Constraining alternatives to the Kerr black hole
aa r X i v : . [ g r- q c ] F e b Constraining alternatives to the Kerr black hole
Rajibul Shaikh, ∗ Kunal Pal, † Kuntal Pal, ‡ and Tapobrata Sarkar § Department of Physics,Indian Institute of Technology,Kanpur 208016, India
Abstract
The recent observation of the shadow of the supermassive compact object M87 ∗ by the Event Horizon Telescope(EHT) collaboration has opened up a new window to probe the strong gravity regime. In this paper, we studyshadows cast by two viable alternatives to the Kerr black hole, and compare them with the shadow of M87 ∗ . Thefirst alternative is a horizonless compact object (HCO) having radius r and exterior Kerr geometry. The second oneis a rotating generalisation of the recently obtained one parameter ( r ) static metric by Simpson and Visser. Thislatter metric, constructed by using the Newman-Janis algorithm, is a special case of a parametrised rotating non-Kerrgeometry obtained by Johannsen. Here, we constrain the parameter r of these alternatives using the results from M87 ∗ observation. We find that, for the mass, inclination angle and the angular diameter of the shadow of M87 ∗ reportedby the EHT collaboration, the maximum value of the parameter r must be in the range 2 . r + ≤ r ,max ≤ . r + for the dimensionless spin range 0 . ≤ a ∗ ≤ .
94, with r + being the outer horizon radius of the Kerr black hole at thecorresponding spin value. We conclude that these black hole alternatives having r below this maximum range (i.e. r ≤ r ,max ) is consistent with the size and deviation from circularity of the observed shadow of M87 ∗ . ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] . INTRODUCTION Understanding the nature of strong gravity is perhaps the key to the formulation of a full theory ofquantum gravity that remains elusive more than a hundred years after Einstein formulated the theory ofgeneral relativity (GR). Ubiquitous in such studies are black holes and the associated null surfaces, i.e., eventhorizons. It is now believed that the centers of most galaxies contain supermassive black holes with massesof the order of 10 − M ⊙ . An important observational signature of the event horizon of a black holeis its shadow – a dark region surrounding the central singularity, caused by gravitational lensing, and thecapture of photons at the horizon due to strong gravity. Phenomenal advances in the observational studyof black hole event horizons have come about recently, after the first results on the radio source M87 ∗ werepublished by the Event Horizon Telescope (EHT) collaboration [1],[2],[3]. These results have opened up theexciting possibility of understanding strong gravity near black hole event horizons, which, even a few yearsback, seemed like a purely theoretical aspect.While the phenomenology of black holes continues to be at the focus of attention post the EHT results,horizonless compact objects (HCOs) are fast gaining popularity in the literature. Part of the reason forstudying such objects is that these can be black hole alternatives, whose observational aspects can besimilar to those of black holes. Indeed, since the existence of a singularity usually indicates a pathology ina theory, the existence of black hole singularities in a purely classical version of GR is still debated, withmany advocating that it is removed by quantum effects, a result that is well known in semi classical versionsof GR.Further, as is well known by now, HCOs, which do not have an event horizon, might mimick manyproperties of black holes. Hence, in this EHT era, it becomes all the more important to study black holemimickers, as obseravational signatures from these can be contrasted and compared with the EHT results onM87 ∗ . For example, one such black hole mimicker is the wormhole solution of GR, which contains a throatthat connects two different universes or two distant regions of one universe. Although the presence of thethroat indicates the breakdown of the weak energy condition, it is well known that in dynamical scenarios,or for wormholes in modified gravity, such energy condition violations can be avoided. In [4], Damour andSolodukhin pointed out the close similarities of several theoretical features of black holes with those obtainedfrom wormhole solutions. Apart from wormholes, various other possible HCO geometries have been studiedin the literature, see, e.g., [5] and the references therein.An important aspect of the EHT data is that it can be used to constrain the parameter space of geometrieswhich deviate from Schwarzschild or Kerr black hole (see, e.g., [6], [7], [8], [9]). In this paper, we use thisapproach to put bounds on two Kerr black hole alternatives. The first alternative is a horizonless compactobject (HCO) having radius r and exterior Kerr geometry. For the second one, we construct a metric2y generalising a recently proposed static, spherically symmetric solution of Einstein’s equation, given bySimpson and Visser (SV) [10]. The SV metric is attractive in that it is a minimal one parameter extension ofthe Schwarzschild metric, and can describe a black hole or a wormhole for different choices of the parameter.Here, a rotating version of the SV metric is first derived using the Newman-Janis algorithm, which cancorrespond to a rotating black hole or a rotating wormhole. Using these two alternatives, we study theshadow and compare it with the data for the shadow of M87 ∗ . In the process, we are able to put a boundon the parameter appearing in the solutions.This paper is organised as follows. In Sec. II, we discuss the spacetime geometry of the black holealternatives. In Sec. III, we discuss the detailed method for constructing the shadows of the alternatives.We constrain the parameter of these black hole alternatives using the M87 ∗ results in Sec. IV. We concludein Sec. V. This paper also has an appendix where the relevant details of the Newman-Janis algorithm isdiscussed briefly. Note added :
While this paper was being readied for submission, we became aware of the work of [11],where the authors derive the rotating SV solution and study in details the resulting phase diagram.
II. THE KERR BLACK HOLE AND ITS ALTERNATIVESA. Kerr black hole
The Kerr metric in Boyer-Lindquist coordinates can be written as ds = − (cid:18) − M r Σ (cid:19) dt − M ar sin θ Σ dtdφ + Σ∆ dr + Σ dθ + (cid:18) r + a + 2 M a r sin θ Σ (cid:19) sin θdφ , (1)where M is the mass of the black hole, a is the specific angular momentum defined as a = J/M andΣ = r + a cos θ, ∆ = r − M r + a . (2)For convenience, we define a dimensionless Kerr parameter called spin as a ∗ = a/M = J/M . The horizonradii of the black hole are the roots of ∆ = 0 and are given by r ± = M (1 ± p − a ∗ ) . (3) B. Horizonless compact objects with exterior Kerr metric
We consider a HCO with the exterior being given by the Kerr metric, and its surface outside the outerhorizon of the would-be Kerr black hole, i.e., at r = r > r + , r + being the outer horizon radius of the Kerrblack hole. 3 . Rotating Simpson-Visser metric: a special case of the Johannsen metric Recently, Simpson and Visser constructed a metric where the central singularity is replaced by a minimalsurface of radius r and can acts as a black hole mimicker. The spacetime geometry is given by [10] ds = − − M p r + r ! dt + dr − M √ r + r + ( r + r )( dθ + sin θdφ ) . (4)The above geometry represents a regular black hole when r < M and a wormhole when r > M with r being the wormhole throat radius. In the coordinate system used above, the throat corresponds to r = 0.We now apply the Newman-Janis algorithm to obtain a rotating version of the above metric. After doingthis the metric becomes (see Appendix A) ds = − − M p r + r Σ ! dt − M a p r + r sin θ Σ dtdφ + Σ∆ dr +Σ dθ + r + r + a + 2 M a p r + r sin θ Σ ! sin θdφ , (5)Σ = r + r + a cos θ, ∆ = r + r − M r + a . (6)Note that it reduces to Kerr geometry when r = 0. While this work was in preparation, in [11] whichappeared recently on arXiv, the authors have also obtained the above metric through the same method asours.The above metric can further be simplified using the coordinate transformation ¯ r = p r + r . Performingthe transformation and dropping the bar, we obtain ds = − (cid:18) − M r Σ (cid:19) dt − M ar sin θ Σ dtdφ + Σ∆ ˆ∆ dr +Σ dθ + (cid:18) r + a + 2 M a r sin θ Σ (cid:19) sin θdφ , (7)Σ = r + a cos θ, ∆ = r − M r + a , ˆ∆ = 1 − r r , (8)which is very similar to the Kerr geometry except the term ˆ∆. For r = 0, ˆ∆ = 1 and we have the Kerrblack hole with the horizon radii given by r ± . Note that, the above metric has also horizons at r ± . However,depending on the values of r and the spin a , the horizons may or may not be relevant. For 0 ≤ a/M ≤ r < r + and wormhole when r ≥ r + . However, for a/M > r ± does not exist in this case. In case of wormhole, the throat is given byˆ∆ = 0 and is at r = r . 4t can be shown that the above rotating Simpson-Visser (SV) belongs to a special case of the parametrizednon-Kerr metric constructed by Johannsen. The non-Kerr metric by Johannsen [12] is given by g tt = − ˜Σ[∆ − a A ( r ) sin θ ][( r + a ) A ( r ) − a A ( r ) sin θ ] , g tφ = − a [( r + a ) A ( r ) A ( r ) − ∆] ˜Σ sin θ [( r + a ) A ( r ) − a A ( r ) sin θ ] ,g rr = ˜Σ∆ A ( r ) , g θθ = ˜Σ , g φφ = ˜Σ sin θ (cid:2) ( r + a ) A ( r ) − a ∆ sin θ (cid:3) [( r + a ) A ( r ) − a A ( r ) sin θ ] , (9)where A ( r ) = 1 + ∞ X n =3 α n (cid:18) Mr (cid:19) n , A ( r ) = 1 + ∞ X n =2 α n (cid:18) Mr (cid:19) n , A ( r ) = 1 + ∞ X n =2 α n (cid:18) Mr (cid:19) n , ˜Σ = Σ + f ( r ) , f ( r ) = ∞ X n =3 ǫ n M n r n − . (10)The Kerr metric is recovered when f ( r ) = 0, A ( r ) = A ( r ) = A ( r ) = 1, i.e., α n = α n = α n = ǫ n = 0.The rotating SV metric in Eq. (7) can be obtained as a special case of the above Johannsen metric for f ( r ) = 0, A ( r ) = A ( r ) = 1 and A ( r ) = ˆ∆( r ), i.e., for α n = α n = ǫ n = 0 , α n = 0 ( n = 2) , α = − r M . (11) III. CONSTRUCTING SHADOWS OF THE BLACK HOLE ALTERNATIVESA. Kerr black hole
First, we consider a Kerr black hole. The separated geodesic equations which we need for the purpose ofshadow in this case are given by Σ drdλ = ± p R ( r ) , Σ dθdλ = ± p Θ( θ ) , (12)where R ( r ) = (cid:2) ( r + a ) E − aL (cid:3) − ∆ h K + ( L − aE ) i , Θ( θ ) = K + a E cos θ − L cot θ, (13) E is the energy, L is the angular momentum, and K is the Carter’s constant.The unstable circular photon orbits, which form the boundary of a shadow, are given by ˙ r = 0, ¨ r = 0and ... r >
0, i.e., by R ( r ph ) = 0 , R ′ ( r ph ) = 0 , R ′′ ( r ph ) > , (14)where r ph is the radius of an unstable photon orbit. For the Kerr spacetime, after using the first twoconditions, we obtain the following critical impact parameters ξ ph = 4 M r ph − ( r ph + M )( r ph + a ) a ( r ph − M ) , η ph = 4 M a r ph − r ph [ r ph ( r ph − M )] a ( r ph − M ) , (15)5here ξ = L/E and η = K /E . However, the apparent shape of a shadow is measured using the celestialcoordinates defined by [13] α = lim r o →∞ (cid:18) − r o sin θ o dφdr (cid:12)(cid:12)(cid:12) ( r o ,θ o ) (cid:19) , β = lim r o →∞ (cid:18) r o dθdr (cid:12)(cid:12)(cid:12) ( r o ,θ o ) (cid:19) , (16)where ( r o , θ o ) are the position coordinates of the observer. Using the geodesic equations, we obtain α = − ξ sin θ o , (17) β = ± p η + a cos θ o − ξ cot θ o , (18)The contour of the shadow which is formed by the unstable photon orbits is given by the parametric plot of α ph and β ph where α ph = − ξ ph sin θ o , (19) β ph = ± q η ph + a cos θ o − ξ ph cot θ o , (20) - - - - - α / M β / M FIG. 1. Shadow of the Kerr black hole with a ∗ = a/M = 0 .
94 and θ o = π/ For a rotating Kerr black hole, unstable photon orbits, which form the shadow contour in the α − β plane, consist of both prograde (photons having motion in the direction of the spin) and retrograde (photonshaving motion in a direction opposite to the spin) orbits. The prograde orbits have lesser radii than those ofthe retrograde ones. Figure 1 shows a typical shadow of a Kerr black hole. The spin axis divide the shadowinto two parts in the α − β plane. The contour of the shadow on the left side (i.e., α < ξ > α > ξ < r ph,min is of prograde typeand corresponds to the point ( α ph,min ,
0) of the shadow contour. On the other hand, the unstable photonorbit which has the maximum radius r ph,max is of retrograde type and corresponds to the point ( α ph,max , r ph,min ≤ r ph ≤ r ph,max . The unstable photon orbit radii r ph,max and r ph,min are obtained by β ph = 0,i.e., by η ph + a cos θ o − ξ ph cot θ o = 0 . (21) r ph,max and r ph,min are respectively given by the largest and the second largest roots of the above equation.Note that besides the mass M and the spin a of the black hole, the radii depend on the observation(inclination) angle θ o . B. Horizonless compact object
Let us now consider an horizonless compact object (HCO) with its surface at r = r and with theexterior given by the Kerr solution. If r < r ph,min , then the compact object is completely hidden inside allthe unstable photon orbits of the exterior Kerr metric which take part in the shadow formation. In such acase, the shadow contour of the HCO contour is the same as that of the Kerr black hole. The HCO in thiscase therefore perfectly mimic a Kerr black hole as far as the shadow silhouette is concerned. However, if r > r ph,min , then the unstable photon orbits having radii in the range r ph,min ≤ r ph < r become irrelevantas they lie inside the surface of the compact object. In such a case, the part of the shadow contour, whichis lost due to the unstable orbits lying inside the surface, is given by the photons which have turning pointsat the surface r = r , i.e., by R ( r ) = 0. This gives (cid:2) ( r + a ) − aξ (cid:3) − ∆( r ) h η + ( ξ − a ) i = 0 , (22)where ξ and η denote the impact parameters of photons having turning points at the surface r = r . Afterusing Eqs. (17) and (18), we obtain (cid:2) ( r + a ) + a sin θ o α (cid:3) − ∆( r ) h β + ( α + a sin θ o ) i = 0 , (23)where ( α , β ) denotes the celestial coordinates of photons having turning points at the surface r = r .Therefore, for r ph,min < r < r ph,max , the complete shadow of a HCO is given by the Union of the ( α ph , β ph )curve which is due to the unstable photon orbits and the above ( α , β ) curve. We now describe the detailsprocedure to obtain the shadow. Note that, for r ph,min < r < r ph,max , ( α , β ) curve must intersect the( α ph , β ph ) curve at the points ( α ,max , ± β ,max ) which corresponds to r ph = r , where α ,max = α ph (cid:12)(cid:12) r ph = r β ,max = β ph (cid:12)(cid:12) r ph = r . Therefore, ( α , β ) in Eq. (23) have the ranges α ,min ≤ α ≤ α ,max and − β ,max ≤ β ≤ β ,max . Here, α ,min is obtained by putting β = 0 in Eq. (23). This gives α ,min = r + a ∓ a sin θ o p ∆( r ) ± p ∆( r ) − a sin θ o , (24)where we take the root which is negative and close to α ,max . Therefore, when r ph,min < r < r ph,max , thecomplete shadow contour is given by the union of the ( α ph , β ph ) curve (due to the unstable photon orbits)plotted for r ≤ r ph ≤ r ph,max and the ( α , β ) curve (due to photons having turning points at the r = r surface) plotted between the points ( α ,min ,
0) and ( α ,max , β ,max ). However, when r > r ph,max , then theall the unstable photon orbits become irrelevant as all of them lie inside the surface and the shadow in thiscase is completely given by the ( α , β ) curve. Note that, when plotting the curve ( α , β ) [Eq. (23)], wesimplify it to obtain β = ± q(cid:2) ( r + a ) + a sin θ o α (cid:3) − ∆( r ) ( α + a sin θ o ) p ∆( r ) , (25)and vary α from α ,min to α ,max . C. Rotating Simpson-Visser metric
Let us now consider the rotating SV metric. The separated geodesic equations are the same except thatthe radial equation in this case is modified toΣ drdλ = ± q ˆ∆( r ) p R ( r ) . (26)Note that R ( r ) in this case is the same as that for the Kerr black hole. The construction of shadow contourfrom the unstable photon orbits is bit involved in this case. Therefore, we consider the black hole andwormhole case separately below. However, to this end, we make use of the unstable photon orbits obtainedjust from R ( r ) and its derivative and elaborate what effect the additional factor ˆ∆( r ) have in this SV metriccase. I. Black hole case ( r < r + with ≤ a/M ≤ ): In this case, r ph > r always as the unstable photonorbits which take part in shadow formation lie outside the outer event horizon r + and ˆ∆( r ph ) = 0 at thelocation of a unstable photon orbits. Therefore, the conditions ˙ r = 0, ¨ r = 0 and ... r > R ( r ph ) = 0, R ′ ( r ph ) = 0 and R ′′ ( r ph ) >
0, which is the same asthat for the Kerr black hole. Hence, the unstable circular orbits as well as the shadow contour in this caseare the same as those of the Kerr black hole. As a result, this black hole case acts as a perfect Kerr blackhole mimicker as far as the shadow silhouette is concerned.8
I. Wormhole case (either r ≥ r + with ≤ a/M ≤ or a/M > ): Depending on the relative valuesof r and the minimum radius r ph,min of the unstable photon orbits obtained from R ( r ) and its derivative,this case can be subdivided into two subcases. IIa: If r < r ph,min , then this subcase is similar to the black hole case discussed above. This subcasetherefore mimic the Kerr black hole. IIb: If r > r ph,min , then, for the unstable photon orbits which lies outside the throat, i.e., for r ph > r ,the conditions ˙ r = 0, ¨ r = 0 and ... r > R ( r ph ) = 0, R ′ ( r ph ) = 0 and R ′′ ( r ph ) > r ph = r , ˆ∆( r ph ) = 0 and ˙ r ( r ph ) = 0. Therefore, for such orbits, the conditionsfor them now become R ( r ) = 0 and R ′ ( r ) >
0, which are different from those for the unstable orbitslying outside the throat. Note that the condition R ( r ) = 0 leads to equation (22). Therefore, the shadowcontour and the procedure to obtain it for both the horizonless compact object (HCO) with radius r andthe wormhole in this case will be the same. The only difference in these two cases is that the part of theshadow which gets modified by the r = r surface and is given by R ( r ) = 0 corresponds to photons havingturning point at r = r for the horizonless object with radius r , but, for the wormhole, to unstable photonorbits lying at the throat r . θ o = o θ o = o θ o = o θ o = o θ o = o θ o = o solid = r ph,min ( co - rotating ) dashed = r ph,max ( counter - rotating ) a / M r ph / r + (a) θ o = o θ o = o θ o = o co - rotating0.80 0.85 0.90 0.95 1.001.001.051.101.15 a / M r ph / r + (b)
50 60 70 80 900.991.001.011.021.031.041.051.06 a / M = - rotatingcounter - rotaing0 20 40 60 801.01.52.02.53.03.54.0 θ o ( in degree ) r ph / r + (c) FIG. 2. Minimum (solid) and maximum (dashed) radius of the unstable photon orbits which take part in the shadowformation in the Kerr black hole background for a given spin and observer inclination angle. The minimum and themaximum radius correspond to co-rotating and counter-rotating orbits, respectively.
Figure 2 shows the minimum and the maximum of the unstable photon orbits which take part in theshadow in the Kerr black hole background for a given spin and the observer inclination angle. Figure 3 showssome shadows of a horizonless alternative. Note that the shadow of the object deviate from that of the Kerr9 - - - - α / M β / M (a) r /r + = 1 . - - - - - α / M β / M (b) r /r + = 1 . - - - - - α / M β / M (c) r /r + = 1 . FIG. 3. Shadows cast by the black hole alternatives (solid black plus solid blue) and the Kerr black hole (black dashed).The solid blue part of the shadow contour is due to R ( r ) = 0. Here, the spin is a ∗ = a/M = 0 .
94 and θ = π/ r ph,min /r + = 1 . r > r ph,min . black hole only when r > r ph,min . Therefore, we can define a critical radius of the object by r oc = r ph,min .If the object has a radius greater than this critical value, i.e., if r ≥ r c , then its shadow deviates from thatof the Kerr black hole. Figure 4 shows the dependence of the critical radius on the spin and observationangle. Note that, if the radius of the object lies very close to the would-be event horizon of the Kerr blackhole (say r < . r + for example), then we need both high spin and high observation angle to probe suchmodification through the shadow. IV. CONSTRAINING THE BLACK HOLE ALTERNATIVES FROM THE M87 ∗ OBSERVATION
We now use the results from M87 ∗ observation and put possible constraint on the Kerr black holealternatives. For this purpose, we use the average angular size of the shadow and its deformation fromcircularity. Note that the shadow is perfectly circular for zero spin and start deforming as we increase thespin. With increasing spin, the center of the shadow in the α − β plane also starts shifting from the origin.However, as the shadow has reflection symmetry around the α -axis, its geometric center ( α c , β c ) is given by α c = 1 /A R αdA and β c = 0, dA being an area element. We use this geometric centre to find out the averageshadow radius and deviation from circularity. To this end, we first define an angle φ between the α -axisand the vector connecting the geometric centre ( α c , β c ) with a point ( α, β ) on the boundary of a shadow.Therefore, the average radius R av the shadow is given by [8] R av = 12 π Z π l ( φ ) dφ, (27)10 / r + FIG. 4. Parameter region showing the critical radius r c of the horizonless alternative above which its shadow startsdeviating from that of the Kerr black hole. where l ( φ ) = p ( α ( φ ) − α c ) + β ( φ ) and φ = tan − ( β ( φ ) / ( α ( φ ) − α c )). Following [1], we define the deviation∆ C from circularity as ∆ C = 1 R av s π Z π ( l ( φ ) − R av ) dφ. (28)Note that ∆ C is the fractional RMS distance from the average radius of the shadow.The angular diameter of the shadow is given by ∆ θ sh = 2 R av /D , where D is the distance to M87 ∗ .Following [1], We take D = (16 . ± .
8) Mpc and the mass of the object to be M = (6 . ± . × M ⊙ .The inclination angle is taken to be θ o = 17 ◦ , which the jet axis makes to the line of sight [1]. With this, wecalculate the angular diameter ∆ θ sh and deviation ∆ C for different r and spin a . This is shown in Fig. 5.According to EHT collaboration, the angular size of the observed shadow is ∆ θ sh = 42 ± µ as [1] and thespin lies within the range 0 . ≤ a ∗ ≤ .
94. Note form Fig. 5 that, for the spin range 0 . ≤ a ∗ ≤ .
94, theangular size will be consistent with the M87 ∗ observation, i.e., ∆ θ sh is between 39 to 45 µ as, if the maximumvalue r ,max of r of the black hole alternatives lies within the range 2 . r + ≤ r ,max ≤ . r + . If r is belowthis maximum range (i.e. r ≤ r ,max ), then the shadow silhouette of the alternatives is consistent with theobserved size. For a given spin, say for a/M = 0 . r ,max = 2 . r + , indicating that the angular size ofthe shadow of the black hole alternatives will be consistent with the M87 ∗ observation if r ≤ r ,max . Thedeviation of the observed shadow from circularity is reportedly less than 10%. From Fig. 5, note that thedeviation of the shadow for the black hole alternatives is consistent with the observed values, for the above11entioned spin range and r < r ,max . Δθ sh Δ C FIG. 5. Dependence of the angular size and the deviation of the shadow on r and spin. Here, M = 6 . × M ⊙ and D = 16 . . ≤ a ∗ ≤ .
94. Thetwo horizontal black dashed lines indicate the range of the maximum value r ,max for this spin range, in order that theangular size is within the maximum value of 45 µ as. For a given spin, say for a/M = 0 . r ,max = 2 . r + , indicatingthat the angular size of the shadow of the black hole alternatives will be consistent with the M87 ∗ observation if r ≤ r ,max . The red dashed line shows the critical value r c of r as discussed in the previous section. Note that, in Fig. 5, we took M = 6 . × M ⊙ and D = 16 . ∗ data. Taking D = (16 . ± .
8) Mpc and M = (6 . ± . × M ⊙ , the size of the shadow in dimensionless unit is estimatedto be d sh M = D ∆ θ sh GM = 11 . ± . , (29)where the errors have been added in quadrature. The above quantity must be equal to R av M . Figure 6 showsthe the dependence of the shadow size on different parameters. Note that, after incorporating the errors in M and D , the range of the maximum value of r is modified to 2 . r + ≤ r ,max ≤ . r + . V. CONCLUSIONS
In this paper, we have considered two viable alternatives to the Kerr black hole. The first one is ahorizonless compact object (HCO) having radius r and exterior Kerr geometry. The second one is arotating generalisation of the recently obtained one parameter ( r ) static metric by Simpson and Visser.We have studied shadows cast by these alternatives and compared them with the observed shadow of the12 sh M FIG. 6. Dependence of the size of the shadow on r and spin. Here, M = (6 . ± . × M ⊙ and D = (16 . ± . . ≤ a ∗ ≤ .
94. The twohorizontal black dashed lines indicate the range of the maximum value r ,max for this spin range, in order that thesize is within the maximum value of 12 .
5. For a given spin, say for a/M = 0 . r ,max = 2 . r + , indicating that theangular size of the shadow of the black hole alternatives will be consistent with the M87 ∗ observation if r ≤ r ,max .The red dashed line shows the critical value r c of r as discussed in the previous section. supermassive compact object M87 ∗ , thus constraining the parameter r of the alternatives. We find that,for the mass, inclination angle and the angular diameter of the shadow of M87 ∗ reported by the EHTcollaboration, the maximum value of the parameter r must be in the range 2 . r + ≤ r ,max ≤ . r + forthe dimensionless spin range 0 . ≤ a ∗ ≤ .
94, with r + being the outer horizon radius of the Kerr black hole atthe corresponding spin value. We conclude that these black hole alternatives having r below this maximumrange (i.e. r ≤ r ,max ) is consistent with the size and deviation from the circularity of the observed shadowof M87 ∗ . An increase in precision in the EHT data should improve the constraint that we have reportedhere. Appendix A ROTATING SIMPSON-VISSER METRIC
The Newman-Janis algorithm is a set of steps to construct a stationary, axisymmetric spacetime beginningfrom a static and spherically symmetric spacetime [15, 16]. Staring from a spherically symmetric, staticspacetime written as ds = − f ( r ) dt + dr g ( r ) + h ( r )( dθ + sin θdφ ) , (30)13e make a coordinate transformation to get the metric in the advanced null coordinate defined as du = dt − dr √ f g . (31)Then the above metric of eq.(30) in these coordinates becomes ds = − f ( r ) du − s fg dudr + h ( r )( dθ + sin θdφ ) . (32)The next step is to express the inverse metric g µν in a null tetrad basis in the form g µν = − l µ n ν − l ν n µ + m µ ¯ m ν + m ν ¯ m µ , (33)where the null tetrad is Z µν = (cid:0) l µ , n µ , m µ , ¯ m µ (cid:1) , and a bar denotes complex conjugation. The tetrad vectorsare given by l µ = δ µr , n µ = r gf δ µu − g δ µr , m µ = 1 √ h (cid:18) δ µθ + i sin θ δ µφ (cid:19) . (34)Now, we perform a complex transformation given by r → r ′ = r + ia cos θ , and , u → u ′ = u − ia cos θ , (35)so that, after the complex transformation, the new tetrad becomes l ′ µ = δ µr , n ′ µ = s G ( r, θ ) F ( r, θ ) δ µu − G ( r, θ )2 δ µr , m ′ µ = 1 p H ( r, θ ) (cid:16) ia sin θ ( δ µu − δ µr ) + δ µθ + i sin θ δ µφ (cid:17) . (36)Here, F ( r, θ ), G ( r, θ ) and H ( r, θ ) are the complexified form of the functions f ( r ), g ( r ) and h ( r ) respectively.At this stage, the metric constructed using the new tetrad obtained after complexification contain non-diagonal components other than g tφ . Therefore, the last step is to write the metric in Boyer-Lindquist form(where the only nonzero off diagonal term is g tφ ) using the coordinate transformation du = dt ′ + χ ( r ) dr, dφ = dφ ′ + χ ( r ) dr . (37)We must ensure that the two functions χ ( r ) and χ ( r ) are solely functions of r to define the global trans-formations. Here, χ ( r ) , χ ( r ) are given by [17] χ ( r ) = − q G ( r,θ ) F ( r,θ ) H ( r, θ ) + a sin θG ( r, θ ) H ( r, θ ) + a sin θ , χ ( r ) = − aG ( r, θ ) H ( r, θ ) + a sin θ . (38)The above steps can be used to derive the Kerr metric from the Schwarzschild metric provided we use thefollowing rule for complexifying the metric functions,1 r → (cid:16) r ′ + 1¯ r ′ (cid:17) = r Σ and r → r ′ ¯ r ′ = Σ , (39)14here Σ = r + a cos θ . Now to apply this algorithm to the SV metric, we write down the metric functionsas f ( r ) = g ( r ) = 1 − m ( r ) r + r , h ( r ) = r + r , (40)where m ( r ) = M p r + r and use r → r ′ ¯ r ′ = Σ to obtain the complexified functions as F ( r, θ ) = G ( r, θ ) = 1 − M p r + r Σ + r , H ( r, θ ) = Σ + r . (41)With this χ ( r ) and χ ( r ) become χ ( r ) = − r + r + a r + r + a − m ( r ) , χ ( r ) = − ar + r + a − m ( r ) . (42)Finally, the metric turns out to be ds = − − M p r + r Σ ! dt − M a p r + r sin θ Σ dtdφ + Σ∆ dr +Σ dθ + r + r + a + 2 M a p r + r sin θ Σ ! sin θdφ , (43)Σ = r + r + a cos θ, ∆ = r + r − M r + a . (44) [1] The Event Horizon Telescope Collaboration, First M87 event horizon telescope results. I. The shadow of thesupermassive black hole , Astrophys. J. Lett. , L1 (2019).[2] The Event Horizon Telescope Collaboration,
First M87 event horizon telescope results. V. Physical origin of theasymmetric ring , Astrophys. J. Lett. , L5 (2019).[3] The Event Horizon Telescope Collaboration,
First M87 event horizon telescope results. VI. The shadow and massof the central black hole , Astrophys. J. Lett. , L6 (2019).[4] T. Damour and S. N. Solodukhin,
Wormholes as black hole foils , Phys. Rev. D , 024016 (2007).[5] V. Cardoso and P. Pani, Testing the nature of dark compact objects: a status report , Living Rev. Rel. , 4 (2019).[6] D. Psaltis et al. (Event Horizon Telescope), Gravitational Test Beyond the First Post-Newtonian Order with theShadow of the M87 Black Hole , Phys. Rev. Lett. , 141104 (2020).[7] I. Banerjee, S. Chakraborty and S. SenGupta,
Silhouette of M87*: A New Window to Peek into the World ofHidden Dimensions , Phys. Rev. D , no. 4, 041301 (2020).[8] C. Bambi, K. Freese, S. Vagnozzi and L. Visinelli,
Testing the rotational nature of the supermassive object M87*from the circularity and size of its first image , Phys. Rev. D , no. 4, 044057 (2019).[9] R. Kumar and S. G. Ghosh,
Black Hole Parameter Estimation from Its Shadow , ApJ , 78 (2020).[10] A. Simpson and M. Visser,
Black-bounce to traversable wormhole , JCAP , 042 (2019)
11] J. Mazza, E. Franzin and S. Liberati,
A novel family of rotating black hole mimickers , arXiv:2102.01105 [gr-qc].[12] T. Johannsen,
Regular black hole metric with three constants of motion , Phys. Rev. D , 044002 (2013).[13] S. E. Vazquez and E. P. Esteban, S. E. Vazquez and E. P. Esteban, Strong field gravitational lensing by a Kerrblack hole , Nuovo Cim. B , 489 (2004)[14] R. Shaikh,
Shadows of rotating wormholes , Phys. Rev. D , 024044 (2018).[15] E. T. Newman and A. I. Janis, Note on the Kerr spinning-particle metric , J. Math. Phys. , 915 (1965).[16] E. T. Newman, R. Couch, K. Chinnapared, A. Exton, A. Prakash, and R. Torrence, Metric of a rotating, chargedMass , J. Math. Phys. , 918 (1965).[17] R. Shaikh, Black hole shadow in a general rotating spacetime obtained through Newman-Janis algorithm , Phys.Rev. D , 024028 (2019)., 024028 (2019).