JHEP01(2021)018 b Springer June 29, 2020 : January 5, 2021 : November 5, 2020 November 19, 2020 : : Received Revised Published Accepted and Mokhtar Hassaine a Published for SISSA by https://doi.org/10.1007/JHEP01(2021)018 [email protected] , , Christos Charmousis , a . 3 Eugeny Babichev, 2006.06461 a The Authors. We humbly dedicate this work to the memory of our colleague Renaud Parentani. c Black Holes, Classical Theories of Gravity

Starting from a recently constructed stealth Kerr solution of higher order , [email protected] Casilla 747,Talca, Chile E-mail: [email protected] [email protected] Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France Instituto de Matemática y Física, Universidad de Talca, b a Open Access Article funded by SCOAP ArXiv ePrint: the candidate surface is indeeda an black event horizon hole and quite the distinct disformed from Kerr the metric is Kerr therefore solution. Keywords: from asymptotic infinity wewith find an a enclosed static ergoregion.to limit However, be ergosurface the similar a stationary to timelike limitstationary hypersurface. the of limit Kerr infalling surface. A observers spacetime candidate It israys event which is found are horizon a no is null longer found hypersurface Killing vectors. generated in by Under the a a interior mild null of regularity congruence this assumption, of we light find that scalar tensor theory involvingthe scalar Kerr hair, spacetime with we a analyticallythe constant construct degree disformed of disformal metric disformality versions andKerr, has of a only it regular scalar is a field.consequences found ring While on to singularity the structure be and of asymptotically the neither solution. is Ricci As flat quite we approach nor similar the rotating circular. to compact object Non-circularity has far reaching Abstract: Timothy Anson, Disforming the Kerr metric JHEP01(2021)018 ] ]. 9 5 8 21 19 ], a fact 11 18 3 6 9 ] and the instrument GRAVITY [ 4 , 3 – 1 – 19 11 ], as well as in the case of a cosmological constant [ 6 15 ]. Furthermore, the Kerr geodesics can be computed analyti- 10 ]. Therefore it is fair to say that the Kerr solution is probably the i 7 Q 1 ]. The Kerr metric is the unique vacuum black hole with the afore mentioned ], to the Event Horizon Telescope [ 6 2 , 1 ], whereas the plethora of higher dimensional rotating solutions have only recently 9 For a start, it is already quite a feat that one can analytically solve the partial dif- sion 4.2 A candidate event horizon horizon 4.1 The endpoint of static and stationary observers cally because the relevant Hamilton-Jacobi function is found to be separable [ ferential equations in thisIn case fact [ it istion only [ in 4 dimensionsbeen that worked we out can in analytically full find [ the relevant charged solu- Kerr metric [ symmetries in GR [ most important of solutions togiven the that Einstein equations. it Althoughsymmetries is a and a rather mathematical complex solution metric, propertieseven of that analytically so partial make to physical differential a applications certain equations, tractable extent). it (and has a number of hidden There is a plethora of novel compellingbinaries evidence, [ from gravitational wave emission offor distant supermassive compactRelativity objects, (GR) that a stationary black and holes axially symmetric exist rotating black and hole are is described rotating. by the In General 1 Introduction B Polynomials C Geometry of the null surface D Solving the equation for the candidate horizon via a perturbation expan- 5 Discussion, conclusions A The disformed metric and the scalar field in regular coordinates 3 General properties of the4 disformed Kerr metric From the static to the stationary limit and all the way up to the event Contents 1 Introduction 2 Constructing the disformal transformation of the Kerr metric JHEP01(2021)018 ] 40 , ] (and 39 ], which 24 – 35 – 22 ]”. 33 36 ], it was understood ]. Making use of the 11 12 ] and the Penrose process of 20 – 15 ]. These mathematical properties pave ]. These are part of a wide class of physi- 14 , ]). The object of this article is to construct a 21 13 32 – 2 – – 29 ]). Hence it is fair to say that the Kerr solution is not only 25 ] for some interesting geometrical approaches to construct deviations from 28 ] page 897). Unsurprisingly, this scalar tensor solution has similar properties – ]. The solution may also present pathologies in the scalar perturbations [ 56 26 38 ]. The theory hosting this solution is a particular DHOST/EST theory, whereas 37 Recently, using the geodesic paradigm of Carter’s seminal work [ All observational data up to now agrees with predictions of the Kerr metric. In light of in adequacy with theexample standard [ Hamilton Jacobito method the for GR obtaining Kerr geodesicsperturbations metric. (see is For example, for again the shownversion relevant [ modified to Teukolsky be operator separable for tensor with non observable differences to its GR Taub-NUT spacetime in GR being “a counterexample to almostthat anything one [ could “paint”ion the Kerr [ spacetime withthe well scalar defined field scalar is hairwhere a in (including particular a the Hamilton-Jacobi regular event function fash- horizon). which, crucially, The is kinetic regular term every- of the scalar is therefore constant, solution. The solution we shallcease present to also exist singles once outcan a we be number venture of of away properties theoretical from thatmodynamics, interest GR. simply etc. for Some Even black of from hole thewhich these physics, are point properties for counterexamples of example or to view horizon usual shortcomings of black properties, GR, hole ther- it properties. is This enlightening is to somewhat discuss similar metrics to the starting point will beare degenerate the (DHOST working or prototypefreedom. EST) of scalar Although modified the tensor gravity solution theories theoriesing we [ with to present is this a an class, single exact weas additional believe solution it degree that of gives the of a a solution precise precise can analytic theory be form belong- of of a more different, interest yet than competing the metric theory to itself the prototype GR any particular gravity theory but ratherInteresting maintain hairy certain numerical properties constructions of have the been Kerrforms obtained spacetime). of by matter considering (including scalar or forspecific other example analytic [ solution of modified(stationary and gravity with axisymmetric), which the is same sufficiently spacetime similar symmetries yet as distinctively Kerr different. Our recent/future observational advances, it becomesthe crucial GR to prototype find solution. competing backgrounds However,or it to even is numerically, rather which difficult have to(see a produce e.g. similar solutions, [ analytically geometry whilethe remaining Kerr distinctively metric, different mainly of “phenomenological” origin, i.e. those metrics are not solutions of ist, results in fascinatingextracting effects energy such as from superradiance blackcal [ hole phenomena, rotation which [ alsotheir include acoustic the counterparts [ quantumphenomenologically laser important effects but of also black of holes major [ theoretical interest. Newman-Penrose formalism, linear perturbationsa of the separable Kerr form, spacetime thethe can Teukolsky way be to equation written understanding, [ in amongst(which other are things, important linear forhole stability the and shadows. quasi-normal ring The modes down presence phase of of an binaries) ergosphere and region, then where geodesics static for observers cease black to ex- that is associated to the existence of an additional Killing tensor [ JHEP01(2021)018 , ] 41 45 , (2.1) 44 ]). The 43 . By disformal Kerr , ν φ Killing coordinates as is the Kerr µ φ ϕ ) φ σ ]. and (which is constant on-shell), the resulting disformal Kerr metric φ∂ 47 t σ are not constant) that can venture further X θ φ, ∂ ( – 3 – which can be represented in the following way: B and + disf µν r ˜ g Kerr µν g stationary observers, which is a Killing/event horizon for θ = is a function of ) disf µν X and ˜ g ( r B = B ] for interesting extensions and questions concerning the regularity of solutions). constant, the disformal version of a Schwarzschild type solution is again a mass 46 However, it again has a single ring singularity and is asymptotically very similar to X 1 The paper is organised as follows: in the next section we will proceed to construct the It is known that disformal and conformal transformations of scalar tensor theories [ This seemingly bland statement signifies that a rotating black hole rotating in one direction is not 1 ] are internal maps of the theory, in other words such transformations will take us from necessarily rotating in the opposite direction in its past [ We start by constructing an explicitmetric, example of we a mean a spacetime metric properties of the Kerrmetric solution. in section We 3, will while discussthe in ergosurface, some section the general 4 stationary limit we properties and willin horizons. of study the the We the will 5th special disformed discuss section. our hypersurfaces findings of and the conclude metric: 2 Constructing the disformal transformation of the Kerr metric vector of Kerr is3 now successive a hypersurfaces spacetime meetthat pointing at it vector the is (apart north regular from and there. the south poles poles). of Indeed,disformed the these Kerr rotating solution, solution. so While doing so we will remind the reader of some well known timelike. Therefore, unlike forsurface the in Kerr the ergoregion, spacetime, inside therespecial which is the timelike an Killing observers vectors additional (for are stationaryin spacelike. which limit up In to fact, there theour exist (candidate) (candidate) event event horizon, horizon. is The no longer generator a of Killing this vector. null The surface usual of stationary no Killing return, metric is no longermetric. reflection symmetric in the Kerr. It also hasthe an boundary ergosurface of beyond constant whichKerr, static fails observers to do be so not for exist. the disformed Interestingly, metric. The latter hypersurface is not null but actually metric is geometrically identical with(see also a [ rescaled mass (andAs cosmological we constant) will [ see inchanged. the present paper, The once disformed rotationa is Kerr circular present metric the spacetime, static is as picture no all is completely longer Einstein metrics Ricci are flat. in GR. It In is a furthermore nutshell not this even means that the interesting combination which we believedisformed crucial metric here is is related thatfound to the to the scalar be responsible geodesics surprisingly for of regular the case, Kerr. yet for non As trivial. we In will factrescaled see, Schwarzschild it black the has hole. transformation been With shown is athe that basic disformal disentangling in of factor the coordinates, static and given that modified gravity). The keythose to originating go from further “geodesic” are scalars. disformal transformations, and in42 particular some DHOST Ia theory to some other specific DHOST Ia theory (see for example [ (one expects to make certain starting concessions in order to obtain analytical solutions for JHEP01(2021)018 ]. + r 40 , (2.3) (2.4) (2.2) = 39 r , and , and which 2 θ E r d = 0 2 ρ = tt g r + 2 r . The event horizons d ϕ 2 2 ∂ ∆ ρ ). + 2.1 and 2 ϕ t ϕ d ∂ d t θ , d , and are obtained by solving the  , 2 r θ θ 2 2 2 a cos θ . 2 sin Mr , 2 − 2 a 2 is the angular momentum per unit mass, ρ 2 ∆ sin 2 a − − cos a M is the inner horizon. Extremality occurs for 2 Mar 2 2 4 a a p − − M 2 r – 4 – + + ± − constant). This hypersurface is defined by  p 2 2 constant become null there. Another hypersurface 2 2 t r r a + M θ d = + =  M 2 2 r, θ, ϕ and ∆ = ± r ρ 2 = r  r ρ Mr  2 E r θ ] that a small detuning of the degeneracy condition for the theory can − 2 2 48 ρ 1 sin  − + = ν x d µ , is a nontrivial solution of a subclass of DHOST theory whose metric solution x ) d ] where the authors consider a subclass of DHOST Ia theory, and show that , φ µν is the outer (event) horizon and 37 represents the mass of the black hole, g ∆ = 0 Kerr µν + g , where the two horizons coincide. The black hole event horizon is a Killing horizon, r M ( is the Kerr spacetime. Our starting block will be the stealth black hole solution In order to be as self-contained as possible, let us now detail this construction. We M Note that stealth black holes were found to have a problem with perturbations, namely the scalar 2 = Kerr µν strong that a stationary observer can only rotate along withmode the becomes black non-dynamical hole onHowever, (otherwise such it his background has solutions, beenbe which argued used is in to related [ solve to this strong problem. coupling [ and is accessible to farcoincide away observers. at Note the that poles. theevent two horizon hypersurfaces The is the region ergoregion of ofing spacetime the physical original in effects Kerr between such black as the hole, where the ergosurface one Penrose and can process. have the interest- In outer this region the gravitational pull is so a and the stationary observers with of interest is the ergosurface,is whose the locus endpoint is determined of by static the observers equation ( Killing vectors are adaptedof to the original the metric coordinatesequation are and located read at constant values of where Working with Boyer-Lindquist coordinatesof is off-diagonal mainly components, motivated considerably by facilitating the the minimal calculations. number Furthermore, the where and for clarity we have defined that we will consider to construct the disformal Kerrstart metric from ( the Kerr metric written in Boyer-Lindquist coordinates, g found in [ a nontrivial scalar fieldscalar defined field is on defined the ascongruence some Kerr particular for metric Hamilton the Jacobi solves function Kerr the giving spacetime. a equations regular of geodesic It motion. is precisely The this geometrically induced scalar field where JHEP01(2021)018 2 and r (2.8) (2.7) (2.5) (2.6) t d ), the ∂ ) 2 2.6 ). There r 0 + < 2 (whose proof a ( ϕϕ C g rD ) sign was chosen for 2 D + ∆ being constant ( ] can be written without (1 + ˜ X 37 M , 2 ϕ # − d ) is given by r t d d ∆ , while closed timelike curves can . . Apart from being linear in time, ) 2.2 2 θ 2 2 φ , ρ r 2 q ν = 0 − + + sin ρ φ ∂ 2 2 ∆ = 0 = µ a ϕ ∂ . ∆ ( ], meaning that there is an equal number d φ 2 ˜ 2 ν 2  Mar ρ θ D 11 q θ Mr d D φ∂ 2 2 2 cst, it is natural to consider a disformal function − µ – 5 – . On the other hand, ρ ∂ + p ) and Carter’s separation constant µν 1 + φ + µν X g Z ∆ sin ( r g √ 2 m 4 d → = B a + t ≡ φ t d (where particles can marginally reach timelike infinity), − = − µν ) " X q 2 ˜ g 2 2 q t r B  d = 2 = + a ! m 2 φ + a 2 ˜ ∆ ( , rest mass = 2 ρ Mr r L 2 E ˜  Mr ). Since the DHOST theory admitting the stealth Kerr solution is  − 2 θ 1 2.1 q 2 2

ρ D (to have regularity at the poles). − sin 2 = − + ν = 0 x ], it was shown that the scalar hair painting ( is a constant whose sign is not fixed a priori. In the original coordinates, the C d is constant that we assume positive. Note that the relative µ 37 D = q x d L We now have all the necessary ingredients to perform the disformal transformation In [ µν , while the latter two come from the existence of Killing tensors for the Kerr spacetime. with the same symmetry, i.e. ˜ g , angular momentum ϕ where disformal Kerr metric reads as defined by eq.invariant under ( the transformation B disformal Kerr metric associatedany to loss the of Kerr generality stealth as solution [ ∂ In order for ourinfinity, one scalar must hair take toand be well defined from the event horizon up to asymptotic general HJ functiontions for are Kerr integrable spacetime. forof the conserved Carter Kerr quantities demonstrated spacetime andE that [ spacetime directions. the geodesic Theseof equa- quantities existence are gives energy integrability). at The infinity former two originate from the Killing vectors the scalar field to bethe regular scalar at field the has Kerr a horizons constant standard kinetic term: This last equation is nothing but the Hamilton-Jacobi (HJ) equation determining the most on them. We will encounter some of them in the forthcoming sections. where esting influence of the disformalThere transformation is on the a location single ofdevelop curvature these within singularity this two situated hypersurfaces. region (where at theare axial of Killing vector course becomes many timelike otherKerr solution, important and properties it (physical proves interesting or to mathematical) study the underlying impact the of the disformal transformation future pointing tangent vector could not be timelike). We will see later the rather inter- JHEP01(2021)018 t (2.9) , 2 ) ) D is a given r, θ ( D 1 . )(1 + Q 2 ), starting from 2 2 a Ω 2.8 M d + ]). 4 2 2 r a , and the coordinate ). This is one of the r . In these coordinates, ) 49 2 ( 0 + 3 D D . Although the regularity 12 2 ρ r B 4 d D > (1+ 1 = − 1+ ! . A non-trivial property of the M/ µν a ] (see also [ ˜ ˜ r , R r = M d 2 µν 45  ˜ , 3 M ˜ ˜ R − r M 2 1 ˜ 44 g . , Mr

− − 2 , assuming 39 1 + √ √ D  2 , ) can be removed by the following coordinate D = T ∆ )] – 6 – is singular. Nevertheless, as in GR, this latter ˜ g d 1 + θ 2.8 , − ± − √ ! 2 r ) T p ˜ t/ in ( 6 r M d D = ρ 2 tr ) can be found in the appendix → ) r = ˜ g 2 t − D t Q 1 r, θ d )(1 + ( [1 + 3 cos(2

2 2 a (1 + as the metric’s disformality. Indeed, although Q − and , which cannot be eliminated by a coordinate change without 2 Mr + 1 2 tr = D 2 ˜ M g Q r ν ( Da x d 12 − µ ρ the scalar field is again a HJ function for the disformed metric. x d = = ) ˜ µν D R ˜ g µναβ ˜ , the off-diagonal term R (1 + / 2 q = 0 µναβ − a ˜ R Before entering the details of the properties of the disformal metric, we would like to = ˜ where the expressions of of scalar invariants does not constitute a necessary and sufficient condition for regularity the singularities. Anthat initial the inspection hypersurface of definedhypersurface the by is metric merely may a naivelychoice coordinate lead of singularity, coordinates. to which can the One can be conclusion start removed by by computing an metric appropriate scalar invariants such as case of Schwarzschild-de-Sitter metric in refs. [ 3 General properties ofLet the us disformed analyze Kerr some metric of the properties of the disformal Kerr metric ( Hence, in the Schwarzschildonly static to case, rescale the the mass net parameter. effect Note of that the a disformal similar observation transformation has is been noted in the case transformation: and the resulting metric is nothing but the Schwarzschild spacetime with a rescaled mass: field conjugated with thenew non-zero metric angular is momentum theintroducing term other off-diagonal elementsmain differences (see with discussion the Kerr inoff-diagonal metric, section term and will we will have see significant below consequences. that On the the presence other of hand, this in extra the static limit constant parametrising a given (DHOST Ia)it theory, in can a be more phenomenologicalX thought approach, of as an additional parameteremphasize of that its the nontrivial disformed character is metric. mainly due to Given the that time dependence of the scalar has been conveniently rescaled as the determinant of the disformal metric is the same as that of the Kerr metric: We will be referring to For simplicity, we have introduced the rescaled mass JHEP01(2021)018 , . ) η (for 2.8 .A = 0 . In (3.1) ) ± and ϕθ cannot r = 0 µν ˜ g k tr = K ρ ˜ g α ( r and ϕ , ∇ d ϕr ∧ ˜ g , one can try to , θ 1 d tθ ˜ g , ∧ , ϕ  r ∂ tr ), our disformal metric ˜ g d ϕ D ϕ ∂ ]) but it is fair to say that ∧ d and t 51 which satisfies the linearized d ∧ t are manifest since the metric and ∂ θ θ µν . ϕ 3 t d , ∂ ∂ ν ∧ sin x r = 0 of the Kerr metric. DδK θ d d ]. This means that locally the metric and η where the disformal spacetime ( + µν ϕν d ∧ t 50 g t A ∂ µν K ) cos ∧ d ) again represents a stationary and axially 2 K = ˜ η r  ∧ = 2.8 + tϕ 4 k ˜ g 2 ρ ∂θ , η µν – 7 – a ∂ which verifies the Killing equation ( ˜ = surfaces orthogonal to the Killing fields ν K tϕ x k − ˜ g µν ˜ d d 2 Mr . In addition to these Killing vectors, the Kerr metric K tν 2 − ∧ ϕ g . We will deal with possible horizon hypersurfaces and q tt η ). It can be shown that the system of equations is not ˜ g = ˜ ∂θ ∧ ∂ D ˜ = 0 and k Mr k t 2 ρ ϕϕ a ˜ g 4  D ]). In particular, this implies that the off-diagonal term tr − g 53 , = = ˜ 52 k d ∧ η ∧ k ” reflection isometry, which is the simultaneous change of the time and of the rotation It is well known that Einstein metrics belonging to the class of stationary and axisym- It is clear that the disformal Kerr metric ( ϕ − t and hence the disformalconditions Kerr are metric equivalent is toour not the choice circular. eliminability of of adapted Incan the coordinates not addition, to cross-terms be since the cast in Killing themomentum the vectors tensor integrability Lewis-Papapetrou [ form as in vacuum GR (or with a circular energy- In our case, an explicit calculation yields are integrable. According to Frobenius’sto theorem, the the circularity following of integrability the conditions: metric is equivalent example, may fail to bemost a known circular spacetime solutions (see in forspacetime the example is [ literature said are torespectively be indeed the circular circular. one-forms if associated the Rigorously, to an the axisymmetric Killing vectors metric spacetimes are circular spacetimes,is see not e.g. only [ independent of" the time and ofangle. the A rotation rotating angle solution but such also as invariant a under rotating the neutron star with toroidal magentic field, for integration of theconstruct geodesic a symmetric equation. tensor of theKilling For form equation small (to deformations orderconsistent. 1 This in means that ifbe a written Killing tensor as does a exist deformation for of the the disformal metric, Killing it tensor cannot coordinate singularites in the next section. symmetric spacetime, whose Killingcoefficients vector are fields independent of possesses a non-trivial Killing tensor This gives rise to the conservation of Carter’s constant along geodesics, and enables the there are no physicalmore singularities rigorous apart argument from isis the given re-written standard in in ring the a singularitythis appendix at Kerr coordinate coordinate system, systemexcept both making at the apparent the metric ring its singularity and regularity the at scalar field are manifestly well-defined of the metric it is nevertheless a good starting point. The above expressions suggest that JHEP01(2021)018 . . ]). M and # (3.4) (3.3) (3.2) j 45 , ϕ  x d 44 r i sin x θ d ij β sin . r ! 2 2 i r ˜ a β , = φ

in order to compare y α  , φ O 2 / ϕ 1 M 7 + r i µανβ  x cos  ) is in general not Ricci flat. d R θ r O T 2.8 − r , d sin i d . It is then convenient to rewrite ) r ∼ αν α →∞ r 2 r φ r ! ) µ = ) 2 ˜ α 2 + / r M x 3 2 φ r 2 2 / , the Ricci tensor of the disformal metric ˜ 7 a − + M − ( r 2 2 ), which reads ˜ a µν a = 0 ˜ ( Mr

∆ ∆(1 2 2.1 φφ µν 2 ˜ – 8 – O ρ q  Mr R ) h 2 + ˜ D ) M 2 q ), which does not in general vanish (it can do only 2 θ D − T 2 d 3.2 − T D d r ! cos ( (1 + ˜ 2 2 = 3 M q r t 2 ˜ ˜ a d − Ma

D = 2 O " µν = R ]. In our case, the lack of circularity will turn out to be fundamental. D T r − D 55 ˜ g 1 + µν ˜ R + . Keeping only the leading order corrections, the line element of the disformal 2 Kerr θ cannot be eliminated, it is possible to do so in the asymptotic region ]. This is the familiar BBMB black hole with secondary scalar hair. It is however s d 54 tr cos ˜ g = r Let us study the asymptotic region at large distance The circularity property is twofold: it is known to be fundamental for proving im- 2 ˜ s = d z Kerr metric in the asymptotic region reads and one can seethe that line it element decays rapidly in enough terms at of large Cartesian coordinates Indeed, through a redefinition of the time coordinate the cross term becomes the disformal metricimportant with for the astrophysical Kerr applications,the spacetime in supermassive particular in black hole by in this observing theterm region. center orbiting of our stars galaxy. This around Although in comparison general the may off-diagonal be is given by thefor right certain hand scalars side sourcing ofOnce ( highly we symmetric have rotation cases the suchproperty disformal as as metric well. spherical is This no symmetrybasic longer inevitably [ properties Ricci leads flat of to and GR fundamental loses black differences the holes, and circularity as to we the will loss now of see. usual of the original and disformal Ricci tensors ( Hence, since the Kerr metric is Ricci flat, exists [ proven, under the crucialtotically hypothesis flat of black circularity, holes thattrivial in scalar stationary-axisymmetric hair this asymp- [ theoryFor are a start, only it those is describedIf easy to it by see were, the that circularity the Kerr would disformal metric of Kerr and metric course ( follow. A simple argument comes from the relation break circularity. portant theorems such asrestrictive. the An constancy illustrative ofconformally example surface coupled of gravity, but scalar this can where latter also a fact prove non are to trivial black be static holes too and in spherically a symmetric theory solution with a be removed by a coordinate change without introducing other off-diagonal elements that JHEP01(2021)018 . ), 0 (1) ] has 2.8 O 60 r > T d ’s are itself. Note ] ij x becomes null. ) . β d
4 y 0) t, r ]. It is important , ( /r − 0 φ 1 57 , [ y and rescaled angular 0 d ’s and , i O x ˜ [ α M is stably causal, in other = (1   a > M 5 ) µν 1 t r µ ˜ ( g l  ), the scalar is regular for and the O ]: + A.3 Da 56 ˜ . 3 M r a i 1 + 2 4˜ z  √ d − = + 2 2 – 9 – ˜ a T y d d According to ( ]. This function can be thought of as a global time. +   3 3 2 58 1 r x , than those of the Kerr metric, . As can be seen from the above expressions, the effect d   ) is the line element for the Kerr metric with parameters a ] to show that a black hole with an accreting k-essence field [ h 2 / O 59 7 3.4  + and /r r 1 ) itself. It is known that in GR there exist solutions that 1  ˜  r M M 2.8 2 O O − , it can be written in the form [ 1 1 + r   − + , while for the next-to-leading order corrections the disformal off diagonal (in particular outside the event horizon) is causally disconnected from the ) it is indeed timelike. a = r are larger, (where closed timelike curves are present, similarly to the Kerr case). 2.6 , rather than i 0 ˜ a x 2 Kerr s d . For large d ˜ T ˜ to smaller radii studying interesting surfaces on the way, until we meet a perspective r < M r ]. Finally, we finish this section with an important property that stems from the con- horizon The same argument was used in [ 59 3 and ˜ The outermost interesting hypersurfacethe in limiting the hypersurface case where of the the timelike Kerr Killing metric vector is theno ergosurface, closed timelike curves.in Similarly [ to our case, the k-essence field was identified as a global time function large horizon or hit the ringparallel singularity. to In order that to of make the it Kerr more intuitive, metric, we4.1 which make has our been analysis summarized The in endpoint section of 2. static and stationary observers 4 From the static to the stationary limit andWe are all now the ready way to upwhich discuss include to the the the properties timelike of event static importantsurface(s), and hypersurfaces which stationary in limiting if the surfaces present metric as are ( well candidate as the event null horizon(s). hyper- We will move gradually from that due to ( Therefore our spacetime issome globally positive causal, provided thatregion the region of the spacetime for is achieved by showingwords that it the remains spacetime causal withbe under the stably a metric small causal, perturbation itfuture of is directed the timelike sufficient vector light to field cone.In show [ our For that a case there spacetime there exists to is a function such whose a gradient function is by a construction, the scalar field terms d struction of thecontain metric time machines, ( an exampletherefore being to the Kerr show metric that with our disformed Kerr metric does not contain such a pathology. This Notice that the Kerr partparameter of the metric contains theof rescaled the mass disformalthe transformation parameter at leading order is merely a rescaling of the mass and coefficients. The first term ina ( In the above expression we have defined JHEP01(2021)018 t a ∂ (4.6) (4.3) (4.4) (4.5) (4.1) (4.2) ), we ) is a 2.8 4.3 ) has the ) can move 2.8 4.3 , or equivalently ). Using ( = 0 2.8 ) t,ϕ ( is the same as for the Kerr . We can thus see that the , ˜ a . ) , is modified with respect to t t µ (  ∂ l = 0 → , . ϕϕ sections of ( a θ ˜ g ) ) 2 θ tt ( = 0 ˜ g , and therefore the vector ( ) sin . For some region of space-time inside t, ϕ + tt − ˜ ( θ r ˜ g r , Mr r, θ 2 ) with , direction with respect to ( 2 tϕ ( ) can still be null or timelike. Therefore ϕ = 2 ˜ g r ) is positive. This discriminant is nothing = 2 and ρ 4.2 r ⇔ 4.3 = 0 ω∂ q θ r ˜ of the ν MDa 4.5 at 2 + l ± ) becomes null: 2 ) as ergosurface, similarly to the Kerr case, or t µ with the solutions ), the locus of the ergosphere for the two does θ ) l – 10 – ∂ t,ϕ tϕ t cos ( µ ( ˜ g ω 3 = 0 l 4.2 µν 2 ) + = ) ˜ g − a r t l ν (  l + ) ˜ ∆( t µ 2 ( 1 ϕϕ l ) ˜ g ≡ θ µν ( ) ˜ g r = r, θ ± ( ω P as the Kerr metric, we can define the ergosurface in a similar way, ) is zero: ) does not depend on t µ ( 4.5 l ω . This implies, like for Kerr, that the ergosurface is not an event horizon. ). Therefore, if the disformed and Kerr metrics are matched at leading order r (see the discussion in section 4.2 ), can no longer exist in its interior. Since the disformed metric ( r ϕ : . Moreover, is real only when the discriminant in ( ϕ + ∂ r Following the same reasoning as in the case of the Kerr metric, one can write The next step is to consider a combination of the two independent Killing vectors and ω = but the two dimensional determinant Det find an equation forthe the discriminant hypersurface of where ( the determinant Det which results in a quadratic equation in This second order algebraic equation is fullyfor analogous to its Kerr counterpart. The solution r Killing vector as well, so that the event horizon is also a Killing horizon. the ergosurface, which isobservers timelike, that the have vector a ( smallto perturbation increasing in the One can then lookcase for of the the surface Kerr inside metric, which this stationary hypersurface observers is cease null to and exist. it In turns the out to be the event horizon as static limit. and which defines stationary observers at constant for large not coincide. Indeed,disformed the metric equation at for large the radii)locus ergosphere is of of for the the ergosphere, the form correspondingthe Kerr ( to Kerr metric the case. (matching Killing the vector We refer to the surface ( From the above we have so that the locus ofspacetime the with surface a related to rescaledenters the mass. eq. Killing ( vector Note, however, that the non-rescaled Kerr parameter r, θ same Killing vector and it corresponds to the surface where This surface is often called the static limit, since static timelike observers (with constant JHEP01(2021)018 , = 0 = 0 (4.8) (4.9) (4.7) , thus (4.10) P D r , the surface . While in the D r , which depends ). Let us define a , and its norm is ϕϕ and ˜ g ) can also be found 0) / 2.8 , θ ) tϕ 2.3 θ . This implies that the ˜ g ), the null hypersurface ( θ 0 0 − R 4.10 − , ) = , ) θ 1 , θ ( , ). First of all, by taking ) is that it is not null. Indeed, ( 0 , 0 ω , θ > 4.7 R . 4.7 )
can be seen from its construction: it = (0 depends on θθ = 0 ˜ g , ) r µ R, θ ) yields θ 2 ( 0 0 θ ( N = 0 ( 0 P 0 R R P , − R = 0 ⇒ is time-like, as we saw above. Therefore we + ) − θ q = P , ( rr ± 1 r g is time-like. ) is a fourth order algebraic equation in R – 11 – , = 0 ) 0 = as a function of = ˜ = θ 4.6 P ) = 0 ( ν r  r 0 = , θ N R in order to find a candidate horizon. The candidate ) µ µ dθ dR θ n r ( N =  hypersurface is 0 r µν R ˜ g ( . Note that the same equation can be obtained by requiring P = 0 ˜ Mr P 2 ). A solution to the above equation ( ), although a combination of Killing vectors, is not a Killing vector − ) taken at the surface 4.6 ) is not a horizon. By definition, it has the meaning of a stationary 2 4.3 all Killing vectors are spacelike (except at the poles)! r 4.5 , however, the solution 4.7 + , unlike in the case of Kerr. Note also that for arbitrary 2 = 0 6= 0 a of the Kerr metric. Eq. ( P , which is also similar to the Kerr case, indeed, eq. ( D 6= 0 ) = , is our candidate event horizon for the disformed Kerr metric. The above equation r defined in ( is given in ( = 0 D = 0 ) ) reduces to the equation determining the locus of the horizon in the Kerr case, µ θ ˜ l coincides with the (outer) ergosurface at the poles. Moreover, in the interior of this P ∆( rr ( . For rr g for ˜ g R 4.6 The physical meaning of the hypersurface Another important property of the hypersurface ( Let us comment on the hypersurface given by eq. ( θ = 0 = Asking that the above vector is null, the hypersurface we seek verifies the equation where R horizon has to beindependent a and null axisymmetric hypersurface, according isnormal to situated vector the inside to metric the such symmetries stationarity a limit ( hypersurface and is time as the stationary limit. 4.2 A candidateThe event last horizon stationary surfacehave defined to by proceed to even smaller is the last surface of stationarycase of observers, the i.e. Kerr observers metric with it constant coincidesthe with hypersurface the horizon ( (since it islimit null), (cf. for the the disformal static metric limit). Therefore from now on we will refer to the hypersurface a normal vectoreasily to evaluated, the implying that the hypersurface vector itself. Indeed, eq. ( on P hypersurface but we do nothave give multiple it roots, here and we since aresituation it always with interested is the in not Kerr the especially metric, outermost one. where informative. there This The are resembles the above ateq. equation most ( can two solutions. ∆ = 0 that from one can write an analytic solution for where JHEP01(2021)018 . c ) 2 θ ( ) is π/ such D a R (4.11) (4.12) (4.13) ζ ) these . This = = 4.10 and the . To do , similar 1 + | θ r are given 0 √ D D 4.10 = 1 | a ). Note that ˜ . In addition, hypersurfaces, . In this case, D > M . For the sake ˜ a < 2 and ) D 4.10 , we have to also θ a ( θ π/ ) = 0 R − π = ), one can show that ( 0 . Then, requiring that π = θ 2 R r is evaluated at 4.11 → ), one has π/ , θ R and . In terms of the angular 3 = 2 1 a (0) = θ 0 into ( 3 . = 0 + R 2 , which agrees with the fact that R 0 , 2 ) θ θ 0 π/ R ( 2 there are more physical requirements R ) and the Kerr parameter = . The above equation can be solved in ) , but not for negative coincide at the poles and the equator. θ Da θ . The same horizon equation ( yields a necessary condition, ) ( B D 8 ζ ) = θ 2 R θ ( at 0 , D > ) also agrees with the equation defining the 0 π/ − ≥ − R , D < a priori R D 2 π ) is a generator of null geodesics skimming the and 4.9 = ( which verifies a fourth order polynomial equa- – 12 – is shown in figure ! ) is of first order, therefore we only need to specify ) θ = R 2 1 ) and θ 2 ( 4.10 D ) = 0 ) R D 1 + 4 0 R c ( 4.10 θ Da a R c ( √ ( a is twice differentiable at ) with ( R ) is of first order and has two branches. In each branch, 3 − . This is done by defining a new radial variable = 1 Q R c C a 4.10 . It is possible to see analytically why one of the conditions − 4.10 is given in appendix D R ) numerically, this indeed becomes a problem for some ranges 3

) satisfying ( Q (as well as , which is not automatic. Note that by virtue of eq. ( and ) cannot be satisfied for large enough values of 4.10 . Eq. ( is by definition monotonous. Depending on the sign of in a similar way to Kerr in Boyer-Lindquist coordinates. One can r 4.9 ) a to be real, where in the above equation ) θ as a function of , ( θ yield 4.10 ( ˜ c 2) R 2) = 0 = 0 M a R ], albeit for other Kerr deformations. π/ = π/ ζζ = 0 ( = ( g 0 28 00 measured by an observer at infinity (see section . We assume that θ , R R R R ˜ , one has to choose a particular branch among the two in ( ˜ a , a Taylor expansion around M (one can show that there is only one positive real solution). For 27 θ , ) is automatically satisfied for positive D ) is invariant under the change 2 c a 2) = 0 4.11 4.10 On the other hand, the equation ( The differential equation ( π/ ( cannnot exceed a critical value 0 terms of arguments at The solution for momentum substituting the relevant solutiona for tion in where the expression for in order for Eq. ( this, we examine theof behaviour convenience, of we the discuss solutionmeans around that the the results radius inin terms terms of of ‘natural’R units setting one boundary condition. Thison implies the that solution thanwe the will available see freedom by inof solving the the ( parameters choice ofon the the boundary solution conditions. of As ( to be smooth, itssince derivatives we should ask vanish for atrequire the the that solution poles, to beconditions symmetric mean with that respect the to surfaces horizon in [ the solution interval of eq. ( our solution should be symmetric with respect to the equator. For the solution as we demonstrate inthat appendix the horizonthen is given situated by atcheck some that constant the vector ( hypersurface can also be found by requiring that the metric is degenerate on JHEP01(2021)018 . π in . So 6= 0 ) ≤ 1 , and θ θ ( 0 2) by ex- 0 . The ≤ p > ) R 2 π/ 2 θ . We are ( , meaning ( 0 D < reaches its π π/ R R ) when ≤ 6= 0 D θ are extremely = ) ( . p θ ) when . The same is true D 0 r R 3 θ (2 2 ( , and ) is increasing for ≤ R 0 c is allowed. = π/ 6= 0 2 R r = do not give a smooth D > when 4.10 2) for this approach. π/ θ ) 1 a < a π/ and D for D 2) = 0 ( or ( 0 ) c θ R π/ 2 ˜ a < ( ( (0) 0 0 R π/ 2 R R a > a ≤ θ (0) = 0 , we have , etc. are real at ≤ c R (6) 0 do not give additional conditions. Indeed, if we ,R 2 , the curves a > a (4) , so that the solution a π/ 1 π R = θ ≤ . For – 13 – , one can check that in the Taylor expansion is linear in . The physical branch of ( to a high precision when θ D p 0 2 ), we used two different techniques. The direct nu- ≤ 4.13 2 4.10 D < 2) = 0 ) for the numerical results. π/ and c , it is convenient to choose the boundary condition either π/ a ( D 0 = 1 1.0 0.8 0.6 0.4 0.2 This does not guarantee, however, that the solution is smooth R 4.12 ˜ 4 For small values of ) with a given boundary condition by the Runge–Kutta method M . as a function of , then the order 5 c . = 0 a 2 4.10 θ in our numerical integration. Having this in mind, we only have to π/ . For points in the shaded region, 2) = 0 2 0 ≤ π/ -1 ( π/ θ D > +1) ) in either one of the two intervals = ≤ and decreasing for ) in an iterative series, where at each order a solution can be found knowing p θ (2 . Similar results are obtained when evaluating 0 2 c . We could only verify this numerically, as can be seen from figure R 4.10 c 4.10 or when π/ . Critical value ≤ ). a > a 6= 0 = 0 θ Let us first consider the case Depending on the sign of As for finding the solution of ( Note that this study allows us to verify that values of a < a Higher orders of the Taylor expansion around Note that our numerical integration fails when we try to integrate in the other range, for 4.10 θ 5 4 (0) ≤ 0 assume that we do not have additionalfor constraints the to expansion ensure around that We believe thatin this ( is due to the numerical instability exploding when the negative branch is chosen 0 maximum at the equator.the In interval this case we numerically integrate the equation for the result at the previous order. We refer theat reader to appendix solve eq. ( again using natural units ( numerical integration yields when merical integration of ( is discussed below. Anpanding alternative ( approach is to search for the solution for physical solutions. Using that the disformed metric looks like a sub-extremal Kerr solutionsolution to for an the observer null at surface. infinity. for Figure 1 R JHEP01(2021)018 = D θ (orange). , which is 1 . 2 π , respectively 6= 0 D = 0 (red) and a 2) D is decreasing for 3 . 1.0 0 ) π/ θ ( − 0 ( R R = = and varying D 9 r . 8 3π (purple) and 0.9 ). = 0 05 3 . , we have a c (black), for = 0 75 . + D 0 a > a R 0.8 − − 0 = . Thus the null surface has a minimum at R π (blue), 4 π D 1 ≤ . – 14 – and 0 θ for becomes large. − + 0.7 | ≤ R = 2) D | 2 − π/ D ( R 0 π/ at the equator (see figure R , and the numerical results are consistent with the condition 0 . For higher values c 2] R (red), a 0.6 D=-0.75 D=-0.3 D=-0.1 + 8 π 3 + 6= . = , π/ -R 0 0 R [0 a , the physical branch of the solution − R-R R 0 = D=-0.3 D=-0.1 D=0.05 D=0.1 D=-0.75 D 2 π D > ′ R 0.1 0.4 0.3 0.2 and increasing for . This remains true as we increase the rotation, until the rotation parameter 2 (black), 0 π/ . Numerical value for . Numerical integration of 75 0.0 0.2 0.4 . -0.2 ≤ 0 2) = 0 (blue). − θ In the case 1 π/ . = ( ≤ 0 0 R reaches a critical value clear from the fact that 0 Figure 3 D The solution becomes unphysical when close in the whole range Figure 2 − JHEP01(2021)018 , ) ) ] θ 0 ( are , π ζ 2 a R ζ < ( term in π/ [ 0 and tφ g both ways. . Note that D ) 3 ζ > θ ( ζ const surfaces in coming from the are spacelike for R a = ) , which is identical can be constructed cannot be satisfied. θ r r ( ζ = 0 in the interval , see figure , while for R , i.e. the surfaces ) 2 ζζ and θ D 2) = 0 large, there is no longer a g ( C π/ θ R π/ D ( = 0 θ R null (and timelike) geodesics can and ) coordinates) is spacelike inside the θ is analogous to a ( ) r R θ . Intuitively, this can be understood ( ) ζ θ , such that the . These coordinates are adapted to the non-constant ( ). On the other hand, if both R and 0 ) within the stationarity limit. results in the disappearance of the root of R θ ζ , let us comment on the causal structure of a < ) 4.13 ) is smooth at , since the numerical integration breaks down = term compared to the Kerr metric in Boyer- θ , crossing the hypersurfaces 0 ( π ζ – 15 – r ζ tr R t, ζ, θ, ϕ ˜ = g 4.10 . Apart from the constraint on ( 0 θ and , whose equation is simply introduced in the appendix to ) . Indeed, for both θ exists, our candidate horizon is indeed an event horizon D < 2 ζ ( discussed above, in this case there is an additional bound D = 0 ) 0 θ π/ decreasing corresponds to the null surface 2 ζ R ( , an increase of . Similarly to the case of negative is a constant labeling a given hypersurface. The parameter = R π/ and 2 D = ζ = 0 θ and = a = π/ r ζ , which means that the condition θ R to 0 ) at is outside (inside) the candidate horizon. Under the mild assumption 2) = 0 , where ζ ) (non-extremality for Kerr), it can be shown that these surfaces are timelike θ 4.10 ( r, π/ ζ ˜ ) + ( M R θ P ( ) (which we defined with constant , and that there exists small . This construction shows that outside of > R 0 0 const surfaces in coordinates is weaker than the one obtained in ( ) 4.3 ). In other words the set of hypersurfaces θ ( = 2) R ) = It may seem paradoxical that inside the surface of last stationary observers, there is still Having found the null hypersurface ζ ζ > θ π/ 4.10 ( ( at the equator. However, one can show that the bound coming from the existence of < ζ < corresponds to the coordinate ζ 0 0 5 Discussion, conclusions In this article wemetric. have Crucially, considered the a disformal disformal directions transformation were of given the with standard respect GR to Kerr derivatives of the stationary limit. However, timelikein vectors with between the surfaces from the fact thatLindquist coordinates. there This is additional a termKerr, plays which non-zero a distinguishes complementary static role fromtimelike to stationary observers the observers. moving towards Here, increasing it allows the existence of that a regular solution shielding the ring singularity. The disformed Kerr metric isa therefore region a of black space-time hole. vector from ( which light and particles can escape. Indeed, the stationarity ζ travel towards increasing On the other hand, insideonly the move candidate in horizon one direction. surface,greater null This than (and picture the timelike) is geodesics inner fully can horizon analogous radius. to the We Kerr therefore black conclude hole that for radii under the assumption to ( the Kerr case. Therefore the surface that for spacetime. We consider theR following continuous oneζ parameter family of hypersurfaces, are candidate horizon situated at R not too large, the numericalin solution of this ( case wein solve from the interval from we believe that this is related to a growth of the numerical instability. consistency of ( on the possible valuessolution of of In other words, forP a given the equator, contrary to the case JHEP01(2021)018 . ), and = 0 2.8 3 ], and tt ˜ A g 37 (solution , ) G θ ( R , just like Kerr. = , inherent to the ) R = 0 t, r ( ρ φ , and one can show that ˜ a surface, we have therefore ) θ ( R = R which is regular at the equator (no knee ) θ ) as a one-parameter family of Kerr deforma- ]. Such theories are constrained from gravity ( . Furthermore, asking for the event horizon to R 43 2.8 ˜ a – 16 – = ) using timelike geodesics. R 2.8 ]. The resulting disformal metric is a stationary and ] associated to dark energy and one has to be rather careful when The solutions we have discussed here are asymptotically 37 , the disformal metric resembles the Kerr solution with a 62 ]) assuming that the scalar is varying at vast cosmological r 7 61 observers. This latter stationary limit surface is given by an ), and is located inside the ergosurface, which is given by ) are chosen so that the scalar is well defined from the event θ dependent metric ( 6 4.6 D ,( and r and angular momentum , meaning that the disformed solution will look like a sub-extremal Kerr ˜ ) = 0 M theories where our spacetime is identical to the GR Kerr solution [ 6= 0 r, θ ( D P = 1 if )). This hypersurface was shown to be situated in the interior of the stationary T c ˜ M The disformal transformation is an internal map within DHOST Ia theories. We start In summary, we have found necessary conditions and numerical evidence for the ex- 4.10 Here, we include theThere metric itself have which been is criticisms trivially on a Killing such tensor effective for theory a metric calculations connection. that recent data from LIGO/Virgo parameters in the notation of DHOST [ 6 7 1 ˜ tions, which may be tested by present and future gravity experiments. In the metricare ( within the strongmaking coupling stringent scale claims. [ scales ie., a darkflat energy and locally field. influence theries. speed Independently of of gravity gravity waves wave forinterest constraints, these of the particular solutions these scalar discussed particular tensor hereright theo- theories go as beyond and simple, the we analytic, believeone benchmark that may alternatives consider they to the are the interesting prototype in Kerr their solution. own Indeed, from map to a disformedA Kerr metric for somewave tests DHOST (see Ia for theory example with [ some given tionary spacetime). Under the assumptionshown of that a our regular candidate horizonstrating is indeed that the our event horizonrather spacetime of nice is a feature black free hole dueconstruction while of to of demon- time the the disformal machines presence metric (causally of ( a stable). global time The function latter is a spacetime to an observer at infinity. istence of a regularsingularity). null hypersurface We showedtrapped that interior when region this from hypersurface where is no present lightlike it (or timelike) is signals the can boundary escape of (in a a sta- equation We have shown that forrescaled mass large be physical results in ana upper < bound for the rotation parameter spacetime geodesics. We haveEinstein shown that (unlike the Kerr), resulting butWe spacetime that have is it found non circular compelling hasthere and a evidence exists non single (an a important ring regular number singularity nullof hypersurface, of at ( our necessary event conditions) horizon,limit that situated of at constant metric. The scalar isconserved a parameters of particular the Hamilton Kerrthe Jacobi spacetime function two (originating for Killing from Kerr thehorizon tensors geodesics. two to Killing The vectors asymptotic four and axisymmetric infinity spacetime, [ like Kerr, while the scalar field is again related to the disformed scalar field which are tangent vectors to a regular geodesic congruence of the spacetime JHEP01(2021)018 ], the authors claim ] fail due to the non 64 ], where, amongst other , therefore one may look 67 , 69 D 66 can also be instructive when tr ˜ g ]. Additionally, one could include ] and references within). It would 37 68 ]. For example, in ref. [ 63 – 17 – ]. 46 ], where it is claimed that the related spinning black hole should break 64 A study on a similar subject appeared recently [ ). This question might be rather non-trivial and requires a separate study. ]. Breaking the circularity hypothesis constitutes a milder approach to the 65 2.8 Note added. Last but not least, the solutions described here are interesting on purely theoretical The presence of the non circular off-diagonal term or what is thesome effect of the of interesting having questionsconstruction. an that extra one can ergosurface consider on starting the from this Penrose relatively process simple findings, ? the authors These confirmed are some of the results presented here. can one go aboutextending the studying notion the of thermodynamicsstudied surface by of gravity several for these authors horizons solutionsbe in which ? different interesting are contexts no to (see The longer studydefinitions [ Killing possibility the provided has of for thermodynamics been surface ofhave gravity. an this additional specific Other Killing solution interesting tensor questions under for include: the this differing do disformed we metric still and are geodesics integrable, spacetime ( Furthermore, there are issueshorizon. related Classical to black hole thehorizon theorems failure which for of assert an that the ancircularity axially event event horizon of symmetric horizon to is and spacetime. also be stationary a a spacetime Therefore, Killing Killing [ how can one define surface gravity here and how most probably not modified gravity.for Therefore axially circularity symmetric should and not stationary be metrics taken beyond for the granted realmgrounds of as GR. counterexamples toquestions usual which GR we black have hole left metrics. unanswered, starting Indeed, with there the are global numerous causal structure of the non-circularity have also been foundblack numerically hole in [ the caseone of concluded a in DGP [ Horndeskieither rotating the stationarity or axisymmetryhints hypothesis that (or the both). circularity hypothesis Our is relatively very simple much analysis tied in with Einstein metrics and GR, but This observation may beso-called pertinent Chern-Simons for modified the gravity searchthat [ of stationary a and axisymmetric rotating solutions blacknot of hole exist, Chern-Simons solution but modified of gravity theyto probably the limit do this their issue, analysis one to must clearly circular include configurations. metric In contributions that order are to not bring circular. light Signs of conformal transformations which willregularity conditions not [ alter the null cones butlooking may for yield approximate solutions, interesting fordisformal example solution in the the resulting slowly off-diagonal rotatingnates term limit. in can the Indeed, first be for order eliminated the approximation, by but a this change is of no longer coordi- the case from the next order on. either for constraining deviationsity from modifications. GR or, Furthermore, oninclude given the effects their other beyond simple probable hand, originsense strong for related we coupling smoking think to scales gun it of geodesics, grav- wouldtuning particular even they solutions EFT be may starting very theories. from interesting the In to regular this study solutions disformations in of [ dark energy self deviations from GR are encoded in the disformality coefficient JHEP01(2021)018 ). 2.8 (A.1) (A.2) (A.3)  a priori ϕ d θ 2 ) to the met- sin , we repeat the a 2.7 ± r . While it is well + ± r = r d r ]: = 6   ) may suggest otherwise. r ϕ d 2.5 θ , 2   sin 2 a a r , ), one can straightforwardly obtain + Mr + 2 2 d r r v , A.3 d d q ∆ Mr r  ∆ 2 d , although eq. ( 1 + ), this singularity is not physical, we Z + 2 Z ± r 2 a Z − 2.2  = – 18 – r − + ϕ r d r ϕ − θ v 2 − v , → → − sin   t a  ϕ q ) inherits the problem from the Boyer-Lindquist presen- 2 + ϕ = d 2.8 v θ φ d 2   , starting from a regular form of the Kerr metric. The metric of a 2 2 + sin ρ Mr 2 2 θ d −  1 2 ρ  − + . The connection to the Boyer-Lindquist coordinates is made via = ± ) and using the expression for scalar field ( 2 r s d = A.1 r and one sees thatric it ( is regular. Applying the disformal transformation ( In the Kerr coordinates, the scalar field reads which, unlike the same metricat in Boyer-Lindquist coordinates, does not have a singularity calculations of section rotating black hole in GR can be written in the Kerr coordinates [ known that in the casedo of the not GR know solution ( whetherMoreover, this since the is scalar also fieldto a is establish coordinate a that part singularity the of scalar inTo the is see the modified regular explicitly disformed gravity at that theory, metric both we would ( the also metric like and the scalar are regular at A The disformed metric andThe the disformed scalar Kerr field metrictation in ( of regular the coordinates Kerr solution, namely that it has a singularity at the CNRS grant 80PRIME andThessaloniki warmly for thanks hospitality during the the Laboratorybreak. course of The of Astronomy this authors of work AUTh alsoC18U04. and in gratefully in acknowledge particular the the kind virus support out- of the ECOSud project We are very happy toeslav thank Dokuchaev, Eloy Yury Eroshenko, Ayón-Beato, Victor EricRobertson Berezin, Gourgoulhon, and Karim Marco Alexey Smirnov Noui, Crisostomi, for George Vyach- CNRS/RFBR interesting Pappas, discussions. Cooperation Scott The program work was forholes: supported 2018-2020 by consistent the n. models 1985 and “Modified experimental gravity signatures”. and CC black acknowledges support from Acknowledgments JHEP01(2021)018 , D 2 2] r ] (A.4) , π/ D [0 2 . In fact, . We set θ ∈ 253 θ 0 d in order to θ 2 − − ≤ ρ + 4 . π r ) for negative R + ] 2 486 4 c a → ϕ ϕ D a − d R, θ d θ (
r ) i 4  θ d P (0) = r θ θ 6 R 2 . is 2076 r o cos sin d , 2 c − 124 + 243 )] + 2 ] ) cos(2 v 6 a a ) θ . d a D D θ − D 4.10 8 c )] 6 a )   − + 2 (2 θ D 2 2 − θ a ϕ appear in the expressions for the solutions, which respectively read 4 13 d + Mr sin + 562 ) + 15 2 2 v R r 4 − . We choose 2 6 c d cos a ) + 3 D a ± ) cos(2 θ r, θ 2 D q θ ( [3 + 2 cos(2 r (627 + 100 2 R )] (50 D 2 4 2 : a D r − Q ) + 18 D sin 1 + 2 2 c ) 2 a ρ a θ r ( − ) cos(2 + 2324 3 (0) = + 15 + and )] 3 4 D D aMr 2 – 19 – θ Q θ θ R [39 + r ) 4 D ρ 2 2 6 2 )] D r θ + 1 + r, θ sin 343 (414 + 517 cos ( )] 2   1  ) + 4 (48 + 25 4 θ r 138) cos(4 D θ r + 32 d Q 2 D Mr − 2 ) + cos(4 2 ) + (4 + 6 − a 2 + 2 θ θ a   D 2 D 15 2 v r 15 + d ) cos(4 ) + 9 cos(4 − + 4 256 h 2 [15 + Mr θ 2  D 6 2 θ ) + ( [cos(4 a r − + − θ 2 1 θ  2 q . These correspond to the horizons of the Kerr solution with a ) is compatible with constant 2 ρ ) i Mr 2 2 cos 2 a ) = 160 [4 + 3 cos(2 a Q 4 c cos 1 + n a a − 4.10 − 6 [33 + 14 cos(2 + ( 2 2 [3 cos(6 2 a [3 (52 + 3 3 2 − 2 D 2 4 2 a ˜ r θ Q 1 a , which has two solutions M Da a  2   , eq. ( p + 12 + 1 + + 4 − − the polynomial equation determining the critical value of the two polynomials sin in order for the solution to be regular at the north pole. This implies that D  ± 4 3 = 0 + − − 0) = 0 ) = 48 ) = [127 + 56 cos(2 ˜ , M D = r, θ r, θ 2 (0) = ( ( ˜ s 2 1 R (0) = 0 d ( 0 ± Q Q rescaled mass. AR necessary condition to haveP a solution to have a common locus with thesince ergosphere. the We differential can equation focus on defining half the of horizon the is interval, symmetric under C Geometry of theWhen null surface R In section appears with the following expression for For completeness, weIn list section here thecurvature polynomials invariants: that we encountered in the main text. B Polynomials the disformal metric in the Kerr-like coordinates, JHEP01(2021)018 . . , ) θ 0 2 2 dθ D R ) and π/ θ sin 4.10 (C.2) (C.1) ( ) 2 2 1 + θ = ρ ( we have H √ . θ R 2 2] is however ˜ − . In other ϕ/ coordinate. MDRa t ) = 2 π/ d θ , π/ , i.e. at some r 2 dr r 2 → [0 = ρ ] . Putting it all ϕ sin θ = 2 ˜ = 0 B 2 ∆ + . We can however ) 2 2) = 0 ρ π/ R = 0 D dζ dζ π/ and . Indeed, at = ( ν 0 ˜ MDRa θ L 2 R branch via our unique initial µ (1 + = 0 , L θ −  2) = 0 2 = 0 3 µν , since we have already used ) ˜ ∆ + ) 0 2 g ( at ϕ π/ a R in the half interval ( − d 0 2 is not at constant + q R 2 π/ 2) = 0 ) , − B θ R = has been rescaled as = ( t ( has even more cross-terms and is not π/ 2 ( d θ R 0 ϕ ρ = 0 ζ H such that labels the branch fixing the sign of . In between ) coordinates our horizon equation ( R B ] MR r ! ) . The condition = and } 1 by [2 φ , π D D aMR r ± r 2 2 2 q = 0 (2 ,L  R θ θ = π/ θ ˜ [ – 20 – (1 + 2 Ma −  (0) = 0 ,L 0 θ 1 − 8 d R { . (instead of B ˜ + sin M D ζ 2 , and the angle = occur at 2 , and θ but there we can choose the 0 D − 0 a Θ 2 ) ζ R d 2 . This hypersurface is obtained by setting L R  = 0 sin R = θ + 2 = 2 = 0 + ζ 2 2 3 R ∆ 2 2 coefficient is positive, which is to be expected since we are within ζζ a dθ ds d ( at the horizon surface. We can now define g 2 MRa t

is spacelike. The determinant of the metric is found to be zero and 0 so that our junction at the equator has a continuous first derivative to  t +2 MR , ∂ 2 ≤ 2 2 ρ dθ dR ) s π/ 2 H  a 2) = 0 = . By direct substitution and after some calculation, the metric reduces to a + θ Θ = π/ 2 dθ and write down the 3-dimensional hypersurface representing the event horizon ( is non trivial, monotonous and therefore we must have that ) d 0 R θ ) ( R θ ) at ( = 0 = ( H R and the magnitude and sign of B Unlike for the Kerr metric, the horizon dζ 4.10 A similar branching occurs at = changes sign in the second half interval as we change branch in the differential equation. 8 a which is adapted to the horizon. We can set 0 the ergoregion and we have a null hypersurface.null The vector (perfect generators square) we form of take the metric suggests that forcondition. the The only possible zeros for where and We can note that the d perfect square: The coordinate transformed metricparticularly replacing helpful given that weset do not explicitly knowgeometry. the It function is easy to verifyis that accordingly in given by dr This is of course coordinateζ related and in thewords following we we have will defineas a a radial new coordinate radialconstant coordinate radial which coordinate is such that the horizon is now at but not unique. Ifa the knee solution singles singularity at out the theof second horizon’s branch equator. pictured As above, we there will will see be this depends on the magnitude up our initial conditioncompatible upon with setting the differentialof equation, ( as one can show by calculating the derivative In order to havewe a need regular solution, onethe must second also branch have in athat regular fashion for is strictly negative. Note that we cannot impose R JHEP01(2021)018 . + ), ˜ M R 4.6 (C.3) (D.3) (D.1) (D.2) = 2) = 0 , which ? , where ) R π/ 2 ( a 0 = 0 + 2 R 2 ? and ζζ R at each order, which is a null ( g Da 2 ? − ) } R 2 ? θ ) = 0 φ ( R ) as a perturbative D n D ,L , at the equator, while θ , where our generator (1+ D.1 . , δR ) for the null hypersur- π ,L . If we set 2 0 = = a = 0 , B is some parameter for the 4.10 = 0 θ ? 1 θ MRa − 6= 0 { θ λ 2 L θ 1 is a solution of the differential 2 D 2 =  ) √ MR = 2 ) θ 2 a , cos ( ϕ θ a sin n L 2 2 R where R − a 2 δR 1 = sin } + n 0 2 the last term on the left-hand side can ) √ 2 Da R 2 φ and hence a ,L 2 2 R + 2 − ) Da 1 + + Da B ( φ  = 0 B ) 2 θ R =1 : D ∞ B a X 2 2 2 n , λL – 21 – ) + (2 D 0 + − 2 + θ sin Da 2 a 2 . In particular, the first correction in the expan- ∆ + + ) 2 a + R − θ ρ R ( ∆ + ( θ ( 2 2 ? + 0 2 n . For small = 2 and also at the poles for R ρ 2(1 2 R , λL δR R − 0 = 1 ∆ + MRD = 0 − ˜ 2 2 λ, ζ M and ) = ), we obtain an algebraic equation for  { D θ is the outer horizon for the Kerr metric with rescaled mass s ) ( 1 2 θ dθ = dR = D.1 ( a for n a  θ δR } − L without loss of generality and Y ] for a similar discussion on the Kerr metric). Then, switching to the 2 δR 1 a 70 ) in ( + a √ 2 + = 1 R with is not a Killing vector, however it coincides with the Kerr Killing generator t ). There is a set of null curves, defined on the hypersurface D.2 , L . Then we see that, a a 0 . This is consistent with the Kerr case, for which we have ? L , L , we discussed the numerical integration of equation ( a 2 = 1 + ) R 1 4.10 a 4 { D = + ) can be rewritten as follows: + 2 ? R = a R (1+ 2) = expansion dλ 4.10 dY = π/ ( a Kerr φ Substituting ( expressed through sion reads series with the perturbation parameter where where we use natural units be considered as a perturbation. We can then write the solution of ( D Solving the equation for theIn section candidate horizonface. via In this a appendix, weeq. perturbation attack ( the same problem with a different approach. Using ( (see for example [ second branch, we can move toagain the smooth second and half well interval defined andagain until the becomes we three Killing. reach dimensional the metric south is pole at The vector L at the equator, asR we need aL smooth solution, we have are tangent to the fourvector dimensional defined trivial throughout extension this of via hypersurface. These 4-dimensional nullcurve. curves These curves are describe defined thefalling last photon in orbits nor skimming falling the null out. hypersurface This without is the candidate event horizon for our disformal black hole. where we set equation ( together we get: JHEP01(2021)018 ) ]. is θ , ( 1 n (D.4) R , δR Phys. , where SPIRE ]. , Metric of IN 4 [ . 4 SPIRE ) around IN ][ D.1 (1975) 905 ]. (2018) L15 34 ]. . Therefore, it is natural 3 615 ]. SPIRE . IN ) (1968) 399. SPIRE θ ]. ( ][ IN n 26 SPIRE R ][ , one should use another approach. arXiv:1602.03837 = IN ) [ R [ θ SPIRE , see figure

( ) ) IN 0 Phys. Rev. Lett. θ ) The general Kerr-de Sitter metrics in all ( R , ][ R ( 0 ]. R P ]. + . Similarly to the approach considered above, Astron. Astrophys. Phys. Lett. A (1965) 918 0 n 2 , hep-th/0404008 ∂P/∂R SPIRE 0 , GWTC-1: a gravitational-wave transient catalog of Observation of gravitational waves from a binary ( [ R 6 -th order in the approximation, the solution at IN – 22 – SPIRE R (2016) 061102 . Since in this case the would-be small parameter n [ ) IN θ − arXiv:1811.12907 [ ( is close to [ n 0 116 ) R R θ ( (2005) 49 = arXiv:1608.03593 ). Notice that the sign of the leading correction R ]. ), which permits any use, distribution and reproduction in 2 [ (1963) 237 53 +1 collaboration, collaboration, ]. = Detection of the gravitational redshift in the orbit of the star S2 n (1968) 280 Da J. Math. Phys. r 11 R with respect to , . This is motivated by our numerical results of section SPIRE 10 ) ) , which is in accordance with our findings in section (2019) 031040 θ IN 0 n SPIRE ( Testing general relativity with accretion-flow imaging of Sgr A* D 9 0 R Imaging an event horizon: submm-VLBI of a super massive black hole ][ IN [ Phys. Rev. Lett. R − CC-BY 4.0 , Uniqueness of the Kerr black hole (2016) 091101 +1 This article is distributed under the terms of the Creative Commons 0 n (for small collaboration, J. Geom. Phys. | R , ( 2 Gravitational field of a spinning mass as an example of algebraically special A new family of einstein spaces Hamilton-Jacobi and Schrödinger separable solutions of Einstein’s equations 117 Phys. Rev. Lett. Da , Phys. Rev. X , , i.e. around ∼ | arXiv:1807.09409 2) = 0 is already present in the zeroth-order solution a rotating, charged mass dimensions Commun. Math. Phys. metrics Rev. Lett. arXiv:0906.3899 GRAVITY near the Galactic centre massive[ black hole compact binary mergers observed byruns LIGO and Virgo during theLIGO first Scientific, and Virgo second observing black hole merger LIGO Scientific, Virgo Yet another version of the same approach is to make an expansion around the solution E.T. Newman, R. Couch, K. Chinnapared, A. Exton, A. Prakash and R. Torrence, R.P. Kerr, D.C. Robinson, B. Carter, S. Doeleman et al., T. Johannsen et al., G.W. Gibbons, H. Lü, D.N. Page and C.N. Pope, B. Carter, 2 π/ P ( [9] [6] [7] [8] [4] [5] [2] [3] [1] [10] [11] Attribution License ( any medium, provided the original author(s) and source areReferences credited. The last expression is obtained byand performing neglecting a Taylor expansion ofwe eq. ( obtain an algebraic equation at eachOpen step. 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