Journal of Cosmology and Astroparticle Physics

PAPER Generating rotating spacetime in Ricci-based gravity: naked singularity as a black hole mimicker

To cite this article: Wei-Hsiang Shao et al JCAP03(2021)041

View the article online for updates and enhancements.

This content was downloaded from IP address 140.112.95.133 on 29/03/2021 at 09:35 JCAP03(2021)041 hysics P le ic a,b,d t ar strop A and Pisin Chen https://doi.org/10.1088/1475-7516/2021/03/041 c Che-Yu Chen osmology and and osmology a,b C 2011.07763 modified gravity, astrophysical black holes, gravity Motivated by the lack of rotating solutions sourced by matter in General Relativity rnal of rnal ou An IOP and SISSA journal An IOP and 2021 IOP Publishing Ltd and Sissa Medialab Department of Physics andTaipei Center 10617, for Taiwan Theoretical Physics,Leung National Center Taiwan University, for CosmologyTaipei and 10617, Particle Taiwan Astrophysics, NationalInstitute Taiwan University, of Physics, AcademiaTaipei Sinica, 11529, Taiwan Kavli Institute for ParticleSLAC Astrophysics National and Accelerator Cosmology, Laboratory, StanfordStanford, University, CA 94305, U.S.A. E-mail: [email protected], , [email protected] [email protected] b c d a c Generating rotating spacetime in Ricci-based gravity: naked singularity as a black holeWei-Hsiang mimicker Shao, J Received November 21, 2020 Accepted February 1, 2021 Published March 15, 2021 Abstract. as well as in modifiedof gravity theories, the we extend minimal a Einstein-scalargravity recently discovered coupled theory exact to rotating to a solution itsof Born-Infeld counterpart scalar a in field. well-developed Eddington-inspired Thisand mapping Born-Infeld General is between Relativity. accomplished solutions The with newInfeld of the solution coupling implementation is Ricci-Based constant parametrized Palatini apart bythe the from theories spacetime scalar the of prior charge mass and to gravity and theInfeld the spin Born- mapping, scale of we the are find compact thatnaked able the object. null to high-energy singularity. Compared modifications suppress to modifications at Depending but may the on even Born- not give the rise removeIn to sign the spite an of additional of curvature timelike the that, divergence singularitycasting both Born-Infeld exterior of shadows, of to coupling the and the the constant, null naked as original be one. these singularities a identical before consequence as and of seen afterfield the by the induces mapping a mapping a relation, are distant peculiarendpoints capable observer their oblateness of held on shadows to the fixed, turn thebound equatorial out the on appearance plane. to closedness the of absolute Even condition the valueshadow though of for closely shadow the the resembling with the scalar scalar those charge, its shadow of which left contour leads aKeywords: to and Kerr sets observational black right features a hole. of small the upper ArXiv ePrint: JCAP03(2021)041 – 1 – manipulations, thus the field equations are not guaranteed off-shell and even structural changes in the Cauchy horizon as well as the central 1 A more precise disclosure of the nature of these compact objects urges us to not just The scalar-tensor setup of gravity has sparked interest due to the natural emergence of scalar fields in 4.1 Curvature4.2 and singularities Shadow 11 13 2.1 Ricci-Based2.2 Gravity theories4 Mapping2.3 between RBGs and Eddington-inspired GR Born-Infeld with7 gravity a scalar6 1 supergravity [18] and in the low-energy limit of string theory [19]. settle for vacuum solutionsmatter but degrees to of freedom. also take Indeed,the into besides Einstein-Maxwell theory, the account the well-studied the inclusion electrically of couplingsstars charged scalar black between [6–8], fields holes gravity has in hairy and been shown blackization to holes [16, give rise [9],17], to boson dilaton effects in black holes [10–15], spontaneous scalar- to be preserved, and this operation does not necessarily reproduce the expected results [4,5]. singularity of both Kerr and Reissner-Nordström black holes [20, 21]. The simplest among 1 Introduction The conservation of angularof momentum physics. plays an In extremely particular,conservation at important is astrophysical that role almost scales, in all a many stellarsuch direct objects aspects as consequence that stars have of and been black angular formed holes,of momentum via carry gravitational those nonzero collapse, spins. stellar objects Typically,on an Einstein’s that accurate General physical are description Relativity endowed (GR).linearity However, with of due a the to Einstein strong theconfigurations substantial equations, gravitational is complexity finding field a and the daunting should non- include challenge. solutions be angular describing A based momentum few such into tricks rotatingNewman-Janis a have algorithm spacetime been general [1] (see static put2] [ forward solution, orin [3] in the for “rederiving” an a most the comprehensive attempt renowned description) Kerr to rithm that one was solution hinges being successful entirely using the on the Schwarzschild metric as its seed. Yet the algo- 3 An exact rotating solution4 of EiBI9 gravity Properties of the solution 5 Conclusions and outlook A Derivation of the mapping B Degeneracy of the shadows on both sides of the mapping 21 11 18 20 Contents 1 Introduction1 2 Theoretical4 framework JCAP03(2021)041 – 2 – Ricci tensor, with the latter defined in terms of the affine connec- and the scalar-deformed Zipoy-Voorhees (ZV) [26, ]27 and Erez-Rosen [ 28] 2 symmetrized The point of emphasis is that without considering any simplifying ansatz, perturbative To make matters worse, numerous unsolved mysteries in the Universe, such as the dark In recent years, a particular class of gravitational theories termed Ricci-Based Grav- The FJNW solution and the Wyman solution were shown to be the same [25]. 2 tion. Besides GR itself, the family of RBGs encompasses quite a variety of modified gravity treatments of slowly rotatingcounterpart scenarios, of a or non-vacuum numerical solutionFJNW approaches, remains solution, constructing a naive nontrivial application the task of rotating that the in fails Newman-Janis GR. to algorithm Take fulfill for [32] the leads instanceup field to the equations in a [4, order line30]. element tosuch More as arrive sophisticated the methods at Clément have exact to transformationrotating be rotating [34] MES conjured that solutions solution. assisted (see, in e.g., the [30, discovery of]33 the andenergy aforementioned references and therein), darkthe matter inability puzzles, to the consistentlythe inevitable incorporate possibility occurrence GR of of andgravity alternative spacetime theories quantum theories singularities, contain theory, of seem more and more gravity to complicated beyond [35, structures, thrust reach and upon36]. thanthe their us the other vacuum ones Needless hand, solutions in in toHorizon are GR, the Telescope say, even let (EHT) wake these Collaboration of alone modified LIGO/Virgo [37] the the Collaboration and recent non-vacuum the (see imaging rotating detection for of solutions.which of example a the [38, gravitational large On waves39]), M87 number by a of galactic the put modified promising center gravity to new by theories era test along the has as withsaid, Event just their observations spacetime begun probe without solutions in deeper precise can into be theories, knowledge the there of strong will the gravity still regime gravitationalextent. be [40, field difficulties41]. Therefore, configurations in That efficient surveying producedwould being methods the be in details designed highly these of desirable to their from obtain predictions a exact to theoretical the analytic standpoint. full ity (rotating) (RBG) solutions theories [42]pursuing has exact caught solutions considerable of attention,is as RBGs formulated it and assuming has alsomulation), that stimulated GR and the thoughts in the on metric a gravitationalmetric and systematic Lagrangian and the is the way. constructed connection This out are class of independent of contractions (Palatini theories between for- the these models is GR withscalar a theory, minimally or coupled MES real theory masslessstatic for scalar solutions field of short) (the the which MES minimal providesmetric theory Einstein- additional have FJNW long-range been solution effects. found obtained in Several by by the Wyman past, Fisher, [24]), including Janis, the Newman, sphericallymetrics and sym- Winicour presented [22, in]23 parameter (and [29]. later which characterizes These theRotating solutions effective solutions coupling have of strength in the of theorytating common the have generalization also a gravitating of been scalar dimensionless the discovered, scalar-deformed field. point and scalar for ZV one this charge metric of work. [30] them This that isishing rotating multipole will a MES moments be Kerr-like solution induced ro- carries by taken the oblate asspherical scalar deformation the symmetry field, due starting in so to even a the in way nonvan- theoften similar static shared to limit by the it asymptotically vacuum departs flat ZV from a spacetimes metric with naked [26, a singularity27]. massless [31], An scalara and interesting source the curvature feature is singularity rotating the at presence MES of the solution surface is where no the exception. event In horizon fact, of it the possesses Kerr metric is located. JCAP03(2021)041 grav- ) R ( f event horizon of the new spacetime (a null – 3 – would-be Given that only the symmetric part of the Ricci tensor contributes to the action, RBGs This work is based on the mapping between RBGs and GR with a scalar source, and singularity). The corrections introducedat in accordance most with tame theincapable the Born-Infeld of prescription curvature can fully divergence resolving of it. the Not naked only singularity that, if to the become EiBI milder coupling constant but is are positive, these thus possess projective symmetry.higher-order gravity They theories have the arepropagating merit generally degrees of of prone freedom avoiding to,nection ghost accompany since field the instabilities is projective connection that non-dynamical,] [57 symmetryformulation so (see of ensures RBGs also GR, no [ 58, in whichspeed allows59]). vacuum of these are The gravitational theories in con- waves to essence inferredinvolving naturally possible nothing from lie nonminimal observations but within [60, couplings the the61]. tograted Palatini constraints gravity out When of are bosonic to included, the matter arrive thecontains fields at new connection interactions the can and be Einstein isauxiliary inte- frame now metric coupled representation to is of an RBGs relatedbraic auxiliary mapping, where to Einstein-frame and metric the the its [57]. metric matter physical fieldsourced The sector spacetime equations by are metric the formally stress-energy in equivalent tensor tomathematical the the of point Einstein the RBG equations of matter frame sector view, viato in the the an some mapping Einstein alge- relation matter frame. thenby source From connects a different with purely a interactions. that solutionexploit of of Hence, RBG this GR rather coupled mapping coupled thanpool to to tackling of find the the solutions classical same field onwell solutions matter the equations established other source on for directly, side. cases but one one where Suchelectromagnetic described can side the a fields matter simply correspondence sector [63]. between by consists RBGsgenerating of borrowing solutions Furthermore, and scalar of from the GR fields different the has [62], RBGs mapping [64, ing fluids been has65], a [42], with rotating and been a charged more shown blackobtained recent hole to and from in intriguing be its EiBI example be- gravity counterpart, successful coupled the in to Kerr-Newman Born-Infeld solution, electrodynamics in [66] GR. the goal is tocounterpart extend solution the within newly discovered thetheory rotating framework that naked of we singularity will an solutionto be RBG of its aiming MES theory. ability for to to with Moreof its ameliorate the those specifically, spacetime mapping contained the singularities is in RBG insolution astrophysical EiBI the objects as gravity, early which [45, the Universe is67–72]. seed asEiBI motivated and We well due gravity. will implement as start the a As with mappingon handful we the to the shall rotating obtain MES GR see, amatter this side new sector mapping exact to on rotating brings a the solutionto the scalar EiBI of the free field side. matter canonical modified sectorinteractions massless Similar characterized by can to scalar by the the the be field EiBI nonlinear structureEiBI thought mass Born-Infeld of coupling of scale. constant EiBI corrections as Their which gravity, ramifications in effectivelyparameter the enters are aside an the into represented modifications from by infinite the the the mass, generated seriessolution, the spacetime of it spin, solution and will higher the as be scalar derivative over an charge. shown to additional that With the regard the to EiBI naked the singularity side generated in and the lies original at MES the solution is carried theories that have madeity their [43], quadratic appearances Palatini inand gravity the other [44], literature, extensions Eddington-inspired of including Born-Infeldhigh Born-Infeld Palatini (EiBI) energy gravity gravity scales, theories [ 45], andwidely [46–49]. their investigated impact [50–56]. These on theories cosmological modify and GR astrophysical at scenarios have been JCAP03(2021)041 (2.1) is the speed of light, c Ψ] , µν That is, causal geodesics can g [ 3 m S +  (Γ) singularities in the sense that they result ) µν ( true R , µν – 4 – g  G L g − √ x 4 d Z = for the metrics. S +) , is the Einstein gravitational constant. We will also adopt the mostly plus + 4 , + , for the units will be used throughout this paper, where πG/c − ( 8 = 1 ≡ 2 2 As is widely known, hosting a naked singularity does not prevent the object from show- The paper is organized as follows. We begin our discussion in2 section by recapping the κ κ See [75] for an explicit example in which spacetimes plagued with curvature divergences are geodesically 3 = complete. 2 Theoretical framework 2.1 Ricci-Based GravityRicci-Based theories Gravity (RBG) [42]full is action a of particular the family form of gravitational theories in which the casing interesting optical propertiesvations to [86–102], a some high ofthe degree. new which EiBI nonetheless In spacetime agreeas fact, have with potential their we obser- barriers naked shall singularities forsky see cloaked of impacting that by the photons. both photon observerphotons the regions that The [103], which encloses rotating photon act an and MES region area theTheir spacetime defines which existence apparent and a is of shadow essentially boundary it images theidentical, in enables cross namely will the section the that be for two thecan investigated, capturing naked appearance be and singularities is traced under to insensitive backexplore certain cast to to later. situations shadows. the the they Born-Infeld mappingas The are modifications. relation a peculiar that spin-dependent This ties oblateness upper togetherconsistency, of bound the will the on two also the solutions, shadowsignificance be scalar as induced of charge discussed. we both by parameter will solutions the imposedto Most by by scalar that showing importantly, observational that of field, we their a assigns will shadow Kerr of well contours highlight black them bear the hole being close with possible astrophysical resemblance the candidates same for black amountmain hole of elements mimickers. mass that and layEinstein the spin, frame foundations thus representation of displaying of the early the the recipe matter field for equations source generating of beingIn solutions, RBGs, section3, a including the we scalar, form the briefly ofrecently, followed and depict the by go the mapping an on rotating with to explicitIn solution construct application its section4, of counterpart of the we in it MES thesubsection examine EiBI theory to devoted the framework to that EiBI through spacetime a was the gravity. remarks primitive properties discovered mapping. study on of of prospective the thec directions shadow generated cast for EiBI by future it. solution, research The with follow summary a and in several section5. The convention high-energy corrections trigger anof additional the divergent scalar behaviorabovementioned matter of singularities on in the the a stress-energy solution are tensor timelike hypersurface outside thereach these previous singularities in null finite singularity.beyond affine them. parameters The Although but the are formationby obstructed of from the naked extending singularities cosmic further are censorship hypothesizeddemonstrating to conjecture that be [76], forbidden it numerous isgravitational studies collapse possible have under for provided suitable such counterexamples initial singularities conditions to]. [77–85 take place as the end product of and signature in geodesic incompleteness of the spacetime manifold [73, ]. 74 JCAP03(2021)041 , ) is λ µν g ( Γ and and (2.5) (2.3) (2.4) (2.6) (2.2) µν ) (Γ)] R g µν ) ( g , µν µν (  R R βν , µν q g µν and the stress- Moreover, since , which itself is g αβ ) [ α = : µν G 4 µν T . g ( ) L µ λ Ψ g R δ ( . R − α µσ is the covariant derivative Γ µν which depends only on the q , ρ να ) is introduced solely for math- (Γ) µ and the affine connection Ψ] Γ , etc. αλ , α ) µν ∇ q µν q − µν ( T µν q , g α g µν [ ∂ ) α νσ + R m Γ G µν − ( S αν µν L . In other words, the connection which ρ µα q q R ∂ µα µν ∂ q = : q λα ν µ + Γ g ) ∂ q T − (  ρ µσ + defined by [62] R , √ q Γ , 2 ν ) να − – 5 – µν µν ∂ q q ( q ≡ √ T is the torsion tensor. In the above equations, we µ − ∂ µν ] , which is constructed solely from the independent ( = = µν R in terms of the spacetime metric µν ρ νσ  λ [ λα q q Γ µν (Γ) q ) αν g µ µν metric − 1 2 ∂ q , where the Riemann tensor is defined as G µν q q √ ( = 2Γ = − R depends generally on the Ricci tensor µαν − L √ α λ µν µν  λ (Γ) = µν Γ G R µν q T (Γ) α L auxiliary = σµν ∂ ∂g ∇ ρ is the stress-energy tensor of matter, 2 µ λ µν R δ R , and m µν − S δ δg . The affine connection (2.6) is then completely determined by algebraic λ µν Ricci tensor  g Γ 2 µν . Though µν − , i.e. T − 5 . √ λ µν q q ) g along with its Levi-Civita connection (rather than the independent connection = 0 Γ − = ( √ µν µν  µν µν being its determinant. There is no hypermomentum sourcing the connection field g λ R T ) (Γ) λ T q By performing independent variations of the action (2.1) with respect to the metric and g symmetrized , etc., and those of the auxiliary metric: ( ∇ ) that defines the true covariant derivative of the geometry [104], and the actual phys- We will set aside theNotice case the of distinction spinor between fields the and deal symbols with used just for minimally curvature coupled quantities bosonic of fields the in physical this work. metric: ) 4 5 g µν ( λ µν solves eq. (2.4) can be written as with equations (2.4) duethe to the action matter (2.1)projective fields contains invariance. not only This coupling allows the usof to to symmetric projective be the transformation part able connection. [57], to of whereasthe gauge the away Levi-Civita the the (symmetric connection torsion Ricci part of by of a the tensor, the) suitable auxiliary the choice connection metric is RBGs given by possess where with defined in terms of thefield connection, equations we can (2.3) removeenergy and this dependence tensor express by resorting to the metric ematical convenience. Dynamics ofmetric the RBG theory (2.1) areΓ still governed by theical spacetime quantities are theR ones that are associated with them, such as R equations and thus carriesfrom no projective dynamical invariance ensures degrees the offorward, absence freedom. it of is ghosts This in worth crucial these pointing theories feature out [58]. stemming that Before the moving auxiliary metric is considered in thetreated Palatini as formulation (with independent the field metric variables). The gravitational Lagrangian associated with have also introduced the connection a scalar functionthe built out of contractions involving the inverse spacetime metric the connection respectively, we obtain the following two field equations [57]: As for the matter sector, we consider a matter action metric and minimally coupled matter fields collectively represented by JCAP03(2021)041 (2.8) (2.9) (2.7) in favor guarantees µν g . Therefore, ν µν µ is the trace of q T , which generally µν and the auxiliary Note that at this T µν g µν 6 µν . g g µν q . = # , T ν µ # δ ν µ  being functions of contractions δ 2 T , and  which links ν purely in terms of ν µ µ 2 + T ν Ω µ Ω G is the Einstein tensor of the auxiliary + Ω L
. G )  ν and q L α ( on-shell −  G Ω R ν L µ − αν µα T q ν g µ " – 6 – 1 2 | T = " Ω − | 1 | ) µν q q Ω p | 1 ( [57]. At the level of the field equations, one finds that p αν µν R q Ψ) = ) = , q µα ( µν q ν q µ ( ν G µ ) = q ˜ in general, they can be algebraically related to the matter content, T ( ) αν µν G ( R µα , i.e. q that is minimally coupled to the auxiliary metric denotes the determinant of the matrix . However, by the field redefinition (2.7), we can always replace ν | µ of the RBG theory will become mathematically equivalent to a problem in the ≡ ν ˜ and µ αν T Ω ) ν ˜ | T T µ q through the relation , ( µν ˜ T µα ν g µν µν q µ q q and the matter fields such that the right-hand side of eq. (2.8) takes on the same G ) = = It is useful to introduce the deformation matrix : q µν ( The tilde symbol will be used throughout this paper to denote quantities in the Einstein frame associated q 6 ν ν µ µ ˜ structure as appears in of with the auxiliary metric. the stress-energy tensor.between As mentioned, with and the right-hand side ofeq. eq. (2.8) (2.8) can is be equivalent written formulation to of the GR metric with fieldright-hand a equations side modified (for of matter the the sector auxiliaryin whose equation. metric) vacuum stress-energy of is Written tensor the in consistentadditional is Palatini degrees this with given of form, by freedom the the in it earlierones. the is gravitational statement sector now that other also than these the obvious two RBG that massless tensorial eq. theories2.2 (2.8) propagate no Mapping betweenThe RBGs Einstein and frame representation GR (2.8) withcoupled of matter the a fields metric scalar field suggests equations theprovided of idea RBGs that of with identifying we minimally itT as reinterpret the the Einstein field right-handpoint equations side the of right-hand GR, of side of it eq. as (2.8) still a contains modified the spacetime stress-energy metric tensor where metric This algebraic equation relates theframe. matter sectors Once in the the EinsteinG correspondence frame and between the the original RBG twoframework frames of is GR, established, andsolutions the eq. of field) (2.7 RBGs equations would andof then GR GR serve or, (and as better vicescalar a versa). fields yet, portal [62, obtain for This64], solutions us idea and of electromagnetic to has RBGs fields correlate been from [63, spacetime implemented65, known in66 ]. solutions previous works for fluids [42], metric Given a specific RBG theory,the the fact deformation that matrix the can connection be is worked Levi-Civita out with from respect eq. to (2.5). the Now, auxiliary metric eq. (2.3) can be brought to the form [63] the existence of anin Einstein the frame matter representation for sector the coupled action to (2.1), with new interactions JCAP03(2021)041 . As have (2.17) (2.10) (2.11) (2.12) (2.14) (2.15) (2.16) (2.13) m  ˜ S , where, µν q and m in the Einstein S φ ν φ ∂ α , ∂ i , µα g is the trace of
q . , . − m ) ) X ν ≡ √ µ λ δ ν − L ) ˜ µ X, φ X, φ ( ( ˜ m X | − ) m L m X, φ ˜ L L X ( µν ( φ . q g m ν , R − − X∂ L  m √ 2 1 φ ∂ √ , + L + α x x m ∂ G X − 4 4 µν L d d L ν is coupled to the auxiliary metric µα g φ µ 2 – 7 – g X ∂ ∂ ) Z Z −| X | | | 2 1 2 1
≡ q Ω Ω Ω m | | | 1 1 1 ˜ − − ν h X, φ µ L ( x p p p 4 X X m ) = ) = d Various constraints on the value of the parameter ˜ ∂ = = L ) = 7 Z . m m . Now, in order for the correspondence to be self-consistent, = ˜ ˜ ˜ 2  1 X, φ X, φ L L / ( / ( φ ν ˜ ˜ ) X µ X, φ ∂ m = m 1) ν ( since we are aiming for asymptotically flat solutions, hereafter we will still ˜ T S µ S m ˜ − δ X ∂ ˜ L m λ EiBI = 1 ˜ L S λ ( − Λ = ( ν µ ˜ X ) ,(2.17) reduces to the Einstein-Hilbert action with an effective cosmological m is some arbitrary function of its arguments, where ˜ / L 1 m ˜ X is a constant length squared parameter associated with the additional Born-Infeld L ∂  Let us briefly review the correspondence between the matter sectors of the two frames |  = ( in the Einstein frame in terms of quantities in the RBG frame. Although we can set 7 µν ν µ m ˜ ˜ leave it explicit in our calculations. constant given by where mass scale at which|R large-curvature corrections become relevant. At small curvature scales which describes adensity general minimally coupled non-canonical scalar field. The Lagrangian not only doesthat the the algebraic scalar equation field (2.9) solution have is to compatible be with the satisfied, field one equations also of has both to make sure presented in [62] for thedetailed derivation. case of The a starting singleby point scalar is field. the RBG We theory refer (2.1) the with reader the to matter appendixA actionfor given a 2.3 Eddington-inspired Born-InfeldLet us gravity now dive intoEddington-inspired a specific Born-Infeld case (EiBI) of the gravityThe mapping theory action where [45] for the RBG the (see under EiBI [56] consideration gravity for is theory the a is thorough given by review). The stress-energy tensor of this scalar matter source reads The other side ofthe the matter correspondence action is of GR the with RBG a theory minimally in coupled the Einstein scalar frame field representation, defined which by we write as frame. As in (2.12),T the stress-energy tensor of the scalar in the Einstein frame has the form which are key toL the mapping, as they allow us to construct the matter Lagrangian density discussed in appendixA, thisLagrangian leads densities to in the the following two frames: necessary relations between the matter The modified matteranalogous Lagrangian to (2.11) in the RBG frame, we have defined JCAP03(2021)041 (2.19) (2.18) (2.21) (2.22) (2.23) (2.24) (2.25) (2.26) (2.20) . With / associated
) λ | − X, φ ( Ω | m L p using quantities in the . = . By applying the defini- ν . m ν µ ˜ µ  L , G . In this case, we start with Ω ˜ Ω m ν X L m µ ˜ L . . L ˜  ∂ X ) ) . − ) − , ν µν µ worked out, the metric field equa- m  ν = µ ˜ λ L ˜ , T 2 T ˜ X, φ ˜ µν ˜ X X ) f ( q  2 − − | − ˜ µν ˜ f ˜ ( X − ν X∂ Ω , µν µ |  ν + R . Using this notation, the EiBI Lagrangian  | )  µ q ν λδ λg δ ) = m µ ( ( Ω αν  + | ˜ ( δ | L g =  2 ) ˜ q R X – 8 – Ω ˜ µν − ν | 1 X, φ g and the stress-energy tensor by inserting eq. (2.7) µ ( + √ µα  p ˜ ˜ ) = ) m = X ∂ X, φ G g µν ˜ = ( ˜ L X g = 1 L − ˜ ( µν + f ˜ ν 1 µν q X, φ ˜ ν µ X ( − ] µ =  1 q q δ m Ω ) = 2 ν  −  2 − L µ ] = 1 [Ω √ − ν ˜ − X, φ µ ( in terms of Ω [Ω m = 1 ν L 1 µ ˜ f Ω Before proceeding to discuss the correspondence between EiBI gravity and GR, the With an eye to generating a solution of EiBI gravity in this work, we focus on the inverse We can now solve for tion (2.5) to the EiBI action (2.17), we obtain the relation from which it follows that density can actually be written in a more compact form as in the metric field equations, yielding first and foremost task is to determine the deformation matrix mapping from GR withcoupled a to minimally a scalar coupled field free described massless by scalar the field Lagrangian density back to EiBI gravity been obtained from solar and109], cosmological including observations the, [67 most105]–107 stringent toexperiments bound nuclear110]. [ to physics date [108, that has been set in consideration of collider where [62] in the Einstein frame,with the and EiBI find theory. the Thismaking can corresponding use be of Lagrangian done eq. density by (2.15) suitably to rearranging arrive terms at in eq. (2.14), and then Substituting the EiBI Lagrangian and eq. (2.23) into the expression above, we get For the scalar field modelhas (2.23) the that we form are dealing with here, the deformation matrix (2.21) the form (2.18) oftions the (2.3) connection-compatible of this metric theory can then be expressed as For our purposes, itEinstein frame, turns which out can be toWe achieved with therefore be the rewrite aid more eq. of practical the (2.20) mapping to as relations express (2.14) and2.15). ( the scalar Lagrangian density JCAP03(2021)041 as a , (3.3) (3.1) (3.4) (3.2) ) (2.29) (2.27) (2.28) m L The line r, θ ( ζ 8 θ 2 sin 2 ∆ . The determinant a ) − , X 2 ∆ , ϕ , ) . d 2 j (1 + a θ ) = x , µν − 2 d g i . 2 i X/ ) , r, θ x λ M ( φ ( d =  . The effect of the scalar field is µ / . = 0 − θ ij 2 λ ∂ 2 ˜ 2 Σ h X φ √ )( , and as a result, we obtain − − 1 µ  X , ,H φ ) + ∆ sin − cos ! ∇ µ 2 θ via X f θ θ 2 µν µ X ∂ θ 2 2 ( a 2 ˜ 1 + X + ∇ X  √ − 2 sin + sin sin 1 + ) 2 2 − R − + ∆d ϕ 2 √ = 1 + a r h and a d  2 | g µν ) − r µν – 9 – ω q  2 ˜ aMr φ , − − X ∆ 2 X X ν ) = ∆ 2 − = ( )(d √ t x − Ω ) = M φ ∂ r, θ 4 | X µν (d ( µ r, θ d g ( 2 ∂ f 1 ρ ) = Z X, φ 1 + H − 1 + = (

1 2 √ r, θ = = m ( µν and j L = = R x 2 ) = 2 GR S d a s i µν , ω d x r, θ T + ( θ d ζ 2 ij h Mr sin 2 2 2 ρ a − 2 − r ∆ ) = r ) = can then be computed to give ∆( r, θ ν ( µ f In their notation, the scalar field is normalized by a factor of Ω 8 encoded in the function with the various functions given by of function of the quantities in the EiBI frame [62]: where This Born-Infeld typewormholes of scalar [112] fieldphase has and [113–115]. been investigated, cosmological for solutions instance, in with the3 context a of late-time An exact accelerating rotatingIn expanding solution this of section, EiBIsolution we gravity of take EiBI advantage gravity ofconsidering by the mapping the from machinery MES a theory exploredby known that so seed was far metric touched in to upon GR. construct in First, an the let prior exact us section. continue Its actionand is the given field equations are which results in the relation as well as allowing us to convert between which possesses thematter square-root [111]. structure thattensor According is to characteristic2.12), ( of the Born-Infeld scalar theories field of in the EiBI frame has a stress-energy Previously, an exact rotatingelement of solution the of geometry the has MES the form system was obtained in [30]. JCAP03(2021)041 , , ) 2 ) a 2 to a ∞ − , the π/ (3.7) (3.8) (3.6) (3.5) , µν 2 + q + only in = r r ( M = µν , θ √ ∈ q r -metric given γ r 2 ± r = 0 d M and the spin r # 2 = 2 M Σ ± -metric) in the MES  ∆ r γ 2 , . . 2 − ! 2 ϕ . 2 2 ) 2 ) d a a Σ 2  ∆ θ a r, θ 2 − − 2 ( − 2 2 2 ζ − , on the other hand, denotes the sin M 2 ) M M ( ∆ ρ / µν ! 2 √ √ " g θ r, θ induced by the scalar field. Instead, Σ ( 2 + − + ) ! ζ . θ ) ϕ , the line element does not boil down to 2 sin 2 M M 2 ∆ r, θ d 0 ρ a ( r t − − − ζ 2 , 2 sin d = → 2 r r 2 θ r M ρ a ( rr

a 2 Ma / q 6= 2 2 − i Σ ∆ sin 2 log – 10 – ν 2 2 + − ) ρ , 1 ) , besides the “ring” singularity (at φ 2 µ 2 M − 1 r a a ∂ ,(3.3) reduces to the Kerr line element as expected. Mar ∆ ( ≤ + 2 4 for − rr 1 + 2 − 2 Finally, the solution of the scalar field is [30] dependence in r Σ ρ −

q . 2

θ 9 2 M 2 Σ = 0 t a/M Σ ∆ − /M d 2 + 2( µν 1 2Σ ρ ≤ ∼ 2 q h  p θ 0 = Σ R = = = 2 ∗ ) d ρ Mr γ R ) = rr 2 µν r and g g r, θ ( ( − and φ ζ acquired using the relation (2.27) differs from the GR metric 1 2 6= 0  ρ = 1 µν Σ g γ − + , = = 1  is the scalar charge parameter, which, together with the mass 2 EiBI s d Σ -component: With everything in place, it is now straightforward to employ the mapping discussed in rr To be more specific, the non-rotating limit of (3.3) corresponds to the scalar-modified 9 in [29] with inherited from the original Kerr solution, the event horizons located at Consequently, the EiBI counterpart of the rotating MES solution (3.3) takes the form Once again, one canBorn-Infeld check scalar explicitly source that (2.28) are the satisfied field by equations the (2.19) solution of (3.8). EiBI The gravity with mapping2.7) ( a that where that of FJNW since therewe recover is a still particular casetheory of that the was generalized obtained Zipoy-Vorhees solution in ( [29]. We see that for parametrizes the solution. For and it can bemore, verified the explicitly Ricci that scalar of the the field solution equations reads (3.2) are indeed satisfied. Further- are now replaced by curvaturefield singularities as on well these due to hypersurfaces. the divergent Hence, behavior the of the solution scalar (3.3) is well defined only for One should note that in the non-rotating limit section 2.3 to findrotating the solution spacetime (3.3) solution of ofscalar the EiBI MES field gravity theory. in that It GR isin was in is shown the correspondence mapped there with EiBI that into the a frame. a free Born-Infeld canonical scalar For massless field clarity with of Lagrangian illustration, density we (2.28) follow earlier notations and use and it describesfeaturing a a scalar-deformed naked Kerr singularity]. [116 spacetimeRicci scalar As produced (3.6) we by come scales a close as to compact the rotating naked object singularity at denote the metric associated with the line element (3.3). corresponding EiBI metric in question.the Owing EiBI to metric the scalarthe field being spherically symmetric, JCAP03(2021)041 , + 0 r ) is 2 (4.2) (4.1) =  > r π/ and the ) = 2 a θ − compared to becomes null 2 is still present ϕ M ∆ ( ∂ /  2 ) Σ 2 . In this case, the a − , the right-hand side + 0 , ∆ + ∆ = 0 2 2 − r ρ (  < r < r → ∞ ∼ a/  λ  R 4 + . . For the EiBI parameter t on which 2 + 2 ∂ is a null singularity. In addition, it 0 Σ Σ +  + =  → r r 2 F on the equatorial plane ( 2 K = − = . or r ∆ . We see that the curvature singularity 2 r shows that it diverges on the hypersurfaces a ∆ ∆ = 2 r ∆

ζ ) ζ ∆) − ) 2 2 g 2 2 2 ( ρ ρ ρ ( Σ M , thus µν  / = – 11 – / 2 2 √ R Σ rr ∆ , i.e. − −  µν g ζ hypersurface is determined by the norm of its normal 2 2 g 2 ∆ ∆ = 0 ρ + = M gets “softened” by roughly a factor of ζ − 4 r 2 µ ) = 0 . A closer evaluation makes it clear that the curvature = ) = ζ ρ + 2 n − = g r ) − µ ( 2 r 2 r r, θ ∆( ≡ n ( R Σ = ζ = ) 2Σ  2 F r r 2 ρ . This explains why there is an additional curvature singularity r, θ p − ( + , with the former case diverging as r F 2 = ∆ ζ Σ Nevertheless, unlike in GR where the divergence of the stress-energy 2 T  ρ ( , which is given by 10 / r 2 ∂ Σ ∆ = 2  is already a curvature singularity, we will only focus on regions exterior to it and ignore the ,(3.8) reduces to the Kerr line element. This is consistent with the property = ζ : 2 + ∼ r n ρ = R Σ = 0 r as well due to the vanishing of the scalar field, which then guarantees that the metric and r The causal structure of the Since 10 after the mapping. tensor is directly transferred toinstead, the curvature the through matter the and Einsteinthe geometric equations, in sectors curvature EiBI couple divergence gravity, in at a different manner (cf. (2.19)) such that and does not concern us. vector field that of (3.6).depicted in The figure1, which Riccithe captures scalar Ricci the scalar as major occursthe traits a when right-hand of side function its is of behavior. positive,coordinate and Another we greater can divergence than infer of thatexterior the to condition will the bebecomes previous satisfied one, at negative, a as radial andadditional is the curvature apparent singularity, condition from if can it figure1. exists, only is For be covered by met the within previous singularity at turns out that the Killing vector of the combination interior structure at, for example, of the MES solution (3.3) at the hypersurface The expression vanishes exactly when was applied did notscalar involve fields any on manipulationthat both of for the sides matter of fields the at correspondence all, have meaning identical that configurations the (3.5). Note ∆ = 0 where we have defined of RBGs thatare their thus deviations absent from inlarge GR vacuum. are Likewise, entirely) (3.8 satisfies attributed asymptotically weak-field tests. approaches to the Analysis thewill with Kerr matter be regard solution to sector, performed at the afterwards and strong-field by properties studying of its the shadow. spacetime 4 Properties of the4.1 solution Curvature andWe singularities embark on thecomputation analysis of of the Ricci the scalar EiBI solution) (3.8 by inspecting the curvature. Explicit latter as singularities are causedscalar by field the being trace divergent: of the stress-energy tensor (2.29) of the Born-Infeld JCAP03(2021)041 ,  N (4.3) (4.4) (4.5) . The = 2 | Σ | . So far, the r ) d r, θ and the coordinate ( r F vanishes, it is evident p 2.4 = 0 ) has a normal vector rr x , 2 ϵ= 0.01 ϵ=-0.01 ϵ=0 = . g : d Σ ϕ  ∂θ x . ∂ a ∆ / d 2 a ∆ +  2.3 ∂F ϕ + of the spacetime (3.8), which are ∆ = 2 ∂θ r ∂F ζ -component of the metric. To begin µ ± ∂ 2 l  = d ), it suffices to compute the norm of : ρ ) rr 0 are the Killing vectors associated with θθ ϕ g ϕ 2.2 r, θ d ¯ > (or ( 1 ∂ + F 2 θθ is “defined” by g  p and x , ) = 0 ± t x , ∂r – 12 – ∂F t ∂ 2.1 d ∂ r, θ  = 0 2 ( 2 a rr F a rr g g + ∆ + ∆ submanifold on which 2 = singular hypersurface. Since there is no signature change 2 r 2.0 µ r and write + N at which the additional curvature singularity appears. µ = − t ) of the Ricci scalar on the equatorial plane for different values of µ l ) = 0 r + µ N ∂ 0) = 0 = d : µ r, θ ± r, l ( 1.9 ( v r > r F d F R (see [118] for a similar situation). Fortunately, we have no intention . The idea is to perform a coordinate transformation in order to remove ) or N 5000 10 000 30 000 25 000 20 000 15 000 r, θ ( are two new coordinates, and F , which is set equal to one, and we have chosen the scalar charge to be ¯ ϕ 2) = 0 corresponding to the curvature (3.6) of the rotating MES solution. The radial distance is GM . Radial profile ( and is also a Killing horizon, just as in the case of its GR counterpart (3.3). r, π/ v = 0 ( through + The other singular hypersurface  F r θ = given by that our currentill-defined choice norm of of coordinates is not adequatewith, and we has find to the be (normalized) altered principal to null avoid vectors an relations in (4.5) doon not constitute a validof coordinate carrying transformation out due further tocausal analysis the character based dependence of on the thein transformed the line element metric except on to this extract hypersurface the ( the coordinate singularity caused by the problematic with norm To examine the causal structure of this hypersurface on which Paralleling the techniques ofwe bringing take the the Kerr ingoing metric direction into the Eddington-Finkelstein form, where the Figure 1 on this hypersurface as well [117], where with in units of vertical line marks the value of the stationarity and ther axial symmetry of the metric (3.8). As a result, the hypersurface transformation) (4.5 is well defined. The induced metric on the equatorial plane can then be JCAP03(2021)041 (4.9) (4.7) (4.8) (4.6) (4.10) (4.11) . 2 ϕ d ¯ in) (3.8 are components ! 2 ϕ belongs to the ϕϕ g r , and the energy the hypersurface Ma τ and 0 2 t + , and 2) = 0 ,  > 2 tϕ i a g z , + r, π/ . The same conclusion can ( tt aL 2 + , g r F − ) ,

r ) ( i 2 + z r > r a R ϕ 1 aL + d ¯ − 2 x − ! r d ( 2 ) a 2 ∆ E Σ 2 2 a 2  h ∆ r r 2 − 2 + a = ϕ 2 − r d ¯ η , µ + ∆ ( v eq ¯ 2 ζ N d – 13 – E 2 eq 1 r h µ of the particle are conserved quantities associated ζ

submanifold. Therefore, for ¯ , respectively. These two equations have the same N z a r ∆ ϕ = = Ma ) + L ∂ 4 2 2 , and the geodesic equations for the + ˙ r 2 ˙ aE δθ − r 2 0) = 0 2 x aE ! and − = d ! 2 r, z t ( E − 2 v r 2 ∂ L E r ρ z F (

a L

, we see that the surface satisfying + 2 d = = x 2 ˙ t ˙ ϕ v 2 2 d r r  . The majority of the orbits are certainly not confined to the equatorial r M δθ 2 components, which are given by constant hypersurfaces on the equatorial plane. The vector field normal to . This scheme was considered in [119] when studying the shadow of a rotating is in fact a timelike singularity. θ − 2 2 + = 1 2 according to (4.6), which is clearly positive for  Σ π/ x π/ 2  and − ≈ r = π/ θ = θ On the equatorial plane, = 2 eq ∆ = 2 θ and azimuthal angular momentum s ζ 2 d read black hole whose metricthat functions the analytic have results rather obtainedregarding complicated in how the forms. this shadow way is are Inneeding affected still this by to able the resort work, to presence to shed we of light a the expect on parameters full-blown in some ray-tracing the key analysis. model, features without be drawn by consideringρ the plane. That said,on such the equatorial an plane approximationangle far works away well from the enough object when where the photons observer essentially is arrive at located a polar where the dot denotesE the derivative with respect to an affine parameter 4.2 Shadow Now let us turn ourThe attention vital to the ingredient shadow inphotons cast by determining undergo the the EiBI unstable shadow rotatingequations spherical compact is of scalar motions the the object. around surrounding spacetimethe the photon (3.8) absence central region can of object. be in anfor derived which analog the using The radial the of null and Hamilton-Jacobi the geodesic render polar approach. them Carter so angular constant With by components for onlywith are considering this photon in spacetime, trajectories general close the not to geodesic the decoupled; equatorial equations however, plane we and can work with the Killing vector fields family of at forms as those of a Kerr black hole, as the metric components these hypersurfaces has norm rewritten in an Eddington-Finkelstein form such that the line element becomes From our “definition” of free of Born-Infeld andfor scalar the corrections. The corrections come into play in the equations JCAP03(2021)041 , is = 0 ph being r (4.14) (4.15) (4.16) (4.12) (4.13) ph r , Q | ) r ph ( r 0 = r R

[120]. Another , with  2 r 0 and Y , i.e. /E ≥ 2 2 ) − π/ ph ) ≡ Q , it further restricts the r ) = 0 . , r = ( 0 η ) ph η # θ 2 . r 2 a ( ) far away from the object, it 2  > Y − − a 2 ( R 0 and Σ r − M M ( − never reaches zero, and thus the )∆ . These conditions demand that ξ / in the observer’s sky to describe r /E . Since the spacetime (3.8) is only 2 2 ) z ) M Σ r ) + 2 ( + ( L − 2 M − 6= 0 r # represent the apparent perpendicular eq r eq ≡ r ) η  ζ 2 α, β Y − ( ζ ) ) r β ( , 2 ξ ( r " , Y + ( M 4 eq − r ∆ ∆ 1 2 ζ ph ) r − and , whereas when − = r − 0 = r = r r α r ( 2 (

0 and ) r " )  – 14 –  < 0 Y − 2 aξ eq ( + ) a > 2 /ζ − , which in turn requires M ) 2) = ∆ 2 − Y (∆ a ∆) when ξ of these photon orbits are parametrized solely by the 2 + 2 2 = 0 ( eq + r + 2 r, π/ ζ η − ( . That way, when light originating from a distant source ( − r 2 ph / r r 0 ζ r 2 | 2 ( ≡ ) ¨ r 2 Σ ) ≡ > a and r > r  r aξ ) ( 2 q ) = ( r ξ ph and − ( Y r r r − ( | 2 eq ) eq a R ζ r ζ ) + ( = 0 r + 00 2 ph R r r | ( , the range of which is dictated by the condition ˙ r Y − ( 1 ph ∆ r  Ma eq is the function (3.4) evaluated on the equatorial plane 2 ζ ) r ( ) = ) = eq Equations (4.10) and (4.11) govern the propagation of light near the equatorial plane Given that the spacetime (3.8) is asymptotically flat, we can set up a coordinate system ζ ph ph r r ( ( ξ η second requirement does notis trouble automatically us satisfied in for any way. As argued earlier, the first requirement of the spacetime (3.8).that constitute We the are photon particularly region. interested A in photon unstable orbit of spherical constant lightlike Boyer-Lindquist orbits radius as in figure 2, and adopt the celestial coordinates discussion of geodesics specificallyWe will to come regions back exterior to to this the point additional later timelike as singularity. a consistency check. where The kinematic quantities orbital radius significant property of theperturbations, orbits in i.e. the photontravels region near is that these they unstableescape are orbits, and unstable reach it under an radial willregion observer thus either at defines spiral infinity, a insky depending bright of and ring on the on hit the observer. the impact the boundary parameter. naked of singularity, The an or unilluminated photon shadow region inthe the outline of thedistances shadow. from the The boundaryequatorial coordinates of plane, the respectively. shadow For an to observer the at symmetry a axis distance of the object and to the well defined outside the naked singularity at characterized by where the prime standsfor for spherical the photon derivative orbits with the respect constants to of motion must satisfy where and We have introduced twothe dimensionless Carter-like parameters constant for orbitsmake near sense the only equatorial when plane. Note that eqs. (4.10) and (4.11) JCAP03(2021)041 -axis. (4.17) (4.18) z ,   0 , θ can be obtained by 0 r The intrigued reader

푥 ) plane, which is normal to /dr 11 δθ ) β dr ( - d δθ α ( 2 0 d r observer   and →∞ 0 η . r √ = lim ± dϕ/dr 0 푟 = yields 0 . , β r 2 . Now, – 15 – π/   2 0 ≈ ξ , β , θ π/ θ 0 − r

= 0 = 0 dr 휃 dϕ θ α 0 푧 θ 푎 sin 2 0 r plane) − plane, and the symmetry axis of the object to be aligned with the 훽   - z - 훼 ( x does not enter into these parameters, signalling that the appearance of observer’s sky observer’s →∞ 0  r = lim α is the inclination angle between the symmetry axis of the object and the direction . Schematic illustration of the celestial coordinates. Due to the azimuthal symmetry of 0 θ them to be so by fixing the polar angle Recall that we were able to work with separable null geodesic equations (4.10) and (4.11) since we had 11 forced the direction from the object to the observer. can be shown that [121] where That is, thedetermined position by eqs. of (4.14)Born-Infeld the and scale image (4.15) of for unstable the spherical shadow photon contour orbits. on Notice the that celestial the plane is fully is referred to appendix B for an explicit proof. This degeneracy applies to other solutions the shadow will notscale be at affected which they bystudying manifest, the the in higher-order shadow (2.17) image derivative andSuch of terms, a (2.28). both as degeneracy the Therefore, can well MES be we asthe solution accounted are the photon for (3.3) in by length and region fact the the unchangedfield fact simultaneously new that so configurations EiBI the long for solution mapping relation (3.8). as (bosonic) (2.7) the matter will are null leave geodesic spherically symmetric. equations are separable and the of the observer (seeequatorial2). figure plane, In so our we case, shall we take are limited to light rays propagating near the The observer sees the shadow of the object as being projected onto the Figure 2 the object ofobserver interest, to which be is on situated the at the origin of the coordinate system, we can choose the combining the geodesic equations (4.9)–(4.11).and then Substituting retaining them into only the the expressions leading above terms in JCAP03(2021)041 8 6 To an exceeds | 13 4 Σ | 0.3, 0.4, 0.45 2 α/M Σ = 0 0.9, = fixed, the boundary of M / -2 the MES and the EiBI a a -4 that lead to closed shadow 6 4 2 0

-2 -4 -6

both M / β | Σ | 6 . The dashed black curve illustrates the | 4 Σ | -axis bears great resemblance to that of a 2 0.3, 0.5, 0.7 α and , but reduces to an open arc once | α/M Σ = 0 Σ | – 16 – a/M 0.7, ), the boundaries of their corresponding photon = -2 ), whereas the red, blue, and green curves correspond to M / a -4 Σ = 0 Σ = 0 . A precise description of the overall shape of the shadow

4 2 0 6

-6 -2 -4

M / β δθ 2 + 6 . To our surprise, for values of cr π/ Σ 4 ≈ θ 2 0.3, 0.5, 0.7 takes the values above each panel sequentially from left to right. | Σ α/M Σ = | 0 . In figure3, it is clear that with the rotation parameter 0.4, Σ = -2 . Shadow cast by both naked singularities (3.3) and (3.8) as seen by a distant observer M / Still, in our case the structure of the shadow will be influenced by the scalar charge a Figure3 offers a visualization of the apparent shape of -4 12 A more complete treatment incorporating numerical ray tracing followed by a quantitative analysis of the This type of naked singularity is dubbed a strongly naked singularity (SNS) [86], as opposed to the ones

2 0 6 4

-4 -6 -2 13 12 M / β size and distortion of the shadow will be carried out elsewhere. that we are consideringthey in are covered this by paper, photonblack which regions. holes fall [86], WNSs into and can the can in general thus category give possibly of rise act weakly to as naked observational black features singularity hole similar (WNS) mimickers. to since that of Figure 3 on the equatorial plane for different values of shadow contour of a Kerr black hole ( the cases where generated through the RBG/GR mappingseveral involving preceding spherically works, symmetric e.g. mattershadow [64], sources displays in and its is insensitivity to just deformations one of of the many metric. instances [ 122–124] in which the rotating scalar objects. We cannaked see singularities, from they the figure areexternal that, still photon despite the capable region. underlying ofbeen entities That casting being the known shadows shadow (see, thanks is e.g.,candidates to not that]), [125 the a can and existence produce signature it102]. of them unique has [87, to an , 90 been black96 , holes establishedparameter 97], has that even long naked when singularities lacking a are photon also sphere [100– the shadow region is closed for small enough a certain criticalcontours value (including the Kerr case regions intersect the equatorial planethese in photon circles regions of have the inequatorial exact common plane, same are radii. which the is progradein In and other figure3 why words, retrogradecoincide. the what circular leftthe orbits Therefore, image on and as of the right far a endpoints as closed the shadow of contour observer the nearby on closed the the shadow equatorial contours plane is concerned, Kerr black hole, notplane to approximation mention that thisas part seen of from the image othernot is inclination dwell accurate angles too under is much theobserver beyond on on equatorial the the it, equatorial scope other plane, of than besides the to the approximation, make fact a so that the few we scalar will qualitative objects comments and below. the Kerr black JCAP03(2021)041 and takes M Σ → , there is no a of the photon (can be either = 0 η . a Σ M 1.0 , so the scalar charge = 3 Σ r 0.8 , we find that the prograde circular cr ), we can naturally expect that the 1 Σ 0.6 , while parameters lying in the shaded region < a | cr will deviate only a little from that of a Kerr Σ Σ | a/M so that the prograde circular photon orbit can 0.4 – 17 – cr Σ for each spin as a whole cr of the naked singularity spacetimes (3.3) and3.8). ( The dashed Σ is the equatorial circular orbit at ) 0.2 | Σ | δθ 2 + a/M, ( π/ 0.0 ≈ , which was purposely excluded from the parameter space in figure4, 0.3 0.2 0.1 0.0 0.7 0.6 0.5 0.4 limits, since the closedness requirement alone forces the scalar charge to θ , the photon region gets elongated radially inwards, but at the same time, 0 = 0 | Σ  Σ a | is just slightly above the critical value → | cease to exist. For this reason, a tiny segment of the shadow contour near the a Σ | η . Parameter space The point When has to be below the critical value | Σ exist, which is ascalar necessary charge condition that for allows thethe a shadow closed scalar contour shadow charge to contourcorresponding be to is closed occur closed. constrained shadow is contour The to shown range in be of figure4. small the Seeing ( non-rotating that is a special caseturns that out that deserves more theon scrutiny. only the consistent assumption When spherical the photon spin orbit parameter that is we exactly can zero, work it with premised black hole. In particular, they willbe practically be zero. nearly identical in both the extremal photon orbit onwith the small equatorial planeleft and endpoint a is few nowfurther of enlarge missing, its and neighboring thean spherical boundary increasing photon of number orbits of theshadow contour prograde shadow to is spherical be no photon| longer chipped orbits closed. away fail from to As the remain, we left thus (see causing figure3). the Hence, the scalar charge prograde or retrograde). This orbit happens to exist for all values of critical scalar charge since the circular orbit on the equatorial planehole exists have for similar all shadow sizes,less notice deformed from vertically when figure3 thethatpart object in the picks the shadow up metric of a inpassing the scalar a object through charge. way is the that The more equatorial it reasonon or plane tends is the (cf. to that shadow (4.14) suppress in the and termsthe polar (4.15)), past of angular where which its the speed then “oblateness”. shadow has usuallyon This receives a is the a notable to variation left in be effect due sizethe contrasted as to NUT with well known as charge parameters a examples [129], such horizontal in or distortion as others the in electric various extensions charge of [126, the127], Kerr the solution tidal [130–136]. charge [128], give rise to closed shadow contours when observed from the equatorial plane. For Figure 4 line indicates the critical scalar charge JCAP03(2021)041 . − a . It ph | r 

| ∆] ζ 2 ρ [ , it can be confirmed that the there is in fact an additional cr 0 Σ | ≤  > Σ | . 0 – 18 –  > parametrize the solution along with the mass and spin holds true for all of the spherical photon orbits characterized Σ 0 > within the equatorial plane approximation. The information of , ensuring that the entire photon region is safe from the additional ph 1 r . | 0 ) = 0 r ( . a 00  R as long as and the scalar charge 0  > Last but not least, for scalar charge in the range In an attempt to achieve a solution free from naked singularity, we turned to study Equipped with the solution, we then revealed that there exists a photon region covering 2 Σ  is unconstrained for just the equatorial photonnon-rotating orbit case. is not The yetstage single enough piece is to of that provide knowledgeSchwarzschild insight its black of into hole left the the when shadow and shadow seen that of right from we the endpoints the can equatorial match deducestability plane. with condition at this those of the shadow produced by a by eqs.4.14) ( and (4.15). Moreover, one can verify that these orbits satisfy the extension ofits the capability rotating to MES smooth out solution singularities in by EiBI modifications gravity, at a the theory Born-Infeld that scale is known for 5 Conclusions and outlook Exact rotating solutions offields either are GR of or pivotal modifiedto importance gravity due serve theories to as in their toy thethese ability presence models to rotating of describe solutions for matter realistic are studyingof compact currently the objects illuminating still interplay and possible scarce, of andtheories signs strong this is of gravity the may new with minimal severely physicsfrom Einstein-scalar hinder matter. a (MES) at the theory massless However, progress astrophysical which scalar scales.and exhibits field. it extra The was A long-range found simplest rotating effects to solution of possess of such a this naked theory singularity. was discovered recently [30], timelike singularity that is present when 2 was shown thatframe a representation theory where the of fieldfor equations EiBI correspond an gravity formally auxiliary coupled toLagrangian. the metric to Einstein sourced equations This a by correspondence matterGR and the provides source EiBI a same gravity admits such mappingon an matter that the between the Einstein but solution the other on describedwith spacetime one side by side the solutions simply can rotating a of be MES throughsolution obtained different solution a from of matter as its transformation EiBI the counterpart seed of gravitywhere metric, coupled the we to metric. generated a an Born-Infeld-type exact In scalar analytic particular, rotating field starting via the correspondence, Unfortunately, the naked singularity in thebehind MES solution a is neither horizon resolved norstill after is resides it mapping at concealed the what wouldits solution be curvature into the event divergence the horizon suppressed. Born-Infeld hypersurface of setting. Furthermore, the new for The spacetime, only singularity with the naked singularity without comingto in draw contact connections with with it.work, observations we This by have set limited analysing the ourselves stagefar the to for away shadow from considering us cast the to a by object proceed stationary the where observer object. light on rays the In that equatorial reach this plane can be approximated as propagating at timelike singularity exterior to theusing previous null eq. singularity. (4.10) On that top radialsingularities of null in all rays a that, propagating one finite on canextended. amount the verify of This equatorial affine is plane sufficient time, reachencounter to beyond both in lead which of this us the these new to geodesics solution the cannot conclusion are be that geodesically further the incomplete. naked singularities one may JCAP03(2021)041 ), 1 < | Σ | as seen by the closed . The scalar charge must, at Σ) a, ( – 19 – Therefore, the necessity of their existence rules out a huge 14 . For such an observer, one would merely need the unstable spherical 2 π/ ≈ θ This work can be viewed as being complementary to [64–66], which all together build Strictly speaking, our line of sight is far from being on the equatorial plane of the M87 black hole [137]; 14 nevertheless, we highly doubtthe that boundary there suddenly will breaking off) be when any viewed drastic from change different in inclination the angles. structure of the shadow (such as observer on the equatorial plane,Event which is Horizon inconsistent Telescope with [37]. theportion shadow image of unveiled possible by the configurationsthe in very least, the be parameter below space ity the solutions. critical value The in critical order scalar to charge ensure the was validity found of to the be naked singular- a small number in general ( photon orbits near the equatorialdecent plane accuracy. to describe The the calculations approximate weretain shape thus immediate of considerably the analytic simplified, shadow results with andwe that we showed capture were that able under a to circumstances fewmatter ob- where prominent fields the are features null spherically of geodesic symmetric, thethe equations the mapping, shadow. are shadows independent are separable First, completely of and identical the ter the on sector. high-energy both Born-Infeld So, sides corrections we of haveseed to essentially GR MES studied and the solution approximate to and shadowthe the corresponding the prograde mat- to generated spherical both photon EiBI the orbitsdisappear solution that at as remain the the on same scalar orWithout close time. charge these to exceeds unstable Next, the a orbits, we equatorial critical the plane found shadow value begin that boundary that to will depends no on longer the be spin parameter. up a familyIn of short, spacetime not solutions onlygenerating in have we exact the confirmed solutions framework once withobjects of again a EiBI the in scalar reliability gravity extended of field,Regarding with the theories the but matter new mapping of also sources. solution, procedure investigatinginitial gravity enriched in its conditions the by formation as through discussion contributing the well ofray collapse a as tracing from compact its new would reasonable be stability solutionstudies the and to natural to the explore follow-ups entire the whether to structureor the this literature. of not solution work. there its truly It are qualifies shadow isbe extraordinary as in compelling observable. deviations a detail to black in carry via hole On the outsearch mimicker shape for a these and other of wider whether possible the level,predictions non-vacuum shadow rotating it that that solutions might are is to potentially available both expandwork for our along appealing ongoing toolbox this and and of direction future theoretical necessary is observational underway. to tests carry of on gravity. withAcknowledgments Further the The authors wouldcomments like and valuable to discussions. thank WHS and Hsu-Wen PC Chiang are supported and by Yun-Chung Ministry Chen of Science for and well-aimed a polar angle which implies that the full exactwill shadow, only even when deviate observed a from little differentthe inclination from effect angles, that of of the a scalarthe Kerr field black shadow hole. on vertically the Moreover, inwards we originalthis also Kerr while is demonstrated shadow true leaving that is, so the qualitativelya far speaking, left novel only to type and deform from of rightby the modification endpoints the viewpoint to intact. parameters of the in anshadow shadow Although other equatorial-plane could that Kerr-like possibly observer, is metrics. serve it as distinct is Such a from nonetheless a rather the direct highly ones indicator non-degenerate of typically imprint scalar induced on field effects the in observations. JCAP03(2021)041 by ) , and (A.3) (A.2) (A.1) (A.5) (A.4) ν µ ˜ X, φ X ( X and = ν . After deducing µ X δ µν , it can be formally , we find that q ν λ µ α T Ω .  ν µ , δ . , ν ν  µ ν µ µ 2 T ˜ : X X , X ν ) i m ) µ + n m L X T  G ) L X is a combination of 4 L X, φ X T Λ (  ν ∂ are proportional to itself, or to be more µ 2 ( X ∂  X, φ f − ( | | T ) φ 2 ν ν + Ω Ω f µ | 1 | 1 ν T µ φ ∂ X, φ p p  ( δ ) + α – 20 – ) | n ∂ = = a Ω in favor of the auxiliary metric | 1 ν µα m X, φ µ g =0 ˜ ∞ X, φ ( p L X µν ˜ n ( 1 X ˜ g 1 X f ≡  f = h ν m on ˜ µ for a scalar field described by the action). (2.10 Note that, = = ˜ Ω X ∂ L Ψ) = ν ν 2 , X ˜ ν µ X µ µ / ∂ ) µν are functions dependent on the scalar field model. Now, we can in terms of the matter field so that eq. (2.7) can be put to use. Ω X  X ν q ) ( µ ν δ µ ν µ m Ω ˜ T X, φ L . Therefore, any power of from eq. (2.7) and then substituting (A.2) for ( ( 2 X φ f 1 − λ − ν ∂ n µ λ and α X X ) ) Ω = m µα L n q is generally a nonlinear function of the stress-energy tensor X, φ is the mass scale that characterizes the high-energy corrections in the RBG, and X ( X ν = ∂ 1 µ Λ f φ Ω α = ( ∂ ν µ µα specific, With this relation attaking hand, its trace. we are Let us then now able return to to obtain solving an eq. (2.9): expression for where get rid of the dependence of g by construction, all powers of where T as a result, eq. (A.1) can be simplified as Equating the off-diagonal terms on both sides, we get which, together with eq. (A.3), yields the relation Technology (MoST) grant No.108-2811-M-002-682 109-2112-M-002-019. and Institute CYC of is Physics,Department supported Academia of Sinica. by Energy MoST PC under is grant contract also No. No. supported by DE-AC03-76SF00515. U.S. A Derivation of theWe mapping review the steps thatare were relevant carried to the out mapping inbetween discussed [62] RBG in to this and arrive work. at GRnamely the The coupled (2.10) goal relations and to is)2.14)–(2.16 ( (2.13). to that thethe establish To same field a do equations correspondence scalar so, of we field theas must but RBG stated solve theory in described the are section algebraic by in2.2.the equation different the deformation (2.9) But actions, same matrix in prior form order to as that, that the we Einstein would equations like of to GR, first find a general expression for Since expressed (in matrix notation) as a series expansion in JCAP03(2021)041 have (B.1) (B.2) (A.9) (A.6) (A.7) (A.8) ) (A.10) r, θ ( µν g coming from the φ φ . . , α ) m ∂ m m ˜ L is the action function. Let us L . µα φ φ ) q ∂ ∂ A θ − L  ( g coordinates. We will focus on the q θ m m , ) − 2 − 2 ˜ A L , L ν √ √ ˜ p X X m ∂ µ ) + = L =  p r φ X∂  i ( q ∂ µν t, r, θ, ϕ r φ φ ( − | g + α α . In fact, it follows from eqs.) (A.3 and ( A.5) A ∂ ∂ 1 2 Ω √ G m | 1 + L − L µα µα = p g ϕ q X (2 – 21 – z φ = ) ∂ |  = α L m ∂ Ω m A | 1 m ˜ L + ∂τ ∂ L ˜ µα and L p ˜ X X g φ . Moving on to identifying the diagonal terms in eq. (2.9), ∂ Et m ∂ ) ∂ ( ν ˜  0 = L − coordinates of a light ray. By utilizing the Hamilton-Jacobi m g ) = ˜ is the affine parameter, and = q X L θ = − ∂ τ − X µ √ ˜ , ∂ A √ X, φ  ( ν and ( h ˙ µ x g µ r m have to satisfy for the mapping to work. Note that the entire ∂ with respect to the scalar field give ˜ ∂ − L µν ) m g √ ˜ S = ˜ X, φ µ ( and m ˜ L /∂x m S A ∂ and = ) µ p X, φ ( m Hence, the equivalence of the two scalar field equations (A.7) and (A.8) further demands that between the partial derivatives it is then clear that So far, we have obtainedL two conditions from eq. (2.9) that the Lagrangian density functions two matter actions (2.10)Variations and of (2.13) are equivalent so that they admit the same solutions. and respectively. Using eqs. (2.7)the and two (A.3), equations we above find are that in the fact terms equal: inside the square brackets in and this completes the derivation of the mapping relations (2.14)–(2.16). B Degeneracy of theIn shadows this on appendix, both wetime investigate sides under metrics of what related the by conditions (2.7)a the mapping will shadows stationary, be corresponding axially degenerate. to symmetricbeen space- Let spacetime brought us solution to begin the whose on Boyer-Lindquistgeodesic metric form the equations in components RBG for the the side andformulation, consider the desired nullequation geodesic equations can be obtained from the Hamilton-Jacobi where procedure of mapping betweeneq. solutions (2.7) of GR involves only andthis solutions the to of metric be the while consistent, given the we RBG should scalar by also field means make configuration of sure is that left the field unchanged. equations For of that eq.) (A.4 holds even when also assume that theJacobi spacetime action we can are be considering written admits in the a form separable solution, so that the JCAP03(2021)041 , ) = r . , we # µν (B.7) (B.5) (B.4) (B.6) (B.3) (B.8) ) q θ −Q ( 2 Ψ = Ψ( -dependent )Φ θ will respect at r, θ can be written for (nonlinear) ( is anisotropic even ) 1 ν µν α - and ν q µ r r, θ F − F ( Ω , α ) 2 µ , µν θ  q F ( θ 2 Ψ) θ = ∂ A d ( d ν ν , + Φ µ µ  elements are nonvanishing in  2 ) ) K K ,  r θ ( ( θθ  θ 2 1 ) . θ Ψ) A r  d , leading to identical shadows. The R ( d ) ) )Φ 15 2 , ∂ . θ and  + (  R 2 ) ) r, θ r, θ (Ψ Q θ ( ( θ 2 rr +  ( being the electromagnetic field strength θ ( F 1 θ ) 2 µν ] + Φ r Q q Φ Ω ν in the matter sector, it is always possible to (  + " 1 ) Q − ) A ) ν + Φ  r r µ µ R − [ ( ) ( and 1 ) δ 1 ∂ 1 θ Q – 22 – r, θ r -components as (  ) = ) 1 ( F R θ 1 r ) r ) = depends only on the matter fields, and the matter Ψ) ( θ r Φ r − 2 ( r, θ = 2 1 ν ( Ω , ∂ 1 ( 2 µ R = = Φ − µν Ω R (Ψ 2 2 - and µν Ψ = Ψ( + g ) ˙ ˙ 1 r r θ F = = 2 F 2 2 2 rr 2  θθ r, θ ˙ ˙ r θ g g ( r on the GR side of the mapping can be expressed as r 1 . Then, by equating both sides to the Carter constant 2 A ) = rr 2 θθ z d F q q d µν L are functions of their respective arguments, and they contain the q  + r, θ is basically proportional to the kinetic term of the matter fields, 2 ( ) 2 ν Φ r ν  µ and ( µ 1 r Ω r K E A R d for scalar fields as in eq. (A.2), whereas , thus it is clear that only its d 2 , and ν 1  µ Ψ) ) Φ r X ∂ , ( ( 2 1 = R R ) ν , | ∼ µ 1 . For bosonic matter fields ν K µ r, θ R ν ( α r K r | Ω Though it is possible that general RBGs admit branches of solutions where Ω 15 µα g organize the deformation matrix into the form in the presence ofaddition, the isotropic anisotropic matter branches were sourcesin shown [138], physical to applications. contain the pathological EiBI behaviors [138] gravity and theory should treated be discarded in this work does not. In The counterpart solution One can see that if the solution for the matter fields are spherically symmetric, i.e. get the geodesic equations for the where then this leads to the following null geodesic equations: Now suppose that, after inserting) (B.2 into eq. (B.1) and separating the where as As discussed in section orbits 4.2 areof completely theTherefore, main determined with text, by the the above the locations equationsfactor function in differing of front inside from the of eqs. the the unstablesame (B.4) square spherical square brackets, and for we photon (B.5) both brackets can merely conclude spacetime inequivalence by that solutions among eq. the an the photon overall (B.7). shadows regions of aremapping spherically exactly symmetric follows the spacetimes trivially related from by our the discussion. RBG/GR constants of motion tensor. The tensor field configurations are unchanged by the mapping (2.7), the GR metric terms, we arrive at an equation of the form electromagnetic fields], [63 with i.e. our consideration. Moreover, since least the time-translational and axialequation symmetries for of null the rays RBG solution. in the The spacetime Hamilton-Jacobi described by the GR metric JCAP03(2021)041 18 ]. 6 , , , 175 , 20 , 380 , (1993) 2411 . 47 (2017) 5 20 (1994) 5155 J. Math. Phys. arXiv:1308.6631 Phys. Lett. B , 50 Zh. Eksp. Teor. Fiz. , ]. , Nucl. Phys. B (2013) 064046 , Phys. Rev. D 88 Class. Quant. Grav. , , arXiv:1306.4866 , ]. Astrophysical signatures of boson ]. Living Rev. Rel. (2013)[ 104020 Phys. Rev. D , , Cambridge University Press (1987). , gr-qc/9807001 88 (1981) 189. Phys. Rev. D , 68 Charged black holes in string theory (1992) 3888] ]. 45 (1993) 2220. (2000) 445[ arXiv:1504.08209 ]. Singularities in Reissner–Nordström black holes – 23 – (1988) 741. 70 Phys. Rev. D arXiv:1902.08323 32 , Superstring theory Phys. Rept. Nonperturbative strong field effects in tensor-scalar Tensor-scalar gravity and binary pulsar experiments , 298 Dynamical Boson Stars Black Holes and Membranes in Higher Dimensional Theories Black holes as elementary particles General relativistic boson stars ]. Asymptotically flat black holes with scalar hair: a review Erratum ibid. gr-qc/9602056 Note on the Kerr spinning particle metric Effective Gravity Theories With Dilatons Black holes with a massive dilaton ]. Uniqueness of the Newman-Janis algorithm in generating the ]. Applicability of the Newman-Janis Algorithm to Black Hole (2015) 1542014[ (2020) 025009[ Supergravity Phys. Rev. Lett. Gen. Rel. Grav. arXiv:1701.00037 , 24 , 37 Nucl. Phys. B . , ] ]. (1991) 3140[ (1996) 1474[ ]. ]. Dilaton black holes with electric charge Towards the rotating scalar-vacuum black holes Singularities in rotating black holes coupled to a massless scalar field 43 54 arXiv:0801.0307 hep-th/9202014 gr-qc/9911008 Scalar mesostatic field with regard for gravitational effects (2017) 19[ Janis-Newman algorithm: generating rotating and NUT charged black holes 3 arXiv:1202.5809 arXiv:1307.4812 hep-th/9209070 hep-th/9310174 [ Int. J. Mod. Phys. D (1986) 409. with Dilaton Fields [ arXiv:1905.04613 Solutions of Modified Gravity Theories (2003) R301[ stars: quasinormal modes and inspiral resonances Phys. Rev. D (1992) 447[ [ Class. Quant. Grav. (1948) 636 [ (1965) 915. Universe [ theories of gravitation Phys. Rev. D Kerr-Newman metric [9] C.A.R. Herdeiro and E. Radu, [8] S.L. Liebling and C. Palenzuela, [5] D. Hansen and N. Yunes, [6] F.E. Schunck and E.W. Mielke, [1] E.T. Newman and A.I. Janis, [7] C.F.B. Macedo, P. Pani, V. Cardoso and L.C.B. Crispino, [2] S.P. Drake and P. Szekeres, [3] H. Erbin, [4] Y.F. Pirogov, [10] D.G. Boulware and S. Deser, [11] G.W. Gibbons and K.-i. Maeda, [20] P.M. Chesler, [21] P.M. Chesler, R. Narayan and E. Curiel, [19] M. Green, J. Schwarz and E. Witten, [12] D. Garfinkle, G.T. Horowitz and A. Strominger, [13] C.F.E. Holzhey and F. Wilczek, [22] I.Z. Fisher, [15] M. Rakhmanov, [14] R. Gregory and J.A. Harvey, [17] T. Damour and G. Esposito-Farese, [18] P. Van Nieuwenhuizen, [16] T. Damour and G. Esposito-Farese, References JCAP03(2021)041 , 33 , ]. 19 , ]. 100 (2011) ]. (1985) , (2017) 31 Int. J. Mod. 509 Int. J. Mod. , Phys. Rev. D 119 , , (1970) 2119. 2 Phys. Rev. D Class. Quant. Grav. arXiv:1705.11098 ]. gr-qc/9611067 , , (1966). 1137 (2018) 084039 Phys. Rept. 7 , Phys. Rev. D 98 , arXiv:1602.03837 Commun. Math. Phys. (2019) L1 , (2017) 1[ Phys. Rev. Lett. Phys. Rev. D 875 , , Universality of Einstein equations (1998) 43[ ]. ]. Theories and Beyond 692 ) 15 R ]. Axially symmetric and static solutions of ( f Phys. Rev. D arXiv:1801.10406 Jo. Math. Phys. , (2016) 061102[ , Mapping Ricci-based theories of gravity into Modified Gravity Theories on a Nutshell: GW170814: A Three-Detector Observation of Observation of Gravitational Waves from a Astrophys. J. First M87 Event Horizon Telescope Results. I. 116 , Phys. Rept. Reality of the Schwarzschild Singularity gr-qc/9710109 , ]. ]. – 24 – arXiv:1501.07274 Extended Theories of Gravity (2018) 021503[ Class. Quant. Grav. ]. ]. arXiv:2001.02936 Gravitation without black holes , 97 collaborations, collaborations, collaboration, (1998) 4885[ (1959) 47. Phys. Rev. Lett. Generation of rotating solutions in Einstein-scalar gravity ]. 57 , gr-qc/9701021 8F Black Hole Based Tests of General Relativity arXiv:1101.3864 (2015) 243001[ Virgo Virgo The gravitational field of a particle possessing a multipole moment ]. 32 (1968) 878. ]. ]. and and (2020) 124006[ Phys. Rev. D 20 Janis-Newman-Winicour and Wyman solutions are the same , New method to generate exact scalar-tensor solutions Static axially symmetric gravitational fields arXiv:1602.02413 arXiv:1812.04934 Testing General Relativity with Present and Future Astrophysical Observations 102 From Schwarzschild to Kerr: Generating spinning Einstein-Maxwell fields from Topology of some spheroidal metrics Phys. Rev. D Static Spherically Symmetric Scalar Fields in General Relativity (2011) 413[ Event horizons in static scalar-vacuum space-times (1997) 4831[ Palatini Approach to Modified Gravity: , 20 12 arXiv:1709.09660 arXiv:1108.6266 (1981) 839. arXiv:1906.11238 arXiv:1810.01460 Phys. D for the Ricci squared Lagrangians general relativity (2016) 054001[ Class. Quant. Grav. Gravitational Waves from a141101[ Binary Black Hole Coalescence LIGO Scientific Binary Black Hole Merger (2019) 024051[ Event Horizon Telescope The Shadow of the[ Supermassive Black Hole LIGO Scientific Inflation, Bounce and Late-time Evolution static fields 1280. 167[ (1970) 276. [ Phys. Rev. D Einstein equations with self-gravitating scalar field Bull. Res. Counc. Israel Phys. A Phys. Rev. Lett. 24 [44] A. Borowiec, M. Ferraris, M. Francaviglia and I. Volovich, [42] V.I. Afonso, G.J. Olmo and D. Rubiera-Garcia, [43] G.J. Olmo, [41] K. Yagi and L.C. Stein, [40] E. Berti et al., [39] [37] [38] [34] G. Clement, [33] B. Chauvineau, [35] S. Capozziello and M. De Laurentis, [36] S. Nojiri, S.D. Odintsov and V.K. Oikonomou, [32] A.G. Agnese and M. La Camera, [31] J.E. Chase, [30] I. Bogush and D. Gal’tsov, [29] B. Turimov, B. Ahmedov, M. Kološ and Z. Stuchlík, [28] G. Erez and N. Rosen, [27] B.H. Voorhees, [26] D.M. Zipoy, [23] A.I. Janis, E.T. Newman and J. Winicour, [24] M. Wyman, [25] K.S. Virbhadra, JCAP03(2021)041 , , , ]. , 90 ]. ]. (2014) 024033 ]. Phys. Rev. Lett. Gravitational 90 , ]. ]. Phys. Rev. D The trivial role of , Born–Infeld inspired ]. Dynamical generation arXiv:1907.04183 ]. ]. ]. (2019) 044040 gravity arXiv:1409.0233 arXiv:0707.2664 ) Phys. Rev. D ]. collaborations, , 99 R ( ]. arXiv:1507.00028 f ]. (2019) 149[ New scalar compact objects in Correspondence between modified arXiv:2004.11357 ]. arXiv:1006.1769 11 ]. ]. (2014) 004[ arXiv:1704.03351 Infrared lessons for ultraviolet gravity: the ]. Born-Infeld gravity and its functional arXiv:1906.04623 arXiv:1707.06101 Accelerated cosmological models in Ricci hep-th/0407090 Semiclassical geons as solitonic black hole (2016) 40[ 11 (2007) 104047[ JHEP Modified Eddington-inspired-Born-Infeld INTEGRAL Phys. Rev. D Born-Infeld- , , 76 76 arXiv:1406.1205 (2020) 585[ and arXiv:1207.6004 (2018) 1[ JCAP – 25 – 80 , arXiv:1705.03806 (2014) 119901] [ (2019) 044[ Ricci-Based Gravity theories and their impact on Bounding the speed of gravity with gravitational wave 727 Ghosts in metric-affine higher order curvature gravity arXiv:1710.05834 The Cosmology of Ricci-Tensor-Squared gravity in the Reissner-Nordstr\’om black holes in extended Palatini Nonsingular black holes in quadratic Palatini gravity 12 (2017) 161102[ 113 (2004) 103503[ arXiv:1112.0475 ]. arXiv:1306.2504 arXiv:1901.08988 Eddington’s theory of gravity and its progeny Phys. Rev. D 119 Instabilities in metric-affine theories of gravity with higher 70 (2014)[ 044003 , JCAP Eur. Phys. J. C , , (2012) 044014[ 90 Phys. Rept. (2017) 235003[ (2017) L13[ , Eur. Phys. J. C 86 , (2013) 011[ 34 (2019) 656[ (2012) 2098[ Erratum ibid. ]. 848 ]. 07 79 72 Phys. Rev. D arXiv:1403.7409 , Phys. Rev. Lett. , Phys. Rev. D JCAP , , Phys. Rev. D , (2010) 011101[ arXiv:1810.04239 arXiv:1403.0105 Ricci-based gravity theories [ Maxwell and nonlinear electromagnetic models gravity and general relativity with scalar fields observations Astrophys. J. Lett. theories of wormholes with charged[ fluids in quadratic Palatini gravity modifications of gravity torsion in projective invariantClass. theories Quant. of Grav. gravity with non-minimally coupled matter fields Eur. Phys. J. C order curvature terms LIGO Scientific, Virgo, Fermi-GBM Waves and Gamma-rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A remnants Eur. Phys. J. C Palatini variational approach Gravity with a Trace Term extensions case of massive gravity and Born-Infeld squared gravity (2014)[ 024066 105 [64] V.I. Afonso, G.J. Olmo, E. Orazi and D. Rubiera-Garcia, [63] A. Delhom, G.J. Olmo and E. Orazi, [62] V.I. Afonso, G.J. Olmo, E. Orazi and D. Rubiera-Garcia, [61] N. Cornish, D. Blas and G. Nardini, [56] J. Beltran Jimenez, L. Heisenberg, G.J. Olmo and D. Rubiera-Garcia, [54] F.S.N. Lobo, G.J. Olmo and D. Rubiera-Garcia, [57] V.I. Afonso, C. Bejarano, J. Beltran Jimenez, G.J. Olmo and E. Orazi, [59] J.B. Jiménez and A. Delhom, [60] [53] G.J. Olmo and D. Rubiera-Garcia, [55] F.S.N. Lobo, J. Martinez-Asencio, G.J. Olmo and D. Rubiera-Garcia, [58] J. Beltrán Jiménez and A. Delhom, [52] G.J. Olmo and D. Rubiera-Garcia, [49] C.-Y. Chen, M. Bouhmadi-Lopez and P. Chen, [50] G. Allemandi, A. Borowiec and M. Francaviglia, [51] B. Li, J.D. Barrow and D.F. Mota, [47] S.D. Odintsov, G.J. Olmo and D. Rubiera-Garcia, [48] J. Beltran Jimenez, L. Heisenberg and G.J. Olmo, [46] A.N. Makarenko, S. Odintsov and G.J. Olmo, [45] M. Bañados and P.G. Ferreira, JCAP03(2021)041 ]. ]. , ]. , ]. (1969) ]. (2017) 1 Phys. Rev. ]. , 95 Phys. Rev. D (2020) 058 gr-qc/9805071 , ]. ]. Rotating black (1968) 526. 07 ]. arXiv:1311.0815 ]. 48 arXiv:1201.2814 JCAP arXiv:2001.04367 ]. Phys. Rev. D , gr-qc/9303037 Riv. Nuovo Cim. arXiv:1709.03798 , , (1998) 041502[ (2012) 1250066 (2014) 2804[ arXiv:1205.6676 ]. 58 arXiv:1508.03272 21 Annals Phys. 74 New class of naked singularities and , Geodesic completeness in a wormhole (2012)[ 084020 Geonic black holes and remnants in arXiv:2006.08180 arXiv:1208.2146 (1993) 5357[ 85 (2020) 044052[ ]. (2020) 043005[ 47 arXiv:1106.5438 Cosmology with Eddington-inspired Gravity 101 Genericity aspects in gravitational collapse to Multicenter solutions in Eddington-inspired 101 Phys. Rev. D (2012)[ 041501 ]. Equilibrium configurations from gravitational , gr-qc/0109051 (2015) 044047[ Why do naked singularities form in gravitational (2020) 1018[ – 26 – 86 Eur. Phys. J. C Naked singularity formation in the collapse of a 92 (2012) 084018[ , Phys. Rev. D 80 Eddington-inspired Born-Infeld gravity. Phenomenology Compact stars in Eddington inspired gravity , Int. J. Mod. Phys. D Strong curvature naked singularities in spherically Phys. Rev. D 86 , Gravitational lensing by naked singularities , Formation of naked singularities: The violation of cosmic Universe Driven by Perfect Fluid in Eddington-inspired (2011) 235018[ Self-similar scalar field collapse Phys. Rev. D arXiv:1210.1521 (1991) 994. The Singularities of gravitational collapse and cosmology Bouncing Eddington-inspired Born-Infeld cosmologies: an , Phys. Rev. D 28 (1970) 529. , Naked singularities in spherically symmetric inhomogeneous 66 (2002) 101501[ ]. Phys. Rev. D arXiv:1106.3569 314 Phys. Rev. D , 65 , Phys. Rev. D Eur. Phys. J. C ]. , , ]. (2012) 103533[ 86 What is a singularity in general relativity? Gravitational collapse: The role of general relativity Phys. Rev. Lett. Phys. Rev. D , (2011)[ 031101 Class. Quant. Grav. , arXiv:1701.04235 , 107 (2002) 103004. arXiv:1107.3749 arXiv:2006.00761 symmetric perfect fluid collapse 65 their observational signatures 024015[ censorship spherical cloud of counter rotating particles collapse? Born-Infeld Gravity 252 black holes and naked[ singularities Tolman-Bondi dust cloud collapse collapse alternative to Inflation ? Phys. Rev. D Eddington-inspired Born-Infeld gravity Proc. Roy. Soc. Lond. A spacetime with horizons Lett. of non-linear gravity-matter coupling Born–Infeld gravity holes in Eddington-inspired Born-Infeld[ gravity: an exact solution [85] K. Mosani, D. Dey and P.S. Joshi, [86] K.S. Virbhadra and G.F.R. Ellis, [79] T. Harada, H. Iguchi and K.-i. Nakao, [84] K. Bhattacharya, D. Dey, A. Mazumdar and T. Sarkar, [83] N. Banerjee and S. Chakrabarti, [70] I. Cho, H.-C. Kim and T. Moon, [76] R. Penrose, [77] S.L. Shapiro and S.A. Teukolsky, [80] P.S. Joshi, N. Dadhich and R. Maartens, [81] P.S. Joshi, D. Malafarina and R. Narayan, [82] P.S. Joshi, D. Malafarina and R.V. Saraykar, [78] P.S. Joshi and I.H. Dwivedi, [69] P.P. Avelino and R.Z. Ferreira, [71] J.H.C. Scargill, M. Bañados and P.G. Ferreira, [72] G.J. Olmo, D. Rubiera-Garcia and H. Sanchis-Alepuz, [73] R.P. Geroch, [74] S.W. Hawking and R. Penrose, [75] G.J. Olmo, D. Rubiera-Garcia and A. Sanchez-Puente, [68] P. Pani, T. Delsate and V. Cardoso, [66] M. Guerrero, G. Mora-Pérez, G.J. Olmo, E. Orazi and D. Rubiera-Garcia, [67] P. Pani, V. Cardoso and T. Delsate, [65] G.J. Olmo, E. Orazi and D. Rubiera-Garcia, JCAP03(2021)041 , , ]. (2015) 79 92 Phys. Rev. D (2010) 451 ]. , ]. 82 ]. ]. Phys. Rev. D arXiv:1206.3077 (2015) 247001 , (2019) 52 Phys. Rev. D ]. ]. , 32 482 (2008) 124014 (2014) 015002 (2019) 024055 ]. 77 31 arXiv:1011.4127 100 Rev. Mod. Phys. arXiv:1310.8376 Shadows of spherically symmetric ]. , (2012) 063010[ arXiv:1909.10322 Image of the Janis-Newman-Winicour arXiv:2003.06810 86 Shadow of a Naked Singularity without ]. arXiv:2009.07487 ]. arXiv:2004.06525 Testing alternative theories of gravity using Can strong gravitational lensing distinguish Phys. Rev. D Class. Quant. Grav. , arXiv:1201.2544 , Phys. Rev. D How does naked singularity look? (2014) 2983[ (2010)[ 124047 (2019) 064[ Distinguishing black holes from naked singularities , 74 – 27 – 82 10 Can we observationally test the weak cosmic Class. Quant. Grav. (2020) 044042[ arXiv:1109.0249 Shadow of a naked singularity Observational distinction between black holes and naked Phys. Rev. D , Shadows and strong gravitational lensing: a brief review , Theories Of Gravity Mon. Not. Roy. Astron. Soc. Gravitational Lensing by Rotating Naked Singularities (2021) 024015[ (2020) 024022[ Shadow of nulllike and timelike naked singularities without Perihelion precession and shadows near black holes and Time delay and magnification centroid due to gravitational , ]. ) 102 ]. JCAP arXiv:0806.3289 R ( , arXiv:1801.00860 f 103 102 (2012) 104053[ (2012) 15[ ]. Phys. Rev. D Can we distinguish black holes from naked singularities by the 85 Apparent shape of super-spinning black holes Can accretion disk properties observationally distinguish black holes , Eur. Phys. J. C 745 , (2018) 42[ Phys. Rev. D ]. ]. gr-qc/0211061 ]. ]. ]. ]. , (2008)[ 083004 50 Phys. Rev. D Phys. Rev. D arXiv:0812.1328 , , 78 Eddington-inspired Born-Infeld gravity: astrophysical and cosmological Phys. Rev. D , Astrophys. J. arXiv:1505.07017 , (2003) 084002[ arXiv:1905.05273 arXiv:0805.1726 arXiv:1802.08060 arXiv:1401.4227 arXiv:1304.7331 arXiv:0710.2333 photon spheres [ images of their accretion disks? naked singularities Photon Sphere [ the Sun naked singularity with a thin accretion disk constraints [ Gen. Rel. Grav. black holes and naked singularities 044035[ [ through their accretion disc properties singularities: the role of the redshift function naked singularities from black holes? [ censorship conjecture? from naked singularities? (2009)[ 043002 [ Phys. Rev. D lensing by black holes and naked singularities 67 [99] R. Shaikh and P.S. Joshi, [98] G. Gyulchev, P. Nedkova, T. Vetsov and S. Yazadjiev, [97] R. Shaikh, P. Kocherlakota, R. Narayan and P.S. Joshi, [96] N. Ortiz, O. Sarbach and T. Zannias, [94] L. Kong, D. Malafarina and C. Bambi, [92] S. Sahu, M. Patil, D. Narasimha and P.S.[93] Joshi, P.S. Joshi, D. Malafarina and R. Narayan, [95] N. Ortiz, O. Sarbach and T. Zannias, [91] Z. Kovacs and T. Harko, [89] G.N. Gyulchev and S.S. Yazadjiev, [90] C. Bambi and K. Freese, [88] K.S. Virbhadra and C.R. Keeton, [87] K.-i. Nakao, N. Kobayashi and H. Ishihara, [102] D. Dey, R. Shaikh and P.S. Joshi, [100] D. Dey, R. Shaikh and P.S. Joshi, [101] A.B. Joshi, D. Dey, P.S. Joshi and P. Bambhaniya, [103] P.V.P. Cunha and C.A.R. Herdeiro, [105] J. Casanellas, P. Pani, I. Lopes and V. Cardoso, [104] T.P. Sotiriou and V. Faraoni, [106] P.P. Avelino, JCAP03(2021)041 , , 57 94 ]. 42 ]. , 144 metric (2011) γ 28 ]. ]. (2018) 084011 Imaging a boson Int. J. Mod. ]. Phys. Rev. D , 97 , (2021) 10 arXiv:1207.4730 On rotating black holes in arXiv:1510.04170 53 Pub. Astron. Soc. Jpn. Int. J. Theor. Phys. , , hep-th/0606033 ]. Phys. Rev. D Proc. Roy. Soc. Lond. A ]. Shadow of rotating regular black Class. Quant. Grav. , , arXiv:1802.02675 ]. , (2012) 022[ arXiv:1907.05615 11 Does the black hole shadow probe the ]. (2016) 105015[ Gen. Rel. Grav. The Classical wormhole solution and (2007) 2173[ , 33 ]. JCAP 22 (1999) 123. , Constraints on Born-Infeld gravity from the speed (2020) 340[ (2018) 084020[ 17 arXiv:1902.01318 Effective interactions in Ricci-Based Gravity below 80 ]. – 28 – arXiv:2006.07245 97 arXiv:1604.03809 , Cambridge University Press (2019). Cosmology in nonlinear Born-Infeld scalar field theory ]. arXiv:1904.06207 Strong field gravitational lensing by a Kerr black hole gr-qc/0308023 (2019) 627[ (2020) 001[ Shadow of rotating regular black holes and no-horizon spacetimes Class. Quant. Grav. ]. , ]. 795 Phys. Rev. D 11 Int. J. Mod. Phys. A , Eur. Phys. J. C hep-th/0312082 (2016) 104004[ Foundations of the new field theory , , Born-Infeld cosmology with scalar Born-Infeld matter 93 (2004) 489[ ]. ]. Class. Quantum Grav. . (2019) 024014[ , JCAP Comment on ‘Spinning loop black holes’ , arXiv:1605.00820 119 Black hole shadows of charged spinning black holes 100 Black hole mimicker hiding in the shadow: Optical properties of the Spacetime and Geometry Probing gravity at sub-femtometer scales through the pressure distribution inside Eddington-inspired Born-Infeld gravity: nuclear physics constraints and the The apparent shape of a rotating charged black hole, closed photon orbits and the (2005) 355[ gr-qc/0204013 Phys. Lett. B , Phantom cosmology with a nonlinear Born-Infeld type scalar field 14 arXiv:1106.0970 Spherical orbits around a Kerr black hole Phys. Rev. D , arXiv:2007.04022 arXiv:1711.04137 B. Ahmedov, star at the Galactic center event horizon geometry? Phys. Rev. D arXiv:2004.07501 bifurcation set a 4 (2005) 273. [ Nuovo Cim. B (2003) 837[ Phys. D wormhole wave function with a nonlinear Born-Infeld scalar field the proton the non-metricity scale (1934) 425 with negative potentials (2016) 064016[ DHOST theories 148001[ holes validity of the continuous fluid approximation [ of gravitational waves after GW170817 and GRB 170817A [123] P.V.P. Cunha, C.A.R. Herdeiro and M.J. Rodriguez, [124] A.B. Abdikamalov, A.A. Abdujabbarov, D. Ayzenberg, D. Malafarina, C. Bambi and [125] R. Kumar and S.G. Ghosh, [126] A. de Vries, [127] R. Takahashi, [122] F.H. Vincent, Z. Meliani, P. Grandclement, E. Gourgoulhon and O. Straub, [121] S.E. Vazquez and E.P. Esteban, [120] E. Teo, [113] H.Q. Lu, [114] W. Fang, H.Q. Lu and Z.G. Huang, [116] J. Ben Achour, H. Liu, H. Motohashi, S. Mukohyama and K. Noui, [109] P.P. Avelino, [110] A. Delhom, V. Miralles and A. Peñuelas, [111] M. Born and L. Infeld, [112] H.Q. Lu, L.M. Shen, P. Ji, G.F. Ji and N.J. Sun, [119] A. Abdujabbarov, M. Amir, B. Ahmedov and S.G. Ghosh, [108] P.P. Avelino, [115] S. Jana and S. Kar, [117] S.M. Carroll, [118] M. Azreg-Ainou, [107] S. Jana, G.K. Chakravarty and S. Mohanty, JCAP03(2021)041 05 (2013) 85 (2016) ]. 777 94 JCAP , Shadow of (2020) 225013 37 Phys. Rev. D ]. , Astrophys. J. , Phys. Rev. D New method for shadow arXiv:1212.4949 , The Structure and ]. ]. ]. (2013) 429[ Class. Quant. Grav. , arXiv:1907.12575 ]. 344 arXiv:2002.01028 symmetry and their shadow images Anisotropic deformations in a class of (2020) 7[ 2 Shadows of CPR black holes and tests of the Kerr Z – 29 – arXiv:1506.02627 896 A Parametric model for the shapes of black-hole shadows arXiv:1802.06166 arXiv:1707.09451 ]. Shadow casted by a Konoplya-Zhidenko rotating non-Kerr (2020) 084030[ ]. Shadow of a rotating braneworld black hole (2015) 315[ Astrophys. Space Sci. ]. , 101 75 Astrophys. J. (2018) 128[ ]. Asymptotically flat, parameterized black hole metric preserving Kerr , (2017) 051[ ]. 855 10 arXiv:1112.6349 Photon Rings around Kerr and Kerr-like Black Holes Rotating black holes without Phys. Rev. D arXiv:2004.01440 , JCAP , arXiv:1607.05767 Eur. Phys. J. C , Astrophys. J. arXiv:1501.02814 , arXiv:2006.07406 (2012) 064019[ Kerr-Taub-NUT black hole (2020) 040[ Dynamics of the SubparsecGHz Jet in M87 Based on 50 VLBA Observations over 17 Years at 43 170[ metric 084025[ black hole projectively-invariant metric-affine theories of gravity calculations: Application to parametrized axisymmetric black holes in non-Kerr spacetimes symmetries [ [129] A. Abdujabbarov, F. Atamurotov, Y. Kucukakca, B. Ahmedov and U. Camci, [137] R. Craig Walker, P.E. Hardee, F.B. Davies, C. Ly and W. Junor, [128] L. Amarilla and E.F. Eiroa, [130] T. Johannsen, [131] M. Ghasemi-Nodehi, Z. Li and C. Bambi, [133] M. Wang, S. Chen and J. Jing, [136] C.-Y. Chen, [138] J.B. Jiménez, D. de Andrés and A. Delhom, [132] Z. Younsi, A. Zhidenko, L. Rezzolla, R. Konoplya and Y. Mizuno, [134] L. Medeiros, D. Psaltis and F. Özel, [135] Z. Carson and K. Yagi,