )/(G+κ ) so before the dangerous point the hole will of the solution (37)-(39). Of course, once radiation pro- begin to emit charged particles, no matter how massive cesses are allowed for, the overall entropy must increase. they are. The consequent decrease of Q may steer the Thus the validity of the generalized second law is not hole away from the limiting point. endangered by growth of α. DDL also warn that systematically growing α will eventually bring the black hole to the point of becom- IX. SUMMARY AND CONCLUSIONS ing a naked singularity. In the RN case (and in their language) this happens when the growing charge reaches A spacetime variable α modifies Maxwellian electro- √GM. In our framework the question is whether in- magnetism. It thus modifies the structure of charged −ψc −1/2 equality (60) will fail because e decreases as α . black holes in general relativity. Here we have derived ab But can the limiting case of the inequality be reached initio the unique family of spherical static charged black in view of the adiabatic invariance of ? According to A holes in the framework of α variability proposed by one Eqs. (61) and (33) of us [9]. This family coincides with a one-parameter ex-

− 2 tension of the dilatonic black holes [13, 14, 15, 24]. In 4G −2 G κ q G+κ2 b G+κ2 contrast with the classic Reissner-Nordstr¨om black holes, 2G . (86) A∝ G+κ2 varying α charged black holes lack an inner horizon; one 1+ q2/8b2 1 − can take the view that variability of α, however weak, p  destabilizes the inner (Cauchy) horizon to a singularity. It may be seen that as long as q2 is finite, b cannot ap- Our charged black hole metric has two additional sec- proach zero while keeping constant. Now the men- tors. One describes the static asymptotically flat space- tioned limiting case is attainedA as b 0 (see discussion time around a charged naked timelike singularity. The following Eq. (60)). Hence, the disaster→ envisaged by last sector represents a static finite spacetime lying be- DDL cannot take place while α is finite. tween two timelike singularities of zero area and bearing 11 opposite charges. This last configuration can occur only ory of varying α. We again restrict attention to static when the κ2 parameter takes on one of an infinite set of spherically symmetric solutions. In this case we expect rational values. the only nonvanishing component of the electromagnetic Charged black holes in varying α theory obey the “no field tensor to be Fθϕ, which corresponds to a radial mag- hair” principle; they are fully determined by the mass netic field. In terms of dual fields this is ∗F tr, so that M and conserved charge Q of the black hole (and by instead of Eq. (3) we have here the asymptotic value of the α field, which is nothing but the coeval cosmological value of α). The allowed range (r2eA+3B ∗F tr)′ =0. (A.1) of the charge-to-mass ratio Q/M is, however, somewhat different from that in the Reissner-Nordstr¨om case, and depends both on the κ2 parameter and on the cosmolog- By analogy with Eq. (8) we have the solution ical α. The area of the horizon of the geometry tends to A+3B 2 ∗ tr zero at the largest allowed Q /M, and this limiting case Fθϕ = e r sinθ F = P sin θ (A.2) − of the solution is not a black| hole.| Nearly extremal black holes have the property that the α field is a power of the where P is an integration constant, the magnetic square of the time Killing vector. For the special value monopole. 2 κ = G/2, an externally sourced magnetic field will be Forming F αβF we now find, instead of Eq. (10), uniform in such a black hole’s vicinity (as judged from αβ infinity). e−2ψ+A−B We have here calculated anew the black hole thermody- (eA+Br2ψ′)′ = κ2P 2 . (A.3) namic functions in the face of varying α. In particular, we − r2 present a trick which enables an otherwise intricate cal- culation of the electric potential to be carried out almost And calculating anew the electromagnetic stress-energy trivially. The α dependence of the black hole thermody- tensor we find the Einstein equations namic functions makes it tempting to suppose that black hole thermodynamics may restrict α variability in the 2A′ tt : A′′ + + A′B′ + A′2 = K (A.4) expanding universe. Such a claim was urged by Davies, r Davis and Lineware [23] because the black hole entropy ′ ′′ ′′ 2B ′ ′ ′ 2G ′ for the Reissner-Nordstr¨om black hole would seem to de- rr : A +2B + A B + A 2 = K ψ 2 (A.5) r − − κ2 crease as α increases. We reiterate, with Fairbairn and ′ ′ A 3B Tytgat [24] and with Flambaum [25], that in view of adi- θθ : B′′ + A′B′ + B′2 + + = K (A.6) abatic invariance of the black hole entropy, the general- r r − e−2ψ−2B ized second law is not endangered by α growth in cosmol- K GP 2 (A.7) ogy. Adiabatic invariance is also sufficient to prevent a ≡ r4 charged black hole from evolving, due to cosmological α growth, into the extremal state of vanishing horizon area. Comparing Eqs. (A.3)-(A.6) with Eqs. (10)-(13) we no- 2 However, we find that when κ > G, even in the absence tice that they differ only by the replacement Q ֒ P and of cosmological α growth, Hawking radiation of neutral ψ ֒ ψ. Accordingly we can immediately write→ down particles tends to drive a charged black hole to the van- the→ solution − for the magnetic black hole: the metric is ishing horizon area state in a finite time. However, it still given by Eqs. (38) and (39) with M and b defined seems likely that late emission of charged particles due by Eqs. (57) and (58) or Eq. (59) with Q ֒ P , while the to a rising black hole temperature will prevent violation scalar field takes the form → of cosmic censorship. κ2 1 σ ψ = ψc 2 ln − − 2 (A.8) Acknowledgments − G + κ 1 σ r b ! − r+b  JDB thanks Oleg Lunin, Victor Flambaum and David where we have turned around again the sign of ψc since Garfinkle for useful remarks. it is by definition the asymptotic value of the scalar field. By the same token, the sign of ψc is to be retained un- changed in places like Eq. (27), (45) and (55). The discus- Appendix: Magnetic black holes sion in Secs. V-VIII can be taken over almost verbatim: it is now P , the conserved monopole, that is restricted The duality principle informs us that there should also by Eq. (60),| | and the observable is − exist purely magnetically charged black holes in the the- e 2ψc P . 12

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