Possible imprints of cosmic strings in the shadows of galactic black holes
aa r X i v : . [ g r- q c ] N ov Possible imprints of cosmic strings in the shadows ofgalactic black holes
Vassil K. Tinchev ∗ , Stoytcho S. Yazadjiev † Department of Theoretical Physics, Faculty of Physics, Sofia University,5 James Bourchier Boulevard, Sofia 1164, Bulgaria
Abstract
We examine the shadow cast by a Kerr black hole pierced by a cosmic string.The observable images depend not only on the black hole spin parameter and theangle of inclination, but also on the deficit angle yielded by the cosmic string. Thedependence of the observable characteristics of the shadow on the deficit angle isexplored. The imprints in the black hole shadow left by the presence of a cosmicstring can serve as a method for observational detection of such strings.
It’s well known that the shadow (or the apparent shape) of a compact relativisticobject encodes information about the nature of this object [1]. That is why the apparentshapes of various black holes and other compact objects, such as wormholes and nakedsingularities, have been intensively studied in the last years. The shadows of the blackholes and naked singularities from the Kerr-Newmann family of solutions of the Einstein-Maxwell equations have been thoroughly investigated in [2]-[8]. The shadow of a blackhole with a NUT-charge has been obtained in [9]. The black hole shadows in Einstein-Maxwell-dilaton gravity, Chern-Simons modified gravity and braneworld gravity havebeen examined in [10], [11], [12]. The apparent shape of the Sen black hole has beenstudied in [13]. The wormhole shadows have been recently investigated in [14], [15].With the advance of technology the experimental observation of the shadows of com-pact objects is now possible. Experiments that allow such observations include theEvent Horizon Telescope [16], which is a system of earth-based telescopes measuring inthe (sub)millimeter wavelength, the space-based radio telescopes RadioAstron and Mil-limetron [17], [18], and the space-based X-ray interferometer MAXIM [19]. In the nextfew years these missions are expected to reach resolution high enough to observe theshadow of the supermassive compact object at the center of our galaxy or those locatedat nearby galaxies [18]. The results of these experiments should be compared with thetheoretical models. In this way the observations will reject some of the models or willmake it possible to distinguish between different types of compact objects [20]. Evenmore, the mentioned observations can be used for detection of theoretically predicted ob-jects and effects not observed so far. Such a theoretical prediction is the possible existence ∗ E-mail: [email protected]fia.bg † E-mail: [email protected]fia.bg
1f cosmic strings. These objects are expected to have formed during phase transitions inthe early universe through spontaneous symmetry breakings [21]. It has also been shownthat cosmic strings generally form at the end of inflation within the framework of varioussupersymmetric grand unification theories [22].A cosmic string makes the space-time around it a conical space-time with a deficitangle δ = 8 πGµ/c where µ is the string tension and G and c are the gravitationalconstant and speed of light, respectively. The deficit angle manifests itself physicallyby giving rise to interesting phenomena and effects [21], [23]-[27] which can be used fordetection of cosmic strings. For example, the gravitational lensing phenomena serve asdirect evidence for cosmic strings although none have been detected yet. However, it isunlikely that the pure gravitational lensing of a single string could be measured in theviolent astrophysical conditions. It is much more natural to search for signs of cosmicstrings in the gravitational lensing by the whole system consisting of the cosmic stringand the matter surrounding it. In a galactic context we can consider a model configurationconsisting of a central galactic black hole pierced by a cosmic string. The gravitationallensing of such configurations has been recently investigated in [27] and it has been shownthat the presence of a cosmic string leaves observable imprints. Then, in view of theseresults, it is very natural to expect that the presence of a cosmic string would also leaveimprints in the shadow of the black hole pierced by it.With this motivation in mind, we continue the theoretical analysis of the black holeshadows. The aim of the current paper is to investigate the apparent shape of a rotatingblack hole pierced by a cosmic string, and compare the results with the case of the Kerrblack hole. Then we consider the deviations from the Kerr case as a possible test for theexistence of cosmic strings on a galactic level.The metric of the spacetime describing a rotating Kerr black hole pierced along theaxis of symmetry by a cosmic string is given by [23] ds = − (cid:18) − M rρ (cid:19) dt + ρ (cid:18) dr ∆ + dθ (cid:19) + ζ sin θρ (cid:0) ζ Σ dϕ − aM r dt (cid:1) dϕ, (1)where r , θ and ϕ are the Boyer-Lindquist coordinates and the metric functions ∆, ρ andΣ are defined as usual∆ ≡ r − M r + a , ρ ≡ r + a cos θ, Σ ≡ (cid:0) a + r (cid:1) − a ∆ sin θ, (2)where M , a and ζ are parameters. The parameter ζ (0 < ζ ≤
1) describes the influence ofthe string on the metric and it is related to the deficit angle of the string by δ = 2 π (1 − ζ ).In the particular case when ζ = 1, the metric reduces to the well-known metric of Kerr.The parameter a is the angular momentum per unit mass. M , however, does not coincidewith the physical mass of the black hole. The physical mass M phys and the physicalangular momentum J phys of the black hole pierced by a cosmic string are given by [23] M phys = ζ M, J phys = ζ J. (3) For numerical solutions describing rotating black holes with cosmic string hair we refer the reader to[28]. − ∂S∂λ = g αβ ∂S∂x α ∂S∂x β , (4)where S is the particle action, λ is the affine parameter along the geodesics of the metric g αβ . Since our spacetime described by the metric (1) is stationary and axisymmetric wehave two conserved quantities - the energy of the particle E and its angular momentum L z about the axis of symmetry. As in the case of pure Kerr spacetime, we have anotherconserved quantity, namely the Carter constant K , leading to the separability of theHamilton-Jacobi equation, which then has a solution of the form S = 12 m λ − Et + L z ϕ + S r ( r ) + S θ ( θ ) , (5)where m is the mass of the test particle. Using (5) the Hamilton-Jacobi equation reducesto the following equations for S r ( r ) and S θ ( θ ):( S ′ r ) = ( a + r ) ∆ E + a ζ ∆ L z − aMrζ ∆ EL z − m r + K ∆ ≡ R ( r ) , ( S ′ θ ) = K − m a cos θ − E a sin θ − ζ sin θ L z ≡ Θ( θ ) . (6)Then (5) takes the form S = 12 m λ − Et + L z ϕ + Z p R ( r ) dr + Z p Θ( θ ) dθ. (7)Hence by using the standard procedure we find the null geodesics (i.e. m = 0) in thespacetime of a rotating black hole pierced by a cosmic string, namely ρ dtdλ = h ( a + r ) − aMrζ ξ i − a sin θ,ρ drdλ = q ( a + r ) + a ζ ξ − aMrζ ξ − ∆ η ≡ √ R,ρ dθdλ = q η − a sin θ − ζ sin θ ξ ≡ √ Θ ,ζ ρ dϕdλ = ξ sin θ − a ∆ (cid:16) aζ ξ − M r (cid:17) , (8)with ξ ≡ L z /E and η ≡ K/E being the impact parameters. We have also redefined theaffine parameter Eλ → λ .The photon orbits are in general of two types - orbits falling into the black hole andothers scattered away from the black hole to infinity. An observer far from the blackhole will be able to see only the photons scattered away from the black hole, while those3aptured by the black hole will form a dark region. This dark region observed on theluminous background is the shadow of the black hole.The boundary of the black hole shadow is the critical orbit that separates the escapeand plunge orbits. In order to find the shadow boundary we reformulate the problem asone-dimensional problem for a particle in an effective potential by rewriting the radialgeodesic equation in the form (cid:18) ρ drdλ (cid:19) + U eff ( r ) = 0 , where U eff ( r ) = − R ( r ). In this formulation, it is clear that the critical orbit betweenescape and plunge, which is obviously unstable and circular, corresponds to the highestmaximum of the effective potential. The conditions therefore for the critical sphericalorbit determining the boundary of the black hole shadow are U eff = 0 , U eff dr = 0 , d U eff dr ≤ , (9)or, equivalently, R = 0, dRdr = 0 and d Rdr ≥
0. Since the effective potential U eff (orequivalently R ) depends on r as well as ξ and η , the conditions (9) give in fact a parametricrelation between the impact parameters that should be satisfied on the shadow boundary.Of course, in addition to the conditions above, the impact parameters should be such thatΘ( θ ) ≥ R in our case, the solution tothe conditions R = 0 and dRdr = 0, that also satisfies Θ( θ ) ≥
0, is given by ξ = − ζa ( r − M ) [ r ( r − M ) + a ( r + M )] ,η = r − M ) [ r ( r − M ) + a ( r + M )] . (10)The condition d Rdr ≥ r + a − r − M ) (cid:2) r ( r − M ) + a ( r + M ) (cid:3) ≥ . (11)Equations (10) and (11) define the boundary of the shadow in parametric form. Itis clear from the derivation that the boundary of the shadow is determined only by thespacetime metric and does not depend on the details of the emission mechanisms.In real observations, however, what is seen, is in fact the projection of the shadow onthe observer’s sky defined as the plane passing through the black hole and normal to theline of sight. Taking this into account, it is more natural to present the shadow boundaryin the so-called celestial coordinates α and β . The celestial coordinates are defined by[29] α = lim r → ∞ (cid:18) − r sin θ dϕdr (cid:19) , (12)4 = lim r → ∞ (cid:18) r dθdr (cid:19) , (13)where the limit is taken along the null geodesics and θ is the inclination angle betweenthe axis of rotation of the black hole and the line of sight of the observer. From thedefinition of the celestial coordinates and using the null geodesics equations (8), we get α = − ξζ sin θ ,β = q η − a sin θ − ξ ζ sin θ . (14)After substituting (10) in (14), we find α = a ( r − M ) sin θ [ r ( r − M ) + a ( r + M )] ,β = q r ( r − M )+ a ( r + M )]( r − M ) − a sin θ − [ r ( r − M )+ a ( r + M )] a ( r − M ) sin θ . (15)These two equations give in parameteric form the boundary of the shadow in celestialcoordinates. Formally (15) coincide with the celestial coordinates for the shadow of thepure Kerr black hole. However, although the string parameter ζ does not enter explicitlythe equations for the shadow boundary in celestial coordinates, it indeed influences theblack hole shadow through the parameter M , which is related to the physical mass of theblack hole via eq. (3), i.e. M phys = ζ M .The shadow of the Kerr black hole pierced by a cosmic string is presented in Figs. 1–4for several inclinations angles, spin and string parameters.There are two observables that characterize the shadow, namely the radius R s of theshadow and the so-called dent D s [8]. The radius of the shadow is defined as the circlepassing through the three points located at the top, bottom and right end of the shadow(see Fig. 5). The other important characteristic, the dent D s , is defined as the differencebetween the left end points of the circle and the shadow. It is also useful to define adistortion as δ s ≡ D s /R s (see Fig. 5).For small deficit angles (i.e. for ζ close to 1) the shadow of the black hole is very closeto that of the Kerr black hole. However with the increase of the deficit angle the radiusof the shadow boundary increases as it is seen in Fig. 6. The characteristic deformationsof the shadow are also more pronounced for large deficit angles as one can see in Fig. 7.In general, for a fixed mass M phys , the shadow radius R s and the distortion δ s dependon three parameters - the spin parameter a ∗ = J phys /M phys , the string parameter ζ andthe inclination angle θ . For fixed inclination angles, the dependence of R s on the spinparameter a ∗ for different values of the string parameter ζ is shown in Fig. 6. It is seenthat the radius R s depends weakly on the spin parameter, while the dependence on thestring parameter is more strongly expressed.For fixed M phys and θ the dependence of the distortion δ s on the spin parameter fordifferent values of the string parameter is shown in Fig.7All these results show that the presence of a cosmic string piercing the black holesleaves observable imprints in the shadows of the black holes. If we can measure themass, the spin parameter and the inclination, then our results and more precisely thedependences shown in Fig. 6 and Fig. 7, allow for a determination of the string parameter ζ which is equivalent to the detection of the presence of a cosmic string if ζ = 0.5 cknowledgements The support by the Bulgarian National Science Fund under Grant DMU-03/6 and bythe Sofia University Research Fund under Grant 33/2013 is gratefully acknowledged.
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D85 , 064044 (2012).[28] R. Gregory, D. Kubiznak and D. Wills, JHEP , 023 (2013).[29] S. V´asquez and E. Esteban, Nuovo Cim., , 489 (2004)7 - Α Β - - - - - - Α Β - - - - - - Α Β - - - - Α Β a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . - - Α Β - - - - - - Α Β - - - - - Α Β - - - - Α Β a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . - - Α Β - - - - - - Α Β - - - - - Α Β - - - - Α Β a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . - - Α Β - - - - - Α Β - - - - Α Β - - - Α Β a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . Figure 1:
The shadow of the Kerr black hole pierced by a cosmic string (solid line) and the Kerr blackhole (dashed line) with inclination angle θ = π/ rad for different rotation and string parameters. Thephysical mass of both solutions is set equal to 1. The celestial coordinates ( α, β ) are measured in theunits of physical mass. - Α Β - - - - - - Α Β - - - - - - Α Β - - - - Α Β a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . - - Α Β - - - - - - Α Β - - - - - Α Β - - - - Α Β a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . - - Α Β - - - - - - Α Β - - - - - Α Β - - - - Α Β a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . - - Α Β - - - - - Α Β - - - - - Α Β - - - Α Β a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . Figure 2:
The shadow of the Kerr black hole pierced by a cosmic string (solid line) and the Kerr blackhole (dashed line) with inclination angle θ = π/ rad for different rotation and string parameters. Thephysical mass of both solutions is set equal to 1. The celestial coordinates ( α, β ) are measured in theunits of physical mass. - Α Β - - - - - - Α Β - - - - - - Α Β - - - - Α Β a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . - - Α Β - - - - - - Α Β - - - - - - Α Β - - - - Α Β a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . - - Α Β - - - - - - Α Β - - - - - Α Β - - - - Α Β a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . - - Α Β - - - - - Α Β - - - - - Α Β - - - Α Β a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . Figure 3:
The shadow of the Kerr black hole pierced by a cosmic string (solid line) and the Kerr blackhole (dashed line) with inclination angle θ = π/ rad for different rotation and string parameters. Thephysical mass of both solutions is set equal to 1. The celestial coordinates ( α, β ) are measured in theunits of physical mass. - Α Β - - - - - - Α Β - - - - - - Α Β - - - - Α Β a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . - - Α Β - - - - - - Α Β - - - - - - Α Β - - - - Α Β a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . - - Α Β - - - - - - Α Β - - - - - Α Β - - - - Α Β a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . - - Α Β - - - - - Α Β - - - - Α Β - - - - Α Β a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . ; a/M phys = 0 . , ζ = 0 . Figure 4:
The shadow of the Kerr black hole pierced by a cosmic string (solid line) and the Kerr blackhole (dashed line) with inclination angle θ = π/ rad for different rotation and string parameters. Thephysical mass of both solutions is set equal to 1. The celestial coordinates ( α, β ) are measured in theunits of physical mass. s R s R s D s - - - Α Β Figure 5:
The shadow of Kerr black hole (solid line) and the circle (dashed line) passing through thethree points located at the top, bottom and rightmost end of the shadow. The radius of this circle is R s .The difference between the left end points of the circle and the black hole’s shadow is D s . The definitionof the distortion parameter is δ s ≡ D s /R s . Ζ= Ζ= Ζ= Ζ= Ζ= R s J phy s (cid:144) M s Ζ= Ζ= Ζ= Ζ= Ζ= R s J phy s (cid:144) M s θ = π/ rad ; θ = π/ rad ; Ζ= Ζ= Ζ= Ζ= Ζ= R s J phy s (cid:144) M s Ζ= Ζ= Ζ= Ζ= Ζ= R s J phy s (cid:144) M s θ = π/ rad ; θ = π/ rad Figure 6:
The spin parameter a ∗ of the Kerr black hole pierced by a cosmic string as a function of thecircle radius R s for a few different inclination angles θ and string parameters ζ . = Ζ= Ζ= Ζ= Ζ= ∆ s J phy s (cid:144) M s Ζ= Ζ= Ζ= Ζ= Ζ= ∆ s J phy s (cid:144) M s θ = π/ rad ; θ = π/ rad ; Ζ= Ζ= Ζ= Ζ= Ζ= ∆ s J phy s (cid:144) M s Ζ= Ζ= Ζ= Ζ= Ζ= ∆ s J phy s (cid:144) M s θ = π/ rad ; θ = π/ rad Figure 7:
The spin parameter a ∗ of the Kerr black hole pierced by a cosmic string as a function of thedistortion parameter δ s for a few different inclination angles θ and string parameters ζ ..