How do Black Holes Spin in Chern-Simons Modified Gravity?
aa r X i v : . [ g r- q c ] N ov How do Black Holes Spin in Chern-Simons Modified Gravity?
Daniel Grumiller and Nicol´as Yunes Center for Theoretical Physics, Massachusetts Institute of Technology,77 Massachusetts Ave., Cambridge, MA 02139, USA Institute for Gravitation and the Cosmos, Department of Physics,The Pennsylvania State University, University Park, PA 16802, USA (Dated: October 25, 2018)No Kerr-like exact solution has yet been found in Chern-Simons modified gravity. Intrigued bythis absence, we study stationary and axisymmetric metrics that could represent the exterior field ofspinning black holes. For the standard choice of the background scalar, the modified field equationsdecouple into the Einstein equations and additional constraints. These constraints eliminate essen-tially all solutions except for Schwarzschild. For non-canonical choices of the background scalar, wefind several exact solutions of the modified field equations, including mathematical black holes andpp-waves. We show that the ultrarelativistically boosted Kerr metric can satisfy the modified fieldequations, and we argue that physical spinning black holes may exist in Chern-Simons modifiedgravity only if the metric breaks stationarity, axisymmetry or energy-momentum conservation.
PACS numbers: 04.20.Cv,04.70.Bw,04.20.Jb,04.30.-w
I. INTRODUCTION
General relativity (GR) is one of physics’ most success-ful theories, passing all experimental tests so far with everincreasing accuracy [1]. Nevertheless, modifications toGR are pursued vigorously for two main reasons: from atheoretical standpoint, we search for an ultraviolet (UV)completion of GR, such as string theory, that would leadto corrections in the action proportional to higher powersof scalar invariants of the Riemann tensor; from an ex-perimental standpoint, observations in the deep infrared(IR) regime suggest the existence of some form of darkenergy [2, 3, 4]. One possibility to accommodate dark en-ergy is to consider an action with non-linear couplings tothe Ricci scalar [5, 6], similar in spirit to the correctionsthat we expect from a UV completion of GR.UV and IR corrections entail higher derivatives of thefundamental degrees of freedom in the equations of mo-tion, which on general grounds tend to have disastrousconsequences on the stability of the solutions of the the-ory [87]: the so-called Ostrogradski instability (for a re-view cf. e.g. [7]). A few loopholes exist, however, thatallow to bypass this theorem (for example, if the non-linear corrections can be converted into a representationof a scalar-tensor theory). Along these lines, special com-binations of scalar invariants that play the role of a topo-logical term, such as the Euler or Pontryagin term, canin general be added safely to the action.In this paper, we study Chern-Simons (CS) modifiedgravity [8], where the Einstein-Hilbert action is modifiedby the addition of a parity-violating Pontryagin term.As described by Jackiw and Pi [8], this correction arisesthrough the embedding of the 3-dimensional CS topolog-ical current into a 4-dimensional spacetime manifold. CSgravity is not a random extension of GR, but it has physi-cal roots in particle physics. Namely, if there is an imbal-ance between left- ( N L ) and right-handed ( N R ) fermions,then the fermion number current j µ has a well-known gravitational anomaly [9], ∂ µ j µ ∝ ( N L − N R ) ∗ R R , anal-ogous to the original triangle anomaly [10]. Here ∗ R R is the Pontryagin term (also known as the gravitationalinstanton density or Chern-Pontryagin term) to be de-fined in the next section. CS gravity is also motivated bystring theory: it emerges as an anomaly-canceling termthrough the Green-Schwarz mechanism [11]. Such a cor-rection to the action is indispensable, since it arises asa requirement of all 4-dimensional compactifications ofstring theory in order to preserve unitarity [12].CS gravity has been studied in the context of cosmol-ogy, gravitational waves, solar system tests and Lorentzinvariance. In particular, this framework has been usedto explain the anisotropies in the cosmic microwave back-ground [13, 14, 15] and the leptogenesis problem [12, 16](essentially using the gravitational anomaly describedabove in the other direction). CS gravity has also beenshown to lead to amplitude birefringent gravitationalwaves [8, 17, 18, 19], possibly allowing for a test of thistheory with gravitational-wave detectors [20]. Moreover,CS gravity has been investigated in the far-field of a spin-ning binary system, leading to a prediction of gyromag-netic precession [18, 19] that differs from GR. This pre-diction was later improved on and led to a constraint onthe magnitude of the CS coupling [21]. Finally, CS grav-ity has been studied in the context of Lorentz-invarianceand -violation [22] and the theory has been found to pre-serve this symmetry, provided the CS coupling is treatedas a dynamical field. For further studies of these and re-lated issues cf. e.g. [8, 15, 17, 19, 21, 22, 23, 24, 25, 26,27, 28, 29, 30, 31, 32, 33, 34, 35].CS gravity introduces the following modification to theaction [8]: S = S EH + S mat + S CS , where S EH is theEinstein-Hilbert action, S mat is some matter action, andthe new term is given by S CS ∼ Z dV θ ∗ R R . (1)In Eq. (1), dV is a 4-dimensional volume element, ∗ R R is the Pontryagin term and θ is a background scalar field (we shall define this action in more detail in the next sec-tion). This scalar field, sometimes called a gravitationalaxion, acts as a CS coupling function that can be inter-preted either as an external or a dynamical quantity. Inthe former case, CS gravity is an effective theory thatderives from some other, more fundamental gravity the-ory that physically defines the scalar field. In the lattercase, the scalar field possesses its own equation of mo-tion, which could in principle contain a potential and akinetic term [21].The strength of the CS correction clearly dependson the CS coupling function. If we consider CS grav-ity as an effective theory, the coupling function is sup-pressed by some mass scale, which could lie between theelectro-weak and the Planck scale, but it is mostly un-constrained [21]. In the context of string theory, thecoupling constant has been computed in very conser-vative scenarios, leading to a Planck mass suppression[16]. In less conservative scenarios, there could existenhancements that elevate the coupling function to therealm of the observable. Some of these scenarios arecosmologies where the string coupling vanishes at latetimes [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46], or wherethe field that generates θ couples to spacetime regionswith large curvature [47, 48] or stress-energy density [20].The CS correction is encoded in the modified fieldequations, which can be obtained by varying the mod-ified action with respect to the metric. The divergenceof the modified field equations establishes the Pontrya-gin constraint ∗ R R = 0, through the Bianchi identitiesfor a vacuum or conserved stress-energy tensor. Notonly does this constraint have important consequenceson the conservation of energy, but it also restricts thespace of solutions of the modified theory. For example,although this restriction is not strong enough to eliminatethe Schwarzschild solution, it does eliminate the Kerr so-lution. Since astrophysical observations suggest that su-permassive black holes (BHs) at the center of galaxiesdo have a substantial spin (cf., e. g. [49] and referencestherein), this raises the interesting question of what re-places the Kerr solution in CS gravity.In this paper, we search for solutions to the CS modi-fied field equations that could represent the exterior grav-itational field of a spinning star or BH. We find that solu-tions cluster into two different classes: GR solutions thatindependently satisfy both the vacuum Einstein equa-tions and the modified field equations; non-GR solutionsthat satisfy the modified field equations but not the vac-uum Einstein equations. We carry out an extensive studyof solutions by looking at three groups of line elements:spherically symmetric metrics; static and axisymmetricmetrics; and stationary and axisymmetric metrics. Thefirst group contains GR solutions only, independently ofthe choice of the CS scalar field. The second group leadsto a decoupling of the modified field equations for ’nat-ural’ choices of the scalar field, which again reduces to trivial GR solutions. In fact, we show here that staticand axisymmetric line elements are forced to be spatiallyconformally flat if such a decoupling occurs. The thirdgroup also leads to the same decoupling for the canon-ical choice of the scalar field, and we argue against theexistence of non-trivial solutions.This paper suggests that stationary and axisymmet-ric line elements in CS gravity probably do not admitsolutions of the field equations for the canonical choiceof the CS scalar field. However, solutions do exist whenmore general scalar fields are considered, albeit not rep-resenting physical BH configurations [88]. We find twotypes of solutions, mathematical BHs and ultrarelativis-tically boosted BHs, which, to our knowledge, are thefirst examples of BH and BH-like solutions in CS grav-ity, besides Schwarzschild and Reissner-Nordstr¨om. Thefirst type arises when we consider a subclass of station-ary and axisymmetric line elements (the so-called vanStockum class), for which we find both GR and non-GRsolutions for non-canonical scalar fields. For instance, weshall demonstrate that the line-element ds = − ρ (cid:16) − m √ ρ (cid:17) dt − ρ dt dφ + 1 √ ρ (cid:16) dρ + dz (cid:17) , (2)together with the CS scalar field θ = 2 √ ρ z/
3, satisfiesthe modified field equations but does not arise in GRas a vacuum solution. The metric in Eq. (2) representsBHs in the mathematical sense only: it exhibits a Killinghorizon at √ ρ = 2 m = const . , but it contains unphysicalfeatures, such as (naked) closed time-like curves. Thesecond type of solutions with a non-canonical scalar fieldarises when we consider scalar fields whose divergence isa Killing vector. These fields lead to exact gravitationalpp-wave solutions of GR and non-GR type. One partic-ular example that we shall discuss in in this paper is theultrarelativistically boosted Kerr BH, ds = − du dv − h δ ( u ) ln (cid:0) x + y (cid:1) du + dx + dy , (3)with the CS scalar field θ = λv , where h and λ areconstants.Although we did not find a Kerr analogue by searchingfor stationary and axisymmetric solutions, spinning BHsdo seem to exist in the theory. This suggestion is fueledby the existence of two different limits of the Kerr space-time that are still preserved: the Schwarzschild limit andthe Aichelburg-Sexl limit, Eq. (3), which we shall showpersists in CS gravity. These limits, together with theexistence of a non-axisymmetric far-field solution [19],indicate that a spinning BH solution must exist, albeitnot with the standard symmetries of the Kerr spacetime.Unfortunately, spacetimes with only one or no Killingvector are prohibitively general and their study goes be-yond the scope of this work. Nonetheless, the possibilityof constructing such solutions by breaking stationarityor axisymmetry is discussed and a better understand-ing of solutions in CS gravity is developed. Finally, weshow how to recover the Kerr solution by postulating, inan ad-hoc manner, a non-conserved energy momentum-tensor and deduce that it violates the classical energyconditions.This paper is organized as follows: Sec. II reviews somebasic features of CS modified gravity and exploits two al-ternative formulations of the Pontryagin constraint, onebased upon the spinorial decomposition of the Weyl ten-sor and one based upon its electro-magnetic decomposi-tion, to reveal some physical consequences of this con-straint; Sec. III revisits the Schwarzschild, Friedmann-Robertson-Walker and Reissner-Nordstr¨om solutions inCS modified gravity and addresses the sensitivity of thesesolutions to the choice of CS coupling function; Sec. IVstudies static, axisymmetric line elements in great detail,while Sec. V investigates stationary, axisymmetric met-rics and provides the first non-trivial exact solutions toCS modified gravity, including mathematical BH solu-tions; Sec. VI addresses metrics that break axisymmetryor stationarity and concentrates on non-trivial solutionsfor pp-waves and the Aichelburg-Sexl boosted Kerr met-ric; Sec. VII concludes and points to future research.We use the following conventions in this paper: wework exclusively in four spacetime dimensions with sig-nature ( − , + , + , +) [50], with Latin letters ( a, b, . . . , h )ranging over all spacetime indices; curvature quanti-ties are defined as given in the MAPLE GRTensorIIpackage [51]; round and square brackets around indicesdenote symmetrization and anti-symmetrization respec-tively, namely T ( ab ) := ( T ab + T ba ) and T [ ab ] := ( T ab − T ba ); partial derivatives are sometimes denoted by com-mas, e.g. ∂θ/∂r = ∂ r θ = θ ,r . The Einstein summationconvention is employed unless otherwise specified, andwe use geometrized units where G = c = 1. II. CS MODIFIED GRAVITYA. ABC of CS
In this section, we summarize the basics of CS modifiedgravity, following the formulation of [8]. Let us begin bydefining the full action of the theory [89]: S = κ Z d x √− g (cid:18) R − θ ∗ R R (cid:19) + S mat , (4)where κ = 1 / (16 π ), g is the determinant of the metric,the integral extends over all spacetime, R is the Ricciscalar, S mat is some unspecified matter action and ∗ R R is the Pontryagin term. The latter is defined via ∗ R R := ∗ R abcd R bacd , (5)where the dual Riemann-tensor is given by ∗ R abcd := 12 ǫ cdef R abef , (6)with ǫ cdef the 4-dimensional Levi-Civita tensor [90]. ThePontryagin term [Eq. (5)] can be expressed as the diver- gence ∇ a K a = 14 ∗ R R (7)of the Chern-Simons topological current (Γ is theChristoffel connection), K a := ǫ abcd (cid:18) Γ nbm ∂ c Γ mdn + 23 Γ nbm Γ mcl Γ ldn (cid:19) , (8)thus the name “Chern-Simons modified gravity” [91].The modified field equations can be obtained by vary-ing the action with respect to the metric. Exploiting thewell-known relations δR bacd = ∇ c δ Γ bad − ∇ d δ Γ bac (9)and δ Γ bac = 12 g bd ( ∇ a δg dc + ∇ c δg ad − ∇ d δg ac ) , (10)the variation of the geometric part of the action leads to δS − δS mat = κ Z d x √− g (cid:18) R ab − g ab R + C ab (cid:19) δg ab − κ Z d x √− g ∗ R R δθ + Σ EH + Σ CS . (11)Here, the tensor C ab stands for a 4-dimensional Cotton-like tensor, which we shall refer to as the C-tensor [92],given by C ab := v c ǫ cde ( a ∇ e R b ) d + v cd ∗ R d ( ab ) c , (12)where v a := ∇ a θ , v ab := ∇ a ∇ b θ = ∇ ( a ∇ b ) θ (13)are the velocity and covariant acceleration of θ , respec-tively [93]. We shall always assume that v a does notvanish identically, because otherwise the model reducesto GR [94].Surface terms are collected in the third line of Eq. (11)and arise due to repeated integration by parts and appli-cation of Stokes’ theorem. In particular, Σ EH and Σ CS arise from variation of the Einstein-Hilbert and CS sec-tor of the action, respectively. The former expression iswell-known, while the latter contains a term with δ Γ,Σ CS = κ Z d x √− g ∇ d (cid:0) θ ∗ R abcd δ Γ bac (cid:1) + . . . (14)It is worthwhile pointing out that one cannot just im-pose Dirichlet boundary conditions on the induced metricat the boundary by adding the Gibbons-Hawking-Yorkterm, as it is the case in GR [52, 53]. There is no obviousway to cancel the term containing the variation of theconnection, δ Γ, in Eq. (14), except by imposing suitablefall-off conditions on the scalar field θ or Dirichlet bound-ary conditions on the connection. Even though we shallneglect boundary issues henceforth, we emphasize thatthese considerations are relevant in many applications,such as BH thermodynamics.The modified field equations are then given by the firstline of Eq. (11), provided the second line vanishes. Thevanishing of ∗ R R is the so-called Pontryagin constraintand we shall study it in Sec. II B. The modified fieldequations in the presence of matter sources are then givenby G ab + C ab = 8 πT ab , (15)where G ab = R ab − g ab R is the Einstein tensor and T ab is the stress-energy tensor of the source. In this paper, weare primarily concerned with the vacuum case, T ab = 0,for which the modified field equations reduce to R ab + C ab = 0 , (16)due to the tracelessness of the C-tensor, C aa = 0. Likein GR, vacuum solutions in CS gravity satisfy R = 0 . (17) B. Pontryagin Constraint
Let us now discuss the Pontryagin constraint ∗ R R = 0 , (18)which then forces the second line in Eq. (11) to vanish.One route to obtain the Pontryagin constraint is to treat θ as a dynamical field (or rather a Lagrange multiplier).By varying the action with respect to θ , we obtain theequations of motion for the scalar field that dynamicallyenforce the Pontryagin constraint.Another route to obtain the Pontryagin constraint isto treat θ as an external quantity. In this case, there areno equations of motion for the scalar field. Nonetheless,by taking the covariant divergence of the equations ofmotion and using the contracted Bianchi identities, oneobtains ∇ a C ab = 18 v b ∗ R R = 8 π ∇ a T ab . (19)Usually, it is desirable to require that the stress-energybe covariantly conserved. However, in CS modified grav-ity this need not be the case because a non-vanishingcovariant divergence ∇ a T ab = 0 could be balanced by anon-vanishing Pontryagin term – this is, in fact, how theterm arises in some approaches in the first place, cf. [27].We shall come back to this issue at the end of Sec. VI,but for the time being we shall set T ab = 0, which thenleads to the Pontryagin constraint.The Pontryagin constraint is a necessary conditionfor any vacuum spacetime that solves the modified field equations, but what does it mean physically? We shallattempt to answer this question by providing two al-ternative formulations of this constraint, but before do-ing so, let us discuss some general properties and conse-quences of Eq. (18). First, notice that setting the ∗ R R term to zero leads to the conserved current K a [Eqs. (7)and (8)], which is topological in nature, and thus impliesthis quantity is intrinsically different from typical con-served quantities, such as energy or angular momentum.Second, when the CS action is studied on-shell [Eq. (4)with ∗ R R = 0] it reduces to the GR action, an issue thatis of relevance for stability considerations, e.g. thermo-dynamic stability in BH mechanics.The first physical interpretation of the Pontryagin con-straint can be obtained by considering a spinorial decom-position. Let us then consider the useful relation ∗ R R = ∗ C C , (20)which we prove in appendix A. In Eq. (20), C is theWeyl tensor defined in (A2) and ∗ C its dual, defined in(A3). This identity allows us to use powerful spinorialmethods to map the Weyl tensor into the Weyl spinor[54], which in turn can be characterized by the Newman-Penrose (NP) scalars (Ψ , Ψ , Ψ , Ψ , Ψ ). In the nota-tion of [55], the Pontryagin constraint translates into areality condition on a quadratic invariant of the Weylspinor, I , ℑ ( I ) = ℑ (cid:0) Ψ Ψ + 3Ψ − Ψ (cid:1) = 0 . (21)Such a reality condition is particularly useful for theconsideration of algebraically special spacetimes. For in-stance, it follows immediately from Eq. (21) that space-times of Petrov types III , N and O obey the Pontryaginconstraint, since in the latter case all NP scalars van-ish, while in the former cases (in an adapted frame)only Ψ or Ψ are non-vanishing. Moreover, all space-times of Petrov types D , II and I are capable of vio-lating Eq. (21). For example, for spacetimes of Petrovtype II one can choose an adapted tetrad such thatΨ = Ψ = Ψ = 0, which then reduces Eq. (21) tothe condition that either the real part or the imaginarypart of Ψ has to vanish.The reality condition of Eq. (21) can also be useful inapplications of BH perturbation theory. For instance, inthe metric reconstruction of the perturbed Kerr space-time [56], the NP scalars Ψ = Ψ = 0 vanish. In thiscontext gravitational waves are characterized by Ψ , ,while Ψ is in general non-vanishing. In a tetrad thatrepresents a transverse-traceless frame, these scalars aregiven by Ψ , = ¨ h + ∓ i ¨ h × , (22)where h + , × are the plus/cross polarization of the wave-form, and the overhead dot stands for partial time deriva-tive [57]. Obviously, Ψ Ψ = ( ¨ h + ) + ( ¨ h × ) is real, whichagain reduces Eq. (21) to the condition that either thereal part or the imaginary part of Ψ has to vanish. Nei-ther of these possibilities is the case for the Kerr BH orperturbations of it [56].Another interpretation of the Pontryagin constraintcan be obtained by exploiting the split of the Weyltensor into electric and magnetic parts (cf. e.g. [58]).Given some time-like vector field u a , normalized so that u a u a = −
1, one can define the electric and magneticparts of the Weyl tensor as( C abcd + i ǫ abef C ef cd ) u b u d = E ac + iB ac . (23)Then, the Pontryagin constraint is equivalent to the con-dition [95] E ab B ab = 0 . (24)This leads to three possibilities: either the spacetime ispurely electric ( B ab = 0) or purely magnetic ( E ab = 0)or orthogonal, in the sense that Eq. (24) holds. Equa-tion (24) is a perfect analogue to the condition ∗ F F ∝ E · B = 0, which holds for specific configurations inelectro-dynamics, including purely electric ( B = 0),purely magnetic ( E = 0) and electromagnetic wave con-figurations ( E = 0 = B , E · B = 0). This suggeststhat there could be single shock-wave solutions in CSgravity compatible with Eq. (24), which we shall indeedencounter in Sec. VI. In light of this electro-magneticanalogy, the Pontryagin constraint can be rephrased as“the gravitational instanton density must vanish,” sincethe quantity ∗ F F is sometimes referred to as the “instan-ton density.”The electromagnetic decomposition of the Pontryaginconstraint also allows for a physical interpretation interms of perturbations of the Schwarzschild solution. InBH perturbation theory (cf. e.g. [59]), the metric per-turbation is also decomposed through the electromag-netic Weyl tensor. The electric and magnetic parts canthen be related to the flux of mass and angular momen-tum across the horizon. Suffice it to say that for a bi-nary BH system in the slow-motion/small-hole approxi-mation [59], these tensors are of order E ab ∼ O (Φ) and B ab ∼ O ( v Φ), where the Newtonian potential Φ is of O ( v ) via the Virial theorem, with v ≪ O ( v ). Within thepost-Newtonian (PN) approximation [60], these conclu-sions imply that the PN metric for non-spinning point-particles in the quasi-circular approximation violates thePontryagin constraint at 2 . O ( v/c ) ], which isprecisely the order at which gravitational waves appear.Even for non-canonical choices of the scalar field, suchas θ = z proposed in [33], the far field expansion of theKerr metric does not satisfy the Pontryagin constraint toall orders. This is so because obviously ∗ R R is indepen-dent of θ . In fact, one can show that violations of theconstraint for the metric considered in [33] occur alreadyat second order in the metric perturbation, which ren-ders this metric hopeless as an exact CS solution. This E CSP ab R = 0 ab C = 0 R + C = 0 ab ab
FIG. 1: Space of solutions of Einstein gravity E and CS mod-ified gravity CS . observation is concurrent with the role the Pontryaginconstraint may play for gravitational waves [61].Finally, we can employ the electromagnetic analogy toanticipate the answer to the question we pose in the ti-tle of this paper. Namely, we are looking for a “rotatingcharge” configuration (where E = 0 = B ), which si-multaneously is an “electromagnetic wave” configuration(where E · B = 0). We know that no such solutions ex-ist in electrodynamics, except for two limits [96]: if therotation (and thus B ) approaches zero or if the chargeis infinitely boosted (and thus B becomes orthogonal to E ). The first case corresponds to a static configuration,while the second one to an ultrarelativistic limit. Weshall indeed find below both analogues as solutions of CSmodified gravity, but we stress that the naive analogywith electrodynamics does not yet rule out other possi-ble spinning configurations in CS modified gravity. C. Space of Solutions
Before discussing some specific solutions to the CSmodified field equations, let us classify the space of so-lutions. Figure 1 presents a 2-dimensional depiction ofthis space. The set E denotes the Einstein space of so-lutions, whose members have a vanishing Ricci tensor,while CS denotes the CS space of solutions, whose mem-bers satisfy the CS modified field equations [Eq. (16)],without necessarily being Ricci flat. The intersection ofthe Einstein and the CS space defines the Pontryaginspace, denoted by P := E ∩ CS , whose members satisfyboth the Einstein and the modified field equations in-dependently. Therefore, solutions that live in P possessa vanishing C-tensor and automatically satisfy the Pon-tryagin constraint, while those living in E \ P satisfy thevaccum Einstein equations but not the Pontryagin con-straint. Moreover, solutions that live in
CS \ P are notRicci-flat but do satisfy the Pontryagin constraint be-cause they solve the modified field equations. Solutionsof class P shall be referred to as GR solutions, while so-lutions of class CS \ P shall be referred to as non-GRsolutions.To date, only one non-GR solution has been foundperturbatively [19] by assuming a far field expansion forpoint-particle sources in the PN weak-field/slow-motionapproximation. We shall show in the next sections thatnon-GR solutions exist only in scenarios with a suffi-cient degree of generality, but not in highly symmetriccases. In the language of dynamical systems theory, the P space acts as an “attractor” of highly symmetric solu-tions, emptying out the CS space.In view of this, let us discuss some properties of solu-tions that live in the P space. In this space, the C-tensorsimplifies to C ab | R ab =0 = v cd ∗ R d ( ab ) c = v cd ∗ C d ( ab ) c = 0 , (25)where C abcd is the Weyl tensor and ∗ C its dual, defined inEqs. (A2) and (A3). Equation (25) leads to three distinctpossibilities:1. The (dual) Weyl tensor vanishes. However, sinceclass P members also have a vanishing Ricci ten-sor, this condition reduces all possible solutions toMinkowski space.2. The covariant acceleration of θ vanishes. This con-dition imposes a strong restriction on the geometry(cf. e.g. [55]), which leads to spacetimes that areeither flat or exhibit a null Killing vector.3. Only the contraction of the covariant accelerationwith the dual Weyl tensor vanishes.Moreover, for solutions in P , the vanishing of the Riccitensor forces the Weyl tensor to be divergenceless, via thecontracted Bianchi identities. These observations are aclear indication that the solutions inhabiting P must bespecial – for instance, exhibit a certain number of Killingvectors. Conversely, one may expect that solutions in-habiting CS \ P cannot be “too special.” We shall putthese expectations on a solid basis and confirm them inthe next sections.
III. PERSISTENCE OF GR SOLUTIONS
In this section, we study some solutions of GR that areknown to persist in CS gravity [8, 22], using the insight onthe Pontryagin constraint gained so far. In the languageof Sec. II C we look for solutions that inhabit P , cf. Fig. 1. A. Schwarzschild Solution
The Schwarzschild solution, ds = − (cid:18) − Mr (cid:19) dt + (cid:18) − Mr (cid:19) − dr + r (cid:0) d Θ + sin (Θ) dφ (cid:1) , (26) is also a solution of the CS modified field equations if [8] θ = tµ → v µ = [1 /µ, , , . (27)We refer to Eq. (27) as the canonical choice of the CSscalar field [8]. In that case, the C-tensor can be in-terpreted as a 4-dimensional generalization of the ordi-nary 3-dimensional Cotton tensor. Moreover, spacetime-dependent reparameterization of the spatial variables andtime translation remain symmetries of the modified ac-tion [8].We investigate now the most general form of θ = θ ( t, r, Θ , φ ) that will leave the Schwarzschild metric a so-lution of the modified theory. The Pontryagin constraintalways holds, regardless of θ , because the spacetime isspherically symmetric, but C ab = 0 yields non-trivialequations. Since we have chosen the Schwarzschild lineelement, we cannot force the (dual) Weyl tensor to vanish(option 1 in Sec. II C), where the only linearly indepen-dent component is C tr Θ φ = 2 Mr sin Θ . (28)Another possibility is to force the scalar field to have avanishing covariant acceleration (option 2 in Sec. II C).This condition then yields an over-constrained systemof partial differential equations (PDEs), whose only so-lution for M = 0 is the trivial one: constant θ . Weare thus left with the remaining possibility (option 3 inSec. II C), namely that only the contraction of the co-variant acceleration with the dual Weyl tensor vanishes.This possibility yields the following set of PDEs θ ,t Θ = θ ,tφ = ∂∂r (cid:18) θ , Θ r (cid:19) = ∂∂r (cid:18) θ ,φ r (cid:19) = 0 , (29)the solution of which is given by θ = F ( t, r ) + rG (Θ , φ ) . (30)Note that this scalar field possesses a non-vanishing co-variant acceleration, namely v tt , v tr , v rr , v ΘΘ , v Θ φ and v φφ are non-vanishing, e.g. v tt = ∂ rr F − Mr (cid:18) − Mr (cid:19) ( ∂ r F + G ) . (31)For the choice of θ given in Eq. (30) the Schwarzschildsolution is always a solution of the modified theory. Notethat Eq. (30) reduces to the canonical choice for G = 0and F = t/µ , for which the only non-vanishing compo-nent of the covariant acceleration is v tr = − M/ ( r f µ ).This simple calculation of the most general form of thescalar field that respects the Schwarzschild solution leadsto two important consequences: • The existence of specific solutions depends sensi-tively on the choice of the scalar field. • The satisfaction of the Pontryagin constraint is anecessary but not a sufficient condition for the C-tensor to vanish.In order to illustrate the second point, let us considerthe scalar field θ = m CS sin Θ, with m CS a constant.Then the Pontryagin constraint is still satisfied, but theC-tensor has one non-vanishing component, C tφ = 3 M m CS r sin Θ (cid:18) − Mr (cid:19) , (32)and the Schwarzschild line element [Eq. (26)] is no longera solution to the modified field equations [Eq. (16)]. B. Spherically symmetric metrics
Let us now pose the question whether there can benon-GR solutions in CS modified gravity that preservespherical symmetry. Any line element respecting thissymmetry must be diffeomorphic to (cf. e.g. [62]) ds = g αβ ( x γ ) dx α dx β + Φ ( x γ ) d Ω S , (33)where g αβ ( x γ ) is a Lorentzian 2-dimensional metric withsome coordinates x γ , Φ( x γ ) is a scalar field (often called“dilaton” or “surface radius”) and d Ω S is a line elementof the round 2-sphere, with some coordinates x i . Forsuch a line element, one can show straightforwardly thatthe Pontryagin constraint is always satisfied (cf. e.g. ap-pendix A of [63]), and that the only non-vanishing com-ponents of the Ricci tensor are R αβ and R ij . On the otherhand, for the most general scalar field θ , the only non-vanishing components of the C-tensor are of the form C αi .Remarkably, the C-tensor and the Ricci tensor decoupleand both have to vanish independently as a consequenceof the modified field equations. In other words, for spher-ically symmetric line elements there cannot be solutionsthat live in CS \ P . Instead all solutions are pushed to P ,which then uniquely leads to the Schwarzschild solutionby virtue of the Birkhoff theorem [97].We have just shown that for all spherically symmet-ric situations the vacuum solutions to the CS modifiedfield equations live in P , and therefore are given uniquelyby the Schwarzschild solution. For non-vacuum solu-tions with the same symmetries, similar conclusions hold,since the field equations still decouple into non-vacuumEinstein equations and the vanishing of the C-tensor.Therefore, all solutions are again pushed to P and spher-ically symmetric solutions of GR (such as the Reissner-Nordstr¨om BH or Friedmann-Robertson-Walker space-times) persist in CS modified gravity, provided θ is ofthe form θ = F ( x γ ) + Φ( x γ ) G ( x i ) . (34)This result is completely analog to Eq. (30). In all spher-ically symmetric scenarios, the solutions to the CS mod-ified field equations live in P and the expectations ofSec. II C hold. C. Losing the Kerr solution
As an example of a relevant GR solution that does notpersist in the modified theory we consider the Kerr solu-tion. The Kerr metric yields a non-vanishing Pontryaginterm [98], which in Boyer-Lindquist coordinates ds = − ∆ − a sin ΘΣ dt − aM r sin ΘΣ dtdφ + ( r + a ) − a ∆ sin ΘΣ sin Θ dφ + Σ∆ dr + Σ d Θ (35)can be written as ∗ R R = 96 aM r Σ cos Θ (cid:0) r − a cos Θ (cid:1)(cid:0) r − a cos Θ (cid:1) , (36)with Σ = r + a cos Θ and ∆ = r + a − M r . In lightof the physical interpretations of Sec. II B, one wouldexpect this result since the Kerr spacetime possesses acomplex Newman-Penrose scalar Ψ .The Pontryagin constraint is satisfied in certain limits.For example, as the Kerr parameter goes to zero, a →
0, the Schwarzschild solution is recovered and the right-hand side of Eq. (36) vanishes. Similarly, in the limit asthe mass goes to zero, M →
0, the right-hand side ofEq. (36) also vanishes. However, for any finite a and M the Pontryagin term is non-vanishing and, thus, the Kerrspacetime cannot be a solution to the CS modified fieldequations [33].What line element then replaces the Kerr solution inthe modified theory? A reasonable attempt to constructa spinning BH in CS gravity is to consider generic ax-isymmetric and either static or stationary line elements,which we shall investigate in the next sections. IV. STATIC, AXISYMMETRIC SOLUTIONS
Before embarking on a tour de force through genericstationary and axisymmetric solutions, we shall first con-sider the simpler case of static and axisymmetric solu-tions. Following [64], the most general static and ax-isymmetric line element is diffeomorphic to ds = − V dt + V − ρ dφ + Ω (cid:0) dρ + Λ dz (cid:1) , (37)where we have three undetermined functions of two coor-dinates: V ( ρ, z ), Ω( ρ, z ) and Λ( ρ, z ). The two commutingKilling fields, ξ a = ( ∂ t ) a and ψ a = ( ∂ φ ) a , are associatedwith stationarity and axisymmetry respectively. How-ever, since there is no cross-term dtdφ , the line elementof Eq. (37) is not just stationary but also static. Thecomponents of its Ricci tensor are given by R tφ = R tρ = R tz = R φρ = R φz = 0 , (38) R tt = 12Ω " V ,ρρ + V ,zz Λ + V ,ρ ρ − V ,ρ V − V ,z V Λ+ V ,ρ Λ ,ρ − V ,z Λ ,z (cid:21) , (39) R φφ = 12Ω " ρ V ,ρρ V + ρ V ,zz V Λ + ρV ,ρ V − ρ V ,ρ V − ρ V ,z V Λ − ρ Λ ,ρ V Λ + ρ Λ ,ρ V ,ρ V Λ − ρ Λ ,z V ,z V (cid:21) , (40) R ρρ = V ,ρ ρV − V ,ρ V − Ω ,ρρ Ω − Ω ,zz ΛΩ + Ω ,ρ ρ Ω + Ω ,ρ Ω + Ω ,z ΛΩ − Λ ,ρρ
2Λ + Λ ,ρ − Ω ,ρ Λ ,ρ ,z Λ ,z Ω , (41) R zz = − V ,z V − ΛΩ ,ρρ Ω − Ω ,zz Ω − Ω ,ρ Λ ρ Ω + Ω ,ρ ΛΩ + Ω ,z Ω − Λ ,ρρ − Λ ,ρ ρ + Λ ,ρ − Ω ,ρ Λ ,ρ
2Ω + Ω ,z Λ ,z , (42) R ρz = − V ,z V ,ρ V + V ,z ρV + Ω ,z ρ Ω , (43)and exhibit only five non-vanishing components. Withthe canonical choice of the CS scalar field [Eq. (27)] it isnow straightforward to check that the five correspondingcomponents of the C-tensor vanish, C tt = C φφ = C ρρ = C zz = C ρz = 0 . (44)As in the spherically symmetric case, we are faced withthe remarkable consequence that the field equations[Eq. (16)] decouple into the vacuum Einstein equationsplus the vanishing of the C-tensor, viz. R ab = 0 , C ab = 0 . (45)In other words, using the classification of Sec. II C, allstatic and axisymmetric solutions live in P , which againconfirms previous expectations.With these considerations in mind, we can now sim-plify the line element of Eq. (37). From [64], the functionΛ can be chosen to be constant, e.g. Λ = 1, and thereforethe line-element reduces to the Weyl class, ds = − e U dt + e − U (cid:2) e k ( dρ + dz ) + ρ dφ (cid:3) . (46)The vacuum Einstein equations then simplify to∆ U = 0 , k ,ρ = ρ ( U ,ρ − U ,z ) , k ,z = 2 ρU ,ρ U ,z , (47)where ∆ = ∂ /∂ρ + 1 /ρ∂/∂ ρ + ∂ /∂z is the flat spaceLaplacian in cylindrical coordinates. The function U thussolves a Laplace equation, and for any such solution thefunction k can be determined by a line integral [55].The Pontryagin constraint is fulfilled automatically forall line elements diffeomorphic to Eq. (46), but as we have seen in the previous sections, this is not sufficientto achieve C ab = 0. For example, with the choices [65, 66]( m is constant) U = − m p ρ + z , k = − m ρ ρ + z ) , (48)the vacuum Einstein equations hold and the Pontrya-gin constraint is fulfilled, but the C-tensor has the non-vanishing components C ρφ = 2 m ρ zµ ( ρ + z ) exp " m ρ ( ρ + z ) , C zφ = 12 ρz C ρφ . (49)Since the C-tensor must vanish independently fromthe Einstein equations, once more we are faced withthree distinct possibilities, identical to those describedin Sec. II C. The first possibility (option 1 in Sec. II C) isto demand that the Weyl tensor vanishes, but since alsothe Ricci tensor vanishes, the spacetime would have tobe flat. The second possibility (option 2 in Sec. II C) isto demand that the covariant acceleration of θ vanishes, i.e. , ∇ a θ is a covariantly constant vector. However, aswe have mentioned already, a vacuum solution with a co-variantly constant vector field must be either flat, or thevector must be a null-vector. The first alternative is triv-ial, while the second one is not particularly interesting inthe context of static axisymmetric spacetimes. We shalldiscuss the latter possibility further in Sec. VI A.The only remaining possibility (option 3 in Sec. II C)is for the contraction of the covariant acceleration andthe dual Weyl tensor to vanish. The C-tensor can thenbe simplified to C ab ∝ Γ tρt ∗ C t ( ab ) ρ + Γ tzt ∗ C t ( ab ) z = 0 , (50)which has only two non-vanishing components. Usingthe Einstein equations to simplify these expressions weobtain a set of nonlinear PDEs, U ,ρz U ,ρ + U ,zz U ,z = 2 ρ (cid:0) U ,z U ,ρ + U ,ρ U ,z (cid:1) − ρ U ,ρ U ,z ,U ,ρz U ,z − U ,zz U ,ρ = ρ (cid:0) U ,ρ − U ,z (cid:1) − ρ U ,ρ . (51)We used Maple to obtain some solutions to these PDEs.The Schwarzschild solution U = 12 ln p ρ + ( z + M ) + p ρ + ( z − M ) − M p ρ + ( z + M ) + p ρ + ( z − M ) + 2 M (52)of course solves these PDEs. Some other simple so-lutions are U = U , U = U ± ln ρ and U = U +ln ( p ρ + z + z ) /
2, where U is a constant. Not onlydo these solution yield a vanishing Ricci tensor, but theyalso yield a vanishing Riemann tensor, which shows theyare Minkowski spacetime in disguise. In addition to thesetrivial solutions, there exist exactly two more: ds = − z dt + zdz + z ( dρ + ρ dφ ) (53)and ds = − (cid:18) mz − (cid:19) dt + (cid:18) mz − (cid:19) − dz + z ( dρ + sinh ρ dφ ) (54)While these solutions certainly are non-trivial, neitherthe first [Eq. (53)] nor the second [Eq. (54)] solution isphysically relevant. The former has a naked singularityat z = 0, while the latter, whose singularity at z = 0is screened by a Killing horizon at z = 2 m = const . ,possesses a Killing vector k a = ( ∂ t ) a that is spacelike inthe “outside” region z > m , i.e. g ab k a k b = 1 − m/z > i.e. g ij = Φ δ ij ,where the conformal factor Φ is a function of the coordi-nates and δ ij is the spatial part of the Minkowski metric.We may then exploit a result by Luk´acs and Perj´es [67]that the line elements of Eqs. (53), (54) and (26) are theonly static and axisymmetric solutions that are spatiallyconformally flat. Therefore, it follows that these equa-tions are the only solutions to the modified field equa-tions.The above considerations also apply to more generalCS scalar fields. All simplifications hinge on the decou-pling of the modified field equations, which occurs if andonly if θ ,tφ = 0. We can solve this PDE to obtain θ = θ ( t, ρ, z ) + θ ( ρ, z, φ ) . (55)For all scalar fields of this form, the modified field equa-tions decouple and the C-tensor has five non-vanishingcomponents, which define a system of PDEs for one ofthe two arbitrary functions k or U . However, we donot expect more solutions to arise in this way, since thiscase leads to the same constraints as the canonical one,plus three extra PDEs, which essentially compensate thefreedom to tinker with the two arbitrary functions inEq. (55).The most general CS scalar field, however, does notallow for a decoupling of the type described above. Ifthe scalar field has θ ,tφ = 0, then the ( ρ, ρ ), ( ρ, z ) and( z, z ) components of the modified field equations do notdecouple. However, the ( t, t ) and ( φ, φ ) components stilldo decouple because the corresponding C-tensor compo-nents vanish. The equation R tt + R φφ = −
12 Λ ,ρ Λ ρ Ω = 0 , (56)forces Λ to be a function of z alone. Through a diffeo-morphism, this function can be set to unity, as argued in[64].The modified field equations are too difficult to solveanalytically with Maple, so in order to study solutions that do not lead to a decoupling of the modified fieldequations, we shall assume for simplicity θ = ˜ θ ( t, φ ).From the Ricci sector of the field equations ( R tt = 0 = R φφ ) we find that U is again a solution of ∆ U = 0. Wecan use this relation to simplify the C-tensor, and the en-suing equations C tφ = C tρ = C tz = C φρ = C φz = 0 leadto a system of second order PDEs for θ and k . We inves-tigated this system with Maple and found that solutionsexist if and only if θ is a function of only one variable, i.e. θ = θ ( t ) or θ = θ ( φ ). These results indicate thatthere are no solutions of the modified field equations if θ is bivariate.In summary, we have shown in this section that thefield equations decouple if the CS scalar field solves θ ,tφ =0, and their solution is the Schwarzschild BH and two ad-ditional (unphysical) solutions [Eqs. (53) and (54)]. ForCS fields that satisfy θ ,tφ = 0, the modified field equa-tions do not seem to have a solution. Therefore, thereare no static and axisymmetric solutions in CS gravity,apart from the Schwarzschild BH and some unphysicalsolutions, irrespective of the CS scalar field. V. STATIONARY, AXISYMMETRICSOLUTIONSA. General line elements
Equipped with the tools from the previous section, wedrop the requirement of staticity and replace it by theweaker one of stationarity. In essence, this means thatwe shall allow the gravitomagnetic sector of the metricto be different from zero. The most general, stationaryand axisymmetric line-element is diffeomorphic to [55] ds = − V ( dt − wdφ ) + V − ρ dφ + Ω (cid:0) dρ + Λ dz (cid:1) , (57)where the functions V , w , Ω and Λ depend on ρ and z ,only. This line element is identical to Eq. (37) as w → w can be identified with the angularvelocity. The Ricci tensor for this line element is similar0to R static ab [Eqs. (38)-(43)] and its components are R tρ = R tz = R φρ = R φz = 0 , (58) R tt = R static tt + (cid:16) w ,ρ + w ,z Λ (cid:17) V ρ Ω , (59) R tφ = w h − V ,ρρ − V ,zz Λ − V ,ρ ρ + V ,ρ V + V ,z V Λ − V ,ρ w ,ρ w − V ,z w ,z w Λ − w ,ρρ Vw − w ,zz Vw Λ+ w ,ρ Vρw − w ,ρ V ρ − w ,z V ρ Λ − Λ ,ρ V ,ρ
2Λ + Λ ,z V ,z − Λ ,ρ w ,ρ V w Λ + Λ ,z w ,z V w Λ i , (60) R ρρ = R static ρρ + w ,ρ V ρ , (61) R zz = R static zz + w ,z V ρ , (62) R ρz = R static ρz + w ,ρ w ,z V ρ , (63) R = R static + (cid:16) w ,ρ + 1Λ w ,z (cid:17) V ρ Ω . (64)The somewhat lengthy component R φφ can be deducedfrom R and the other components. The quantity R static , R static = 12Ω h V ρ V Λ − V ρ V − V z V Λ − ρρ Ω − zz ΛΩ + 4 Ω ρ Ω + 4 Ω z ΛΩ − ρρ Λ − ρ ρ Λ + Λ ρ Λ − ρ Ω ρ ΛΩ + 2 Λ z Ω z Λ Ω i , (65)is the trace of Eqs. (38)-(43).As before, let us begin with the canonical choice forthe CS scalar field, namely Eq. (27). Then the only non-zero components of the C-tensor are C ρt , C zt , C ρφ and C zφ . As in the previous cases, there is a decoupling of thefield equations that allows us to set Λ = 1 and to considerthe slightly simpler line element (Lewis-Papapetrou-Weylmetric) ds = − e U ( dt − wdφ ) + e − U (cid:2) e k ( dρ + dz ) + ρ dφ (cid:3) , (66)where again the functions U , k and w depend on the co-ordinates ρ and z only. With this line element, the lastlines vanish in the multi-line expressions for the Ricci ten-sor, Eqs. (39)-(42), (60) and (65), because Λ = 1. Thevacuum Einstein equations simplify considerably withΛ = 1. Essentially, they are similar to Eq. (47) but witha complicated source and an additional equation for w .Even within GR, explicit solution to this set of PDEs canonly be found in certain special cases [55].The Pontryagin constraint for the line element ofEq. (66) is not satisfied in general. This constraint yieldsa complicated second order PDE for w , U and k , pre-sented in appendix B, which of course is trivially satisfied as w →
0. Certain solutions to the PDE in appendix Bcan be obtained, e.g. ( ¯ w := e U w ) k = k ( ρ, z ) ¯ w = c e U , U = U ( ρ, z ) , (67a) k = k ( ρ, z ) ¯ w = ± ρ , U = U ( ρ, z ) , (67b) k = ln( ρ ) + ˜ k ( z ) , ¯ w = ˜ w ( z ) ρ , U = 12 ln( ρ ) + c , (67c) k = k ( ρ ) , ¯ w = ¯ w ( ρ ) , U = U ( ρ ) , (67d)where c is a constant. The first line reduces to static so-lutions upon redefining t ′ = t − c φ . The second line leadsto metrics of Petrov type II , the so-called van Stockumclass, which we shall discuss in Sec. V B. The third line ofEq. (67) cannot be made to solve the modified field equa-tions. The last line implies cylindrical symmetry, whichagain via the field equations leads to flat spacetime. Wehave thus been unable to find non-trivial solutions eitherby hand or using symbolic manipulation software [99].Unlike the previous section, we cannot provide herea truly exhaustive discussion of all solutions of the de-coupled field equations. This is because C ab = 0 doesnot necessarily imply spatial conformal flatness for thestationary case. Based on the evidence found so far, itseems unlikely that there are other non-trivial and phys-ically interesting solutions besides the static ones. Thisis because the vacuum Einstein equations [ R ab = 0] al-ready determine the function k uniquely up to an inte-gration constant, and also impose strong restrictions onthe functions U and w [55]. The constraints C ab = 0 im-pose four additional conditions on these functions thatcan be found in [68]. Since the system of partial dif-ferential equations is over-constrained, it is unlikely thatadditional solutions exist. Therefore, whenever the fieldequations decouple into R ab = 0 = C ab we do not expectphysically relevant solutions besides the Schwarzschildone and its flat space limit.The decoupling exhibited by the modified field equa-tions does not occur only for the canonical choice of theCS scalar field. In order for such a decoupling to occur,the following system of PDEs must be satisfied: θ ,tt = θ ,φφ = θ ,tφ = θ ,ρ = θ ,z = 0 , (68)which yields the solution θ = tµ + φν , (69)with constant µ , ν . The canonical choice is recovered as ν → ∞ .But what if the scalar field is not of the form ofEq. (69)? In this case, the field equations do not decoupleand solving the entire system is much more complicated.However, we can deduce from Eq. (58) that still the fourC-tensor components C ρt , C zt , C ρφ and C zφ have to van-ish. Therefore, even though no decoupling occurs, thesame issue of an over-constrained system of equationsdoes arise, analogous to the one in Sec. V A. Even with1this generalization, it is still quite difficult to find solu-tions to the coupled system. In general, one might be ableto find solutions both of class P and class CS \ P becausenon-canonical CS fields allow for general θ , which entailsa new degree of freedom. We shall see in Sec. V B thatfor a simplified subclass of stationary and axisymmetricline elements, which automatically satisfy the Pontrya-gin constraint, solutions can indeed be found, includingmathematical BHs. B. Van Stockum line element
We study now a slightly less general line element thatstill is stationary and axisymmetric, namely the vanStockum line element [55] ds = ρ Ω dt − ρdtdφ + 1 √ ρ (cid:0) dρ + dz (cid:1) , (70)where the only arbitrary function is Ω = Ω( ρ, z ). Themetric is different from that considered in Eq. (66) sinceit does not possess a dφ component. The only non-vanishing component of the Ricci tensor for such a space-time is R tt = − ρ / , (71)where again ∆ is the flat space Laplacian in cylindricalcoordinates.The Pontryagin constraint is automatically satisfied forthe van Stockum line element even though it is of Petrovtype II , precisely because of the vanishing dφ term. The tt component of the modified equations then determinesΩ, and this forces all other components of the C-tensorto vanish, except for C tφ and C φφ that are automati-cally zero. These constraints act as a system of PDEsfor the scalar field, whose unique solution is θ = θ ( ρ, z ).Note that the canonical choice for θ is not compatiblewith the van Stockum line element. The remaining PDE R tt + C tt = 0 can be solved for θ and Ω, where C tt nowsimplifies to C tt = ρ h ( θ ,ρρ − θ ,zz ) (cid:0) Ω ,ρz + 34 ρ Ω ,z (cid:1) + θ ,ρ (cid:0) Ω ,zzz + Ω ,ρρz + 32 ρ Ω ,ρz + 38 ρ Ω ,z (cid:1) − θ ,z (cid:0) Ω ,ρρρ + Ω ,ρzz + 94 ρ Ω ,ρρ + 34 ρ Ω ,zz + 38 ρ Ω ,ρ (cid:1) − θ ,ρz (cid:0) Ω ,ρρ − Ω ,zz + 32 ρ Ω ,ρ (cid:1)i . (72)Combining this with R tt from Eq. (71) we find two simplesolutions of Eq. (16):Ω = c, θ = θ ( ρ, z ) , (73)where c is a constant andΩ = c + d √ ρ , θ = 23 √ ρ z + ˜ θ ( ρ ) , (74) where c and d are constants [100]. Equation (73) leadsto zero Ricci and C-tensor separately and it is thus a GRsolution that belongs to the subspace P . The ensuingmetric is exceptional in that it has a third Killing vector, t∂ t − φ∂ φ + ct∂ φ . Some of the non-vanishing Riemanntensor components for this geometry are R tρtρ = c ρ , R tρρφ = 18 ρ , R tφtφ = 14 √ ρ . (75)On the other hand, Eq. (74) is perhaps even more inter-esting since it is not Ricci-flat, but has one non-vanishingcomponent of the Ricci tensor, R tt = − d ρ = − C tt . (76)This solution is thus a non-GR solution and it belongs tothe subspace CS \ P . Some of the non-vanishing compo-nents of the Riemann tensor for this solution are R tρtρ = d + 2 c √ ρ ρ / , R tρρφ = 18 ρ , R tφtφ = 14 √ ρ . (77)Notice that such a solution can represent a mathematicalBH, provided Ω vanishes for some ρ , i.e. a Killing hori-zon emerges. We call these configurations “mathematicalBHs” because they are physically not very relevant: theKilling vector generating axial symmetry is light-like, asevident from (70), and the spacetime admits closed time-like curves which are not screened by a horizon [55]. For c = 1 and d = − m we recover (2).Let us then summarize the most important conclusionsof this section. We have investigated stationary and ax-isymmetric solutions to the modified field equations. Wefound that, for the canonical choice of θ , it is unlikely thatsolutions can be found that differ from Minkowski andSchwarzschild. Nonetheless, for non-canonical choices ofthis scalar, solutions must exist. This conclusion derivesfrom the investigation of a slightly less general stationaryand axisymmetric metric, namely that of van Stockum.For this line element we found a solution to the modifiedfield equations that lives in P and a family of solutionsthat live in CS \ P , both with non-canonical CS scalarfields. To our knowledge, this is the first time an exactnon-GR solution is found for CS modified gravity, whichin particular can represent mathematical BH configura-tions.
VI. BEYOND THE CANON
We have failed in finding an exact, stationary and ax-isymmetric solution to the CS modified field equationsrepresenting a physical spinning BH. A solution, how-ever, already exists for a similar line element, albeit in aperturbative sense. In [19] and later in [21], a far-field so-lution to the CS modified field equations with a canonicalCS scalar field was found in the weak-field/slow-motionapproximation. This solution is identical to the far-field2expansion of the Kerr solution, except for the additionof two new components in the gravitomagnetic sector ofthe metric g i . These components vanish in GR, sinceonly one component is required and it is aligned with theangular momentum of the spinning source. In CS grav-ity, the remaining components of g i are proportional tothe curl of the spin angular momentum, thus breakingaxisymmetry but preserving stationarity. Such a sta-tionary, but non-axisymmetric BH will not emit grav-itational waves, but it will possess a non-trivial multi-polar structure, with probably more than just two non-vanishing multipoles. Such a far-field structure suggeststhat perhaps the only way to obtain an analog to theKerr solution in CS gravity is to relax either the assump-tion of axisymmetry or stationarity. Alternatively, thevan Stockum example suggests that another possibilityis to allow for a general CS scalar field. In this case,however, the line element must significantly differ fromthe Kerr metric such that it satisfies the Pontryagin con-straint [101]. We shall explore these possibilities in thissection. A. Killing embedding
We study now the possibility that the ’embedding co-ordinate’, i.e. , the velocity of the CS scalar field θ , is aKilling vector. Then, v a is covariantly conserved becauseof the Killing equation ( ∇ ( a v b ) = 0) and the fact thatthe connection is torsion-free ( ∇ [ a v b ] = ∇ [ a ∇ b ] θ = 0).This puts a strong restriction on spacetime, which for atime-like v a yields line elements that are diffeomorphicto ds = − dt + g ij ( x k ) dx i dx j , (78)where i and j range over all coordinates except time. Ac-tually, Eq. (78) describes a special class of static space-times. When studying static solutions to the CS modifiedfield equations with timelike v a in Sec. IV, we found nophysically relevant solution besides Schwarzschild. Thesame conclusions hold here, except that we do not evenrecover Schwarzschild, so this route is not a promisingone. A similar discussion applies to spacelike Killing vec-tors.A more interesting situation arises if the vector field v a is a null Killing vector, v a v a = ∇ ( a v b ) = 0. In this case,we get in an adapted coordinate system the line element ds = − dvdx + g ij ( x k ) dx i dx j (79)Once again, the Pontryagin constraint is immediatelysatisfied, the Ricci tensor has non-vanishing R ij com-ponents, but no components of the C-tensor vanish.Even when we pick a simple null Killing embedding,e.g. v a = (0 , χ, ,
0) with χ = const . , the C-tensor hascomplicated spatial non-vanishing components and themodified field equations are too difficult to solve in fullgenerality. Therefore, we focus instead on an interestingspecial case in the next subsection. B. pp-waves and boosted black holes
As suggested at the end of Sec. II B, it might be pos-sible to find solutions to the modified field equations ifone considers line elements that represent exact gravita-tional wave solutions (pp-waves [69]). The line elementfor these waves is ds = − dvdu − H ( u, x, y ) du + dx + dy , (80)which is simply a special case of the line elements con-sidered in the previous subsection [Eq. (79)]. Particularexamples of physical scenarios that are well-representedby Eq. (80) are the Aichelburg-Sexl limits [70] of var-ious BHs. In essence, this limit is an ultrarelativisticboost that keeps the energy of the BH finite by takinga limit where its mass vanishes while the boost velocityapproaches the speed of light. In particular, Eq. (80) canbe used to represent ultrarelativistic boosts of the KerrBH [71, 72].Is it conceivable that a Kerr BH that moves ultra-relativistically solves the modified field equations, eventhough the Kerr BH does not? One of the main prob-lems with the Kerr metric is that it does not satisfy thePontryagin constraint, cf. Eq. (36), but that constraintis trivially satisfied as M →
0. Nonetheless, the satis-faction of the Pontryagin constraint is only a necessarycondition, but not a sufficient one, to guarantee that themodified field equations are also satisfied.In order to study this issue, let us find the appropri-ate expressions for the Ricci and C-tensors. The onlynon-vanishing component of the Ricci tensor for the lineelement of Eq. (80) is given by R uu = ∆ H , ∆ := ∂ ∂x + ∂ ∂y . (81)In general, the components C ux , C uy , C xx , C yy , C xy areall non-vanishing and form a system of PDEs for H and θ . The C xx , C yy and C xy components are given by C yy = − C xx = θ ,vv H ,xy , C xy = 12 θ ,vv ( H ,xx − H ,yy ) . (82)Let us first look for GR-solutions of class P , such that R ab = 0 and C ab = 0 independently. Since C ab = 0, thereare two possibilities here: either θ ,vv = 0 or H ,xy = 0 = H ,xx − H ,yy . In the latter case, H is constrained to H = 12 (cid:0) x + y (cid:1) A ( u ) + x B ( u ) + y C ( u ) + D ( u ) , (83)which also forces C ux and C uy to vanish. The onlycomponent of the field equations left is ( u, u ), whichupon simplification with Eq. (83) yields C uu = 0 and R uu = 2 A ( u ), so that A ( u ) = 0. We have then found thesolution H = x B ( u ) + y C ( u ) + D ( u ) , θ = θ ( u, v, x, y ) , (84)3to the modified field equations. However, this solution isnothing but flat space in disguise.Another possibility to find GR-solutions is to pick θ such that C xx , C yy and C xy vanish, i.e. θ ,vv = 0. Thiscondition leads to θ = λ ( u, x, y ) v + ˜ θ ( u, x, y ) , (85)The remaining non-( u, u ) components of the C-tensorlead to C ux = 0 → λ ,x H ,xy = λ ,y H ,yy (86) C uy = 0 → λ ,x H ,yy = λ ,y H ,xx (87)where we have used R uu = 0. The solution to this systemof PDEs leads either to flat spacetime or to λ ( u, x, y ) = λ ( u ) . (88)Choosing Eq. (88), the remaining modified field equation[the ( u, u ) component] becomes∆ H = 0 , (89)2 H ,yy ˜ θ ,xy = H ,xy (˜ θ ,yy − ˜ θ ,xx ) . (90)For some H that solves the Einstein equations [ i.e. theLaplace equation in Eq. (89)], the C-tensor yields a PDEfor ˜ θ [Eq. (90)]. Thus we conclude that we can lift anypp-wave solution of the vacuum Einstein equations to app-wave solution of CS modified gravity (of class P ) bychoosing θ such that Eqs. (85), (88)-(90) hold.Let us give an example of this method to generate CSsolutions by studying ultrarelativistically boosted KerrBHs, for which H = h δ ( u ) ln (cid:0) x + y (cid:1) (91)satisfies Eq. (89). In Eq. (91), δ ( u ) is the Dirac deltafunction and h is a constant. Inserting this H intoEq. (90) we find˜ θ = xα (cid:16) yx (cid:17) + β (cid:0) x + y (cid:1) , (92)where α and β are arbitrary functions of their respectivearguments ( y/x ) and ( x + y ). Equation (92), togetherwith Eq. (85) and (88), give the full solution for the CSscalar field. We have therefore lifted the boosted Kerr BHto a solution of the modified field equations of class P bychoosing the CS scalar field appropriately. For θ = λv we recover Eq. (3).Let us now search for non-GR solutions to the modi-fied field equations. Since all equations decouple exceptfor the ( u, u ) component, we must enforce that the non( u, u )-components of the C-tensor vanish, i.e. θ ,v = 0,which leads to θ = ˜ θ ( u, x, y ) . (93) With Eq. (93), the only component of the modified fieldequations left is again the ( u, u ) one, which simplifies toa linear third order PDE:(1 + ˜ θ ,y ∂ ,x − ˜ θ ,x ∂ ,y )∆ H == (˜ θ ,xx − ˜ θ ,yy ) H ,xy − ( H ,xx − H ,yy )˜ θ ,xy . (94)For simplicity, we choose˜ θ = a ( u ) x + b ( u ) y + c ( u ) , (95)and Eq. (94) reduces to the Poisson equation∆ H = f . (96)The source term f solves a linear first order PDE bf ,x − af ,y − f = 0 , (97)whose general solution [assuming b ( u ) = 0] f ( u, x, y ) = e x/b ( u ) φ [ a ( u ) x + b ( u ) y ] (98)contains one arbitrary function φ of the argument a ( u ) x + b ( u ) y . We shall assume this function to be non-vanishingso that R ab = 0. We can now insert Eq. (98) into thePoisson equation and solve for H ( x, y, u ). We need twoboundary conditions to determine H from the Poissonequation [Eq. (96)] and another one to determine thearbitrary function φ in Eq. (98). Let us then providean example by assuming that b ( u ) < φ remainsbounded. In this case, we must restrict the range of thecoordinates to the half-plane, 0 ≤ x < ∞ , −∞ < y < ∞ .We impose a boundary condition H ( u, y ) := H ( u, , y )and appropriate fall-off behavior for | y | → ∞ . We thenobtain the particular solution H ( u, x, y ) = 1 π ∞ Z −∞ xH ( u, η ) dηx + ( y − η ) − π ∞ Z ∞ Z −∞ e − ξ/ | b ( u ) | φ [ a ( u ) ξ + b ( u ) η ] × ln (cid:20) ( x + ξ ) + ( y − η ) ( x − ξ ) + ( y − η ) (cid:21) dξdη (99)where the double integral extends over the half-plane.The exponential behavior in Eq. (98) is particularly in-teresting, since it resembles the gravitational wave solu-tions found in Refs. [8, 17, 18, 20]. Moreover, as x → ±∞ [depending on the sign of b ( u )] the source term in Eq. (96)diverges, indicating a possible instability. Since we weremainly concerned with the existence of solutions we havenot attempted to construct solutions for more general θ than Eq. (95). C. Losing a Killing Vector
From the analysis so far, it is clear that stationaryand axisymmetric solutions in CS gravity do not seem4to be capable of describing physical spinning BHs. Thefar-field solution has guided us in the direction of loss ofaxisymmetry, which in essence corresponds to losing the( ∂ φ ) a Killing vector. Analogously we could conceive oflosing stationarity instead of axisymmetric by droppingthe ( ∂ t ) a Killing vector. The general idea is then that bylosing one Killing vector we gain new undetermined met-ric components that could allow for a physical spinningBH solution in CS gravity. However, our attempts havenot revealed any interesting exact solution correspondingto a spinning BH, so we confine ourselves to a couple ofgeneral remarks.Spinning BHs that break axisymmetry or stationar-ity would be radically different from those considered inGR. On the one hand, non-axisymmetric spinning objectswould have an intrinsic precession rate that would not al-low the identification of an axis of rotation. Such preces-sion would possibly also lead to solutions with more thantwo non-zero multipole moments, thus violating the no-hair theorem. On the other hand, non-stationary spin-ning objects would unavoidably lead to the emission ofgravitational radiation, even if the BH is isolated. Theseconsiderations could be flipped if we take them as predic-tions of the theory, thus leading to new possible tests ofCS gravity. Work along these lines is currently underway.The results of [73] for the Pontryagin constraint may behelpful here.
D. Adding matter
The inclusion of matter sources is of relevance in thepresent context for several reasons. First, the Kerr BHhas a distributional energy momentum tensor [74], so weneed not set the stress-energy tensor strictly to zero toconstruct a Kerr-like solution. Second, in Ref. [16] thePontryagin-term in the action arises from matter cur-rents, so the inclusion of the latter would actually bemandatory within that framework.Two conceptually different approaches are possible tothe problem of finding exact solutions of the modifiedfield equations in the presence of matter. These ap-proaches essentially depend on whether we require theenergy-momentum tensor to be covariantly conserved, ∇ a T ab = 0, or not. If this tensor is conserved, thenthe Pontryagin constraint must be satisfied and the KerrBH cannot be a solution. Basically, this route leads toonly a slight generalization of the discussion presented sofar, with solutions of class P , R ab − g ab R = 8 πT ab , C ab = 0 , (100)and solutions of class CS \ P that solve Eq. (15). Relax-ing covariant conservation of the stress-energy tensor, wecan promote the Kerr BH to a solution of the modifiedfield equations, provided that R ab − g ab R = 8 πT dist ab , C ab = 8 πT ind ab . (101) Here T dist ab = 0 except for the usual distributional con-tributions for Kerr [74], while T ind ab provides the non-conserved matter flux. The induced matter fluxes forthe Kerr BH are given by T ind tr = am πµ Σ ∆ cos Θ (cid:0) r − a cos Θ (cid:1)(cid:2) a cos Θ (cid:0) r − a (cid:1) + r (cid:0) r − a (cid:1)(cid:3) , (102) T ind t Θ = − am r πµ Σ sin Θ (cid:2) r − r a cos Θ + a cos Θ (cid:3) , (103) T ind φr = − a m πµ Σ ∆ sin Θ cos Θ (cid:2) cos Θ a (cid:0) a − r (cid:1) + cos Θ r a (cid:0) a + 12 r (cid:1) − r a − r (cid:3) , (104) T ind φ Θ = − a sin Θ T ind t Θ . (105)Of course, with such a method any GR solution can bepromoted to a solution of the modified field equations.The crucial issue here is whether or not the inducedmatter flux can be regarded as physically acceptable.In order to shed light on this issue, we analyzed if theinduced stress-energy given by Eqs. (102)-(105) obeysthe energy conditions of GR [75]. Because T ind ab is al-ways traceless, the strong and weak energy conditions areequivalent and reduce to the statement that T ind ab ξ a ξ b ≥ ξ a . This, however, is not thecase, as we can show by considering for instance ξ t = √ ξ r = 1, which is timelike for sufficiently large r : ξ a ξ b g ab = − m/r + O ( m/r ) . The only relevantcomponent of T ind ab is given by Eq. (102), but since T ind tr is proportional to cos Θ, this quantity is negative in halfof the spacetime, and thus the weak energy condition isviolated. While this might be tolerated close to the hori-zon, we stress that this violation arises also in the asymp-totic region. This violation is somewhat attenuated bythe fall-off behavior of T ind ab , where its components decayat least as 1 /r and the scalar invariant T ind ab T ab ind as1 /r as r → ∞ . Thus, if ordinary matter is added thenthe induced exotic fluxes might not be detectable afterall for a far-field observer.There is another approach capable of circumventingthe Pontryagin constraint that also relies on new mattersources. Namely, if the field θ is considered a dynamicalfield, instead of an external field, it is natural to studymore general actions than Eq. (4) with S mat = 0, suchas [21] S = κ Z d x √− g (cid:18) R + 12 ( ∇ θ ) − V ( θ ) − α θ ∗ R R (cid:19) . (106)Then the Pontryagin constraint (18) is replaced by ∗ R R = − α [ (cid:3) θ + V ′ ( θ )] , (107)where α is a constant. This provides a natural general-ization of the model considered in our paper. However,it also introduces an amount of arbitrariness, since V is a5free function and, in fact, more general couplings between θ and curvature might be considered.We conclude that allowing GR solutions to be also CSsolutions by inducing a stress-energy tensor via Eq. (101)can lead to unphysical energy distributions. In particu-lar, the Kerr solution induces an energy momentum ten-sor given by Eqs. (102)-(105), which violates all energyconditions, even in the asymptotic region. The alterna-tive approach described above lifts θ to a genuine dynami-cal field with a kinetic term and possibly self-interactions,at the cost of introducing an arbitrary potential. VII. CONCLUSIONS AND DISCUSSION
No exact solution has yet been found that could pos-sibly represent a spinning BH in CS modified gravity.In particular, the Kerr solution is found to be incom-patible with the constraints imposed by the modifiedfield equations. Previously, only perturbative solutionsof CS gravity had been considered, which might repre-sent the exterior of a BH. The first study was carriedout by Alexander and Yunes [18, 19], who performeda weak-field parameterized post-Newtonian analysis tofind a non-axisymmetric Kerr-like solution. This studywas later extended by Smith, et. al. [21] to non-point likesources, finding that the Israel junction conditions areeffectively modified by the C-tensor. Another study wascarried out by Konno, et. al. [33], but this analysis wasrestricted to a limited class of perturbations that did notallow for the breakage of stationarity or axisymmetry.Within that restricted perturbative framework, a Kerr-like solution was found only for non-canonical choices of θ , concluding that BHs cannot rotate in the modified the-ory for canonical θ . This conclusion of Konno et. al. is atodds with both the results of Alexander and Yunes andSmith et. al.In order to address these issues, in the current pa-per we attempted to determine what replaces the Kerrsolution in CS modified gravity. We thus studied ex-act solutions of the modified theory, comprising spher-ically symmetric, static-axisymmetric, and stationary-axisymmetric vacuum configurations, as well as somegeneralizations thereof.We began our analysis in Sec. II A by considering theCS action in detail and rederiving the equations of mo-tion, together with the resultant surface integral terms.We continued in Sec. II B by rederiving the Pontrya-gin constraint from the equations of motion and provid-ing two alternative interpretations of it. One of them[Eq. (21)] is a reality condition on a quadratic curvatureinvariant of the Weyl tensor, while the other [Eq. (24)]is a null condition on the contraction of the electric andmagnetic parts of the Weyl tensor. Before consideringspecific line elements, in Sec. II C we classified all solu-tions into two groups: GR-type (class P ), which containssolutions of both the vacuum Einstein equations and themodified field equations; non-GR type (class CS \ P ), which contains solutions of CS gravity that are not solu-tions of the vacuum Einstein equations (cf. Fig. 1).After these general considerations, we began a system-atic study of line elements, starting with general spher-ically symmetric metrics in Sec. III. This class of lineelements [Eq. (33)] is particularly important since it con-tains the Schwarzschild, Friedmann-Robertson-Walkerand Reissner-Nordstr¨om solutions. We showed that, forthe canonical choice of the CS scalar field [Eq. (27)] andmore general choices [Eq. (34)], the modified field equa-tions decouple and any possible solution is forced to beof class P .We continued in Sec. IV with an analysis of static andaxisymmetric metrics [Eq. (37)]. We showed that, for thecanonical choice of the CS scalar field and more generalchoices [Eq. (55)], the modified equations decouple oncemore. We also showed that any static and axisymmet-ric line element is forced to become spatially conformallyflat, provided the field equations decouple. Exploitingthis result, we found three different solutions [Eqs. (26),(53) and (54)], only one of which was physically relevant,namely the Schwarzschild solution. For the most generalCS scalar field, however, the field equations do not decou-ple, but we have shown that fields with such generalitydo not seem to allow for a solution to the field equationsapart from trivial ones. Thus, we may conclude that CSgravity does not allow for static and axisymmetric solu-tions, apart from flat space, the Schwarzschild solutionand two additional (unphysical) solutions, irrespective ofthe choice of the CS scalar field.Static line elements then gave way to the central pointof this paper: stationary and axisymmetric solutions ofCS gravity, discussed in Sec. V. As in the previous cases,we showed that, for the canonical choice of the CS scalarfield and slightly more general choices [Eq. (69)], the fieldequations again decouple. In this case, however, the Pon-tryagin constraint does not hold automatically and weused it to constrain the class of possible metric func-tions, cf. appendix B. In essence, the decoupling requiresnot only that solutions must obey the Einstein equa-tions, but also the fulfillment of additional constraints(cf. Ref. [68]), which leads to an overdetermined systemof PDEs. Therefore, we concluded that non-trivial sta-tionary and axisymmetric solutions do not seem to existfor canonical CS fields.When a completely generic CS scalar field is consid-ered, the modified field equations do not decouple andsolutions are not easy to find, even with the simplifica-tions derived from the Pontryagin constraint. However,generic CS fields increase the degrees of freedom of theproblem and thus might allow for stationary and axisym-metric solutions. We proved this statement by providingan example in Sec. V B, through a sub-class of station-ary and axisymmetric metrics [Eq. (70)], belonging tothe van Stockum class. In that case, we showed that theonly possible CS field compatible with the field equationsexcludes the canonical choice. Moreover, we found both,non-flat solutions of class P [Eq. (73)] as well as non-flat6solutions of class CS \ P [Eq. (74)] To the best of ourknowledge, this is the first time an exact solution in CSmodified gravity is constructed that is not also a solutionof GR. One of these solutions [Eq. (74)] represents math-ematical BHs, in the sense that although they exhibita Killing horizon, they are not physically relevant, be-cause the Killing vector generating the ’axial’ symmetryis light-like and closed timelike curves arise. We con-cluded that it is unlikely that stationary, axisymmetricsolutions exist that represent a spinning physical BH.Finally, in Sec. VI we considered the possibility of con-structing solutions beyond the set of stationary and ax-isymmetric spacetimes. We began in Sec. VI A by con-sidering CS scalar fields whose velocity is a Killing vectorof the spacetime and found that the only interesting casearises if that vector is null. Naturally, such consider-ations led to exact gravitational shock-wave spacetimes[Eq. (80)]. Within this pp-wave scenario, in Sec. VI Bwe constructed a generating method through which anypp-wave solution of GR can be lifted to a solution of CSmodified gravity with an appropriate choice of the CSscalar field. We also built a solution of class
CS \ P [Eq. (99)] that is not a GR pp-wave solution but doessatisfy the CS modified field equations.Through this detailed study of solutions in CS gravitywe have ascertained that at least two different limits ofthe Kerr BH are solutions to the modified field equations,even though the Kerr BH is not: the Schwarzschild limitand the Aichelburg-Sexl limit. The former was alreadyknown to be a solution to the CS modified field equa-tions, but the latter, which includes ultrarelativisticallyboosted BHs, was not. The existence of these solutionsconcurs with the naive expectations expressed at the endof Sec. II B. Moreover, such expectations, together withthe non-axisymmetric far-field solution, point to the ex-istence of a physical spinning BH solution in CS gravity,provided spacetimes with only one Killing vector are con-sidered. We addressed this possibility briefly in Sec. VI C,but unfortunately such spacetimes are so general that themodified field equations become prohibitively difficult,even with the use of symbolic manipulation software.Other possibilities of bypassing the Pontryagin con-straint were discussed in Sec. VI D, since this constraintis in essence responsible for the absence of interestingstationary and axisymmetric solutions. First, we statedthat obviously any (GR or non-GR) solution formallycan be lifted to a solution of the modified field equationsby allowing for arbitrary matter sources, and we demon-strated the nature of these matter sources for the KerrBH. We found that the induced energy momentum ten-sor [Eqs. (102)-(105)] is exotic even in the asymptoticregion, but drops off rapidly with the radial coordinate.Second, we mentioned the possibility that the CS scalarfield θ might acquire a kinetic term and self-interactions.In this case, the Pontryagin constraint ceases to hold andis replaced by a dynamical condition [Eq. (107)], relatingthe gravitational instanton density to the (generalized)Klein-Gordon operator acting on θ . We now conclude with a list of possible directions forfuture research to which our current work may providethe basis. • The number of physical degrees of freedom in CSmodified gravity is not known yet. Various consid-erations appear to lead to contradictory expecta-tions. On the one hand, the appearance of higherorder derivatives in the action [Eq. (1)] suggeststhat additional degrees of freedom should emerge.On the other hand, the appearance of an additionalconstraint [Eq. (67)] suggests that fewer degreesof freedom should arise. Actually, the lineariza-tion procedure suggests that these competing ef-fects cancel each other and that there are two po-larizations of gravitons, just like in GR, albeit withproperties that differ from GR [8, 20]. • The role of boundary terms induced in CS grav-ity for BH thermodynamics could be investigatedmore thoroughly [76]. Also here general consid-erations lead to contradictory expectations. Onthe one hand, new boundary terms that arise inCS gravity [Eq. (14)] differ qualitatively from thosethat arise in GR or in scalar tensor theories. Suchboundary terms suggest modifications of BH ther-modynamics, even for solutions whose line elementscoincide with GR solutions, like the Schwarzschildspacetime. On the other hand, the Pontryagin con-straint eliminates the CS contribution [Eq. (1)] tothe on-shell action, which suggests that BH thermo-dynamics is left unchanged, at least in the classicalapproximation. • Both previous issues can be addressed by a thor-ough Hamiltonian analysis, which is also of interestby itself and for exhibiting the canonical structureas well as the classical constraint algebra. Such astudy would also be useful for numerical evolutionsof BH binary spacetimes in CS gravity, which iscurrently being carried out [77]. • While our discussion of stationary and axisymmet-ric solutions was quite comprehensive, a few issuesare still open, which may be an interesting topicfor mathematical relativists. For instance, whilewe were able to provide a proof that there are onlythree types of solutions for static and axisymmet-ric spacetimes (with the canonical choice for the CSscalar field), we could only provide good evidence,but no mathematical proof, that no further solu-tions exist for spacetimes that are stationary andaxisymmetric. • Combining the evidence found in this paper withthe far field solutions found previously, we con-cluded that spinning BHs should break either sta-tionarity or axisymmetry (or both) in CS modi-fied gravity. Perturbations away from axisymmetrywere neglected in [33], although non-axisymmetric7solutions can still represent spinning BHs, albeitwith an inherent precession induced by the CSmodification. Therefore, future work could focuson finding exact spacetimes with a smaller amountof symmetries. • A manageable implementation of the Pontryaginconstraint could be useful in many CS gravity ap-plications. The brute force methods that led us tothe formulas in appendix B will render any general-ization unintelligible. The considerations presentedin Ref. [73] provide such an implementation, but ithas not been exploited so far in the construction ofexplicit solutions. • Far-field solutions of CS gravity that break sta-tionarity could also be studied. These solutionscould then be used as tests of the modified theory,through comparisons with gravitational-wave andastrophysical observations [78]. • Perhaps it is feasible to apply the method ofmatched asymptotic expansion for caged BHs [79,80] to the construction of spinning BH solutions inthe present context. To this end, one would needan asymptotic expansion and a near horizon ex-pansion of that BH. The former exists already, soit remains to construct the latter and perform theasymptotic matching. • Finally, it is worthwhile to consider not just vacuumsolutions, but also solutions with matter sources, asoutlined briefly in Sec. VI D.Certainly the range of issues that can be addressed hasbeen extended in a non-negligible way. Only through abetter understanding of the consequences and predictionsof CS gravity will we be able to determine the viabilityof the modified theory.
Acknowledgments
We are grateful to Stephon Alexander and RomanJackiw for encouraging us to study this problem in thefirst place and for enlightening discussions. We wouldalso like to thank Abhay Ashtekar, Henriette Elvang,Alexander Hariton, Scott Hughes, Ralf Lehnert, BenOwen, Richard O’Shaughnessy, Carlos Sopuerta, MaxTegmark and Richard Woodard for discussions and com-ments. Most of our calculations used the computer al-gebra systems MAPLE v.11 in combination with theGRTensorII package [51].DG is supported in part by funds provided by theU.S. Department of Energy (DoE) under the coopera-tive research agreement DEFG02-05ER41360. DG hasbeen supported by the project MC-OIF 021421 of theEuropean Commission under the Sixth EU FrameworkProgramme for Research and Technological Development(FP6). NY acknowledges the support of the Center for Grav-itational Wave Physics funded by the National Sci-ence Foundation under Cooperative Agreement PHY-01-14375 and support from NSF grant PHY-05-55-628.
APPENDIX A: PROOF OF ∗ R R = ∗ C C
The equality ∗ R R = ∗ C C (A1)relates the Pontryagin term expressed as in Eq. (5) tothe Weyl tensor C abcd := R abcd − δ [ a [ c R b ] d ] + 13 δ a [ c δ bd ] R (A2)and its dual ∗ C abcd := 12 ǫ cdef C abef . (A3)Equation (A1) is quite simple to prove, but not entirelyobvious. Indeed, we were not able to find it in any ofthe standard textbooks, review articles or papers on CSmodified gravity. Therefore, we provide here a proof bystraightforward calculation. Proof.
Let us begin by inserting the definitions (5), (A2)and (A3) into (A1), ∗ R R = ∗ R abcd R bacd = ∗ C abcd C bacd + ∆ . (A4)where ∆ is precisely the violation of Eq. (A1). Thus, ifwe can show that ∆ vanishes in Eq. (A4) we have provenEq. (A1). The quantity ∆ contains eight terms. Four ofthem are linear in the Weyl tensor. Two of these termsare proportional to C cdef and two are proportional to C cdeb . Since ǫ cdef C cdef = ǫ cdef C cdeb = 0 , (A5)these terms vanish. Each of the remaining four termscontains at least two Kronecker δ . These terms alwayslead to a contraction of the Levi-Civita tensor, e.g. of theform ǫ cdcf = 0. Therefore, also these four terms vanishand establish ∆ = 0 . (A6)8 APPENDIX B: PONTRYAGIN CONSTRAINT
For the line element Eq. (66) the Pontryagin constraint Eq. (18) is given by ( ¯ w := e − U w )0 = A ¯ w + A ¯ w + A ¯ w ,ρ + A ¯ w ¯ w ,ρ + A ¯ w ,ρ + A ¯ w ,z + A ¯ w ¯ w ,z + A ¯ w ,z + A ¯ w ,ρ ¯ w ,z + A ¯ w ,ρz + A ( ¯ w ,ρρ − ¯ w ,zz ) (B1)with A = − ρ A + 2 ρ (cid:0) U ,ρz k ,ρ − ( U ,ρρ − U ,zz ) k ,z − U ,ρρ U ,z + 2 U ,ρz U ,ρ + 4 U ,z ( U ,ρ + U ,z ) + U ,z ( k ,ρρ + k ,zz )+ 8 U ,ρ U ,z k ,ρ − U ,ρ k ,z + 2 U ,z k ,z − U ,z ( k ,ρ + k ,z ) (cid:1) + 2 ρ (cid:0) U ,ρ k ,z − U ,z k ,ρ + 2 U ,z U ,ρ (cid:1) (B2) A = 8 ρ (cid:0) U ,ρ U ,z ( U zz − U ρρ ) + U ,ρz ( U ,ρ − U ,z ) − U ,ρ k ,z + U ,z k ,ρ + U ,ρ U ,z ( U ,ρ k ,ρ − U ,z k ,z ) (cid:1) + 8 U ,z (cid:0) U ,ρ + U ,z (cid:1) (B3) A = 2 ρ (cid:0) U ,zz U ,z + 2 U ,ρz U ,ρ + 4 U ,ρ U ,z + 4 U ,z + ( U ,ρρ − U ,zz ) k ,z − U ,z ( k ,ρρ + k ,zz ) − U ,ρz k ,ρ − U ,ρ U ,z k ,ρ + 2 U ,z k ,ρ + 2 U ,z k ,z − U ,z k ,z (cid:1) − ρ (cid:0) U ,ρz + 4 U ,z U ,ρ − U ,ρ k ,z (cid:1) − ρ k ,z (B4) A = 4 ρ (cid:0) U ,ρ k ,z + U ,z k ,z − U ,ρ U ,z k ,ρ − U ,ρ U ,z + ( U ,ρρ − U ,zz ) U ,z − U ,ρ U ,ρz − U ,z (cid:1) − U ,ρ U ,z (B5) A = 2 ¯ w (cid:0) ρ U ,z k ,ρ − ρ U ,ρ k ,z + U ,z + 6 ρ U ,ρ U ,z + ρ U ,ρz (cid:1) − ¯ w ,ρ ρ (cid:0) U ,z − k ,z (cid:1) (B6) A = 2 ρ (cid:0) − U ,ρρ U ,ρ − U ,ρz U ,z − U ,z U ,ρ − U ,ρ + ( U ,ρρ − U ,zz ) k ,ρ + U ,ρ ( k ,zz + k ,ρρ ) + 2 U ,ρz k ,z + 4 U ,z U ,ρ k ,z − U ,ρ ( k ,ρ + k ,z ) + 4 U ,ρ k ,ρ (cid:1) + 2 ρ (cid:0) U ,ρρ + 4 U ,ρ − U ,ρ k ,ρ (cid:1) + ρ (cid:0) k ,ρ + 2 k ,z − k ,ρρ − k ,zz (cid:1) − ρ (2 U ,ρ − k ,ρ ) (B7) A = 4 ρ U ,ρ ( U ,ρρ − U ,zz ) + 8 ρ U ,z U ,ρz − U ,ρ − U ,z + 4 ρ (cid:0) U ,ρ ( U ,ρ + U ,z ) − U ,ρ k ,ρ + 2 U ,z U ,ρ k ,z − U ,z k ,ρ (cid:1) (B8) A = ¯ w (cid:0) U ,z − ρ U ,ρz − ρ U ,z U ,ρ − ρ U ,ρ k ,z + 6 ρ U ,z k ,ρ (cid:1) + ¯ w ,z ρ (cid:0) U ,ρ − k ,ρ (cid:1) − ¯ w ,z (B9) A = 2 ¯ w ρ (cid:0) U ,zz − U ,ρρ + 6 U ,z − U ,ρ − U ,z k ,z + 2 U ,ρ k ,ρ (cid:1) + 4 ¯ wU ,ρ + ¯ w ,ρ ρ (cid:0) U ,ρ − k ,ρ (cid:1) − ¯ w ,ρ − ¯ w ,z ρ (cid:0) U ,z − k ,z (cid:1) (B10) A = 2 ρ (cid:0) U ,zz − U ,ρρ + 2 U ,z − U ,ρ + 2 U ,ρ k ,ρ − U ,z k ,z (cid:1) − ρ k ,ρ − w ρ (cid:0) U ,ρ − U ,z (cid:1) + 4 ¯ w ρ (cid:0) ¯ w ,ρ U ,ρ − ¯ w ,z U ,z (cid:1) − ( ¯ w ,ρ − ¯ w ,z ) ρ (B11) A = 2 ρ (cid:0) U ,ρz + 2 U ,ρ U ,z − U ,ρ k ,z − U ,z k ,ρ (cid:1) + ρ k ,z + 4 ¯ w ρ U ,ρ U ,z − w ¯ w ,ρ ρ U ,z − w ¯ w ,z ρ U ,ρ + ¯ w ,ρ ¯ w ,z ρ (B12) [1] C. M. Will, Living Rev. Relativity , 3 (2005), gr-qc/0510072.[2] A. G. Riess et al. (Supernova Search Team), Astron. J. , 1009 (1998), astro-ph/9805201.[3] S. Perlmutter et al. (Supernova Cosmology Project), As-trophys. J. , 565 (1999), arXiv:astro-ph/9812133.[4] M. Tegmark et al., Phys. Rev. D74 , 123507 (2006),astro-ph/0608632.[5] P. Jordan, Z. Phys. , 112 (1959).[6] C. Brans and R. H. Dicke, Phys. Rev. , 925 (1961).[7] R. P. Woodard (2006), astro-ph/0601672.[8] R. Jackiw and S. Y. Pi, Phys. Rev.
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