A novel family of rotating black hole mimickers
PPrepared for submission to JCAP
A novel family of rotating black holemimickers
Jacopo Mazza, Edgardo Franzin, and Stefano Liberati
SISSA, International School for Advanced Studies,via Bonomea 265, 34136 Trieste, ItalyINFN, Sezione di Trieste,via Valerio 2, 34127 Trieste, ItalyIFPU, Institute for Fundamental Physics of the Universe,via Beirut 2, 34014 Trieste, ItalyE-mail: [email protected], [email protected], [email protected]
Abstract.
The recent opening of gravitational wave astronomy has shifted the debate aboutblack hole mimickers from a purely theoretical arena to a phenomenological one. In thisrespect, missing a definitive quantum gravity theory, the possibility to have simple, meta-geometries describing in a compact way alternative phenomenologically viable scenarios ispotentially very appealing. A recently proposed metric by Simpson and Visser is exactlyan example of such meta-geometry describing, for different values of a single parameter,different non-rotating black hole mimickers. Here, we employ the Newman–Janis procedureto construct a rotating generalisation of such geometry. We obtain a stationary, axiallysymmetric metric that depends on mass, spin and an additional real parameter (cid:96) . Accordingto the value of such parameter, the metric may represent a rotating traversable wormhole, arotating regular black hole with one or two horizons, or three more limiting cases. By studyingthe internal and external rich structure of such solutions, we show that the obtained metricdescribes a family of interesting and simple regular geometries providing viable Kerr blackhole mimickers for future phenomenological studies.
Keywords: black hole mimicker, wormhole, regular black hole a r X i v : . [ g r- q c ] F e b ontents Spacetime singularities are a generic prediction of General Relativity (GR) [1], under a setof physically reasonable (yet fairly stringent) assumptions. They represent points at whichgeodesics terminate abruptly and, for this reason, their appearance is usually interpreted asmarking the breakdown of the theory.It is commonly believed that quantum gravitational effects will prevent the formation ofsingularities. A complete description of such small-scale effects is still out of reach; however,one might wonder whether they propagate to larger scales, at which gravity is well describedwithin the framework of differential geometry. By studying the phenomenology of suitablyconstructed effective geometries — hopes are — one may get insight into the process wherebythe singularity is regularised and, therefore, also into the quantum gravity theory responsiblefor it [2].An agnostic analysis [3, 4], almost entirely based on geometric arguments, shows thatthe spectrum of qualitatively different regular geometries is not as wide as one might expect.One alternative emerging from such classification, and particularly relevant to the presentdiscussion, is represented by wormholes: heuristically, the throat of the wormhole, reached atfinite affine distance, defocuses the null geodesics that would otherwise converge (and end)at the singularity. Such defocusing point may lie in an untrapped region, or be cloaked by– 1 –n event horizon: in the former case, the wormhole is traversable (in the sense of Morris andThorne [5]); in the latter, the spacetime contains a regular black hole.Simpson and Visser [6–8] recently proposed a static and spherically symmetric metricthat smoothly interpolates between these possibilities. The line element is (henceforth: SVmetric):d s = − (cid:18) − M √ r + (cid:96) (cid:19) d t + (cid:18) − M √ r + (cid:96) (cid:19) − d r + ( r + (cid:96) ) (cid:2) d θ + sin θ d φ (cid:3) , (1.1)where M ≥ (cid:96) > ; note that r ∈ ( −∞ , + ∞ ). Despite its simplicity— it is a minimal modification of the Schwarzschild solution, to which it indeed reduces for (cid:96) = 0 — this metric is remarkably rich: trimming the value of (cid:96) , it may represent • a two-way, traversable wormhole `a la Morris-Thorne for (cid:96) > M , • a one-way wormhole with a null throat for (cid:96) = 2 M , and • a regular black hole, in which the singularity is replaced by a bounce to a differentuniverse, when (cid:96) < M ; the bounce happens through a spacelike throat shielded byan event horizon and is hence dubbed “black-bounce” in [6] or “hidden wormhole” asper [4].Such variety is particularly appealing. Regular black holes and traversable wormholesrepresent morally distinct scenarios, and have been studied thoroughly, though separately:see for instance [9–15] and [5, 16–21]. However, the discussion in [3, 4] proves that theyare both theoretically motivated. Actually, it suggests more: if one’s aim is to build phe-nomenological models that are agnostic e.g. to the scale of regularisation, one would betternot commit to one particular scenario but should rather prefer a unified treatment. Simpsonand Visser’s proposal fits perfectly in this line of reasoning; and it does so at no expense ofsimplicity — a greatly attractive feature for the sake of phenomenology.The capability of the metric (1.1) to properly describe realistic situations, however, ishindered by the lack of an important ingredient: rotation [22, 23]. Not surprisingly, rotatingregular black holes have been considered before, see e.g. [24–28]; similarly, rotating traversablewormholes have been proposed in [29] and used for phenomenological modelling e.g. in [30–34]. Predictably, the objects considered in the aforementioned references fall in two verydistinct classes. The goal of this paper, therefore, is to construct a spinning generalisationof the Simpson-Visser metric (1.1) suitable for comparison with observations. To achieve it,we employ the Newman–Janis procedure, a method we introduce and describe below.The paper is structured as follows. The Newman–Janis procedure is reviewed andapplied in section 2; the section terminates with our proposal for a spinning SV metric.Section 3 is devoted to the global analysis of the ensuing spacetimes. Section 4 investigatesthe distribution of stress-energy that, assuming GR holds, produces our metric as a solution.Section 5 describes relevant features of the exterior geometry, namely ergoregion, photon ringand ISCO. Finally, section 6 reports our conclusions. We adopt units in which c = G = 1,unless otherwise stated, and metric signature ( − , + , + , +). In [6] a was used in lieu of (cid:96) . However, a is widely used as the spin parameter for rotating spacetimes,thus we renamed the regularisation length scale (cid:96) . – 2 – Building the metric
The Newman–Janis procedure (NJP) [35] is a five-step method to build an axially symmetricspacetime starting from a spherically symmetric one. Its most remarkable application is theconstruction of the Kerr–Newman solution [36]; since then, several extensions to treat e.g.gauge fields or other charges have been studied and successfully employed — see [37] for asummary. Notably, it has been used in [24] to generate spinning regular BHs. Despite itssuccesses, the method encodes some puzzling arbitrariness — on which we will elaboratefurther below — and a true understanding of its working is still lacking. Some solid groundhas been established in [38]; see also [39, 40]. More recently, further insight has come fromthe study of scattering amplitudes [41, 42].For the reader’s convenience, we briefly summarise here the NJP applied to a genericstatic, spherically symmetric seed metricd s = − f ( r )d t + d r f ( r ) + h ( r )[d θ + sin θ d φ ]; (2.1)we will postpone specifying these results to (1.1) until the next subsection.As a step I, write the metric in outgoing (or, equivalently, ingoing) Eddington–Finkel-stein coordinates ( u, r, θ, φ ). Then (step II) introduce a null tetrad { l µ , n µ , m µ , m µ } (theoverline marks complex conjugation) satisfying l µ n µ = − m µ m µ = − l µ m µ = n µ m µ = 0.In terms of this tetrad, the metric can be written as g µν = − l µ n ν − l ν n µ + m µ m ν + m ν m µ . (2.2)As a step III, define r (cid:48) = r + i a cos θ, u (cid:48) = u + i a cos θ, θ (cid:48) = θ, φ (cid:48) = φ, (2.3)where a is a real parameter to be identified, a posteriori , with the spin; even though r (cid:48) , u (cid:48) are complex, these relations define a viable change of coordinates. The usual vector trans-formation law thus yields a transformed tetrad and, therefore, a transformed metric.To obtain a new, axially symmetric metric, one needs (step IV) to replace the oldfunctions f, h with new ˜ f , ˜ h ; the latter are required to be real, though of complex variable,and to coincide with the former when evaluated on the real axis. This replacement, in thestandard NJP, is performed in a rather particular way but is, nonetheless, arbitrary. Forexample, in the Schwarzschild geometry one has f ( r ) = 1 − Mr , h ( r ) = r ; (2.4)to derive the Kerr solution, ˜ f , ˜ h are given by substituting (“complexifing”)1 r → (cid:18) r (cid:48) + 1 r (cid:48) (cid:19) , r → r (cid:48) r (cid:48) (2.5)in f and h — all other prescriptions fail. We would like to stress that, generically, there is noreal justification for the replacement above, except that the result of the NJP when appliedto vacuum and electrovacuum solutions yields vacuum and electrovacuum solutions.– 3 –he complexification usually produces a metric with several non-diagonal components.To eliminate all of them, except for g tφ , one needs to perform an additional transformationto Boyer–Lindquist-like coordinates (step V). The desired change is of the form d t (cid:48) = d u − F ( r )d r, d φ (cid:48) = d φ − G ( r )d r , but is not always possible. Indeed, to integrate the above relationsone needs F, G to be independent from θ : this is the case, for instance, in Schwarzschild andReissner–Nordstr¨om, but examples of the contrary exist.To circumvent at once two problems of the NJP, namely the arbitrariness in the “com-plexification” and the possible lack of a transformation to Boyer–Lindquist-like coordinates,a modified version of the procedure (MNJP) was devised in [43, 44]. In this framework,all the arbitrariness is encoded in a new function Ψ, which enters the resulting metric asan additional degree of freedom; the advantage with respect to the standard NJP is thata Boyer–Lindquist form is ensured (see also [45]) and one can thus set out to constrain Ψinvoking physical arguments. In [43, 44] this was done by demanding that the rotating metricbe a solution of GR for an imperfect fluid, thence deriving a differential equation for Ψ.In practice, however, given the extreme similarity of the metric (1.1) with the Schwarzschildone, in our case all the potential issues that may prevent a successful implementation of theNJP are either easily worked around or outright absent. To apply the NJP to the metric (1.1), i.e. to f ( r ) = 1 − M √ r + (cid:96) , h ( r ) = r + (cid:96) , (2.6)we need first of all to define (step I)d u := d t − d r ∗ := d t − d rf ( r ) . (2.7)The null tetrad that satisfies all requirements of step II is l µ = δ µr , n µ = δ µu − f ( r )2 δ µr , m µ = 1 (cid:112) h ( r ) (cid:18) δ µθ + isin θ δ µφ (cid:19) . (2.8). By changing coordinates to u (cid:48) , r (cid:48) (step III) we get l (cid:48) µ = l ν ∂x (cid:48) µ ∂x ν = δ µr (cid:48) , n (cid:48) µ = δ µu (cid:48) − f ( r )2 δ µr (cid:48) , (2.9) m (cid:48) µ = 1 (cid:112) h ( r ) (cid:18) δ µθ (cid:48) − i a sin θ (cid:0) δ µr (cid:48) − δ µu (cid:48) (cid:1) + isin θ δ µφ (cid:48) (cid:19) , (2.10)where r is now meant as a scalar function (not a coordinate).We now need to provide a prescription for the complexification (step IV) of r yieldingthe new ˜ f , ˜ h . Since the components of the SV metric are derived from those of Schwarzschildby writing √ r + (cid:96) instead of r , there seems to be a natural choice: we can introduce a newcoordinate (cid:37) := √ r + (cid:96) , and complexify it as would be appropriate for Schwarzschild’sradial coordinate. Namely (cid:37) → (cid:37) (cid:48) = (cid:37) + i a cos θ, (2.11)– 4 –o that h ( r ) = (cid:37) → ˜ h ( r (cid:48) ) = (cid:37) (cid:48) (cid:37) (cid:48) = r + (cid:96) + a cos θ (2.12)and f ( r ) = 1 − M(cid:37) → ˜ f ( r (cid:48) ) = 1 − M (cid:18) (cid:37) (cid:48) + 1 (cid:37) (cid:48) (cid:19) = 1 − M √ r + (cid:96) r + (cid:96) + a cos θ . (2.13)Now, the coordinate transformation of step V has the general form (see [24]) F = ˜ h ( r, θ ) + a sin θ ˜ f ( r, θ )˜ h ( r, θ ) + a sin θ , G = a ˜ f ( r, θ )˜ h ( r, θ ) + a sin θ . (2.14)Specifying to our case: F = r + (cid:96) + a r + (cid:96) + a − M √ r + (cid:96) , G = ar + (cid:96) + a − M √ r + (cid:96) ; (2.15)these expressions do not depend on θ and one can safely integrate them to get t (cid:48) ( u, r ) , φ (cid:48) ( u, r ).(From now on we drop the primes.)Thus, the metric obtained by applying the NJP to the SV seed does have a Boyer–Lindquist form. Note that this is obvious, in hindsight, since the functions F, G above arethe same that one would get starting from a Schwarzschild seed, provided one replaces thecoordinate radius r with √ r + (cid:96) , and a Boyer–Lindquist form certainly exists in that case.The metric ensuing from the application of the NJP with the choices above is ourproposal for the rotating counterpart to the SV metric (1.1)d s = − (cid:32) − M √ r + (cid:96) Σ (cid:33) d t + Σ∆ d r + Σd θ − M a sin θ √ r + (cid:96) Σ d t d φ + A sin θ Σ d φ (2.16)with Σ = r + (cid:96) + a cos θ, ∆ = r + (cid:96) + a − M (cid:112) r + (cid:96) ,A = ( r + (cid:96) + a ) − ∆ a sin θ. It reduces to the SV metric when a = 0 and to the Kerr metric when (cid:96) = 0. Formally, itscomponents can be derived from those of the Kerr metric by replacing the Boyer–Lindquistradius r with (cid:37) = √ r + (cid:96) , but without changing d r ; i.e. the metric (2.16) is not related toKerr by a change of coordinates.The result (2.16) obtained with the standard NJP is confirmed by the application ofthe MNJP, provided the arbitrary function Ψ be fixed equal to Σ. This choice is coherentwith requiring that the spinning metric coincides with Kerr when (cid:96) = 0, as it must.The rest of the paper is devoted to characterising the metric (2.16) and the spacetimeit describes. As in the non-spinning case, r may take positive as well as negative values. Our negative- r region, however, should not be confused with the one deriving from analytically extending the– 5 –err spacetime beyond its ring singularity: indeed, the metric (2.16) is symmetric under thereflection r → − r and the spacetime it describes is thus composed of two identical portionsglued at r = 0.Some intuition can be gained by noting that the surface r = 0 is an oblate spheroid ofsize (Boyer–Lindquist radius) (cid:96) . When (cid:96) = 0, the spheroid collapses to a ring at θ = π/ (cid:96) (cid:54) = 0 the singularityis excised and r = 0 is a regular surface of finite size, which observers may cross: the metric(2.16) thus describes a wormhole with throat located at r = 0. The nature of such throat(timelike, spacelike or null) depends on (cid:96) and a . Actually, | a | is the relevant parameter, thus,without loss of generality, we only consider a > (cid:96) and a also determine whether the metric has coordinate singularities.When this is the case, the singularities are given by ∆ = 0 and located at r ± = (cid:20)(cid:16) M ± (cid:112) M − a (cid:17) − (cid:96) (cid:21) / . (3.1)By calling (cid:37) ± := M ± (cid:112) M − a , (3.2)we immediately see that r + is real only if (cid:96) ≤ (cid:37) + and, similarly, r − is real only if (cid:96) ≤ (cid:37) − .Thus, depending on the values of the parameters, we may have two (if a < M and (cid:96) < (cid:37) − ),one (if a < M and (cid:37) − < (cid:96) < (cid:37) + ) or no singularity at all (if a < M and (cid:96) > (cid:37) + , or if a > M ).The cases in which equalities hold are extremal or limiting versions of the above. As theanalysis in the following subsection will prove, these coordinate singularities are horizons ofthe spacetime. For the sake of practicality, we summarise the spectrum of possible cases with the aid ofa “phase diagram” in figure 1: each spacetime structure is associated with a region in (aconstant- M slice of) the parameter space under consideration. We defer a thorough discussionof each case to section 3.3, but lay out our terminology here: WoH traversable wormhole; nWoH null WoH, i.e. one-way wormhole with null throat;
RBH-I regular black hole with one horizon (in the r > r <
RBH-II regular black hole with an outer and an inner horizon (per side); eRBH extremal regular black hole (one extremal horizon per side); nRBH null RBH-I, i.e. a regular black hole with one horizon (per side) and a null throat.A comment is in order, at this point. The parameter (cid:96) represents the spatial extent ofthe throat r = 0. It seems difficult to envisage a single quantum gravity scenario capable ofjustifying all the values of (cid:96) that we consider.Indeed, if for instance one assumes gravitational collapse to proceed as predicted byGR until a certain threshold is met, beyond which quantum effects become dominant, one– 6 – ℓ /M WoHRBH-I RBH-II n W o H n R B H e R B H Figure 1 : Parameter space and corresponding spacetime structure. Acronyms are spelledout in the text.might deduce (cid:96) ∼ L Planck or (cid:96) ∼ L Planck ( M/M
Planck ) / . The two estimates correspond todifferent ways of identifying the threshold: in the first, quantum gravity becomes dominantwhen the radius of the collapsing object becomes of order the Planck length, in the secondwhen the density is of order the Planck density [46, 47]. In either case, for astrophysicallyrelevant masses, one would expect (cid:96)/M to be very small ( ∼ − M (cid:12) /M in one case and ∼ × − ( M (cid:12) /M ) / in the other) and RBH-II to be the only viable geometry, at leastfor a ≤ M .Values of (cid:96)/M = O (1), instead, entail a macroscopic throat and most likely requireadditional ingredients. For instance, a regularisation at the Planck scale might be followedby a dynamical process, after which the structure settles down to the metric (2.16) [48]. If thisis the case, there is no reason for (cid:96) to be linked to the scale of quantum gravity. Such processmight preserve or destroy the horizon, so that the remnant object might correspondinglyconsist of a (regular) black hole or a “naked” wormhole. Note that (cid:96)/M = 1 is also thethreshold above which traversable wormholes, with no horizons, can exist at spins a < M . To check that the singularities at ∆ = 0 are coordinate artefacts, one can introduce ingoingnull coordinates d v := d t + (cid:37) + a ∆ d r, d ψ := d φ + a ∆ d r, (3.3)and notice that the resulting metric is indeed regular at r = r ± except perhaps when v = ±∞ ;equivalently, one could adopt outgoing null coordinatesd u := d t − (cid:37) + a ∆ d r, d ˜ ψ := d φ − a ∆ d r, (3.4)– 7 –nd confirm the same result except perhaps at u = ±∞ . Either patch covers the region onwhich the other is not defined, thus proving that geodesics can be extended beyond r = r ± .The same deduction holds for − r ± .We further investigate the nature of the surfaces r = ± r ± by plotting the null rays v = cst , u = cst, in figure 2. We choose for simplicity θ = 0. The horizontal axes representsthe Boyer–Lindquist radius r , while the time coordinate on the vertical axes t v ∗ is defined byd t v ∗ := d v − d r, (3.5)so that v = cst ⇒ t v ∗ = − r + cst ,u = cst ⇒ t v ∗ = − (cid:90) r (cid:18) − (cid:37) + a ∆ (cid:19) d r (cid:48) + cst . The peeling of outgoing rays shows that the surfaces v = cst and r = ± r ± are indeed horizons: r + and − r − are black-hole horizons, while r − and − r + are white-hole horizons. An analogousanalysis adapted to outgoing rays, in which these appear as straight lines while ingoing rayspresent peeling, shows that the surfaces ± r ± and u = cst have the opposite nature withrespect to their v = cst counterparts: r + is a white-hole horizon, r − a black-hole horizon,etc. The analytical extension of the metric (2.16) across the horizons can be performed by stan-dard methods (see e.g. [49]), by changing to suitable Kruskal-like coordinates
U, V (a redef-inition of φ is also required), defined in terms of u, v by an exponential mapping involvingthe surface gravity of the horizon under consideration and compensating for the peeling ofnull rays off of it.The only practical difference between the textbook case of Kerr and our own lies in thefunctional relation between, e.g. the Boyer–Lindquist radius r and the tortoise coordinate r ∗ .Such difference is inconsequential as far as analytic continuation is concerned; for instance,curves U V = cst correspond to curves r = cst in Kerr as well as in this case.We construct Carter–Penrose diagrams for the maximal extension of the six cases iden-tified in section 3.1 and report them in figures 3 and 4. A detailed description of each casefollows. WoH is a traversable, two-way wormhole with a timelike throat. The Penrose diagram ofthis spacetime is the same as Minkowski’s, provided one distinguishes the regions r > r < a priori they are different. nWoH is a one-way wormhole with a null throat, which is an extremal event horizon. Theanalytically extended diagram continues indefinitely above and below the portion we show.
RBH-I is a spacetime containing an eternal black hole whose singularity is replaced bythe (regular) throat of a spacelike wormhole. The diagram consists of infinitely manySchwarzschild-like blocks stacked one on top of the other and glued at the throats.
RBH-II is a regular black hole with two horizons per side. The Carter–Penrose diagramconsists of two Kerr-like patches glued at the throats. The throats are timelike and can thus– 8 – = / M t * v / M (a) Traversable wormhole (WoH). r = / M t * v / M (b) Null wormhole (nWoH). - r + r = r + r / M t * v / M (c) Regular black hole, one horizon(RBH-I). - r - - r + r = r - r + r / M t * v / M (d) Regular black hole, two horizons(RBH-II). - r + r = r + r / M t * v / M (e) Extremal regular black hole, one hori-zon (eRBH). - r + r = r + r / M t * v / M (f) Null regular black hole, one horizon(nRBH). Figure 2 : Ingoing (red) and outgoing (black) null rays close to ± r ± for the different casesin the phase diagram. Particular values of a and (cid:96) have been picked.– 9 – ur universe"other" universe (a) WoH, corresponding to (cid:96) > (cid:37) + and a < M ,or a > M . The throat r = 0 is a timelikesurface, traversable in both ways. (b) nWoH, corresponding to (cid:96) = (cid:37) + (and a 1, so that G µν = T µν .– 13 – .1 Energy density for infalling observersNull geodesics Consider again the congruence l µ : being null and geodetic, these vectorsare tangent to trajectories that fall towards the centre. Assuming GR holds, the contraction G µν l µ l ν is the energy density measured along these trajectories. A straightforward compu-tation shows that G µν l µ l ν = − (cid:96) Σ . (4.3)This quantity is negative, hence the null energy condition is violated. Thus exoticmatter is encountered everywhere in the spacetime, in an amount that decreases as 1 /r and is maximal at r = 0. Note that the limit (cid:96) → r = 0 , θ = π/ 2, where it is infinite; in fact G µν l µ l ν (cid:12)(cid:12)(cid:12)(cid:12) r =0 = − (cid:96) ( (cid:96) + a cos θ ) . (4.4)Such behaviour is not surprising, since in the limit (cid:96) → r = 0 , θ = π/ Timelike observer Consider now a timelike observer moving along a geodesic with tangentvector u µ . Because of the symmetries of this spacetime, the components u t =: −E and u φ =: L are constants of motion, corresponding respectively to the observer’s energy perunit mass and to the projection along the rotation axis of the observer’s angular momentumper unit mass. Moreover, the existence of the Killing tensor (3.6) yields a third constant ofmotion, in terms of which u θ can be expressed. The remaining component u r is fixed by thenormalisation u µ u µ = − θ = π/ 2, and computeagain the double contraction with the Einstein tensor. We find: ε u := G µν u µ u ν (cid:12)(cid:12)(cid:12)(cid:12) θ = π/ = − (cid:96) (cid:37) (cid:2) M ( L − a E ) − (cid:37) ( L − a E ) − (cid:37) (1 − E ) (cid:3) . (4.5)Thus — assuming GR holds — observers with, say, L = a E measure a negative energy densityat all radii when their energy is such that E > / 2. The weak energy condition is thereforeviolated (and, consequently, the dominant energy condition too). Again, the limit (cid:96) → ε u at r = 0 is the sign of ( x := L − a E ) ε (cid:48) u (cid:12)(cid:12)(cid:12)(cid:12) r =0 = (cid:96) M x − (cid:37) [6 ax E + 3 x + (cid:37) (cid:0) − E (cid:1) ] (cid:37) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:37) = (cid:96) . (4.6)Thus, observers with x = 0 and E > / r = 0 a (local) maximum of the energydensity. To characterise the distribution of stress-energy in an observer-independent way, we diago-nalise the Einstein tensor in mixed form [eqs. (A.4) in appendix A]. We find four distinct Note that G µν is not symmetric, hence it is not guaranteed to be diagonalisable over the real numbers. – 14 –eal eigenvectors, which in Boyer–Lindquist coordinates and up to multiplicative dimensionfulconstants are: v µ = { a + (cid:37) /a, , , } , v µ = { , , , } , v µ = { , , , } , v µ = { a sin θ, , , } . (4.7)We identify as minus the energy density − ε the eigenvalue relative to the timelike eigenvector( v µ when ∆ > v µ otherwise); and as pressures p i ( i = 1 , , 3) the other eigenvalues. Wefind ε = (cid:96) Σ (cid:26) (cid:37) Σ − a M χ (cid:37) − H (cid:0) ∆ (cid:1)(cid:27) , (4.8a) p = (cid:96) Σ (cid:26) a [2 M χ − (cid:37) ( χ − (cid:37) ( (cid:37) − M ) (cid:37) − H (cid:0) ∆ (cid:1)(cid:27) , (4.8b) p = − (cid:96) Σ (cid:26) a (cid:37) [4 M χ + (cid:37) ( χ − a M χ + (cid:37) ( M − (cid:37) ) (cid:37) (cid:27) , (4.8c) p = (cid:96) Σ (cid:26) − M (2 a χ (cid:37) + Σ ) + Σ (cid:37) (cid:37) (cid:27) , (4.8d)where H ( · ) is the Heaviside function. Note that the density and pressures so defined arecontinuous at the horizon, but their derivatives are generically not.In the remainder of this section, we will analyse the null ( ε + p i ≥ 0) and weak (null + ε ≥ 0) energy conditions. Note however that ε + p = − | ∆ | (cid:96) Σ ≤ ε + p i have less wieldy expressions and we therefore not report themhere. Note that expressions (4.8) depend on the polar angle θ in a rather involved way butonly through χ = cos θ ∈ [0 , ε and p i ; in particular, such dependence should be markedonly at small radii and rapidly die out at spatial infinity.We confirm this intuition by studying ε, p i as functions of χ , both analytically andgraphically. The energy density, for instance, has an extremum at χ = 0 (either a minimumor a maximum, depending on the values of the parameters); in addition, it may have at mosttwo more extrema, symmetric with respect to χ = 0. Similar considerations apply to ε + p i : χ = 0 is always an extremum and at most two other extrema, symmetric with respect to χ = 0, can exist. For ε + p , in particular, χ = 0 is the only extremum; at large radii it is aminimum — but can become a maximum at smaller radii, depending on the parameters.Hence, if we aim at characterising the radial distribution of stress and energy, a marginal-isation over the angular variable is justified. Thus, for a generic quantity X , we resolve toconsider (cid:104) X (cid:105) := 12 (cid:90) +1 − d χ X. (4.10)In figure 5 we plot (cid:104) ε (cid:105) and (cid:104) ε + p i (cid:105) for selected values of a and (cid:96) . Notice that the stress-energy content of this spacetime is localised close to the origin: inspection of eq. (4.8) indeedconfirms that energy density and pressures all scale as 1 /r .– 15 – = ℓ = a = a = a = a = r / M 〈 ε 〉 ℓ = - - - ℓ = ℓ = a = a = a = a = - - - - r / M 〈 ε + p 〉 ℓ = - - - ℓ = ℓ = a = a = a = a = r / M 〈 ε + p 〉 ℓ = - - - ℓ = ℓ = a = a = a = a = r / M 〈 ε + p 〉 ℓ = - - - Figure 5 : Energy conditions, averaged over the polar angle, as a function of r . Differentcolours represent different choices for the spin a , while different line styles stand for differentvalues of (cid:96) . Given the difference of scales, one particular value of (cid:96) is plotted in an inset.To quantify the amount of violation of the energy conditions in the whole spacetime, weadopt the strategy proposed in [52–54]. That is, we compute the so-called volume integralquantifier E := (cid:90) d r d θ d φ (cid:112) | g | ε, E + P i := (cid:90) d r d θ d φ (cid:112) | g | ( ε + p i ) , (4.11)where g is the determinant of the four-dimensional metric.We draw contour plots of these quantities for varying values of the parameters a, (cid:96) andreport the results in figure 6. Note that the amount of “effective” matter varies substantiallyas the parameters vary and can be made very small by trimming them carefully. The exterior of a Kerr black hole is rich in noticeable features, which largely determine thephenomenology of these objects. In this section, we focus on the exterior ( r ≥ (cid:96) affects it. Inparticular, we describe the ergoregion and, schematically, the orbits, focusing on equatoriallight ring and ISCO. – 16 – ℓ / M E -35-30-25-20-15-10-50 ℓ / M E +P -60.0-45.0-30.0-15.0-5.0-3.0-1.7 ℓ / M E +P -1012345 ℓ / M E +P -30-25-20-15-10-505 Figure 6 : Contour plots of the volume integral quantifier in the ( a, (cid:96) )-plane. Negative valuesof E + P i entail violations of the (averaged) weak energy condition; negativity of E or E + P i further entail violations of the (averaged) null energy condition. An ergoregion is a region inside of which no static observer can exist. Its boundary, theergosurface, is the locus of points where g tt = 0. In our metric, the roots of this equationcorrespond to values of (cid:37) given by (cid:37) ± erg := M ± (cid:112) M − a cos θ. (5.1)Clearly, this expression coincides with what one finds in Kerr. Contrary to what isusually assumed in that case, however, here we do consider arbitrarily high spins. Therefore,the radicand in eq. (5.1) is not always positive and the ergosurface has markedly distinctshapes in the a > M and a < M cases. An ergosurface exists when at least one of the– 17 – a) WoH with a < M . The ergoregion is lim-ited to a crescent-shaped region about the equa-tor. The dashed blue line represents the would-be outer horizon and is plotted for reference: thewormhole is traversable as long as the throat is“larger than the horizon”, otherwise the object isa regular black hole (figure 8). (b) WoH with a > M . Note that the two branches (cid:37) +erg and (cid:37) − erg , plotted in red and orange respec-tively, only exist for | cos θ | ≤ a/M and joinsmoothly, thus producing an ergoregion of suchpeculiar shape. Figure 7 : Traversable wormhole with ergoregion, corresponding to values of (cid:96) such that (cid:37) + < (cid:96) < (cid:37) +erg . Each plot is a slice at fixed φ , the angle with the vertical axis is θ and thedistance from the centre is (cid:37) . The hatched region is the ergoregion. The black dotted linerepresents the throat r = 0; the grey region is therefore excised from the spacetime.quantities r ± erg = (cid:113)(cid:0) (cid:37) ± erg (cid:1) − (cid:96) (5.2)is real. Note, incidentally, that min θ ∈ [0 ,π ] (cid:0) (cid:37) +erg (cid:1) = (cid:37) + and max θ ∈ [0 ,π ] (cid:0) (cid:37) − erg (cid:1) = (cid:37) − .Hence, if (cid:96) ≥ (cid:37) +erg there is no ergoregion. When this is the case, the object underconsideration is a traversable wormhole (WoH case of section 3), since (cid:37) +erg ≥ (cid:37) + ; note thatthese wormholes may have arbitrary spin.If, on the contrary, (cid:96) < (cid:37) +erg , an ergoregion is indeed present. This eventuality encom-passes all cases of section 3, though with marked differences.Indeed, when the object is a traversable wormhole — i.e. if a > M , or a < M and (cid:96) > (cid:37) + —, the throat intercepts the ergosurface at some angle θ (cid:54) = 0 , π ; therefore, theergoregion is limited to a region that is coaxial with the wormhole and whose longitudinalsection is shaped as a crescent — see figures 7a and 7b. (This is a common feature of otherrotating traversable wormholes, cf. e.g. [29]). Note however that the throat is technicallynot an edge of the ergoregion, which in fact continues in the “other universe” as far as themirrored ergosurface.When instead the object is a regular black hole — i.e. if a < M and (cid:96) ≤ (cid:37) + —, theergoregion extends all the way to the horizon and its external portion is thus tantamount tothat of Kerr.For completeness, however, we describe the structure of the ergoregion inside the hori-zon, too. If (cid:37) − ≤ (cid:96) < (cid:37) + , viz. in the cases nWoH, RBH-I and nRBH, the ergoregion stretchesas far as the throat — see figure 8a. If instead (cid:96) < (cid:37) − , that is in the RBH-II and eRBH cases,the ergoregion has an inner ergosurface; this surface is intercepted by the throat at some an-– 18 – a) RBH-I (nWoH and nRBH are qualitativelysimilar). The ergoregion extends across the hori-zon and to the throat. The innermost dashedblue line represents the would-be inner horizonand is plotted for reference: the case in whichthe throat is “smaller than the inner horizon” isdepicted in figure 8b. (b) RBH-II (eRBH is qualitatively similar: thetwo horizons coincide and the inner and outerergosurfaces touch at the poles). The ergoregionextends across two horizons, and to the throator the inner ergosurface; within the white lobes g tt < 0, hence these regions do not belong to theergoregion. Figure 8 : Ergoregion in regular black holes, i.e. when (cid:96) ≤ (cid:37) + . As in figure 7, each plot isa slice at fixed φ , the angle with the vertical axis is θ and the distance from the centre is (cid:37) .The hatched region is the ergoregion. The black dotted line represents the throat r = 0; thegrey region is therefore excluded from the spacetime.gle θ : thus the only portions of the spacetime, close to the throat, that do not belong to theergoregion are lobes enclosing the poles — see figure 8b. Note that, as before, the throat isnot an edge of the ergoregion, in the sense that g tt does not change sign upon crossing it. The study of orbits can proceed as for the Kerr geometry. More detail can be found inappendix B; here we focus on the most relevant case of equatorial circular motion.Indeed, the t - and φ -motion are readily integrated by exploiting the conservation of thetest particle’s energy E and angular momentum L (per unit mass) along the rotation axis.Thus, setting θ = π/ 2, the problem is effectively one-dimensional and governed by (cid:37) ˙ r = ±√R (5.3)where the dot marks differentiation with respect to the affine parameter along the geodesicand R is the same potential one finds for a Kerr spacetime mapped by the Boyer–Lindquistradius (cid:37) : R = [ E ( (cid:37) + a ) − a L ] − ∆[ µ (cid:37) + ( L − a E ) ] , (5.4)with µ = 0 , +1 for null and timelike orbits, respectively (the expression for generic θ canbe found in appendix B).Circular orbits satisfy simultaneously R = 0 and d R d r = 0 . (5.5)Note that d R d r = d (cid:37) d r d R d (cid:37) = r(cid:37) d R d (cid:37) , (5.6)– 19 –ence Kerr’s circular orbits readily correspond to circular orbits of our spacetime. Since r = 0 does not generically satisfy the above relations, the mapping between Kerr equatorialcircular geodesics and our own is onto.Thus, in particular, to any pair E c , L c there corresponds a solution (cid:37) c of (5.5) as longas (cid:37) c − M (cid:37) c ± a (cid:112) M (cid:37) c ≥ , (5.7)where the plus (minus) sign refers to prograde (retrograde) orbits. When the equality holds,the equation has formally three real solutions for a < M and only one for a > M . In theformer case, however, the smallest of such roots lies inside the horizon and, therefore, doesnot correspond to any orbit; the other two correspond to the familiar unstable circular photonorbits (cid:37) ph , one prograde and one retrograde, of the Kerr spacetime. In the latter case, thecorresponding orbit connects smoothly to the retrograde branch of the a < M case. Theseorbits of the Kerr spacetime translate into orbits of our spacetime, located at r ph = (cid:113) (cid:37) − (cid:96) . (5.8)Notice that for (cid:96) > M we find no prograde photon circular orbit, at any spin; these worm-holes however do have a retrograde circular orbit, if they spin fast enough.Timelike circular orbits are stable as long as (cid:37) c − M (cid:37) c ± a (cid:112) M (cid:37) c − a ≥ . (5.9)Once again, the equality gives rise to two branches of solutions for a < M , the prograde andretrograde branches, one of which (the retrograde) continues to the a > M region. In theKerr spacetime, these solutions represent the innermost stable circular orbit (ISCO). In ourspacetime, they are located at r isco = (cid:113) (cid:37) − (cid:96) . (5.10)Note that, for (cid:96) > M , these wormholes do not present prograde ISCO. They may have aretrograde ISCO, if they spin fast enough. In this paper, we have constructed a rotating generalisation of the Simpson–Visser metric,applying the Newman–Janis procedure. Depending on the values of a and (cid:96) , it may representa traversable wormhole, a regular black hole with one or two horizons, or three limitingcases of the above. The global properties of the ensuing spacetime have been discussedat length. We further characterised our metric by describing the violations of the energyconditions and found that the “exotic” matter is localised in the vicinity of the throat.Finally, we investigated some relevant features of the exterior geometry: an ergoregion existswhen (cid:96) < M , whatever the value of the spin; a (retrograde) circular photon orbit exists for (cid:96) < M and a (retrograde) ISCO for (cid:96) < M , again independently on the spin.The metric (2.16) thus describes a family of reasonable Kerr black hole mimickers,suitable for serious phenomenological inquiry. For instance, one may wonder whether thepresence of an ergoregion affects its stability. Or, again, whether (cid:96) (cid:54) = 0 leads to observationalconsequences e.g. on the electromagnetic shadow. Investigation on these matters is underway. – 20 – cknowledgments We thank Matt Visser for a careful reading of the manuscript and for his precious comments.The authors acknowledge funding from the Italian Ministry of Education and Scientific Re-search (MIUR) under the grant PRIN MIUR 2017-MB8AEZ. A Curvatures Curvature tensors and scalars are readily computed (we employed the xAct bundle [55–57]for Mathematica ). Their expressions are not particularly illuminating, but we report someof them in order to make a few relevant comments.The Ricci scalar reads ( χ := cos θ ): R = 2 (cid:96) (cid:37) Σ (cid:18) a (cid:37) (cid:2) M χ + (cid:37) ( χ − (cid:3) + a M χ + (cid:37) (3 M − (cid:37) ) (cid:19) . (A.1)For a = 0 this expression coincides with Simpson and Visser’s [6]; the limit (cid:96) → r = 0 and θ = π/ 2. Recall that Σ = r + (cid:96) + a cos θ , hence the throat r = 0 is not singular (not even at θ = π/ 2) as long as (cid:96) (cid:54) = 0.Note in particular that r = 0 is an extremum point for the Ricci scalar, i.e. d R d r (cid:12)(cid:12) r =0 = 0 (aminimum, at least for small a, (cid:96) ).The Kretschmann scalar reads: R µνλσ R µνλσ = 48 M Σ K + 16 (cid:96) M Σ K + 4 (cid:96) (cid:37) Σ K , (A.2)with K = ( (cid:37) − a χ ) (cid:104)(cid:0) (cid:37) + a χ (cid:1) − a (cid:37) χ (cid:105) , (A.3a) K = (cid:37) (cid:2) a + (cid:37) (2 (cid:37) − M ) (cid:3) − a (cid:37)χ (cid:2) a + (cid:37) (4 (cid:37) − M ) (cid:3) + a χ (6 (cid:37) − M ) , (A.3b) K = (cid:37) (cid:2) a (cid:37) (2 (cid:37) − M ) + 4 a + (cid:37) (cid:0) M − M (cid:37) + 3 (cid:37) (cid:1)(cid:3) + 2 a (cid:37) χ (cid:2) a (5 M − (cid:37) ) − M (cid:37) + 5 M (cid:37) + (cid:37) (cid:3) + 2 a M (cid:37) χ (6 M − (cid:37) )+ a (cid:37) χ (cid:2) a M + (cid:37) (cid:0) M − M (cid:37) + 3 (cid:37) (cid:1)(cid:3) + a M χ , (A.3c)again, for r = 0 — i.e. (cid:37) = (cid:96) — this expression is finite as long as (cid:96) (cid:54) = 0.The components of the Einstein tensor, in mixed components, are: G tt = (cid:96) (cid:37) Σ (cid:104) a (cid:37) χ [2 M (3 − χ ) − (cid:37) ( χ − a (cid:37) [ M ( χ − 3) + 2 (cid:37) ] − a M χ ( χ − 1) + (cid:37) ( (cid:37) − M ) (cid:105) , (A.4a) G φt = a(cid:96) M (6 a (cid:37) χ + a χ − (cid:37) ) (cid:37) Σ , (A.4b) G tφ = G φt (cid:0) a + (cid:37) (cid:1) (cid:0) χ − (cid:1) , (A.4c) G rr = − (cid:96) (cid:37) − a M χ + a (cid:37)χ (cid:37) Σ , (A.4d) G θθ = (cid:96) (cid:37) − M (4 a (cid:37) χ + a χ + (cid:37) ) + a (cid:37) χ Σ , (A.4e)– 21 – φφ = (cid:96) a (cid:37) χ [ M ( χ − − (cid:37) ( χ − a (cid:37) [ M (3 − χ ) + 2 (cid:37) ] − a M χ + (cid:37) ( (cid:37) − M ) (cid:37) Σ . (A.4f) B A note on geodesics Geodesics can be described by means of the Hamilton–Jacobi method [49, 58], whereby theequations of motion descend from ∂S∂τ = − g µν ∂S∂x µ ∂S∂x ν , (B.1)with S the action and τ an affine parameter along the geodesic. Separability, foretold in 3.4,motivates the ansatz S = 12 µ τ − E t + L φ + S r ( r ) + S θ ( θ ); (B.2)here µ , E and L are arbitrary constants: µ = 0 for null geodesics and +1 for timelikegeodesics; the other two can be interpreted as the energy (per unit mass) and the projectionof the angular momentum (per unit mass) along the rotation axis, respectively.Inserting ansatz (B.2) in (B.1), one obtains relations among the functions t ( τ ), r ( τ ), θ ( τ ) and φ ( τ ); and, differentiating with respect to τ , the following system of first-order,ordinary differential equations:Σ d t d τ = a ( L − a E sin θ ) + (cid:37) + a ∆ [ E ( (cid:37) + a ) − L a ] , (B.3a)Σ d r d τ = ±√R , (B.3b)Σ d θ d τ = ±√ Θ , (B.3c)Σ d φ d τ = L sin θ − a E + a ∆ [ E ( (cid:37) + a ) − L a ] , (B.3d)where R = [ E ( (cid:37) + a ) − L a ] − ∆[ µ (cid:37) + ( L − a E ) + Q ] , (B.4)Θ = Q − cos θ (cid:20) a ( µ − E ) + L sin θ (cid:21) , (B.5)Here Q is the Carter constant already hinted to in section 4.1. Its existence derives from theKilling tensor (3.6) via the constant K := K µν d x µ d τ d x ν d τ , (B.6)as Q := K − ( a E − L ) . Its expression in this spacetime coincides with its Kerr homonym’s: Q = u θ + cos θ (cid:20) a (1 − E ) − L sin θ (cid:21) . (B.7)Note that the system (B.4–B.5) looks almost identical to its Kerr analogue: indeed, theright-hand sides are precisely those one would find performing the same analysis in a Kerrspacetime, charted by the Boyer–Lindquist coordinates ( t, (cid:37), θ, φ ).– 22 –ence, one might expect that our metric and Kerr’s share the same geodesics. Namely,given a Kerr geodesic (cid:0) t ( τ ) , (cid:37) ( τ ) , θ ( τ ) , φ ( τ ) (cid:1) , one could guess that the curve (cid:0) t ( τ ) , r ( τ ) = (cid:112) (cid:37) ( τ ) − (cid:96) , θ ( τ ) , φ ( τ ) (cid:1) might be a geodesic of our spacetime (charted by r as Boyer–Lindquist-like radius) — at least as long as (cid:37) ( τ ) ≥ (cid:96) .This however is not true, in general. 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