Extremal Black Holes in Dynamical Chern-Simons Gravity
EExtremal Black Holes in Dynamical Chern-SimonsGravity
Robert McNees
Loyola University Chicago, Department of Physics, Chicago, IL 60660, USA.Kavli Institute for Theoretical Physics, University of California, Santa Barbara,CA 93106, USA.
Leo C. Stein
TAPIR, Walter Burke Institute for Theoretical Physics, California Institute ofTechnology, Pasadena, CA 91125, USA
Nicolás Yunes
Department of Physics, Montana State University, Bozeman, MT 59717, USA.Kavli Institute for Theoretical Physics, University of California, Santa Barbara,CA 93106, USA.
Abstract.
Rapidly rotating black hole solutions in theories beyond generalrelativity play a key role in experimental gravity, as they allow us to computeobservables in extreme spacetimes that deviate from the predictions of generalrelativity. Such solutions are often difficult to find in beyond-general-relativitytheories due to the inclusion of additional fields that couple to the metric non-linearly and non-minimally. In this paper, we consider rotating black hole solutionsin one such theory, dynamical Chern-Simons gravity, where the Einstein-Hilbertaction is modified by the introduction of a dynamical scalar field that couples to themetric through the Pontryagin density. We treat dynamical Chern-Simons gravityas an effective field theory and work in the decoupling limit, where correctionsare treated as small perturbations from general relativity. We perturb aboutthe maximally-rotating Kerr solution, the so-called extremal limit, and developmathematical insight into the analysis techniques needed to construct solutions forgeneric spin. First we find closed-form, analytic expressions for the extremal scalarfield, and then determine the trace of the metric perturbation, giving both in termsof Legendre decompositions. Retaining only the first three and four modes inthe Legendre representation of the scalar field and the trace, respectively, sufficesto ensure a fidelity of over relative to full numerical solutions. The leading-order mode in the Legendre expansion of the trace of the metric perturbationcontains a logarithmic divergence at the extremal Kerr horizon, which is likely tobe unimportant as it occurs inside the perturbed dynamical Chern-Simons horizon.The techniques employed here should enable the construction of analytic, closed-form expressions for the scalar field and metric perturbations on a backgroundwith arbitrary rotation.PACS numbers: 04.30.-w,04.50.Kd,04.25.-g,04.25.Nx a r X i v : . [ g r- q c ] N ov xtremal Black Holes in Dynamical Chern-Simons Gravity
1. Introduction
Einstein’s theory of general relativity (GR) has passed a plethora of Solar Systemand binary pulsar tests [1], but it has not been tested in depth in the extreme gravity regime [2, 3] where the gravitational interaction is non-linear and dynamical. A numberof new observations will allow us to test this regime of Einstein’s theory: in thegravitational wave spectrum through Advanced LIGO and its partners [4, 5], whencompact objects collide; in the radio spectrum with the Event Horizon Telescope [6],when an accretion disk illuminates its host black hole and creates a ‘shadow’; and inthe X-ray spectrum with the Chandra Telescope [7], when gas heats up and glows asit accretes in the black hole spacetime. Such future observations will either confirmEinstein’s theory at unprecedented levels or reveal new phenomena in the extremegravity regime.Solutions that represent rotating black holes (BHs) in theories of gravity beyondgeneral relativity are an essential ingredient of tests in the extreme gravity regime.Constraining these theories requires a metric with which to calculate observables. Oncea metric is available, one can investigate the modal stability of the solution, calculatethe gravitational waves emitted as two BHs inspiral, compute the ‘shadow’ cast by aBH when illuminated by an accretion disk, and determine the energy spectrum of theradiation emitted by gas accreting into the BH.One beyond-GR gravity theory in which generic rotating BH solutions have notyet been found is dynamical Chern-Simons (dCS) gravity [8]. This theory modifies theEinstein-Hilbert action by introducing a dynamical (pseudo) scalar field that couplesnon-minimally to the metric through the Pontryagin density. The interaction leadsto a scalar field evolution equation that is sourced by the Pontryagin density, andmodified metric field equations with third derivatives. The latter have cast doubt onwhether full dCS is well-posed as an initial value problem [9], and also on whetherstable BH solutions exist. We take the point of view that dCS must be treated as aneffective field theory, since it is motivated from the low-energy limit of compactifiedheterotic string theory [10, 11] (for a review see [8]), from effective field theories ofinflation [12], and from loop quantum gravity [13]. Thus the theory is treated in the decoupling limit : deformations from GR are treated perturbatively, reducing the orderof the field equations.When treated as an effective theory, BH solutions in dCS have been found incertain limits. A non-rotating BH was found by Jackiw and Pi [14], who showed that theSchwarzschild metric is also a solution of dCS. Linear stability of high-frequency wavesabout the Schwarzschild background was suggested by [15]. The first investigations ofrotating solutions assumed a small rotation parameter. The axionic hair on a slowly-rotating BH was found in [16]. Later, the metric solution to linear order in spin wasfound independently by Yunes and Pretorius [17], and Konno, et al. [18]. This solutionwas extended to quadratic order by Yagi, Yunes and Tanaka [19]. More recently, Konnoand Takahashi [20] and Stein [21] investigated the behavior of the dynamical scalarfield about a rapidly rotating Kerr background. Stein also investigated the trace of themetric perturbation, and found that the extremal limit may be singular, which partlymotivated the present work.At present, nobody has succeeded in constructing the full metric of generic rotatingBHs in dCS gravity, despite over two decades of work in that direction [16, 14, 22, 23,24, 17, 18, 19, 20, 21]. The general procedure for obtaining the metric in the decouplinglimit is simple enough. First, one finds the leading order behavior of the scalar field xtremal Black Holes in Dynamical Chern-Simons Gravity ‡ For all of these reasons, as a first step toward finding full rotating BH solutions ineffective dCS gravity, we study the scalar field on a background where the BH spintakes the maximal Kerr value. This extremal limit is of interest not only because themathematics simplify significantly, but also because of the Kerr/CFT conjecture thatposits a dual holographic description in terms of a two-dimensional conformal fieldtheory [25, 26, 27, 28, 29, 30]. Working in the extremal limit, we obtain a general,closed-form expression for the Legendre modes of the scalar field. The radial structureof the scalar field is more complicated than that of the slowly-rotating case. Whereasthe slowly-rotating case only requires a finite polynomial expansion, the rapid-rotationcase is characterized by natural logarithms and arctangents. And unlike the slowly-rotating case, where the scalar field is primarily dipolar, the octupole mode of thefield on the extremal background carries more than half of the field’s ADM energy.We find that retaining the first 3 non-vanishing Legendre modes of the scalar fieldis necessary to achieve a fidelity above in the entire domain relative to the fullnumerical solution. Our results for the scalar field complete the first step in the processoutlined above and establish a viable starting point for either an analytic or numericaltreatment of the dCS metric deformation of the extremal Kerr solution.With analytic, closed-form expressions for the extremal scalar field in hand, we turnour attention to the BH metric. Treating the dCS correction as a small perturbationof the Kerr background and working in a convenient (Lorenz-like) gauge, we show thatthe modified field equations for the trace of the metric perturbation can be solved inquadrature in terms of another Legendre mode decomposition. As in the scalar fieldcase, the radial structure of the metric perturbation is quite different from that ofslowly-rotating BHs. In particular, the dominant monopole mode exhibits a logarithmicdivergence at the extremal Kerr horizon, which confirms a conjecture made in [21] byone of the authors. This divergence, however, may be unphysical because the Kerrhorizon is likely “inside” the perturbed dCS horizon. Away from this region, the angularstructure of the trace of the metric perturbation is predominantly monopolar, withhigher order modes modes playing a more important role than in the slowly-rotatingcase. We find that retaining the first 4 non-vanishing modes of the trace of the metricperturbation achieves a fidelity above in the entire domain relative to the fullnumerical solution.Although the trace of the metric perturbation is not a gauge-invariant observablequantity, and although we have not yet analyzed the full metric perturbation, thecalculation of the trace establishes two useful results. First, the techniques used to solvethe equation of motion for the scalar field are also applicable to the scalar degree of ‡ A more modest approach might be to construct a reliable analytical approximation for the metric.But this requires understanding the scalar field well enough to ensure that the source terms for themetric, which depend on the scalar, are rendered with sufficient detail. And even if the goal is todevelop a purely numerical description of the metric, some basic understanding of the scalar is stillneeded to test the robustness of the simulations. Before we can attack the problem of the metricdeformation, then, we must first make more progress on solving the general problem of the scalar field. xtremal Black Holes in Dynamical Chern-Simons Gravity ( a, b, c, . . . ) in index lists stand for spacetime indices. Parentheses andsquare brackets in index lists stand for symmetrization and anti-symmetrizationrespectively. The metric signature will be ( − , + , + , +) and we choose units in which c = 1 . However, we do not set G or h to unity. All other conventions follow thestandard treatment of [39, 40].
2. The ABC of dCS
Dynamical Chern-Simons gravity [14, 8] is a four-dimensional theory defined by theaction I = I EH + I CS + I ϑ + I Mat . (1)The first term is the Einstein-Hilbert action I EH = (cid:90) d x √− g (cid:18) κ R (cid:19) , (2) xtremal Black Holes in Dynamical Chern-Simons Gravity κ = 8 πG , R is the Ricci scalar associated with the metric tensor g µν , and g isthe metric determinant. The last term in Eq. (1) is the action for all matter degrees offreedom, which couple minimally to the metric tensor and do not couple to ϑ .The Chern-Simons correction is mediated by a canonically-normalized scalar field ϑ , whose kinetic term in the action is I ϑ = (cid:90) d x √− g (cid:18) −
12 ( ∂ a ϑ ) ( ∂ a ϑ ) (cid:19) . (3)This scalar field couples non-minimally to the metric through the term in the action I CS = (cid:90) d x √− g (cid:18) − ακ ϑ ∗ RR (cid:19) , (4)where the Pontryagin density is defined via ∗ RR := ∗ R abcd R abcd = 12 (cid:15) abef R ef cd R abcd , (5)and (cid:15) abcd is the Levi-Civita tensor. Notice that the definition of the Pontryagin densityhere differs from that of [8] by a minus sign, which is compensated by an additionalminus sign in I CS .Variation of the action with respect to the metric yields the field equations G ab + 2 ακ C ab = κ T ab , (6)where G ab is the Einstein tensor, and the traceless ‘C-tensor’ is defined as C ab = ( ∇ c ϑ ) (cid:15) cde ( a ∇ e R b ) d + ( ∇ c ∇ d ϑ ) ∗ R d ( ab ) c . (7)The stress-energy tensor decomposes linearly into a term that depends only onthe matter degrees of freedom and a term that depends only on the scalar field,i.e. T ab = T Mat ab + T ϑab , where the latter is T ϑab := ( ∇ a ϑ ) ( ∇ b ϑ ) − g ab ( ∇ c ϑ ) ( ∇ c ϑ ) . (8)Variation of the action with respect to the scalar field yields its evolution equation (cid:3) ϑ = α κ ∗ RR , (9)where (cid:3) stands for the d’Alembertian operator. Notice that there is no potentialassociated with the scalar field, which implies it is a long-ranged field. This vanishing(or flat) potential means that ϑ retains a global shift symmetry, ϑ → ϑ + const., because ∗ RR is related to a topological invariant [41]. Retaining this shift symmetry may beimportant to protect against certain quantum corrections.The theoretical motivation to study dCS is varied. From a string-theory standpoint,non-minimal scalar couplings of the form of Eq. (4) arise in the low-energy limit ofheterotic string theory upon four-dimensional compactification [10, 11] (for a reviewsee [8]). From a loop quantum gravity standpoint, dCS arises when the Barbero-Immirziparameter is promoted to a scalar field in the presence of fermions [13, 42]. Froma cosmology standpoint, the interaction in Eq. (4) arises as one of three terms thatremain in an effective field theory treatment of single-field inflation [12]. xtremal Black Holes in Dynamical Chern-Simons Gravity κ AY = 1 / (2 κ ) , β AY = 1 and α AY = α/κ . Moreover, we retain all factorsof G , or equivalently of κ , since we do not set G to unity. Without requiring the actionto have any specific sets of units, demanding consistency between I EH , I CS and I ϑ implies [ ϑ ] = [ κ ] − and [ α ] = L , where L stands for units of length. Given some GR solutionwith characteristic length scale L , corrections are then controlled by the dimensionlessparameter ζ := α / L . One can see this by noting that | ∂ ab ϑ | ∝ ( α/κ ) L − from Eq. (9),which implies that | C ab | ∝ ( α/κ ) L − from Eq. (7). Then the fractional corrections toGR are proportional to ( ακ | C ab | / | G ab | ) ∝ α L − = ζ .Current constraints on dCS are rather weak because dCS corrections are relevantonly in scenarios where the spacetime curvature is large. One can see this by noting thatdCS corrections to the gravitational field are sourced by the scalar field, which in turnis only sourced by the spacetime curvature. In fact, one can easily show through theargument given in the previous paragraph that constraints on the α parameter of dCSwill be roughly proportional to a power of L . Let us assume that some observation placesthe constraints | ζ | < δ , where δ is related to the observation and its uncertainties.This constraint can then be mapped to a constraint on α to find (cid:112) | α | < δ / L .Currently, the best constraint on the dCS coupling parameter is (cid:112) | α | (cid:46) km and itcomes from observations of Lense-Thirring precession from satellites in orbit aroundEarth [43]. Such a weak constraint makes sense when one realizes that for these kind ofexperiments the characteristic length scale L = [ R ⊕ / ( GM ⊕ )] / ≈ × km . Binarypulsar observations cannot yet be used to constrain the theory because modificationsto the orbital dynamics are too weak and couple to the spin of the bodies [44].The aforementioned theoretical motivations suggest that one treat dCS as an effective theory valid up to some cut-off scale, i.e., the scale above which higher-ordercurvature terms in the action cannot be neglected [2, 3]. We will here restrict attentionto physical scenarios in which the effective theory is valid, and since we are interestedonly in black holes, this means we restrict attention to those with masses GM (cid:29) √ α .When this is the case, we can work in the decoupling limit of the theory, i.e. we performa perturbative expansion of the field equations and their solutions in powers of ζ .Henceforth, dCS is exclusively treated in the decoupling limit.The decoupling limit can be implemented in practice by expanding the metrictensor and the scalar field in powers of ζ . In this paper, we will expand the metric andthe scalar field as follows: g ab = g (0) ab + ζ / g (1 / ab + ζ g (1) ab + O ( ζ / ) , (10) ϑ = 1 κ ˜ ϑ (0) + 1 κ ζ / ˜ ϑ (1 / + 1 κ ζ ˜ ϑ (1) + O ( ζ / ) , (11)where the superscript denotes the order in ζ of each term. Notice that a factor of κ − in the expansion for the scalar field ensures that ˜ ϑ ( n ) is dimensionless. As weare perturbing about ζ = 0 , our background solution ( g (0) , ˜ ϑ (0) ) must solve the fieldequations for GR and a free massless scalar field. Choosing trivial initial data for ˜ ϑ (0) gives ˜ ϑ (0) = 0 at all times, so we find ( g (0) ab , ˜ ϑ (0) ) = ( g GR ab , at zeroth order, where g GR ab is some known GR solution. If we next examine the system at order ζ / , we find that g (1 / ab satisfies a homogeneous linear equation due to the vanishing of ˜ ϑ (0) . Therefore,again, trivial initial data gives g (1 / ab = 0 at all times. xtremal Black Holes in Dynamical Chern-Simons Gravity g ab = g GR ab + ζ g (1) ab + O ( ζ / ) , (12) ϑ = 0 + 1 κ ζ / ˜ ϑ (1 / + O ( ζ ) . (13)Henceforth, we will focus on BH solutions, with the O ( ζ ) term in the metric, g GR ab ,being simply the Kerr metric. The O ( ζ / ) term in the scalar field, ϑ (1 / , is sourced bythe Kerr metric and, in turn, this sources the O ( ζ ) correction to the metric, g (1) ab . Tobe within the regime of validity of the perturbative expansion, we require ζ (cid:28) , andsince for the Kerr black hole the typical curvature length scale is L = GM , we take ζ = α ( GM ) (cid:28) . (14)Notice that this definition differs from others in the literature [8] in that we do notinclude a factor of /κ in ζ , but rather we factor it out in the scalar field directly.In this paper we are concerned with solutions that represent rotating BHs spinningnear extremality, so in addition to the decoupling expansion we will also perform a near-extremal expansion . Letting the (z-component of the) BH spin angular momentumbe J z , we can define the BH dimensionless spin parameter χ := J z / ( GM ) . We canthen expand all fields in the problem in a bivariate expansion, i.e. a simultaneousexpansion in both ζ (cid:28) and χ ∼ , namely g ( n ) ab = g ( n, ab + ε g ( n, ab + ε g ( n, ab + O ( ε ) , (15) ˜ ϑ ( n ) = ˜ ϑ ( n, + ε ˜ ϑ ( n, + ε ˜ ϑ ( n, + O ( ε ) , (16)where ε := (1 − χ ) / is a near-extremality parameter and ε (cid:28) for near-extremalBHs.
3. Scalar Field: Solution
We wish to solve the evolution equation for the scalar field [Eq. (9)] to leading order in ζ . To this order, the Pontryagin density on the right-hand side of Eq. (9) is evaluatedon the unmodified Kerr spacetime. The wave operator on the left-hand side can alsobe evaluated on the Kerr spacetime, since corrections will be of O ( ζ ) . In polynomialBoyer-Lindquist coordinates, the scalar field evolution equation is evaluated on theline element [45] g (0) ab dx a dx b = − ∆Σ (cid:2) dt − a Γ dφ (cid:3) + Σ∆ dr + ΣΓ dψ + ΓΣ (cid:16) ( r + a ) dφ − a dt (cid:17) , (17)where the usual polar angle θ has been replaced with a coordinate ψ = cos θ , and Γ := 1 − ψ = sin θ . The mass of the black hole is M and it rotates with angularmomentum per unit mass a = J z / ( GM ) , where − GM ≤ a ≤ GM . The functions Σ and ∆ are Σ = r + a ψ (18) ∆ = r − GM r + a , (19) xtremal Black Holes in Dynamical Chern-Simons Gravity ∆ = ( r − r + )( r − r − ) = 0 , are located at r ± = GM ± (cid:112) ( GM ) − a .It will be convenient to replace all quantities with dimensionless variables byscaling out factors of GM : ˜ r = r/ ( GM ) and χ = a/ ( GM ) , so that the rescaledfunctions ˜∆ = ∆ / ( GM ) = (˜ r − − (1 − χ ) and ˜Σ = Σ / ( GM ) = ˜ r + χ ψ .Assuming a stationary and axisymmetric solution for the scalar field, the O ( α ) termin Eq. (9) then takes the form ∂ ˜ r (cid:0) ˜∆ ∂ ˜ r ˜ ϑ (1 / (cid:1) + ∂ ψ (cid:0) Γ ∂ ψ ˜ ϑ (1 / (cid:1) = s (1 / (˜ r, ψ ) (20)where factors of ( α/κ ) and ( GM ) have canceled from both sides of the equation. Thesource s (1 / (˜ r, ψ ) is proportional to Σ ( ∗ RR (0) ) and given explicitly by s (1 / (˜ r, ψ ) = 24 χ ˜ rψ (3˜ r − χ ψ )(˜ r − χ ψ )˜Σ . (21)Equation (20) admits a solution via separation of variables, by expanding thesolution ˜ ϑ (1 / = ∞ (cid:88) (cid:96) =0 ˜ ϑ (1 / (cid:96) (˜ r ) P (cid:96) ( ψ ) , (22)where P (cid:96) ( · ) are Legendre functions of the first kind. The radial modes ˜ ϑ (1 / (cid:96) (˜ r ) thensatisfy the equation ∂ ˜ r (cid:16) ˜∆ ∂ ˜ r ˜ ϑ (1 / (cid:96) (cid:17) − (cid:96) ( (cid:96) + 1) ˜ ϑ (1 / (cid:96) = s (1 / (cid:96) (˜ r ) , (23)with source functions s (1 / (cid:96) (˜ r ) given by the modes in the Legendre decomposition ofEq. (21) s (1 / (cid:96) (˜ r ) = 2 (cid:96) + 12 (cid:90) − dψ P (cid:96) ( ψ ) s (1 / (˜ r, ψ ) . (24)Note that the source function in Eq. (21) is odd in the variable ψ , so its Legendreexpansion (as well as that of the scalar field) will only contain odd modes: (cid:96) = 2 n + 1 for all n ∈ N .The integral in Eq. (24) can be evaluated in closed form in terms of knownfunctions: s (1 / (cid:96) (˜ r ) = ( − (cid:96) +12 Γ( )Γ( (cid:96) + 4)2 (cid:96) Γ( (cid:96) + ) χ (cid:96) ˜ r (cid:96) +4 (cid:104) F (cid:16) (cid:96) +42 , (cid:96) +52 ; (cid:96) + ; − χ ˜ r (cid:17) − ( (cid:96) + 5) F (cid:16) (cid:96) +42 , (cid:96) +72 ; (cid:96) + ; − χ ˜ r (cid:17)(cid:105) , (25)where F ( · , · ; · ; · ) is the ordinary hypergeometric function and (cid:96) is odd. One can show,via identities for hypergeometric functions, that this expression is equivalent to onegiven previously in [46]. Note that the hypergeometric functions go to unity in the ˜ r → ∞ limit, so that the leading behavior at large ˜ r is given by s (1 / (cid:96) (˜ r ) ∼ ˜ r − ( (cid:96) +4) .The solution of Eq. (23) can be obtained through the method of variation ofparameters (see Appendix A). Defining a new variable η = (˜ r − / (cid:112) − χ , the xtremal Black Holes in Dynamical Chern-Simons Gravity ˜ ϑ (1 / (cid:96) is ˜ ϑ (1 / (cid:96) (˜ r ) = P (cid:96) ( η ) (cid:90) η ∞ dη (cid:48) s (1 / (cid:96) (1 + η (cid:48) (cid:112) − χ ) Q (cid:96) ( η (cid:48) ) − Q (cid:96) ( η ) (cid:90) η dη (cid:48) s (1 / (cid:96) (1 + η (cid:48) (cid:112) − χ ) P (cid:96) ( η (cid:48) ) , (26)where Q (cid:96) ( · ) are Legendre functions of the second kind. This solution is regular at ˜ r + ,and approaches zero as ˜ r → ∞ .Our eventual goal is to evaluate Eq. (26) in closed form for the full range of therotation parameter, − ≤ χ ≤ . The slow rotation limit of the field, i.e. the solutionin a | χ | (cid:28) expansion, is already well-understood; it was first derived in [17], verifiedin [18], and extended to second order in rotation in [19]. Similarly, it is also possible tosystematically solve the scalar field equation of motion in the near-extremal expansionintroduced in Sec. 2. Expanding the source functions of Eq. (25) for ε (cid:28) , we find s (1 / (cid:96) (˜ r ) = s (1 / , (cid:96) (˜ r ) + ε s (1 / , (cid:96) (˜ r ) + ε s (1 / , (cid:96) (˜ r ) + O ( ε ) . (27)Recall that the second superscript in each of these terms represents the order in ε atwhich it enters the near-extremal expansion. Because the χ → limit is regular for s (1 / (cid:96) (˜ r ) , s (1 / , (cid:96) (˜ r ) is simply s (1 / (cid:96) (˜ r ) evaluated at χ = 1 .In this paper we will only consider the extremal limit, ε → , which is the leadingterm in the near-extremal expansion. This corresponds to the limit χ → ± of thedimensionless spin parameter χ . The homogeneous solutions are regular in this limit[see Eqs. (A.8)-(A.9)], and the solution for the scalar field [see Eq. (A.3)] at ε = 0 is ˜ ϑ (1 / , (cid:96) (˜ r ) = 12 (cid:96) + 1 (cid:34) (˜ r − (cid:96) (cid:90) ˜ r ∞ d ˜ r (cid:48) s (1 / , (cid:96) (˜ r (cid:48) )(˜ r (cid:48) − (cid:96) +1 − r − (cid:96) +1 (cid:90) ˜ r d ˜ r (cid:48) (˜ r (cid:48) − (cid:96) s (1 / , (cid:96) (˜ r (cid:48) ) (cid:35) . (28)The source functions at leading-order in ε , s (1 / , (cid:96) (˜ r ) , are given by Eq. (24) or Eq. (25)evaluated at | χ | = 1 . The boundary conditions are the same as before: each mode ˜ ϑ (1 / , (cid:96) is regular at ˜ r + = 1 , and goes to zero as ˜ r → ∞ .The integrals in Eq. (28) can be readily evaluated for specific values of (cid:96) . Forexample, the (cid:96) = 1 radial mode is given by ˜ ϑ (1 / , (˜ r ) = 3(˜ r −
1) log (cid:16) ˜ r − √ ˜ r + 1 (cid:17) + 3(˜ r −
1) arccot ˜ r + 3(˜ r − r + ˜ r + 3)2(˜ r + 1) . (29)With more work, we can also give an expression for general values of (cid:96) in terms offinite-order rational polynomials, arccot(˜ r ) , and the log which appears above. Butbefore we can give the general form, we first have to establish a few results for thebehavior of the modes at large ˜ r and at ˜ r = 1 .The far-field behavior of the modes is dominated by the second line of Eq. (28),since the term in the first line decays with a higher power of r . The integral in the xtremal Black Holes in Dynamical Chern-Simons Gravity ˜ ϑ (1 / , (cid:96) ∼ ˜ r − ( (cid:96) +1) as ˜ r (cid:29) . Notice,though, that this asymptotic behavior is not immediately apparent in our initialresult, Eq. (29). In that case the individual terms fall off more slowly than ˜ r − , butcancellations between the terms result in the correct asymptotic behavior. The samewill be true for our result for general (cid:96) : individual terms may not behave as ˜ r − ( (cid:96) +1) atlarge ˜ r , but cancellations between these terms will give the correct result.The near-horizon behavior of the modes is dominated by the first integral inEq. (28), but its asymptotic behavior as ˜ r ∼ cannot be easily discerned from thatequation. Instead, it is easier to return to Eq. (26) and set χ = 1 , remembering thatthe horizon limit ˜ r → is equivalent to η → (this is the case for all values of χ ). Inthis limit, only the first line of Eq. (26) contributes, leading to ˜ ϑ (1 / , (cid:96) (1) = − (cid:96) ( (cid:96) + 1) s (1 / , (cid:96) (1) . (30)The overall (cid:96) -dependent factor comes from the definite integration of Q (cid:96) ( η (cid:48) ) in therange η (cid:48) ∈ (1 , ∞ ) , while s (1 / , (cid:96) (1) is Eq. (25) evaluated at χ = 1 and ˜ r = 1 . As acheck of this result, it is instructive to consider the behavior of the scalar field in theNear-Horizon Extremal Kerr (NHEK) limit [47, 25]. At extremality, regularity of thefield at the horizon insures that the first term on the left-hand-side of Eq. (20) vanishesas ˜ r → . Then the ψ -dependence of the scalar field in this limit is determined by ∂ ψ (cid:0) Γ ∂ ψ ˜ ϑ (1 / , (1 , ψ ) (cid:1) = s (1 / , (1 , ψ ) = 24 ˜ r ψ (3 − ψ )(1 − ψ )(1 + ψ ) . (31)The inhomogeneous solution can be obtained by direct integration, and the full solutionthat is regular on − ≤ ψ ≤ is given by ˜ ϑ (1 / , (1 , ψ ) = − ψ (1 + ψ ) + ψ ψ ) + arctan( ψ ) . (32)It is straightforward to check that this agrees with the Legendre series Eq. (22), withcoefficients given by Eq. (30) for the mode functions at the horizon.With these results, we can now give a general expression for the radial modes ofthe scalar field. They take the form ˜ ϑ (1 / , (cid:96) (˜ r ) = A (cid:96) (˜ r ) + B (cid:96) (˜ r ) arccot(˜ r ) + C (cid:96) (˜ r ) log (cid:18) ˜ r − √ ˜ r + 1 (cid:19) , (33)where the functions A (cid:96) (˜ r ) , B (cid:96) (˜ r ) , and C (cid:96) (˜ r ) are A (cid:96) (˜ r ) = ( − (cid:96) − (cid:96) ( (cid:96) − r − (cid:96) +1 (cid:96) (cid:88) k =0 γ k (˜ r − k + (cid:96) − (cid:88) k =0 α (cid:96),k ˜ r k (34) − (cid:96) + 1(˜ r + 1) + (2 (cid:96) + 1)(4˜ r − (cid:96) ( (cid:96) + 1))4 (˜ r + 1) B (cid:96) (˜ r ) = ( − (cid:96) +12 (cid:96) ( (cid:96) − r − (cid:96) +1 + (cid:96) (cid:88) k =0 β (cid:96),k ˜ r k (35) C (cid:96) (˜ r ) =( − (cid:96) − ( (cid:96) + 1)( (cid:96) + 2)2 (˜ r − (cid:96) . (36) xtremal Black Holes in Dynamical Chern-Simons Gravity Figure 1.
The first five radial mode functions of the scalar field. The verticaldashed line indicates the location of the event horizon of an extremal black hole.The (cid:96) = 1 mode vanishes at the horizon, while modes with (cid:96) > are all non-zeroat ˜ r = 1 . The constants γ k appearing in A (cid:96) (˜ r ) are the first (cid:96) + 1 terms in the Taylor expansionof arccot(˜ r ) around ˜ r = 1 γ k = 1 k ! (cid:18) ∂∂x k arccot( x ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =1 . (37)The remaining (cid:96) + 1 coefficients α (cid:96),k and β (cid:96),k are fixed by imposing the boundaryconditions: each mode falls off as ˜ r − ( (cid:96) +1) at large ˜ r , and takes the value Eq. (30) at ˜ r = 1 . Alternately, the condition at ˜ r = 1 can be replaced with the requirement thatthe leading asymptotic behavior of the mode is given by Eq. (C.25). The coefficients α (cid:96),k and β (cid:96),k for the first several modes are given explicitly in Appendix B.
4. Scalar Field: Properties
Let us now discuss some properties of the scalar field solution obtained in the previoussection. We begin by plotting the first five (odd) modes in Fig. 1. Observe that theintegrated norm of ˜ ϑ (1 / , (cid:96) decays exponentially with (cid:96) . This is because this function isa spectral solution to a differential equation with a C ∞ source, so it must convergeexponentially with mode number. Observe also from Fig. 1 that the (cid:96) = 1 mode of thefield vanishes at the horizon. Modes with (cid:96) > are finite but non-zero at the horizon,with values that scale like (cid:96) / (1 + √ − ( (cid:96) +1) for (cid:96) (cid:29) .By including contributions from a sufficient number of modes, we can construct anarbitrarily accurate approximation of the full, extremal scalar field. An approximation xtremal Black Holes in Dynamical Chern-Simons Gravity - - Figure 2.
The behavior of the scalar field on the extremal background,approximated by its first five Legendre modes. The coordinates ˜ rψ and ˜ r (cid:112) − ψ correspond to ˜ r cos θ and ˜ r sin θ , respectively, in conventional Boyer-Lindquistcoordinates. using the first five modes is shown in Fig. 2, as a function of both radius and polar angle.Notice the similarity to Fig. 3 of Stein [21], resulting from a numerical solution at largebut not extremal ( χ = 0 . ) rotation; we will discuss this more below. The accuracyof this approximation can be characterized using a slicing-independent measure ofthe scalar field energy through the ADM energy. Let u a be a timelike unit vectornormal to a hypersurface S , with γ ab the induced metric on S . Then the scalar field’s xtremal Black Holes in Dynamical Chern-Simons Gravity E = (cid:90) S d x √ γ u a T ϑab t b (38)where t b is the Killing vector ∂/∂t . This energy can be perturbatively expanded inpowers of ζ , E = ζ E (1) + ζ E (2) + . . . (39)The scalar field’s ADM energy at leading order can further be computed via the spectraldecomposition, E (1) = M ∞ (cid:88) k =1 ˜ E (1) k , (40)with the dimensionless ˜ E (1) k functions given by ˜ E (1) k = 14 12 k + 1 (cid:90) ∞ ˜ r + d ˜ r (cid:104) ˜∆( ∂ ˜ r ˜ ϑ (1 / k ) + k ( k + 1)( ˜ ϑ (1 / k ) (cid:105) . (41)The fractional difference between the total energy in the scalar field and the energy inthe first N modes is then δ N = 1 − ME (1) N (cid:88) k =1 ˜ E (1) k . (42)Figure 3 shows the fractional difference δ N for the first seven nonvanishing modes.The contribution from the first five – up to (cid:96) = 9 – differs from the total energy byless than one part in . Observe that the accuracy increases exponentially with N .Observe also that if we wish to capture of the energy in the field, it suffices tokeep only up to the first three odd modes, i.e. N = 5 . Finally, note that the energy inthe scalar field is dominated by the behavior of the scalar field close to the horizon. Ifone is interested in regimes of spacetime outside some two-sphere with radius r (cid:29) M ,then the full scalar field can be accurately modeled using just the dipole ( (cid:96) = 1 ) andoctupole ( (cid:96) = 3 ) modes.Let us briefly compare these analytic results at extremality with numerical resultsaway from extremality, which were computed in Stein [21]. First, we have performed adirect comparison of the analytical calcuation presented here with the numerics in [21].Because of the numerical method of [21], those results must be at values of | χ | < .Specifically, we compared the analytical results presented here with numerical resultscomputed with the highest spin of χ = 1 − − (extremality parameter approximately ε ≈ √ × − ), with N ang = 64 angular and N x = 1024 radial collocation points(high spins require very high radial resolution). Comparisons against lower resolutionsshow that these results are converging. This comparison is displayed in Fig. 4. Wesee that even at the extremality parameter ε ≈ √ × − , the numerical solution isconverging to the regular analytic solution. The nonzero fractional differences in Fig. 4should decrease in the limit as ε → and in the continuum limit N x → ∞ .We can also use the numerical method of [21] to show convergence to the extremallimit, which is regular. This is demonstrated in Fig. 5 as a function of the rotation of xtremal Black Holes in Dynamical Chern-Simons Gravity ● ● ● ● ● ● ● Figure 3.
The fractional difference between the scalar field’s contribution to theADM energy, and the contribution from the field’s first N modes. - - - - - - ε /( r ˜- ) | ( ϑ ˜ ℓ / - ϑ ˜ ℓ , nu m / ) / ϑ ˜ ℓ / | r ⟶ ∞ r ⟶ r + ℓ = ℓ = ℓ = ℓ = ℓ = ℓ = ℓ = ℓ = Figure 4.
Direct comparison between the analytical solution ˜ ϑ (1 / (cid:96) and thenumerical solution at a = 1 − − (approximately ε ≈ √ × − ), using themethod of [21]. The vertical axis is fractional difference between the numericalsolution and the analytical solution for each mode. The extremal solution isevaluated onto the collocation points of the numerical domain x ∈ ( − , +1) where x = 1 − ε/ (˜ r − . The numerical solution was computed with an angularand radial resolution of ( N ang , N x ) = (64 , , and comparisons against lowerresolutions shows that the numerics are converging. The limit ε → is regular, sothe fractional differences will go to zero. the background spacetime, by using the modal contribution to the ADM energy as aproxy. The convergence to the extremal limit is more easily demonstrated by using theextremality parameter as the horizontal axis, as seen in the right panel. We can also xtremal Black Holes in Dynamical Chern-Simons Gravity - - - χ E ˜ ℓ ( ) [ ϑ ˜ ( / ) ] - - - - - ε E ˜ ℓ [ ϑ ˜ / ] Extremal ℓ = ℓ = ℓ = ℓ = ℓ = ℓ = ℓ = ℓ = Figure 5.
Modal contribution to the ADM energy of the scalar field [defined inEq. (41)], as a function of rotation on the horizontal axis. Notice that the extremallimit is regular. This is emphasized by making the horizontal axis the extremalityparameter ε = (cid:112) − χ . Note that each curve becomes horizontal going towardthe left edge of the right panel. easily see that higher (cid:96) modes become fractionally more important as spin increases(this was seen in Fig. 2 of [21]).The essential differences between the structure of the scalar field in the slowlyrotating ( χ → ) and extremal ( χ → ) limits are apparent in Fig. 5. At low spin, χ (cid:28) , the scalar field is almost entirely dipolar, with contributions to the ADMenergy from the (cid:96) ≥ modes amounting to less than 1 part in . The higher modesbecome more important as the spin of the background black hole increases, and theoctupole contribution to the ADM energy exceeds the dipole contribution at around χ ≈ . . At χ = 1 , the dipole contributes about of the total ADM energy,while the octupole contributes just over . This is consistent with the near-horizonbehavior of the modes shown in Fig. 1, and our earlier observation that the scalar fieldADM energy is dominated by the contribution from the near-horizon region.
5. Trace of the Metric Perturbation: Solution
The leading correction to the metric in Eq. (12) is determined by the O ( ζ ) term in themetric equation of motion [Eq. (6)]. We work in a gauge where the covariant divergenceof g (1) ab is proportional to the derivative of its trace g (1) = g ab (0) g (1) ab with respect to thebackground metric: ∇ a g (1) ab = 12 ∇ b g (1) . (43)This gauge leads to simplifications in the O ( ζ ) term of Eq. (6), but its structure is stilltoo complicated to allow for a simple solution. As a first step towards determining g (1) ab , we take the trace of the O ( ζ ) correction to Eq. (6) to find (cid:3) g (1) = − ∇ ϑ (1 / ) . (44) xtremal Black Holes in Dynamical Chern-Simons Gravity (cid:104) ∂ ˜ r ˜∆ ∂ ˜ r + ∂ ψ Γ ∂ ψ (cid:105) g (1) = − ∂ ˜ r ˜ ϑ (1 / ) − ∂ ψ ˜ ϑ (1 / ) . (45)As with the scalar field, we can express g (1) in a Legendre decomposition as g (1) = (cid:88) (cid:96) g (1) (cid:96) (˜ r ) P (cid:96) ( ψ ) . (46)Then, the equation of motion [Eq. (45)] again separates, giving the radial equation (cid:104) ∂ ˜ r ˜∆ ∂ ˜ r − (cid:96) ( (cid:96) + 1) (cid:105) g (1) (cid:96) (˜ r ) = S (cid:96) (˜ r ) . (47)The source functions S (cid:96) (˜ r ) are the Legendre modes of the right-hand side of Eq. (44),i.e. S (cid:96) (˜ r ) = 2 (cid:96) + 12 (cid:90) − dψP (cid:96) ( ψ ) S g (˜ r, ψ ) , (48)where the source function S g is simply S g (˜ r, ψ ) := − (cid:112) − g (0) ( ∇ ˜ ϑ (1 / ) . (49)The solution for the mode functions g (1) (cid:96) (˜ r ) is then given by Eq. (A.3), which in thiscase becomes g (1) (cid:96) = 1 W (cid:96) (cid:32) H + (cid:96) (˜ r ) (cid:90) ˜ r ∞ d ˜ r (cid:48) H − (cid:96) (˜ r (cid:48) ) S (cid:96) (˜ r (cid:48) ) − H − (cid:96) (˜ r ) (cid:90) ˜ r ˜ r + d ˜ r (cid:48) H + (cid:96) (˜ r (cid:48) ) S (cid:96) (˜ r (cid:48) ) (cid:33) . (50)Note that the source is quadratic in the scalar field, which has odd Legendre modes.Thus, both the trace of the metric perturbation and its source function have evenLegendre modes: (cid:96) = 2 n for all n ∈ N .One approach to evaluating the integrals in Eq. (50) is to express the Legendremodes of the source function in terms of the scalar field modes and their radialderivatives. The resulting integrals are significantly more complicated than the oneswe encountered in Sec. 3, so we will opt for a different approach. One can expressEq. (50) in terms of a simpler set of integrals through multiple integrations-by-parts(noting that the source S (cid:96) depends on S g , which in turn is proportional to the squaredderivative of the scalar field) and application of the scalar field evolution equation[Eq. (20)]. Doing so, the modes of the trace of the metric perturbation are given by g (1) (cid:96) (˜ r ) = 2 (cid:96) + 1 W (cid:96) (cid:32) H + (cid:96) (˜ r ) (cid:90) ˜ r ∞ d ˜ r (cid:48) (cid:90) − dψ H − (cid:96) (˜ r (cid:48) ) P (cid:96) ( ψ ) ˜ ϑ (1 / (˜ r (cid:48) , ψ ) s (˜ r (cid:48) , ψ ) − H − (cid:96) (˜ r ) (cid:90) ˜ r ˜ r + d ˜ r (cid:48) (cid:90) − dψ H + (cid:96) (˜ r (cid:48) ) P (cid:96) ( ψ ) ˜ ϑ (1 / (˜ r (cid:48) , ψ ) s (˜ r (cid:48) , ψ )+ H + (cid:96) (˜ r ) (cid:90) ˜ r ∞ d ˜ r (cid:48) (cid:90) − dψ ∂ µ V µ − (˜ r (cid:48) , ψ ) − H − (cid:96) (˜ r ) (cid:90) ˜ r ˜ r + d ˜ r (cid:48) (cid:90) − dψ ∂ µ V µ + (˜ r (cid:48) , ψ ) (cid:33) , (51) xtremal Black Holes in Dynamical Chern-Simons Gravity s (˜ r (cid:48) , ψ ) is the scalar field source given in Eq. (21), and we have defined V µ ± := 12 ( ˜ ϑ (1 / ) (cid:112) − g (0) g µν (0) ∂ ν (cid:2) H ± (cid:96) P (cid:96) (cid:3) − H ± (cid:96) P (cid:96) (cid:112) − g (0) g µν (0) ∂ ν ( ˜ ϑ (1 / ) . (52)The integrals of total derivatives in Eq. (51) can be simplified by noting that (i) √− g (0) g ψψ (0) = Γ , which vanishes when evaluated at the limits of integration ψ = ± ,and (ii) contributions at spatial infinity and at the horizon vanish due to the behaviorof the scalar field modes ˜ ϑ (1 / , the homogeneous solutions H ± (cid:96) , and √− g (0) g rr (0) = ˜∆ .The modes of the trace of the metric perturbation are then g (1) (cid:96) (˜ r ) = 2 (cid:96) + 1 W (cid:96) (cid:32) H + (cid:96) (˜ r ) (cid:90) ˜ r ∞ d ˜ r (cid:48) (cid:90) − dψ H − (cid:96) (˜ r (cid:48) ) P (cid:96) ( ψ ) ˜ ϑ (1 / s (1 / − H − (cid:96) (˜ r ) (cid:90) ˜ r ˜ r + d ˜ r (cid:48) (cid:90) − dψ H + (cid:96) (˜ r (cid:48) ) P (cid:96) ( ψ ) ˜ ϑ (1 / s (1 / (cid:33) − (cid:96) + 12 (cid:90) − dψ P (cid:96) ( ψ ) (cid:16) ˜ ϑ (1 / (˜ r, ψ ) (cid:17) , (53)where in the first and second lines ˜ ϑ (1 / and s (1 / are both functions of ˜ r (cid:48) and ψ . Wehave simplified the last line by extracting a factor of W (cid:96) , defined in Eq. (A.4), whichis a constant.Let us now focus on the extremal limit. With the normalizations definedin Appendix A, the factor W (cid:96) = 2 (cid:96) + 1 and the homogeneous solutions are givenby Eqs. (A.8)-(A.9). We can then write g (1 , (cid:96) (˜ r ) = (˜ r − (cid:96) (cid:90) ˜ r ∞ d ˜ r (cid:48) (cid:90) − dψ P (cid:96) ( ψ ) ˜ ϑ (1 / , s (1 / , (˜ r − (cid:96) +1 − r − (cid:96) +1 (cid:90) ˜ r d ˜ r (cid:48) (cid:90) − dψ (˜ r − (cid:96) P (cid:96) ( ψ ) ˜ ϑ (1 / , s (1 / , − (2 (cid:96) + 1)2 (cid:90) − dψP (cid:96) ( ψ ) ( ˜ ϑ (1 / , ) . (54)This completes the formal solution for the modes of the trace of the metric perturbationin the extremal limit in integral form.The angular integrals in Eq. (54) can be evaluated in closed form using theLegendre decomposition of the scalar field and the source function. From Eq. (22) andEq. (24) we have ˜ ϑ (1 / , (˜ r, ψ ) = ∞ (cid:88) k =1 ˜ ϑ (1 / , k (˜ r ) P k ( ψ ) s (1 / , (˜ r, ψ ) = ∞ (cid:88) j =1 s (1 / , j (˜ r ) P j ( ψ ) , where the sums are over odd integers in both cases. Using the orthogonality of Legendre xtremal Black Holes in Dynamical Chern-Simons Gravity (cid:96) = 0 mode is given by g (1 , (˜ r ) = ∞ (cid:88) k =1 k + 1 (cid:34) (cid:90) ˜ r ∞ d ˜ r (cid:48) ˜ ϑ (1 / , k (˜ r (cid:48) ) s (1 / , k (˜ r (cid:48) )˜ r (cid:48) − − r − (cid:90) ˜ r d ˜ r (cid:48) ˜ ϑ (1 / , k (˜ r (cid:48) ) s (1 / , k (˜ r (cid:48) ) −
12 ˜ ϑ (1 / , k (˜ r ) (cid:35) . (55)For general (cid:96) , the integration over ψ can be expressed in terms of the standard j -symbols. The resulting expression is g (1 , (cid:96) (˜ r ) = (cid:88) k,j (cid:18) (cid:96) k j (cid:19) × (cid:34) (˜ r − (cid:96) (cid:90) ˜ r ∞ d ˜ r (cid:48) ˜ ϑ (1 / , k (˜ r (cid:48) ) s (1 / , j (˜ r (cid:48) )(˜ r (cid:48) − (cid:96) +1 − r − (cid:96) +1 (cid:90) ˜ r d ˜ r (cid:48) (˜ r (cid:48) − (cid:96) ˜ ϑ (1 / , k (˜ r (cid:48) ) s (1 / , j (˜ r (cid:48) ) − (cid:96) + 12 ˜ ϑ (1 / , k (˜ r ) ˜ ϑ (1 / , j (˜ r ) (cid:35) (56)The radial integrals in Eqs. (55) and (56), though still complicated, are more tractablethan the integrals that result from expressing the source function in terms of the modesof the scalar field in Eq. (50).We have not yet obtained a closed-form expression for the trace of the metricperturbation on the extremal background. The main difficulty, apparent in Eqs. (54)-(56), is that the source term depends on the full tower of Legendre modes of the scalarfield. Using our expressions for the modes of the scalar field and its source, Eq. (33)and Eq. (25), it is possible to evaluate individual terms in these sums. However, wehave not been able to perform the sums themselves. Indeed, the analytic results forthe individual terms are sufficiently complicated that we turn to approximations andnumerical analysis, which we discuss in the next section.
6. Trace of the Metric Perturbation: Properties
The results of Sec. 4 suggest that the first three or four modes of the scalar field capturemost of its physics, and should be sufficient for analyzing the behavior of the trace ofthe metric perturbation. But first, let us consider a few important properties of themodes g (1 , (cid:96) that can be extracted from the integral form of the solution.At large radius, ˜ r (cid:29) , the second line of Eq. (54) dominates and the leadingbehavior of the mode is g (1 , (cid:96) ∼ ˜ r − ( (cid:96) +1) . This is because the first line of Eq. (54) decayswith a higher power of ˜ r , while the third line is proportional to ( ˜ ϑ (1 / , ) and thereforedecays as ˜ r − (cid:96) +1) . Near the horizon the first and third line of Eq. (54) dominate. Theasymptotic behavior as ˜ r → is most easily extracted by first evaluating Eq. (50) at ˜ r = ˜ r + = 1 + ε , changing the integration variable to η = (˜ r (cid:48) − /ε , and then takingthe ε → limit, which gives g (1 , (cid:96) (1) = S (1 , (cid:96) (1) (cid:90) ∞ dη Q (cid:96) ( η ) . (57)The overall factor of S (1 , (cid:96) (1) , the source function evaluated at the extremal Kerrhorizon, can be expressed in terms of the source functions for the scalar field. Evaluating xtremal Black Holes in Dynamical Chern-Simons Gravity Figure 6.
The (absolute value of) Legendre modes of the trace of the metricperturbation evaluated at the horizon of the extremal background, on a logarithmicscale, as a function of harmonic number (cid:96) . Eq. (48) at ˜ r = 1 , using lim ˜ r → ˜∆( ∂ ˜ r ˜ ϑ ) = 0 , and performing the angular integralyields S (1 , (cid:96) (1) = − ∞ (cid:88) k,j =1 (cid:96) + 1) (cid:112) j ( j + 1) k ( k + 1) (cid:18) (cid:96) j k (cid:19) (cid:18) (cid:96) j k − (cid:19) s (1 / , j (1) s (1 / , k (1) , (58)where again the result is expressed in terms of j -symbols.For (cid:96) ≥ (recall that (cid:96) is even) the integral in Eq. (57) converges: g (1 , (cid:96) (1) = − (cid:96) ( (cid:96) + 1) S (1 , (cid:96) (1) . (59)In this case the mode is finite at the horizon of the extremal background, just like themodes of the scalar field. The values g (1 , (cid:96) (1) are plotted against (cid:96) in Fig. 6. Ratherthan falling off monotonically with (cid:96) , a feature is observed at (cid:96) = 4 . This mode issuppressed relative to the (cid:96) = 6 and (cid:96) = 8 modes.For the (cid:96) = 0 mode the integral in Eq. (57) does not converge. In this case themode diverges logarithmically as ˜ r → . Its behavior near the horizon is capturedby the first integral in Eq. (55). Expanding the integrand in powers of (˜ r (cid:48) − andextracting the log term from the integral gives lim ˜ r → g (1 , (˜ r ) ∼ S (1 , (1) log(˜ r − , (60)The coefficient S (1 , (1) may be evaluated from Eq. (58), which for (cid:96) = 0 reduces to a xtremal Black Holes in Dynamical Chern-Simons Gravity Figure 7.
The first four radial modes of the trace of the metric perturbation asa function of ˜ r . The dashed vertical line indicates the horizon of the extremalbackground. The (cid:96) = 0 mode exhibits a logarithmic divergence as ˜ r → . single sum S (1 , (1) = − ∞ (cid:88) k =1 k ( k + 1)(2 k + 1) ( s (1 / , k (1)) (61) (cid:39) − . . This is the same value that is obtained from the ratio of the numerical solution for g (1 , (˜ r ) and log(˜ r − , evaluated as ˜ r → . As we will discuss at the end of thisSection, the log divergence in g (1 , (˜ r ) need not imply the existence of a naked curvaturesingularity at the perturbed horizon.As explained at the end of the previous Section, analytical results for the ˜ r -dependence of the modes g (1 , (cid:96) are sufficiently complicated that a numerical analysisis called for. The first four modes of the trace of the metric perturbation are shownin Fig. 7. These are numerical solutions, obtained with closed form expressions forthe source that include contributions from modes of the scalar field with (cid:96) ≤ .It is immediately apparent that the (cid:96) = 0 mode dominates the trace of the metricperturbation, even away from the log-divergent behavior near ˜ r = 1 .We expect, based on the behavior shown in Fig. 7 that g (1 , is well-approximatedby its first few Legendre modes. In the case of the scalar field, a similar conclusionwas justified by examining mode-by-mode contributions to the ADM energy. Butbefore applying this norm to the modes of the trace of the metric perturbation, wefirst consider the fractional difference between g (1 , (˜ r, ψ ) evaluated at ψ = 1 , and an xtremal Black Holes in Dynamical Chern-Simons Gravity Figure 8.
The fractional difference between the trace of the metric perturbationat θ = 0 , and its approximation including only modes with (cid:96) ≤ N . approximation by its first N modes δ N (˜ r ) = 1 − g (1 , (˜ r, (cid:32) N (cid:88) (cid:96) =0 g (1 , (cid:96) (˜ r ) (cid:33) . (62)The fractional difference for N = 0 , , , is shown in Fig. 8. As expected, the log-divergence of the (cid:96) = 0 modes means that the fractional difference δ (˜ r ) → as ˜ r → .Figure 8 shows that if one wishes an accuracy of no more than about , thenretaining only the (cid:96) = 0 mode suffices. To obtain a higher accuracy, more modes areneeded. In particular, since the (cid:96) = 4 mode is suppressed at ˜ r = 1 relative to the (cid:96) = 6 and modes, one must include modes up to (cid:96) = 6 to obtain uniform accuracy of atleast one percent. Observe, however, that if one is interested in the trace of the metricperturbation outside a larger radius, such as for ˜ r (cid:38) , then g (1 , is approximated atbetter than percent precision with only the first two or three modes.We conclude that Legendre modes g (1 , (cid:96) with (cid:96) ≤ capture almost all of thephysics of g (1 , , except near ˜ r = 1 where the (cid:96) = 6 and (cid:96) = 8 modes may be required.Note that the fractional error defined in Eq. (62) puts a bound on the fidelity of ourapproximation of g (1 , at ψ = 1 ( θ = 0 ), where P (cid:96) ( ψ ) = 1 for all (cid:96) . However, sincethe (cid:96) = 0 mode dominates, and − ≤ P (cid:96) ( ψ ) ≤ for (cid:96) ≥ , the fractional error δ N (˜ r ) gives an upper bound on the fidelity of the approximation in the full (˜ r, ψ ) plane. Anapproximation of g (1 , by its first four Legendre modes is shown in Fig. 9.The analysis above uses modes of the scalar field with (cid:96) ≤ to approximate thesource for the trace of the metric perturbation. However, as we saw in Sec. 4, the firstthree modes of the scalar field account for most of its contribution to the ADM energy.Indeed, the behavior of the first few modes of g (1 , is largely unchanged if we includefewer modes of the scalar field. In particular, we achieve comparable results for the xtremal Black Holes in Dynamical Chern-Simons Gravity Figure 9.
The trace of the metric perturbation on the extremal background,approximated by its first four Legendre modes. Near the extremal horizon ˜ r = 1 (the solid line) the logarithmic divergence of the monopole term dominates. mode g (1 , (cid:96) by approximating the scalar field by its first N = (cid:96) + 1 modes.Just as in the case of the scalar field, we can study the convergence of numericalsolutions with ε > in the limit as ε → , using the numerical technique of Stein [21].We use the ADM energy functional, given in Eq. (41), as a norm for the modes g (1) (cid:96) .Despite the fact that g (1) has a logarithmic divergence at the horizon, this divergenceis integrable in the norm of Eq. (41). This results in each of the (cid:96) modes’ energiesbeing convergent as ε → , as seen in Fig. 10.As in Sec. 4, we can again plainly see the differences between the trace of the xtremal Black Holes in Dynamical Chern-Simons Gravity - - - χ E ˜ ℓ ( ) [ g ( ) ] - - - - - ε E ˜ ℓ [ g ] Extremal ℓ = ℓ = ℓ = ℓ = ℓ = ℓ = ℓ = ℓ = Figure 10.
Using the scalar ADM energy functional [defined in Eq. (41)] tomeasure the contributions to the trace of the metric perturbation g (1) in each (cid:96) mode. Notice that the modal energy in the extremal limit is regular, despitethe logarithmic divergence in the (cid:96) = 0 mode. This is emphasized by making thehorizontal axis the extremality parameter ε = (cid:112) − χ . Note that each curvebecomes horizontal going toward the left edge of the right panel. metric perturbation in the slowly rotating and extremal limits, with the higher (cid:96) modesbecoming fractionally more important as rotation increases. In the slowly rotatingcase the monopole mode dominates, with the energy of the modes falling off like χ (cid:28) to a power that is monotonically increasing with (cid:96) . For χ sufficiently close to zero, themonopole and quadrupole modes of the trace of the metric perturbation account forthe ADM energy to better than 1 part in . Near extremality, the monopole modestill dominates, but the contributions to the ADM energy from the first eight modes( (cid:96) ≤ ) span a range of only about 8 decades. As pointed out earlier, approximatingthe trace of the metric perturbation to better than fidelity in the extremal limitrequires at least the first four modes ( (cid:96) ≤ ). Thus, as with the scalar field, modes ofthe trace of the metric perturbation that are not relevant in the limit of slow rotationbecome more important near extremality.Finally, we turn to the question of the significance of the logarithmic divergenceof the (cid:96) = 0 mode, as seen in Eq. (60). Without a full metric tensor perturbationsolution, we cannot determine if this is a true curvature singularity of the perturbedspacetime. However, several pieces of evidence suggest that a divergence in the traceof the metric perturbation need not be a problem.First, note that shifts within the Kerr family of solutions manifest perturbativelyas divergences of the trace of the metric perturbation. Consider shifting the Kerrmetric by M → M + δM and a → a + δa , and expanding the resulting metric in powersof δM (cid:28) M and δa (cid:28) a . This is made formal by the coefficients in the Taylor series,which are given by ( δ M g ) ab ≡ ∂∂M g ab , ( δ a g ) ab ≡ ∂∂a g ab , (63)and so on for higher derivatives, e.g. δ M,M g ab . The perturbed shifted Kerr metric tensorsatisfies the perturbative Einstein equations order-by-order, since they are simply shiftsof the Kerr parameters M and a . A straightforward calculation in Boyer-Lindquist xtremal Black Holes in Dynamical Chern-Simons Gravity Tr[ δ M g ] = 0 , Tr[ δ a g ] = 4 a cos θ Σ , (64) Tr[ δ M,M g ] = 8 r ∆ , Tr[ δ a,M g ] = p ( r, cos θ )∆ Σ , Tr[ δ a,a g ] = p ( r, cos θ )∆ Σ , (65)where p , are certain regular polynomials of r and cos θ , neither of which have zerosat the roots of either ∆ or Σ . Although the Kerr spacetime is regular, as are thelinearly or quadratically shifted spacetimes, we immediately see that the traces of theabove metric perturbations diverge at either the event and Cauchy horizons where ∆ = 0 , or the (ring) curvature singularity where Σ = 0 , or both. Only the blowupat the curvature singularity can be seen as physically significant. When a → M , inBoyer-Lindquist coordinates, the curvature singularity appears on the horizon, at theequator.Next we can bring in arguments that are more specific to the dCS problem. Asnoted by Stein [21], the (inner) Cauchy-horizon divergence of the scalar found by Konnoand Takahashi [20] may result in a divergence of the trace of the metric perturbationat the inner horizon; however, this divergence would be hidden by the outer horizonfor all | a | < M . Indeed, Ref. [21], found that the trace of the metric perturbation isregular in the exterior for all | a | < M . However, in the limit a → M , the inner andouter horizons confluence. Further, in the slow-rotation expansion, the horizon wasseen to be shifted outward to r hor = r hor,Kerr + (915 / ζM χ + O ( χ ) [19]. Thus,one may expect that in the extremal limit, the perturbed horizon may again be exteriorto that of the background, hiding any potential singularity. Finally, it may be the casethat the extremality condition in dCS is shifted away from J = M .All of these arguments suggest that the horizon singularity in g (1 , may not bea physical curvature singularity, but rather a coordinate singularity. A full tensorperturbation solution is needed to confirm this conjecture.
7. Discussion
This paper explored rotating black holes in dCS. Using an effective field theorytreatment of dCS, we worked in the decoupling limit where dCS corrections are smallperturbations from GR solutions. We have further focused on BHs that spin at themaximal Kerr rate, the so-called extremal limit. With these assumptions, we thensolved for the dynamical scalar field in closed analytic form, obtaining a Legendredecomposition that is dominated by its dipole and octupole terms. The radial structureof this decomposition includes natural logarithms and arctangents, unlike the simplepolynomial results obtained in the slow-rotation limit. We then solved for the Legendredecomposition of the trace of the metric perturbation. Retaining Legendre modes with (cid:96) ≤ is sufficient to ensure a fidelity of at least relative to numerical solutions forboth the scalar field and the trace of the metric perturbation.The trace of the metric perturbation in the Lorenz-like gauge exhibits a logarithmicdivergence at the location of the background (extremal Kerr) event horizon. Severalpieces of evidence, discussed in Sec. 6, show that it need not be a physical curvaturesingularity. We conjecture that it is only a coordinate singularity, but a full metrictensor perturbation solution is needed to confirm this conjecture.The techniques we employed rely heavily on Legendre expansions, but our worksuggests that these can be truncated at a finite mode number without losing much of xtremal Black Holes in Dynamical Chern-Simons Gravity Acknowledgments
We would like to thank the Kavli Institute for Theoretical Physics for their hospitalityduring the completion of this work. We would also like to thank Kent Yagi, FransPretorius, and Albion Lawrence for useful discussions. RM acknowledges supportfrom a Loyola University Chicago Summer Research Stipend. LCS acknowledgesthat support for this work was provided by NASA through Einstein PostdoctoralFellowship Award Number PF2-130101 issued by the Chandra X-ray ObservatoryCenter, which is operated by the Smithsonian Astrophysical Observatory for and onbehalf of the NASA under contract NAS8-03060, and further acknowledges supportfrom the NSF grant PHY-1404569. NY acknowledges support from NSF CAREERAward PHY-1250636. The research of RM and NY was supported in part by theNational Science Foundation under Grant No. NSF PHY11-25915. Some calculationsused the computer algebra-system
Maple , in combination with the
GRTensorII package [48]. Other calculations used the computer algebra-system
Mathematica , incombination with the xTensor package [49, 50, 51]. Finally, RM (@mcnees) and LCS(@duetosymmetry) would like to thank Twitter for facilitating discussion during theearly stages of this collaboration.
Appendix A. Solution of the Scalar Equation of Motion
The equations of motion for the scalar field and the trace of the metric perturbationhave the same general form, so let us briefly establish some conventions for the solutionsof such equations. First, consider an equation of the form ∂ ˜ r ( ˜∆ ∂ ˜ r I (cid:96) ) − (cid:96) ( (cid:96) + 1) I (cid:96) = K (cid:96) (A.1) xtremal Black Holes in Dynamical Chern-Simons Gravity K (cid:96) . Denote by H + (cid:96) and H − (cid:96) the solutions of the homogeneousequation ∂ ˜ r ( ˜∆ ∂ ˜ r H ± (cid:96) ) − (cid:96) ( (cid:96) + 1) H ± (cid:96) = 0 , (A.2)with H + (cid:96) regular at ˜ r + and H − (cid:96) → at ˜ r → ∞ . We use the method of variation ofparameters to find the general solution to the inhomogeneous equation. The solutionof (A.1) that is both regular at ˜ r + and goes to zero as ˜ r → ∞ can be expressed interms of the homogeneous solutions and the source as I (cid:96) = 1 W (cid:96) × (cid:32) H + (cid:96) (˜ r ) (cid:90) ˜ r ∞ d ˜ r (cid:48) H − (cid:96) (˜ r (cid:48) ) K (cid:96) (˜ r (cid:48) ) − H − (cid:96) (˜ r ) (cid:90) ˜ r ˜ r + d ˜ r (cid:48) H + (cid:96) (˜ r (cid:48) ) K (cid:96) (˜ r (cid:48) ) (cid:33) . (A.3)Here a factor of ˜∆ has canceled inside each integral, allowing us to pull out a constant W (cid:96) ; this constant depends on the Wronskian of the homogeneous solutions, W (cid:96) ≡ ˜∆ × W [ H − (cid:96) , H + (cid:96) ] (A.4) = ˜∆ × (cid:0) H − (cid:96) ∂ ˜ r H + (cid:96) − H + (cid:96) ∂ ˜ r H − (cid:96) (cid:1) . It is straightforward to verify that this is constant using Eq. (A.2).For the Kerr background, the homogeneous solutions can be written as H + (cid:96) (˜ r ) = c + (cid:96) (1 − χ ) (cid:96) P (cid:96) (cid:32) ˜ r − (cid:112) − χ (cid:33) (A.5) H − (cid:96) (˜ r ) = c − (cid:96) (1 − χ ) − (cid:96) +12 Q (cid:96) (cid:32) ˜ r − (cid:112) − χ (cid:33) . (A.6)where P (cid:96) ( · ) and Q (cid:96) ( · ) are Legendre functions of the first and second kind, respectively.A standard identity for Legendre functions then gives the factor W (cid:96) = c + (cid:96) c − (cid:96) .The factors of (cid:112) − χ in Eq. (A.5)-(A.6) have been chosen so that the extremallimit, χ → ± , is regular. Otherwise, the overall normalization factors c ± (cid:96) are arbitrary.A convenient choice is to set c + (cid:96) = (cid:96) !(2 (cid:96) − , c − (cid:96) = (2 (cid:96) + 1)!! (cid:96) ! . (A.7)Then W (cid:96) = 2 (cid:96) + 1 , and in the extremal limit the homogeneous solutions are simply lim | χ |→ H + (cid:96) = (˜ r − (cid:96) (A.8) lim | χ |→ H − (cid:96) = 1(˜ r − (cid:96) +1 . (A.9)We adopt this normalization throughout Secs. 3 and 5. Appendix B. Expressions for Radial Modes
The radial mode function for general (cid:96) is given in Eq. (33). In this form, each modedepends on (cid:96) + 1 coefficients α (cid:96),k and β (cid:96),k . The coefficients for the modes up to (cid:96) = 9 are given below. xtremal Black Holes in Dynamical Chern-Simons Gravity α ,k β ,k α ,k β ,k α ,k β ,k α ,k β ,k α ,k β ,k − − −
858 3171970 18916 490191260 − − − − −
15 75 − − − − − − − − − − − − − − − − − − Appendix C. Representations of the Scalar Field
Equation (33) provides one representation of the solution to Eq. (23) for arbitraryharmonic number (cid:96) after a Legendre decomposition and in an expansion to leadingorder in ζ (i.e. in the GR deformation) and in ε (i.e., in the extremal limit). This formof the solution depends on (cid:96) + 1 coefficients ( α (cid:96),k and β (cid:96),k ) that are fixed by imposingappropriate boundary conditions on the mode.In this appendix, we present two additional representations of the solution to thescalar field evolution equation that may be preferable in some applications. As in thecase of Eq. (33), these representations will have both advantages and disadvantagesthat we will describe in detail. The solutions start by representing the source functionin the extremal limit in terms of a series: s (1 / , (cid:96) (˜ r ) = ∞ (cid:88) n =0 α (cid:96),n (cid:96) + 1˜ r (cid:96) +4+2 n , (C.1)where we have introduced the constants α (cid:96),n = ( − (cid:96) − ( − n ( (cid:96) + 2 n + 2) Γ( (cid:96) + 4 + 2 n ) Γ( )2 (cid:96) +1+2 n Γ( n + 1) Γ( (cid:96) + n + ) , (C.2)in terms of the Gamma function Γ( · ) . The factor of (cid:96) + 1 in Eq. (C.1) has beenintroduced to simplify some expressions, by canceling a similar factor in the denominatorof Eq. (28). From here on, different representations take different routes to arrive at asolution to Eq. (23) in the extremal limit, so we tackle each of them separately below. xtremal Black Holes in Dynamical Chern-Simons Gravity Appendix C.1. Incomplete Beta Function Representation
Introducing expansion Eq. (C.1) for the scalar source into the solution Eq. (28) for thescalar field: ˜ ϑ (1 / , (cid:96) (˜ r ) = ∞ (cid:88) n =0 α (cid:96),n (cid:34) (˜ r − (cid:96) (cid:90) ˜ r ∞ d ˜ r (cid:48) (˜ r (cid:48) − (cid:96) +1 ˜ r (cid:48) (cid:96) +4+2 n − r − (cid:96) +1 (cid:90) ˜ r d ˜ r (cid:48) (˜ r (cid:48) − (cid:96) r (cid:48) (cid:96) +4+2 n (cid:35) , (C.3)where we have already imposed appropriate boundary conditions. The integrals canbe evaluated in closed-form to obtain ˜ ϑ (1 / , (cid:96) (˜ r ) = β (˜ r ) + β (˜ r ) + β (˜ r ) (C.4)where we have defined β (˜ r ) = − ∞ (cid:88) n =0 α (cid:96),n (˜ r − (cid:96) B / ˜ r (cid:18) (cid:96) + 4 + 2 n, − (cid:96) (cid:19) , (C.5) β (˜ r ) = ∞ (cid:88) n =0 α (cid:96),n (˜ r − (cid:96) +1 B / ˜ r (cid:18) n + 3 , (cid:96) + 1 (cid:19) , (C.6) β (˜ r ) = − ∞ (cid:88) n =0 α (cid:96),n (˜ r − (cid:96) +1 Γ( (cid:96) + 1) Γ(2 n + 3)Γ( (cid:96) + 4 + 2 n ) , (C.7)in terms of the incomplete Beta function B x ( a, b ) (see e.g. Sec. 8.17 of [52]), B x ( a, b ) ≡ (cid:90) x t a − (1 − t ) b − dt . (C.8)Evaluating at x = 1 gives the ordinary Beta function, B ( a, b ) = B ( a, b ) .The sums over n can be evaluated in closed form for β and β . The latter can besummed into β (˜ r ) = ( − (cid:96) +12 (cid:96) Γ( (cid:96) + 1)Γ( )2 (cid:96) +2 Γ( (cid:96) + ) 1(˜ r − (cid:96) +1 (cid:34) (cid:96) (2 (cid:96) + 1) − (cid:96) − (cid:96) + 1) F (cid:0) − , (cid:96) + ; − (cid:1)(cid:35) , (C.9)which gives the leading behavior of ˜ ϑ (1 / , (cid:96) at large ˜ r . The sum over n for β can beevaluated using the series representation of the incomplete Beta function appropriatefor Eq. (C.6), B x (cid:0) m, n (cid:1) = n − (cid:88) j =0 ( − j m + j Γ( n )Γ( j + 1) Γ( n − j ) x m + j . (C.10) xtremal Black Holes in Dynamical Chern-Simons Gravity j and n yields β (˜ r ) = ( − ( (cid:96) +3) / Γ( (cid:96) + 1)Γ(1 / (cid:96) )2 (cid:96) Γ(5 / (cid:96) )(˜ r − (cid:96) +1 (cid:96) (cid:88) j =0 ( − j (3 + j )(5 + j ) 1Γ(1 + j )Γ(1 + (cid:96) − j ) 1˜ r j × (cid:20) (5 + j )(2 + (cid:96) )(3 + 2 (cid:96) ) ˜ r F (cid:18) j , (cid:96) , (cid:96) j , (cid:96) − r (cid:19) − (3 + j )(4 + (cid:96) )(5 + (cid:96) ) F (cid:18) j , (cid:96) , (cid:96) j , (cid:96) − r (cid:19)(cid:21) , (C.11)where P F Q ( · ; · ; · ) is the generalized hypergeometric function. We have not succeededin finding a closed-form expression for the above sum over j , but the sum can beperformed explicitly given any value of (cid:96) .One is then only left with the sum over n for β . To obtain an expression forthis sum, we start with the following representation of the incomplete Beta functionrelevant for Eq. (C.5): B r (2 (cid:96) + 4 + 2 n, − (cid:96) ) = ( − (cid:96) +1 ( (cid:96) + 4 + 2 n )Γ(2 (cid:96) + 4 + 2 n )Γ( (cid:96) + 1)Γ( (cid:96) + 5 + 2 n ) × (cid:20) ln (cid:18) ˜ r − r (cid:19) + 1˜ r − (cid:32) − (cid:96) +3+2 n (cid:88) k =1 k ( k + 1) 1˜ r k (cid:33)(cid:35) (C.12) − Γ(2 (cid:96) + 4 + 2 n )Γ( (cid:96) + 1) 1˜ r (cid:96) +3+2 n (cid:96) − (cid:88) k =0 (˜ r − k − (cid:96) ( − k +1 Γ( (cid:96) − k )Γ(2 (cid:96) + 4 + 2 n − k ) . This allows us to write β (˜ r ) := β (˜ r ) + β (˜ r ) , where we have defined β (˜ r ) = − (˜ r − (cid:96) ∞ (cid:88) n =0 α (cid:96),n ( − (cid:96) +1 ( (cid:96) + 4 + 2 n )Γ(2 (cid:96) + 4 + 2 n )Γ( (cid:96) + 1)Γ( (cid:96) + 5 + 2 n ) × (cid:34) ln (cid:18) ˜ r − r (cid:19) + 1˜ r − (cid:32) − (cid:96) +3+2 n (cid:88) k =1 k ( k + 1) 1˜ r k (cid:33)(cid:35) (C.13) β (˜ r ) = − (cid:96) + 1) 1˜ r (cid:96) +3 ∞ (cid:88) n =0 α (cid:96),n Γ(2 (cid:96) + 4 + 2 n )˜ r n (cid:96) − (cid:88) k =0 ( − k Γ( (cid:96) − k )Γ(2 (cid:96) + 4 + 2 n − k ) (˜ r − k . (C.14)The function β (˜ r ) can be simplified further by performing the sums to obtain β (˜ r ) = ( − ( (cid:96) +1) / (cid:96) (1 + (cid:96) )(2 + (cid:96) )Γ(4 + (cid:96) ) 1˜ r (cid:96) (cid:96) − (cid:88) k =0 ( − k Γ( (cid:96) − k )Γ(6 − k + 2 (cid:96) ) (˜ r − k × (cid:20) ( k − (cid:96) −
5) (3 + 2 (cid:96) ) ( k − − (cid:96) ) ˜ r F (cid:18) (cid:96) , (cid:96) , (cid:96), (cid:96) (cid:96) , − k +2 (cid:96) , − k +2 (cid:96) ; − r (cid:19) − (cid:96) ) (5 + (cid:96) ) (5 + 2 (cid:96) ) F (cid:18) (cid:96) , (cid:96) , (cid:96), (cid:96) (cid:96) , − k +2 (cid:96) , − k +2 (cid:96) ; − r (cid:19)(cid:21) . (C.15) xtremal Black Holes in Dynamical Chern-Simons Gravity β (˜ r ) can also be simplified by performing some of the sums in closedform to obtain β (˜ r ) = ( − (cid:96) Γ( (cid:96) + 1) (˜ r − (cid:96) ∞ (cid:88) n =0 α (cid:96),n ( (cid:96) + 4 + 2 n )Γ(2 (cid:96) + 4 + 2 n )Γ( (cid:96) + 5 + 2 n ) (cid:20) ln (cid:18) ˜ r − r (cid:19) + 1˜ r − (cid:32) − (cid:96) +3+2 n (cid:88) k =1 k ( k + 1) 1˜ r k (cid:33)(cid:35) = ( − (3 (cid:96) +1) / (cid:96) + 3)Γ( (cid:96) + 1) (˜ r − (cid:96) − (cid:20) r −
1) ln (cid:18) ˜ r − r (cid:19)(cid:21) + ( − (cid:96) +1 Γ( (cid:96) + 1) (˜ r − (cid:96) − ∞ (cid:88) n =0 (cid:96) +3+2 n (cid:88) k =1 γ (cid:96),n k ( k + 1) 1˜ r k , (C.16)where we have defined γ (cid:96),n := α (cid:96),n ( (cid:96) + 4 + 2 n )Γ(2 (cid:96) + 4 + 2 n )Γ( (cid:96) + 5 + 2 n ) . (C.17)The last term in Eq. (C.16) can also be represented as follows: ∞ (cid:88) n =0 (cid:96) +3+2 n (cid:88) k =1 γ (cid:96),n k ( k + 1) 1˜ r k = ( − ( (cid:96) +1) / (cid:96) + 3) G (cid:18) r , (cid:96) + 3 (cid:19) ×× ∞ (cid:88) j =0 ∞ (cid:88) n = j +1 γ (cid:96),n (cid:20) (cid:96) + 4 + 2 j )( (cid:96) + 5 + 2 j ) 1˜ r (cid:96) +4+2 j ++ 1( (cid:96) + 5 + 2 j )( (cid:96) + 6 + 2 j ) 1˜ r (cid:96) +5+2 j (cid:21) , (C.18)where we have defined the new function G ( x, N ) := N (cid:88) k =1 x k k ( k + 1) , (C.19)for some x ∈ (cid:60) and N ∈ N . This function is the first N terms of the Taylor series for − log(1 − x ) + x − log(1 − x ) about x = 0 . Notice that the sum in this new functionis finite, and thus G (1 / ˜ r, N ) is simply a polynomial in / ˜ r . Given a particular value of (cid:96) , the remaining sum over j can be performed explicitly. Appendix C.2. Radial Series Representation
Instead of using variation of parameters to solve Eq. (23), we will search for a seriessolution. We thus insert the ansatz ˜ ϑ (1 / , (cid:96) (˜ r ) = σ (˜ r ) + σ (˜ r ) , (C.20)with σ (˜ r ) := ∞ (cid:88) n =0 a (cid:96),n r (cid:96) +1+ n , (C.21) σ (˜ r ) := ∞ (cid:88) n =0 b (cid:96),n r (cid:96) +4+2 n , (C.22) xtremal Black Holes in Dynamical Chern-Simons Gravity a (cid:96),n and b (cid:96),n coefficients.The recursion relations for the a (cid:96),n can be solved to obtain a (cid:96),n = ( (cid:96) + n )! (cid:96) ! n ! a (cid:96), , (C.23)which then leads to σ (˜ r ) = a (cid:96), (˜ r − (cid:96) +1 . (C.24)Since this is the leading behavior of the scalar field at large ˜ r , we can determine thecoefficient a (cid:96), by comparing it with the incomplete Beta function representation ofthe previous subsection: a (cid:96), = − ∞ (cid:88) n =0 α (cid:96),n B (2 n + 3 , (cid:96) + 1) = ( − (cid:96) +12 √ π (cid:96) !2 (cid:96) (cid:20) ( (cid:96) + 2)Γ( (cid:96) + ) F (cid:0) , (cid:96) + ; − (cid:1) − (cid:96) + ) F (cid:0) , (cid:96) + ; − (cid:1)(cid:21) . (C.25)Resumming the coefficients b (cid:96),n is more complicated. We can solve the recursionrelations to express the b (cid:96),n coefficient as finite sums that depend on the coefficients α (cid:96),n in the series expansion of the source: b (cid:96),n = Γ( (cid:96) + 4 + n )Γ(4 + n ) j max (cid:88) j =0 Γ(3 + 2 j )Γ( (cid:96) + 4 + 2 j ) α (cid:96),j − Γ( (cid:96) + 4 + n )Γ(2 (cid:96) + 5 + n ) j max (cid:88) j =0 Γ(2 (cid:96) + 4 + 2 j )Γ( (cid:96) + 4 + 2 j ) α (cid:96),j , (C.26)where j max = n/ if n is even, and j max = ( n + 1) / if n is odd.When one tries to perform the full infinite sum of the b (cid:96),n coefficients over n tofind σ (˜ r ) , one finds a familiar problem: the coefficients of Eq. (C.26) are finite sumswith an upper limit that depends on n , which must then be summed to infinity. To getaround this problem, we can rewrite each finite sum as the difference of two infinitesums: b (cid:96),n = Γ( (cid:96) + 4 + n )Γ(4 + n ) c (cid:96),j max − Γ( (cid:96) + 4 + n )Γ(2 (cid:96) + 5 + n ) d (cid:96),j max , (C.27)where c (cid:96),k = ( − (cid:96) − √ π (cid:96) +1 × (cid:34) ( (cid:96) + 1)(4 (cid:96) − (cid:96) + ) + 2 ( − k (2 k + 3)Γ( k + ) √ π Γ( (cid:96) + + k ) (C.28) + ( (cid:96) − (cid:96) + 9 (cid:96) + 10)2 Γ( (cid:96) + ) F ( − , (cid:96) + ; − − ( (cid:96) + 5 (cid:96) + (cid:96) + 5)2 Γ( (cid:96) + ) F ( , (cid:96) + ; − − k ( (cid:96) −
1) Γ( k + ) √ π Γ( (cid:96) + + k ) F (1 , k + ; (cid:96) + k + ; −
1) + 2 (cid:96) − ( (cid:96) + 2)Γ( (cid:96) + ) F ( (cid:96) − , (cid:96) ; (cid:96) + ; − (cid:21) ,d (cid:96),k = ( − (cid:96) − × (cid:20) −
12 Γ( (cid:96) + 3) + ( − k (cid:96) +2 ( k + 1)( (cid:96) + k + 2)Γ( (cid:96) + k + 3)Γ( k + 2) (C.29) + ( − k (cid:96) +1 ( (cid:96) + 2)Γ( (cid:96) + k + 3)Γ( k + 2) F (1 , (cid:96) + k + 3; k + 2; − (cid:21) . xtremal Black Holes in Dynamical Chern-Simons Gravity b (cid:96),n coefficients expressed in this form, the second sum for the scalar fieldbecomes σ (˜ r ) = ∞ (cid:88) n =0 (cid:20) Γ( (cid:96) + 4 + n )Γ(4 + n ) c (cid:96),j max − Γ( (cid:96) + 4 + n )Γ(2 (cid:96) + 5 + n ) d (cid:96),j max (cid:21) r (cid:96) +4+2 n . (C.30)We have not succeeded in finding closed-form expressions for the sum over n given ageneric (cid:96) value, but the sum can be performed for a given value of (cid:96) . Appendix D. Series Solutions for the Trace of the Metric Perturbation
Instead of truncating the Legendre expansion of the scalar field, it is also possible toconstruct series approximations of the modes g (1 , (cid:96) . We first note that the source term(48) can be expanded in powers of / ˜ r . For the (cid:96) = 0 mode this series takes the form S (1 , (˜ r ) = ∞ (cid:88) n =0 e ,n r n , (D.1)while for (cid:96) ≥ it is S (1 , (cid:96) (˜ r ) = ∞ (cid:88) n =0 e (cid:96),n r (cid:96) +2+ n . (D.2)In terms of the series coefficients for the source, the (cid:96) = 0 mode is g (1 , (˜ r ) = ∞ (cid:88) n =0 e ,n × (cid:34) log (cid:18) ˜ r − r (cid:19) + n +3 (cid:88) j =1 j ˜ r j + 1 n + 3 1˜ r − (cid:18) r n +3 − (cid:19) (cid:35) . (D.3)Note that the coefficient of the log(˜ r − term is (D.1) evaluated at ˜ r = 1 , as inEq. (60). The modes with (cid:96) ≥ can be expressed as a series involving incomplete Betafunctions: g (1 , (cid:96) (˜ r ) = ∞ (cid:88) n =0 e (cid:96),n × (cid:104) − (˜ r − (cid:96) B / ˜ r (2 (cid:96) + 2 + n, − (cid:96) ) − r − (cid:96) +1 B − / ˜ r ( (cid:96) + 1 , n + 1) (cid:105) . (D.4)Since these solutions are obtained directly from Eq. (50) they already satisfy the correctboundary conditions at ˜ r → ∞ and ˜ r = 1 .Given the expansion of the source functions, one can obtain a series solution of theequation of motion Eq. (47) directly. For the (cid:96) = 0 mode this solution takes the form g (1 , (˜ r ) = f , (˜ r −
1) + ∞ (cid:88) n =0 r n n (cid:88) j =0 n + 1 − j ( n + 4)( j + 3) e ,j (D.5)while for (cid:96) ≥ it is g (1 , (cid:96) (˜ r ) = f (cid:96), (˜ r − (cid:96) +1 + ∞ (cid:88) n =0 ( (cid:96) + n )!˜ r (cid:96) +2+ n n (cid:88) j =0 e (cid:96),j (cid:18) j !( n + 1)!( (cid:96) + j + 1)! − (2 (cid:96) + j + 1)!( (cid:96) + j + 1)!(2 (cid:96) + n + 2)! (cid:19) . (D.6) xtremal Black Holes in Dynamical Chern-Simons Gravity f (cid:96), of the leading term can be expressed in terms of oneor more integrals of the source; for (cid:96) = 0 it is f , = (cid:90) ∞ d ˜ r S (1 , (˜ r ) . (D.7) [1] Will C M 2014 Living Reviews in Relativity URL [2] Yunes N and Siemens X 2013
Living Rev.Rel. [arXiv:1304.3473] URL [3] Berti E et al. Class. Quant. Grav. et al. Science
Nucl. Instrum. Meth.
A289
Astrophys. J.
Publ. Astron. Soc. Pac.
Phys.Rept.
Phys. Rev.
D91
Phys. Lett.
B149
Superstring Theory. Vol. 2: LoopAmplitudes, Anomalies And Phenomenology (Cambridge: Cambridge University Press) ISBN9780521357531[12] Weinberg S 2008
Phys.Rev.
D77
Phys.Rev.
D78
Phys.Rev.
D68
Phys. Rev.
D89
Phys. Lett.
B251
Phys.Rev.
D79
Prog.Theor.Phys.
Phys.Rev.
D86
Phys. Rev.
D90
Phys. Rev.
D90
Phys. Rev.
D77
Phys. Rev.
D77
Phys. Rev.
D80
Phys. Rev.
D80
JHEP
045 [arXiv:0901.1670][27] Matsuo Y and Nishioka T 2010
JHEP
073 [arXiv:1010.4549][28] Carlip S 2011
JHEP
076 [Erratum: JHEP01,008(2012)] [arXiv:1101.5136][29] Compere G 2012
Living Rev. Rel. [arXiv:1203.3561] URL [30] Dias O J C, Santos J E and Stein M 2012 JHEP
182 [arXiv:1208.3322][31] McNees R and Stein L C Forthcoming[32] Kanti P, Mavromatos N, Rizos J, Tamvakis K and Winstanley E 1996
Phys.Rev.
D54
Phys.Rev.
D55
Phys.Rev.
D55
Phys.Rev.Lett.
Phys. Rev.
D83
Phys.Rev.
D84
Phys. Rev.
D90
Gravitation (San Francisco: W. H. Freeman)[40] Wald R M 1984
General Relativity (University of Chicago Press)[41] Yagi K, Stein L C and Yunes N 2016
Phys. Rev.
D93
Phys.Rev.
D80
Phys.Rev.
D84
Phys. Rev.
D87
Phys.Rev.Lett. xtremal Black Holes in Dynamical Chern-Simons Gravity [46] Stein L C 2014 [arXiv:1407.0744][47] Bardeen J M and Horowitz G T 1999 Phys. Rev.
D60 http://grtensor.org [49] Martin-Garcia J M, Portugal R and Manssur L R U 2007
Comput.Phys.Commun.
Comput.Phys.Commun.