General parametrization of wormhole spacetimes and its application to shadows and quasinormal modes
GGeneral parametrization of wormholespacetimes and its application to shadowsand quasinormal modes
Kirill A. Bronnikov , a , b , c Roman A. Konoplya , b , d and Thomas D. Pappas d a Center for Gravitation and Fundamental Metrology, VNIIMS, Ozyornaya 46, Moscow 119361, Russia b Institute of Gravitation and Cosmology, Peoples’ Friendship University of Russia (RUDN University),ul. Miklukho-Maklaya 6, Moscow 117198, Russia c National Research Nuclear University “MEPhI” (Moscow Engineering Physics Institute),Kashirskoe sh. 31, Moscow 115409, Russia d Research Centre for Theoretical Physics and Astrophysics, Institute of Physics, Silesian University in Opava,Bezruˇcovo n´am. 13, CZ-746 01 Opava, Czech Republic
E-mail: [email protected] , [email protected] , [email protected] Abstract.
The general parametrization for spacetimes of spherically symmetric Lorentzian, traversablewormholes in an arbitrary metric theory of gravity is presented. The parametrization is similar in spiritto the post-Newtonian parametrized formalism, but with validity that extends beyond the weak fieldregion and covers the whole space. Our method is based on a continued-fraction expansion in terms ofa compactified radial coordinate. Calculations of shadows and quasinormal modes for various examplesof parametrization of known wormhole metrics that we have performed show that, for most cases, theparametrization provides excellent accuracy already at the first order. Therefore, only a few parametersare dominant and important for finding potentially observable quantities in a wormhole background.We have also extended the analysis to the regime of slow rotation. a r X i v : . [ g r- q c ] F e b ontents Wormholes (WHs) belong to a special class of solutions to the Einstein equations representing tunnel-like structures which connect spatially separated regions or even different universes. The first descriptionof such a geometry appeared as early as in 1916 in the paper by Flamm [1] in a study of the spatial partof the Schwarzschild metric. A WH geometry also emerged during the effort of Einstein and Rosen inthe mid-1930s towards a geometric description of elementary particles [2]. However, the foundationsfor our current understanding of WHs have been laid in the seminal work by Morris and Thorne (MT) inthe late 1980s [3] where they investigated the conditions for traversability of these speculative objectsby human travelers. – 1 –s MT’s work and subsequent study has revealed, WHs come with a number of problems, suchas the necessity of exotic matter in order to keep the WH throat open [3], or dynamical instability[4–9] of WHs. By now, there is no noncontradicting model of a traversable, Lorentzian WH, which isdynamically stable, does not require exotic matter for its existence and follows from some fundamentaltheoretical principles. Nevertheless, even a hypothetical possibility to create WHs in a distant futureexperiment looks attractive and justifies the effort towards a further study of various WH solutions.Since there is no fully satisfactory WH model, an appealing question is how to describe thegeometry of WHs in a context as general as possible. A general parametrization of a WH geometrymade in the spirit of the parametrized post-Newtonian (PPN) formalism, but valid in the whole spacefrom the throat to infinity, could be a solution. This would allow one to constrain possible WHgeometries in the current and future experiments via constraints on appropriate parameters of theparametrization.This kind of program has been recently fulfilled by Rezzolla and Zhidenko (RZ) [10] who suggesteda parametrization of arbitrary static, spherically symmetric black hole (BH) metrics convenient for com-parison with observations, independently of a theory of gravity. The RZ parametrization enables one toapproximate any sufficiently smooth BH metric with any prescribed accuracy using a minimum possiblenumber of numerical parameters. This parametrization was further extended to axially symmetric BHsin [11], to higher-dimensional BHs in [12] and applied to analytical representation of various numericalBH solutions in [13–20].The method [10, 11] is based on continued-fraction expansions of the metric functions in theradial direction in terms of a compact coordinate and simultaneous expansion in the polar directionin terms of cos θ around the equatorial plane. The continued-fraction expansion provides the superiorconvergence and clear hierarchy of parameters. The latter is necessary to constrain effectively theallowed geometries of a compact object from experiments.A straightforward way is to formulate a similar kind of parametrization for WHs. Instead of theevent horizon radius used as a natural length scale for BHs, for a WH such a natural length scale isgiven by its throat radius, or, if there are multiple throats (as is the case in some models discussedin the literature), it makes sense to speak of the throat closest to the observers, or, in other words,the external one, outside which we can assume that the spherical radius is a growing function of somereasonably chosen radial coordinate. The radius r of this throat can be well used as a fixed lengthparameter for WHs instead of the horizon radius used when discussing BH metrics.Here we will construct a general parametrization for Lorentzian, traversable, asymptotically flat,spherically symmetric WHs, not necessarily symmetric relative to the throat, which is independent onthe background metric theory of gravity. We will further extend this general parametrization to axialsymmetry in the slow rotation regime. We will show that once the metric functions relatively slowlyapproach their asymptotic values, only a few dominant parameters of the parametrization determinethe behavior of potentially observable quantities around WHs with high accuracy. In the slow rotation– 2 –egime, this general form of the metric, independent of the gravitational theory, has the following form: ds = − f ( r ) dt + h ( r ) dr − m (cid:101) αr sin θdtdφ + r (cid:0) dθ + sin θdφ (cid:1) , (1.1) f ( r ) = − r ( + (cid:15) ) r + r ( a + f + (cid:15) ) r − r a r , (1.2) h ( r ) = − r ( + (cid:15) ) r + r ( b + h + (cid:15) ) r − r b r , (1.3)where r is the location of the WH’s throat, (cid:101) α is the rotation parameter, h , f are the values of themetric functions at the location of the throat, and (cid:15) , a , b are parameters of deformation. Whenevermore accuracy is required, two more parameters of deformation are added. This form of the WH metriccould be used for testing various astrophysical phenomena, such as accretion of matter, shadows,various types of radiation phenomena, and further constraining of the allowed geometries of wormholesvia constraining the appropriate parameters.Another possible application of our parametrization is a derivation of analytical approximationsfor numerical WH solutions [21, 22], and for this kind of work we suggest various approaches tothe construction of alternative parametrizations which take into account a convenient choice of thecoordinate system under consideration and the behavior of the metric near the throat. We calculatethe potentially observable characteristics, such as WH shadows and quasinormal modes, and see thatthose observable values for the parametrized approximation of WH metrics have a relative error (ascompared to exact solutions) which ranges from about a small fraction of one percent to, in the worstcases, a few percent already at the first order of the expansion in the radial direction. The second-order approximation provides always an excellent accuracy if it does not simply coincide with the exactsolution.The paper is organized as follows. In Sec. 2 we summarize the general information about WHs andvarious choices of coordinate systems used for their description. Section 3 is devoted to the constructionof a general parametrization for spherically symmetric and axially symmetric WHs in the slow rotationregime. Section 4 tests this parametrization using a great number of examples of WH metrics that canbe recast to the MT frame. For these cases, the radial coordinate is conceptually identical to the oneused in the parametrization of BH metrics. In Sec 5 we deal with the parametrization of WHs that arenot in the MT frame and develop optimized parametrizations via different choices of a compact radialcoordinate that take into consideration the behavior of the metric near the throat. In Sec. 6 we test theaccuracy of the parametrized description via the calculation of the radii of shadows of WHs and theirquasinormal spectra. Finally, in Sec. 7 we summarize the obtained results and discuss open questions. The general metric describing an arbitrary static, spherically symmetric geometry may be written in theform ds = − f ( r ) dt + h ( r ) dr + K ( r ) dΩ , (2.1)where dΩ = (cid:0) dθ + sin θ dφ (cid:1) is the line element on a two-dimensional unit sphere, and r is anarbitrary radial coordinate, whose specific choice can be made for convenience. Only two of the three– 3 –etric functions f ( r ) , h ( r ) , K ( r ) are independent, and upon using appropriate transformations of theradial coordinate, any metric can be cast in the form where the circumferential radius K ( r ) satisfies K ( r ) = r , albeit this might not always be feasible analytically. In general, the area of the sphere atradial coordinate r is A ( r ) = π K ( r ) .It is said that the metric (2.1) describes a (traversable, Lorentzian) WH if the following twoconditions are met. First, the circumferential radius has at least one minimum K ≡ K ( r ) at somevalue of the radial coordinate r , and K ( r ) should be large as compared to K on both sides from thisminimum. Then, K corresponds to the radius of the WH throat, and r is its location. At the throat,the area of the constant r sphere is minimized, and this allows one to determine r via the condition A (cid:48) ( r ) = → K (cid:48) ( r ) = f ( r ) and h ( r ) are regular and positive in a range of r containing the throatand values of r on both sides from the throat such that K ( r ) (cid:29) K ( r ) . Such a definition includesboth asymptotically flat or AdS WHs and those containing horizons far from the throat, for example,asymptotically de Sitter ones. In this work we will consider asymptotically flat WHs such that, as r tends to some r = r ∞ , f → h (cid:18) dKdr (cid:19) → r , andwe here enumerate three of them: • The curvature coordinate defined by r = K , such that the coordinate is identified with thespherical radius. In this coordinate system, the second condition (2.3) reads h ( r ) → r → ∞ . For WHs this choice of coordinate is unnatural because a minimum of K , i.e., a throat,is a coordinate singularity, h − → ∞ . However, it is often used since it is rather intuitively clearand simplifies the gravitational field equations in the presence of some (but not all) materialsources of gravity. • The “quasiglobal” coordinate r such that f ( r ) /h ( r ) ≡
1. This coordinate is especially convenientfor describing BH horizons but is also used in many WH solutions. We can recall that the mostwell-known solutions of general relativity (Schwarzschild, Reissner-Nordsr¨om, (A)dS) are mostoften written in terms of r which is simultaneously a curvature and quasiglobal coordinate. Sincein (2.3) the first condition requires f →
1, the second one reduces to | dK/dr | → • The Gaussian, or proper length coordinate r = l , which is defined in such a way that h ≡
1. Thesecond condition (2.3) reads: | dK/dl | → l → ∞ .In terms of the curvature coordinate, a very common frame where many WH metrics are written is theone introduced by Morris and Thorne [3] ds = − e Φ ( r ) dt + (cid:18) − b ( r ) r (cid:19) − dr + r dΩ . (2.4)The first metric function Φ ( r ) is the so-called “redshift” function that determines the redshift andtidal forces in the WH spacetime. The absence of event horizons demands that Φ ( r ) should be finite– 4 –verywhere. The second function b ( r ) is called the “shape” function since it indirectly determinesthe spatial shape of the WH in its embedding diagram representation. It should satisfy the so-calledflair-out conditions on the throat, i.e., b ( r ) = r and b (cid:48) ( r ) < b ( r ) < r for r (cid:54) = r , which areactually a reformulation of the condition that K ( r ) has a minimum in the more general representation(2.1). The aforementioned conditions on h ( r ) imply that the location of the throat r in the MT frameis given as a root of the equation h ( r ) = (cid:18) − b ( r ) r (cid:19) = e Φ ( r ) >
0, and the curvature coordinate is defined for r ∈ [ r , ∞ ) . Finally, it should be notedthat asymptotic flatness demands that b ( r ) /r → r → ∞ , which translates to h ( r ) → In this section, we will briefly review the parametrization of spherically symmetric BHs suggested in[10], and then we will see which modifications of this approach are required when going over to WHgeometries.
The Rezzolla-Zhidenko parametrization is build around a dimensionless compact coordinate (DCC)defined by x ( r ) ≡ − r r , (3.1)where r is the location of the outer event horizon of the BH determined via the condition f ( r ) =
0. If K ( r ) = r , i.e., one works with the curvature coordinate, then r is also the radius of the outer eventhorizon. In terms of this DCC, the following parametrization equations are introduced: f ( r ) = (cid:101) A ( x ) , (3.2)1 h ( r ) = (cid:101) B ( x ) (cid:101) A ( x ) , (3.3)where the parametrization functions (cid:101) A ( x ) and (cid:101) B ( x ) are defined as (cid:101) A ( x ) ≡ x (cid:34) − (cid:15) ( − x ) + ( a − (cid:15) )( − x ) + a + a x + a x ... ( − x ) (cid:35) , (3.4) (cid:101) B ( x ) ≡ + b ( − x ) + b + b x + b x ... ( − x ) . (3.5)There are three asymptotic parameters in total, namely, ( (cid:15) , a , b ) , which are determined via theexpansions of the parametrization equations at spatial infinity ( x = a , a , . . . , b , b , . . . ) are the “near-field” parameters and are determined by the corresponding ex-pansions at the location of the event horizon ( x = ( (cid:15) , a , b ) are imposed via the PPNexpansions [23, 24] f ( r ) = − Mr + ( β − γ ) M r + O (cid:18) r (cid:19) = − Mr ( − x ) + ( β − γ ) M r ( − x ) + O (cid:16) ( − x ) (cid:17) , (3.6)and 1 h ( r ) = + γ Mr + O (cid:18) r (cid:19) = + γ Mr ( − x ) + O (cid:16) ( − x ) (cid:17) . (3.7)Notice that the highest-order PPN constraints on the metric are of the order O (cid:16) ( − x ) (cid:17) for theexpansion of the g tt metric component and of O (( − x )) for g rr . Consequently, this means that wecan impose independent constraints for up to three asymptotic parameters in the parametrization of f ( r ) and up to two for h ( r ) . The values of the PPN parameters β and γ are observationally constrainedto be [23, 24] | β − | (cid:46) × − , | γ − | (cid:46) × − . (3.8)The expansion of the parametrization functions (3.4) and (3.5) at x = (cid:101) A ( x ) = − ( + (cid:15) )( − x ) + a ( − x ) + O (cid:16) ( − x ) (cid:17) , (3.9)and (cid:101) B ( x ) (cid:101) A ( x ) = + ( + b + (cid:15) )( − x ) + O (cid:16) ( − x ) (cid:17) . (3.10)Then, the comparison of the expansions (3.6), (3.9) and (3.7), (3.10) imposes the observational con-straints (cid:15) = Mr − a = M r ( β − γ ) , (3.11)and ( + b + (cid:15) ) = γ Mr ⇒ b = Mr ( γ − ) . (3.12)One then concludes that viable BH solutions must have a (cid:39) b (cid:39) We are now going to extend the above prescription to accommodate the parametrization of WH metrics.This can be achieved by appropriately modifying the parametrization equations (3.2), (3.3) and theparametrization functions (3.4), (3.5). As we have discussed in Sec. 2, the location of the WH throat r is defined by the condition K (cid:48) ( r ) = h ( r ) =
0. We are then interested ina parametrization of the WH metric in the region r ∈ [ r , ∞ ) . As in the case of BHs, it is sufficient toparametrize only two metric functions under an appropriate choice of the radial coordinate.– 6 –ollowing the RZ approach, we define the DCC as in Eq. (3.1), thus the DCC maps the interval r ∈ [ r , ∞ ) to the compact range x ∈ [
0, 1 ] . The next step is to introduce two extra near-fieldparameters f , h to account for the fact that the WH metric functions f ( r ) and h ( r ) in principleattain nonvanishing values at r = r . This is to be contrasted to the BH case where f ( r ) = h ( r ) = (cid:101) A ( x ) (3.4) must alsovanish at r or equivalently at x =
0. In terms of the DCC (3.1) we define the following parametrizationequations: f ( r ) = A ( x ) , (3.13) h ( r ) = B ( x ) , (3.14)with the parametrization functions given by A ( x ) ≡ f + x (cid:34) ( − f ) − ( (cid:15) + f ) ( − x ) + ( a − (cid:15) − f )( − x ) + a ( − x ) + a x + a x ... (cid:35) , (3.15) B ( x ) ≡ h + x ( − h ) − ( b + h ) ( − x ) + b ( − x ) + b x + b x ... . (3.16)To impose observational constraints on the asymptotic parameters, we consider the expansions at x = A ( x ) = − ( + (cid:15) ) ( − x ) + a ( − x ) + O (cid:16) ( − x ) (cid:17) , (3.17)1 B ( x ) = + ( + b ) ( − x ) + O (cid:16) ( − x ) (cid:17) . (3.18)A comparison with the PPN expansions (3.6) and (3.7) yields the constraints (cid:15) = Mr − a = M r ( β − γ ) , (3.19)and b = γ Mr − ⇒ b = γ ( (cid:15) + ) − a (cid:39) b (cid:39) (cid:15) . The stationary, axisymmetric generalization of the MT wormhole [3] was found in [25], and the lineelement is given by ds = − fdt + h dr + K (cid:104) dθ + sin θ ( dφ − ωdt ) (cid:105) , (3.21)where the metric functions f , h , K and ω depend only on r and θ and are regular on the symmetry axis θ = π . In terms of the rotation parameter (cid:101) α ≡ J/m , where J is the angular momentum of the WH– 7 –s measured by an asymptotic observer and m is its mass, the slow rotation approximation is obtainedby expanding the metric functions in terms of the dimensionless parameter (cid:101) α/m (cid:28)
1. Up to the linearorder in (cid:101) α we have ds = − f ( r ) dt + h ( r ) dr + K ( r ) dΩ + g ( r ) sin θdtdφ + O ( (cid:101) α ) , (3.22)where it is assumed that all functions depend only on the radial coordinate, and we have defined themetric function g ( r ) ≡ − ω ( r ) K ( r ) , (3.23)that involves the angular velocity metric function ω ( r ) . The latter depends on the rotation parameterlinearly and should exhibit asymptotically the following fall-off behavior: ω ( r ) = Jr + O (cid:18) r (cid:19) . (3.24)Furthermore, asymptotic flatness requires that K ( r ) → r as r → ∞ and thus g ( r ) has the asymptoticbehavior g ( r ) = − Jr + O (cid:18) r (cid:19) , (3.25) = − Jr ( − x ) + O (cid:16) ( − x ) (cid:17) . (3.26)By analogy with the previous sections, in terms of the DCC of Eq. (3.1), a parametrization functioncan be introduced for g ( r ) with a single asymptotic parameter c and a tower of near-field parameters ( g , c , c , . . . ) as C ( x ) ≡ g + x (cid:34) − g − ( c + g ) ( − x ) + c ( − x ) + c x + c x + ... (cid:35) . (3.27)The parametrization equation is defined as g ( r ) ≡ C ( x ) , (3.28)while the asymptotic expansion of Eq. (3.27) at infinity reads C ( x ) = − c ( − x ) + O (cid:16) ( − x ) (cid:17) . (3.29)Consequently, a comparison with Eq. (3.26) reveals that the asymptotic parameter c is associatedwith the angular momentum via c = J/r . In this section, we are going to test our method by parametrizing different types of analytical WHmetrics that can be brought to the MT frame (2.4) and consequently are written in terms of thecurvature coordinate as defined in Sec. 2. Once a parametrization is obtained, we compute the error ofthe parametrized metrics at different orders in the continued-fraction expansion and demonstrate thehigh accuracy of the approximation already at the lowest orders as well as its quick convergence.– 8 – .1 The Bronnikov-Kim II braneworld wormhole solution
As the first example, we consider a braneworld solution [26–28] that exhibits BH and WH brancheswith zero Schwarzschild mass and is described by the following line element: ds = − (cid:18) − a r (cid:19) dt + (cid:18) − a r (cid:19) − (cid:18) + C − a √ r − a (cid:19) − dr + r dΩ . (4.1)The WH branch of the metric should be free of event horizons and this requirement translates to f ( r ) ≡ − g tt ( r ) > ∀ r ∈ [ r , ∞ ) , where r > a is constrained by f ( r ) = (cid:18) − a r (cid:19) (cid:62) ⇒ r (cid:62) a , (4.2)with the equality a = r corresponding to the WH/BH threshold where r is then identified with thelocation of the (double) BH event horizon. Since the metric (4.1) is of the MT type, the location ofthe WH throat is determined by the condition h ( r ) = C should satisfy C = a − (cid:113) r − a (cid:62) C < C =
0. Upon substituting the lastequation into Eq. (4.1) we may express the metric function h ( r ) entirely in terms of r and a as h ( r ) = (cid:18) − a r (cid:19) (cid:32) − (cid:112) r − a √ r − a (cid:33) . (4.4)We begin with the parametrization of the f ( r ) metric function in terms of the dimensionless compactcoordinate x ( r ) given by Eq. (3.1). As anticipated by the polynomial form of the metric function,the comparison of the expansions of the parametrization equation (3.13) at the boundaries x = x = (cid:15) = − a = − a r , f = − a r , a i = ∀ i (cid:62) h ( r ) is not exact, and the corresponding expansions of (3.14) at the boundariesyield b = (cid:115) − a r − h = b = b + a a + r , (4.6) b = a ( + b ) − a ( + b ) r + b r b ( a − r ) − i th order, the continued-fraction expansions in the parametrization functions (3.15)-(3.16) are to be truncated at the i th order by setting the ( i + ) th expansion parameter equal to zero.– 9 –s an example, in the left panel of Fig. (1) we plot the exact metric function h ( r ) as given in Eq. (4.4)and its approximation h app ( r ) at first order for p ≡ a/r = b =
0. In the rightpanel of the same figure, and for the same value of the parameter p , we plot, in terms of the compactcoordinate (3.1), the percentage of the absolute relative error (ARE) given byARE ≡ (cid:12)(cid:12)(cid:12)(cid:12) h ( r ) − h app ( r ) h ( r ) (cid:12)(cid:12)(cid:12)(cid:12) % . (4.8)The radial profile of the ARE is the typical profile obtained in all cases, with the error vanishing at theboundaries of the parametrization range r ∈ [ r , ∞ ) → x ∈ [
0, 1 ] while exhibiting a global maximumat some intermediate value of x ( r ) that emerges at a value of r not far from the location of the throat. Figure 1 : For p ≡ a/r = h ( r ) (4.4) (blue curve) and itsfirst-order approximation (red dashed curve). Right panel: the percentage of the absolute relative errorin terms of the dimensionless compact coordinate x ∈ [
0, 1 ] .To test the convergence of the parametrization, in Table (1) we give the maximum percentagevalue of the ARE for the approximation h app ( r ) of h ( r ) at various orders in the continued-fractionexpansion and for different values of the dimensionless parameter p ≡ a/r (cid:54) p = Table 1 : The maximum absolute relative error in percents between the exact metric h ( r ) (4.4) and itsapproximation at various orders in terms of the dimensionless parameter p ≡ a/r (cid:54)
1. The WH/BHthreshold corresponds to p = p = p = p = p = p = p = O ( − ) p ≡ a/r is smaller than about 0.5, our parametrizationprovides a very accurate approximation of the metric even at the first order, with a maximum absoluterelative error (MARE) below 1%. For larger values of p and as the WH/BH threshold is approached,only the first-order MARE becomes large with the higher-order approximations preserving a negligibleMARE with values well below 1%. Furthermore, we notice that in the limit p →
1, the approximationremains convergent albeit at a slower rate.
In the context of the braneworld scenario, the search for BH and WH solutions led to the following veryinteresting geometry [26, 27, 29, 30]: ds = − (cid:18) − mr (cid:19) dt + − m r (cid:0) − mr (cid:1) (cid:0) − r r (cid:1) dr + r dΩ , (4.9)that describes different objects depending on the values of r > m (we take m > p ≡ r /m > p ∈ ( ∞ ] : Symmetric traversable WH.2. p = p ∈ ( , 2 ) : Regular BH.4. p = : Schwarzschild BH.5. p ∈ ( ) : A Schwarzschild-like BH structure with a spacelike curvature singularity located at r = m .The WH branch of Eq. (4.9) is of the MT type (2.4) with the shape function b ( r ) specified by theidentification h ( r ) = − b ( r ) r = (cid:0) − mr (cid:1) (cid:0) − r r (cid:1) − m r . (4.10)The location of the WH throat r th is then determined by the condition b ( r th ) = r th and so from thelast equation we find that r th = r . Consequently, the parametrization region in this coordinate systemis r ∈ [ r , ∞ ) . In terms of Eqs. (3.1)-(3.14) we obtain the parametrization for f ( r ) and h ( r ) . For theformer it is exact and the values of the expansion parameters are: (cid:15) = mr − a = f = − mr , a i = ∀ i (cid:62) h ( r ) becomes exact at the second order and the expansion parameters havethe values b = m r , h = b = m mr − r , b = m r − m , b i = ∀ i (cid:62) h ( r ) , in Table (2)we give the percentage of the MARE for various values of the dimensionless parameter p ≡ r /m . Table 2 : The percentage of maximum absolute relative error between the exact metric function h ( r ) of Eq. (4.9) and its first-order approximation for various values of the dimensionless parameter p ≡ r /m (cid:62)
2. The WH/BH threshold corresponds to p =
2. The parametrization becomes exact atthe second order.Approx. order p = p = p = p = p = p =
101 45.0447 24.41415 6.41108 1.13354 0.13316 0.0226685For the remainder of this section, we are going to present WH metrics that can be parametrizedexactly in terms of (3.1)-(3.14) with a minimal number of parameters. That is, all parameters thatappear in the continued-fraction expansions a i , b i , ∀ i (cid:62) (cid:15) , a , b and, in some cases, f , h suffice for the parametrization of the metric functions f ( r ) and h ( r ) . A Schwarzschild-like WH metric has been proposed in [31] by deforming the Schwarzschild solution interms of an exponentially small parameter λ ∼ e − πm . The line element reads ds = − (cid:18) − mr + λ (cid:19) dt + (cid:18) − mr (cid:19) − dr + r dΩ , (4.13)and the WH throat is located at r = m . For λ (cid:54) = f ( r ) ≡ − g tt ( r ) is everywherepositive and no event horizon exists unless λ = t → τ according to dt = ( + λ ) − dτ , and then we have f ( r ) = − mr ( + λ ) . (4.14)For this geometry, the EPs which correspond to an exact parametrization via Eqs. (3.1)-(3.14) are thefollowing: (cid:15) = + λ − a = f = λ + λ , a i = ∀ i (cid:62) b = −
12 , h =
12 , b i = ∀ i (cid:62) d s = − (cid:18) − mr (cid:19) dt + (cid:18) − m ( + λ ) r (cid:19) − dr + r dΩ − m (cid:101) αr sin θdtdφ . (4.17)The location of the throat in this case is determined by the roots of the function h ( r ) ≡ − m ( + λ ) r = ⇒ r = m (cid:0) + λ (cid:1) . (4.18)– 12 –o, the DCC (3.1), with r defined as in the above equation, yields an exact parametrization for h ( r ) in terms of (3.16) and (3.14) with all EPs equal to zero. An interesting WH metric was obtained as an exact solution of the Shiromizu-Maeda-Sasaki equations[33] in the context of braneworld gravity [26, 28]: ds = − (cid:18) − mr (cid:19) + (cid:16) − r r (cid:17) − (cid:16) − r r (cid:17) − dr + r dΩ with r ≡ mr r − m . (4.19)The location of the WH throat is at r if r > m and at r if r > m . We consider the case where thethroat is located at r , so the parametric range of interest to us is characterized by p ≡ r / ( m ) > p =
1. Exact parametrizations in terms of Eqs. (3.1)-(3.14) have thefollowing EPs: (cid:15) = mr − a = m r , f = ( r − m ) r , a i = ∀ i (cid:62) b = mr − m , h = b i = ∀ i (cid:62) An interesting metric was suggested by Simpson and Visser (SV) [35] as a toy model which neatlyinterpolates between the standard Schwarzschild BH and a traversable WH in the Morris-Thorne sensethrough the stage of a black bounce. Its extension to axial symmetry was found in [39]. In the slowrotation approximation to linear order in the rotation parameter (cid:101) α , the SV metric is described in termsof a quasiglobal coordinate by the line element ds = − (cid:18) − m √ r + a (cid:19) dt + dr (cid:16) − m √ r + a (cid:17) − m (cid:101) α √ r + a sin θdtdφ + (cid:0) r + a (cid:1) dΩ . (4.22)The location of the throat is identified with the minimum of the circumferential radius K ( r ) , where onehas K (cid:48) ( r ) = → r =
0. In the limit a →
0, a slowly rotating Schwarzschild BH is recovered. When a (cid:54) =
0, we have different branches of the metric depending on the value of the dimensionless parameter p ≡ a/m . At p =
2, there is a WH/BH threshold with the WH branch corresponding to p >
2, whileat p (cid:54) f ( r ) ≡ − g tt ( r ) isdepicted in Fig. (2) for the three branches. According to [35], a black bounce is a minimum of the spherical radius achieved in a nonstatic region of space-timebeyond a BH horizon, in other words, it is a radial bounce in the Kantowski-Sachs homogeneous anisotropic metric in aBH interior. In [35], an example of such behavior was presented as a toy model. Other examples of black bounces areknown in exact solutions of GR in the presence of phantom scalar fields [36–38], where such BHs were named “blackuniverses” because black bounces were followed there by cosmological expansion and isotropization. – 13 – - - Figure 2 : The radial profile of the metric function f ( r ) ≡ − g tt ( r ) in terms of the quasiglobal coordinate r ∈ (− ∞ , ∞ ) in the three branches of the Simpson-Visser metric (4.22). At p ≡ a/m = p = r =
0, and the geometry corresponds to an extremal regular BH, which mayalso be called a one-way WH with an extremal null throat. At p = p →
0, the Schwarzschild BH metric is recovered.Under the radial coordinate redefinition r → ˜ r : ˜ r = r + a , the line element (4.22) can berecast to the MT frame where the metric functions have the following forms: ds = − (cid:18) − m ˜ r (cid:19) dt + (cid:18) − m ˜ r (cid:19) − (cid:18) − a ˜ r (cid:19) − d ˜ r − m (cid:101) α ˜ r sin θdtdφ + ˜ r dΩ . (4.23)The location of the WH throat r in the MT frame is determined by the shape function satisfying b ( ˜ r ) = ˜ r . This condition is equivalent to identifying the roots of the equation h ( r ) ≡ (cid:18) − m ˜ r (cid:19) (cid:18) − a ˜ r (cid:19) = r = m and ˜ r = a , the former is not a suitable choicefor a WH throat because in this case f ( ˜ r ) =
0, and an event horizon emerges. Thus the locationof the throat has to be determined as ˜ r = a , and since ˜ r is the curvature coordinate, the radius ofthe throat is identified with ˜ r . In this frame, the metric (4.23) is parametrized exactly in terms ofEqs. (3.1)–(3.14), and the EPs have the following form: (cid:15) = ma − a = f = − ma , a i = ∀ i (cid:62) b = ma − h = b = − ma , b i = ∀ i (cid:62) Thus far we have considered WH metrics that can be recast in the Morris-Thorne frame where theradial coordinate is the curvature coordinate and the throat is located at some positive value of the– 14 –urvature coordinate r >
0. However, in some cases, such as in construction of numerical solutions,it might be convenient to obtain solutions in more general frames where the radial coordinate is notthe curvature coordinate. In such cases the circumferential radius K ( r ) is not identified with r andconsequently the location of the WH throat r and its radius K ≡ K ( r ) are no longer identified.Furthermore, in the context of such solutions, the throat can be located at zero or even negative valuesof the radial coordinate. When r =
0, the RZ proposal for the compact coordinate (3.1) is no longera suitable choice for a DCC upon which the parametrization can be constructed since it reduces to aconstant x =
1. To this end, the DCC part of the parametrization has to be appropriately modified inorder to accommodate such cases. In the next subsection, we return to the SV metric as given in itsoriginal coordinate system (4.22), prior to its transformation to the Morris-Thorne frame, in order todemonstrate how such a modification of the DCC might be realized.
Let us begin by pointing out that since the SV metric (4.22) is symmetric relative to the throat, once wehave the parametrization in r ∈ [ ∞ ) , we at the same time have the one that is valid for r ∈ (− ∞ , ∞ ) .In the case of a reflection asymmetric metric, if one is interested in the region r (cid:54) r =
0, one canalways perform two independent parametrizations for each side of the throat in a way that is analogousto the one we outline here for the r (cid:62) r = r ∈ [ r , ∞ ) to the compact range x ∈ [
0, 1 ] . To cover the parametrization of metrics in situations like the onedescribed above, let us introduce the following class of optimized dimensionless compact coordinates x ( r ) , to be used instead of (3.1) in the parametrization functions (3.15), (3.16) and (3.27) when thethroat is located at r = x ( r ) ≡ − (cid:18) R x R x + r x (cid:19) x ∞ x , (5.1)where R is an arbitrary “length-scale” parameter which is introduced to make x ( r ) dimensionless, andthe parameters x and x ∞ in Eq. (5.1) are defined via the asymptotics of the metric function we wishto parametrize (symbolically denoted here by g ( r ) ) x ≡ lim r → g (cid:48) ( r ) g ( r ) − g r , (5.2)and x ∞ ≡ lim r → ∞ g (cid:48) ( r ) g ∞ − g ( r ) r . (5.3)Since there are various ways to define the arbitrary parameter R , we will now discuss this feature ofthe optimized DCC (5.1). First of all, notice that in parametrization of BH metrics, the correspondinglength scale r that appears in the RZ DCC (3.1), must necessarily be identified with the size ofthe outer event horizon. This is required by the consistency of the parametrization equation (3.2) atthe horizon, where f ( r ) =
0, and so x ( r ) = r ofthe RZ DCC (3.1) should be identified with the throat radius since in the MT frame one always has h ( r ) =
0. Consequently, there is no freedom (or ambiguity) in the definition of the length scale usedin the construction of the RZ DCC.On the other hand, the length-scale parameter R in the optimized DCC of Eq. (5.1) is not subjectto any kind of “consistency” constraint in terms of the parametrization equations. Different choices for R effectively correspond to different parametrizations, and this in turn means that the possible choiceof R will affect the accuracy of parametrization for a given metric. To this end, when working with theoptimized DCC, one should identify, on a case by case basis, the most convenient way of defining R in terms of the parameters of the metric at hand in order to obtain the most accurate parametrization.In any case, as we will demonstrate, some of the most intuitive ways of choosing the length scale R in a WH spacetime provide parametrizations that are very accurate even at the first order.Returning to the SV metric, when written in the coordinate system of Eq. (4.22), the optimizedDCC (5.1) that is suitable for the parametrization of f ( r ) is obtained via Eqs. (5.2) and (5.3) with x = x ∞ =
1, respectively, and so it reads x ( r ) = − (cid:115) R R + r . (5.4)By comparing both sides of the expansions of the parametrization equation (3.13) at x = x = (cid:15) = mR − a = f = − ma , a = (cid:15) + f − + mR a , (5.5) a = a (cid:18) (cid:15) + f − a − + mR a (cid:19) . (5.6)We now turn to the definition of the length parameter R for the SV metric. Perhaps the most naturaland intuitive way to define a length scale associated with a WH geometry is given by identifying R with the radius of the WH throat, R ≡ K ( r ) = a . (5.7)For the SV metric, this choice provides an exact parametrization in all orders in the expansion. This,however, cannot be expected to hold true for any arbitrary metric function and optimized DCC, andone should, in principle, test different choices for the length scale parameter R constructed by theparameters of the solution at hand in order to obtain the optimal accuracy. In what follows we proposetwo more ways to construct length-scale parameters R from the metric in a systematic way.Since we are interested in the parametrization of asymptotically flat geometries, the metric func-tions asymptote to some finite value (not necessarily the same) at both infinities. Furthermore, theymust be finite everywhere, and consequently (if we exclude the trivial case where the metric functionis constant) we have the emergence of at least one inflection point. Then one may identify R withthe value of the circumferential radius K ( r ) at the location of the inflection point r inf of the metricfunction. For the SV WH this condition yields f (cid:48)(cid:48) ( r inf ) = → R ≡ K ( r inf ) = a (cid:114)
32 . (5.8)– 16 –nother definition of the length scale R in terms of the free parameters of the metric in a systematicway can be via the value of the second derivative of the metric function evaluated at the throat. Inour example we find R ≡ (cid:112) f (cid:48)(cid:48) ( r ) = (cid:114) a m , (5.9)while in cases where f (cid:48)(cid:48) ( r ) =
0, the above definition could be replaced by R = f (cid:48) ( r ) − . In any case,the aforementioned definitions for R do not, by any means, exhaust all possible ways to define R interms of the free parameters of a metric, and when employing the approach of the optimized DCC, onemust inevitably try different values for R .In Tables (3) and (4) with R defined as in Eq. (5.8) and in Eq. (5.9), respectively, we presentthe accuracy of the lowest orders of the approximation of the metric function f ( r ) for various values ofthe parameter p ≡ a/m . Table 3 : The percentage of maximum absolute relative error for various approximation orders andvalues of the parameter p ≡ a/m . Here, the length scale parameter R = a (cid:112) / f ( r ) as in Eq. (5.8).Approx. order p = p = p = p = p = p = Table 4 : The percentage of maximum absolute relative error for various approximation orders andvalues of the parameter p ≡ a/m . Here, the length scale parameter R = a / / √ m has beenidentified in terms of the value of the second-order derivative of the metric function f ( r ) at the locationof the throat, as in Eq. (5.9).Approx. order p = p = p = p = p = p = O ( − ) O ( − ) O ( − ) R we see quick convergence and excellent accuracy of the approximationalready at the first order, even as the WH/BH threshold ( p = ) is approached. Furthermore, aqualitative difference between the two parametrizations is also evident. When R is defined as inEq. (5.8), we see in Table (3) that the parametrization becomes less accurate as the WH/BH thresholdis approached, which is not so for the parametrization with R defined as in Eq. (5.9) according to theentries of Table (4). This verifies that different choices of the length-scale parameter might provide anaccuracy in some regions of the parameter space better than in others. To this end, one is encouraged– 17 –o try different definitions of R in order to identify the one that yields the optimal accuracy in theparameter range one is interested in.In terms of the quasiglobal coordinate of the frame (4.22), the g tφ ( r , θ ) metric component of theslowly rotating SV metric does not explicitly correspond to the Lense-Thirring term as in its MT framerepresentation (4.23). Thus a parametrization by means of Eqs. (3.28) and (3.27) with the optimizedDCC of Eq. (5.4) may be performed. A few first EPs in this case turn out to be c = m (cid:101) αR , g = − m (cid:101) αa , c = c + g + mR (cid:101) αa . (5.10)If, according to Eq. (5.7), we choose to identify the length-scale parameter with the radius of thethroat, i.e., R = a , it turns out that c =
0, then the parametrization function (3.27) reproduces theLense-Thirring term in terms of the compact coordinate exactly.Finally, we mention that our parametrizations (in both frames (4.22),(4.23) and for both choicesof R ) can also be very successfully applied for the regular BH branch of the metric. Using theparametrization approach based on the optimized DCC (5.4) on the left panel of Fig. (3), we plotthe exact metric function f in the regular BH branch ( p = f app in terms of the quasiglobal coordinate. On the right panel of the same figure, we plot the absolutedifference between f and f app in terms of the optimized DCC. - - Figure 3 : The regular black hole branch, for p ≡ a/m = f ( r ) ≡ − g tt ( r ) (4.22) (blue curve) and its first-order approximation f app ( r ) (red dashed curve). Rightpanel: their absolute difference in terms of the optimized dimensionless compact coordinate x ∈ [
0, 1 ] (5.4) for R as given in Eq. (5.9). General relativity with a massless minimally coupled scalar field allows for a solution containing a nakedsingularity, known as the Fisher solution [40]. When the kinetic term has the opposite sign, a solutionemerges which has a WH branch and was called by one of us “the anti-Fisher solution” by analogywith the “anti-de Sitter solution” [41]. The metric in the WH branch of the anti-Fisher solution canbe written as [41, 42] ds = − e u ( r ) dt + e − u ( r ) (cid:2) dr + (cid:0) r + a (cid:1) dΩ (cid:3) , u ( r ) ≡ ma (cid:16) arctan ra − π (cid:17) , (5.11)– 18 –here we assume that the arbitrary parameters satisfy m (cid:62) a >
0. In the case of the lineelement above, we are forced to perform a parametrization in the coordinate frame with K ( r ) (cid:54) = r sincewe cannot analytically move to the MT frame (2.4). This means that, strictly speaking, a differentradial coordinate is used here, and the parametrization in this case is different from the ones discussedin Sec. 4 as was the case with the SV metric in the frame of the previous subsection. On the otherhand, when comparing various parametrized WH geometries, the coordinate choice must be unique.Therefore, this example provides further verification that our proposed parametrization can also providean effective description of WH geometries in other coordinate systems beyond the ones employing thecurvature coordinate.Notice that in the coordinate system (5.11) we have f ( r ) = h ( r ) = e u ( r ) , and so r is a quasiglobalbut not curvature coordinate according to the classification of Sec. 2. Minimization of K ( r ) = e − u ( r ) ( r + a ) reveals that the throat is located at r = m and has a radius K ≡ K ( r ) = e − u ( m ) √ m + a . (5.12)If one is not interested in the limit m → r = m (cid:54) =
0. In the latter case we are going to employ the RZ DCC (3.1) for ourparametrization, keeping always in mind that the radial coordinate is not the curvature coordinate andconsequently the parametrization here is conceptually distinct from the cases considered in the Morris-Thorne frame in Sec. 4. The parametrization of f ( r ) is then obtained in terms of Eqs. (3.15) and (3.13).The expansions of the parametrization equation (3.13) at the boundaries of the parametrization range r ∈ [ r , ∞ ) → x ∈ [
0, 1 ] determine the EPs, the first few of which have the following expressions: (cid:15) = a = f = e u ( m ) , a = f (cid:18) + m a + m (cid:19) − a = f − a − a , a = − f (cid:104) m ( a + m ) − (cid:105) + a [ + a ( + a )] + a a . (5.14)Upon substituting the EPs above into the expression of the parametrization function (3.15), we cancompute the absolute relative error (ARE) and MARE for the approximation of the anti-Fisher solutionat various orders in the continued-fraction expansion.In order to test the convergence of the approximation, in Table (5) we give the percentage ofMARE for the first four orders in the approximation and for various values of the dimensionless parameter p ≡ a/m . It is evident that for all values of p the MARE is decreased when the order of approximationis growing. Table 5 : The percentage of maximum absolute relative error between the exact metric function f ( r ) asgiven in Eq. (5.11) and its lowest-order approximations for various values of the dimensionless parameter p ≡ a/m . Approx. order p = p = p = p = p = p = m → ds = − dt + dr + (cid:0) r + a (cid:1) dΩ . (5.15)At first glance, someone might consider the use of the RZ DCC as a naive choice for the compactcoordinate given the fact that the length scale parameter r = m vanishes in that limit and consequentlywe are seemingly dealing with a situation that has been remedied with the optimization of the DCC inthe previous section. However, the two cases are radically different since in the case of the SV metricone has r = r = m → x = A ( ) =
1. At the same time, all the EPs (5.14) (including the higher-order ones) in ourparametrization of the anti-Fisher metric are finite constants for m = a > m → f ( r ) → (cid:101) α was found, and it is given by ds (cid:101) α = ds − (cid:101) αω ( ) e − u ( r ) (cid:0) r + a (cid:1) sin θdtdφ , u ( r ) ≡ ma (cid:16) arctan ra − π (cid:17) , (5.16)where ds is the line element of the static anti-Fisher WH as given in Eq. (5.11). Here the angularvelocity metric function is ω ( r ) ≡ (cid:101) αω ( ) = (cid:101) α (cid:0) − e u ( r ) (cid:1) (cid:104) + m ( r + m ) r + a (cid:105) a [ − e − πm/a ( + m /a )] . (5.17)The parametrization for the metric function g ( r ) ≡ − (cid:101) αω ( ) e − u ( r ) (cid:0) r + a (cid:1) , (5.18)is obtained in terms of Eqs. (3.27) and (3.28). The asymptotic parameter c and the first few near-fieldparameters are given by c = ae πma ( a + m ) (cid:101) α (cid:104) a (cid:16) e πma − (cid:17) − m (cid:105) , g = c e u ( m ) (cid:2) a + m − e − u ( m ) ( a + m ) (cid:3) ( a + m ) , (5.19) c = c (cid:20) + g c + m e u ( m ) a + m (cid:21) , c = c c + g c (cid:18) − m a + m (cid:19) − c above is exactly equal to the expression 4 J/r = J/m , where J is given by Eq. (26) of [42].Next, in order to test the convergence of the parametrization, in Table (6) we give the MAREfor various values of the parameter p ≡ a/m at the first four orders in the approximation of (5.18).Indeed, we find that the error decreases as more higher-order terms are taken into account, and thisverifies the convergence of the approximation. – 20 – able 6 : The percentage of maximum absolute relative error between the exact metric function g ( r ) / (cid:101) α as given in Eq. (5.18) and its lowest-order approximations for various values of the dimensionlessparameter p ≡ a/m .Approx. order p = p = p = p = p = p = In this section, we turn to computing the radii of shadows and quasi-normal modes as gauge-invarianttests of the parametrization accuracy.
Here, we give a brief overview of the formalism suggested in [44] and generalized in [45] for computationof the shadow radius of an arbitrary static, spherically symmetric compact object, including WH solutionsthat are of interest to us in this work. Consider the general ansatz for a static and spherically symmetricgeometry, ds = − f ( r ) dt + dr h ( r ) + K ( r ) dΩ . (6.1)As is pointed out it in [45], it is possible to derive an expression for the radius of the shadow ofan arbitrary compact object in terms of the general metric functions that appear in Eq. (6.1), andconsequently it may be applied to both BH and WH metrics. The shadow of a compact object isclosely related to the location of its photon sphere which we shall denote here by r ph . It is convenientto introduce the function w ( r ) ≡ K ( r ) f ( r ) , (6.2)in terms of which r ph is determined as the solution to the equation dw ( r ph ) dr = r O , is obtainedvia sin a sh = w ( r ph ) w ( r O ) . (6.4)Finally, under the assumption that the observer is located sufficiently far from the compact object, i.e., r O (cid:29) r , where r is a characteristic length scale that can be identified with the radius of the WHthroat, the following conditions are satisfied: f ( r O ) (cid:39) K ( r O ) (cid:39) r O , (6.5)– 21 –nd one finds that the radius of the shadow is given by R sh (cid:39) r O sin a sh (cid:39) w ( r ph ) = K ( r ph ) (cid:112) f ( r ph ) . (6.6)To test the accuracy of the metric approximations of the previous sections, in each case we arenow going to compute the value of the shadow radius (6.6) using the exact expressions for the metricfunctions and compare its value to the one obtained when employing the corresponding approximationsof the metric at various orders of the expansion. As is evident from Eq. (6.6), the shadow radius iscompletely specified in terms of the metric function f ( r ) and the circumferential radius K ( r ) and isthus independent of the form of h ( r ) . Thus, in the next section we will use the anti-Fisher (5.11) andSimpson-Visser (4.22) metrics as examples for which f ( r ) is not exactly parametrizable. The anti-Fisher solution (5.11) is written in terms of the quasi-global coordinate and the two relevantmetric functions have the following form: f ( r ) = e u ( r ) , K ( r ) = e − u ( r ) (cid:0) r + a (cid:1) , u ( r ) ≡ ma (cid:16) arctan ra − π (cid:17) . (6.7)Thus we have w ( r ) = e − u ( r ) (cid:0) r + a (cid:1) , (6.8)and the condition d ( w ( r ) /dr = ) yields the photon sphere radius r ph = m . Then via Eq. (6.6) weobtain the exact expression for the shadow radius R sh = e − u ( r ph ) √ m + a . (6.9)In terms of the dimensionless parameter p ≡ a/m , we have compared the value obtained by Eq. (6.9)with the one obtained via the approximations of the metric (5.11) at various approximation orders,and the results are given in Table (7). We point out that the value of r ph , which we used to obtainthe approximate values for the shadow radius, is obtained for each order in the approximation viaextremization of the function w ( r ) . Table 7 : The percentage of absolute relative error between the exact value of the shadow radius (6.9)and its value as obtained via Eq. (6.6) for various approximation orders of the metric function f ( r ) .The dimensionless parameter p ≡ a/m .Approx. order p = p = p = p = p = p = .3 Shadows of the Simpson-Visser wormhole The two metric functions which are relevant to the computation of shadows from the line element ofEq. (4.22) are the following: f ( r ) = (cid:18) − m √ r + a (cid:19) , K ( r ) = (cid:0) r + a (cid:1) . (6.10)Thus we have w ( r ) = ( r + a ) (cid:16) − m √ r + a (cid:17) , (6.11)and the photon sphere radius corresponds to r ph = (cid:112) ( m ) − a . This means that the SV WH branchof the metric has a photon sphere when 2 m (cid:54) a (cid:54) m with the WH/BH threshold corresponding to a = m . Then via Eq. (6.6) we obtain the exact expression for the shadow radius R sh = √ m , (6.12)where we can see that it does not depend on the parameter a and it is identified with the shadow radiusof the Schwarzschild BH. For a detailed discussion on the degeneracy of shadows in the SV spacetimesee [46]. For various values of the dimensionless parameter p ≡ a/m , we have compared the valueobtained by Eq. (6.12) with the one obtained via the approximations of the metric at various orders inthe approximation. The obtained percentage of the absolute relative error are given in Table (8) for R as defined in Eq. (5.8) and in Table (9) for R as defined in Eq. (5.9). Table 8 : The percentage of absolute relative error between the analytic value of the shadow ra-dius (6.12) of the Simpson-Visser wormhole and its value as obtained via Eq. (6.6) for various approx-imation orders of the metric. The WH/BH threshold corresponds to p =
2, while for p > R = a (cid:112) / p = p = p = p = p = p = O ( − ) O ( − ) O ( − ) O ( − ) Table 9 : The percentage of absolute relative error between the analytic value of the shadow ra-dius (6.12) of the Simpson-Vissser wormhole and its value as obtained via Eq. (6.6) for various approx-imation orders of the metric. The WH/BH threshold corresponds to p = p > R = a / / √ m according to Eq. (5.9).Approx. order p = p = p = p = p = p = O ( − ) O ( − ) O ( − ) O ( − ) O ( − ) O ( − ) O ( − ) O ( − ) O ( − ) O ( − ) – 23 –irst, notice that in both Tables (8) and (9), the accuracy in the approximation of the shadowradius is in all cases better than the accuracy of the corresponding order of the approximation for themetric as given in Tables (3) and (4), respectively. This is a consequence of the fact that the locationof the photon sphere radius r ph , which is important for the calculation of the shadow radius, is notidentified with the location of the MARE of the metric r MARE , as can be seen in Fig. (4). Close to theWH/BH threshold, i.e., in the limit p →
2, the photon sphere lies in the region r ph > r MARE . Then,gradually, as the value of p increases, it attains a critical value for which r ph is identified and with r MARE and consequently, as p further increases, we end up with r ph < r MARE .Second, notice that according to the entries of Tables (8) and (9), close to the limit where the WHhas a photon sphere ( p = r ph shifts towards the lower boundary of the parametrizationregion ( x =
0) where the throat is located, see the right panel of Fig. (4), and as we have alreadydiscussed, the error between the exact metric and its approximation at any order is exactly zero at both x = x = R of the parametrization,for a given approximation order, the obtained values of the shadow radius error at some p can beunderstood as an interplay between the following criteria: The relative location of r ph w.r.t. r MARE ,the value of MARE at that p and, finally, how far is r ph located from the throat. Figure 4 : The percentage of the absolute relative error of the first-order approximation of the metricin terms of the optimized compact coordinate x ∈ [
0, 1 ] (5.4) with R as given in Eq. (5.9). Alsodepicted are the location of the MARE r MARE (purple line) and of the photon sphere r ph (orange line)as obtained with the first-order approximation of the metric. Left panel: near the wormhole/black holethreshold with p ≡ a/m = p = Here, as another example of observable quantities, we will calculate the fundamental quasinormalmodes of the electromagnetic field propagating in a wormhole background. Quasinormal modes arecharacteristic frequencies of a compact object (be it a black hole or a wormhole) which are independentof the initial conditions of perturbations and are fully determined by the parameters of the compact– 24 –bject [47–49]. The real part of the quasinormal mode represents a real oscillation frequency, whilethe imaginary one is proportional to the damping rate. Quasinormal modes of various wormholes werestudied in a great number of publications [7, 9, 50–58]. Therefore, here we will concentrate on thoseexamples of wormholes which include the parametric transition between the black hole and wormholestates. This way we will be able to understand whether the suggested parametrization provides areasonable and compact approximation near the threshold of transition, that is, whether it allows oneto describe wormholes as possible black hole mimickers [28, 31, 59–61]. The general covariant equationfor an electromagnetic field has the form1 √ − g ∂ µ (cid:0) F ρσ g ρν g σµ √ − g (cid:1) = F ρσ = ∂ ρ A σ − ∂ σ A ρ , and A µ is a vector potential. After separation of the variables Eq. (6.13)for the spherically symmetric static spacetime (Eq. (6.1)) takes the following general wave-like form d Ψdr ∗ + (cid:0) ω − V ( r ) (cid:1) Ψ = r ∗ is defined in terms of the general metric functions f ( r ) and h ( r ) ofEq. (6.1) by the relation dr ∗ = dr (cid:112) f ( r ) h ( r ) , (6.15)and the effective potential is V em ( r ) = f ( r ) (cid:96) ( (cid:96) + ) r . (6.16)Notice that although the effective potential depends only on one of the two metric functions, thequasinormal modes will depend on both because the tortoise coordinate includes both.The effective potential for wormholes may have different forms. For example, it can have a singlemaximum located at the wormhole throat, or a couple of peaks, which are situated symmetrically relativeto the throat. A common feature is that at “minus infinity” and “plus infinity” the effective potentialapproaches some constant values representing the asymptotic flatness of the spacetime. Therefore theboundary conditions for finding quasinormal modes of a wormhole are similar to those for a black holein the tortoise coordinate [62]: purely outgoing waves are required at both infinities.Here we shall use the time-domain integration method, which does not depend on the form of apotential barrier and can be applied to all cases under consideration. We will integrate the wave-likeequation rewritten in terms of the light-cone variables u = t − r ∗ and v = t + r ∗ with the help of theGundlach-Price-Pullin discretization scheme [63]: Ψ ( N ) = Ψ ( W ) + Ψ ( E ) − Ψ ( S )− ∆ V ( W ) Ψ ( W ) + V ( E ) Ψ ( E ) + O (cid:0) ∆ (cid:1) , (6.17)where we used the following designations for the points: N = ( u + ∆ , v + ∆ ) , W = ( u + ∆ , v ) , E = ( u , v + ∆ ) , and S = ( u , v ) . The initial data are given on the null surfaces u = u and v = v .– 25 –o extract the values of the quasinormal frequencies, we will use the Prony method which allows oneto fit the signal by a sum of exponents with some excitation factors. This method was used in a greatnumber of works (see, for instance, the recent papers [64–67] and references therein) and showed agood agreement with accurate results obtained by other methods.Quasinormal modes of the Bronnikov-Kim braneworld wormhole with zero Schwarzschild mass,the CFM wormhole and the Simpson-Visser metrics are presented in Tables 10, 11, 12. Table 10 : Fundamental quasinormal modes ( (cid:96) = n =
0) of the electromagnetic field for Bronnikov-Kim II wormhole.order C = − C = − C = − − i − i − i − i − i − i − i − i − i Table 11 : Fundamental quasinormal modes ( (cid:96) = n =
0) of the electromagnetic field for CFMwormhole ( m = / r = m r = m r = m exact 0.50177 − i − i − i − i − i − i Table 12 : Fundamental quasinormal modes ( (cid:96) = n =
0) of the electromagnetic field for Simpson-Visser wormhole ( m = / a = a = a = − i − i − i − i − i − i − i − i − i The Prony method does not allow for extracting quasinormal frequencies with guaranteed accuracyhigher than a fraction of one percent. This happens because the time at which the beginning and endof quasinormal ringing process occurs is conditional. Therefore, when using time-domain integration,there is no point in looking at high orders of the parametrization, because the error of the Prony methodwill be evidently larger than that of the parametrization. Nevertheless, this method allows us to judgeabout the convergence of the parametrization at the first few orders. From the above tables we see thatfar from the threshold of the wormhole/black hole transition, already the first order of expansion makesa relative error about one percent or less. At the same time, the period of quasinormal ringing is veryshort near the transition and is “contaminated” by consequent echoes, as can be seen, for example,from Fig. 2 in [28]. Therefore, we are unable to see whether the first-order expansion is really bad nearthe transition point, but we see that already at the second order (for the Bronnikov-Kim II metric) thesituation is remedied. – 26 –
Conclusions
In this work we have developed a general parametrization for static, spherically symmetric and asymp-totically flat wormhole geometries in an arbitrary metric theory of gravity. We have also extended theparametrization to a slow rotation mode. We used the previously developed general parametrizationof black holes [10], which we adopted for wormholes by a number of appropriate modifications. Thecontinued-fraction expansion in the radial direction in terms of a compact coordinate provides superiorconvergence, and the various examples of analytic wormhole metrics, which we considered for testingour parametrization, have shown that a very good accuracy is achieved already at the first order ofthe expansion, while the second order guarantees that the relative error is less than one percent in allconsidered cases. Remarkably, the method is convergent also at a transition between the black holeand wormhole states, so that wormholes mimicking black holes’ behavior can also be well described bythe above parametrization. The latter is confirmed by our calculations of the radii of shadows cast bythe wormholes and by their quasinormal spectra.Our paper could be further extended to arbitrary rotation in a similar fashion with the black holecase [11], that is, by expansion in terms of cos θ around the equatorial plane. However, apart fromthe evident lack of a sufficient number of examples in the literature, apparently this most generalparametrization of axially symmetric wormholes will not share the elegance and simplicity of interpre-tation of the case considered here, and therefore deserves a separate consideration. Acknowledgments
R. K. and T. D. P. acknowledge the support of the grant 19-03950S of Czech Science Foundation(GAˇCR). K.B. was supported by the RUDN University Strategic Academic Leadership Program. Theresearch of K.B. was also funded by the Ministry of Science and Higher Education of the RussianFederation, Project “Fundamental properties of elementary particles and cosmology” N 0723-2020-0041, and by RFBR Project 19-02-00346.
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