Accretion Disk Around the Rotating Damour–Solodukhin Wormhole

Eur. Phys. J. C (2019) 79:952 https://doi.org/10.1140/epjc/s10052-019-7488-7

Regular Article - Theoretical Physics

Accretion disk around the rotating Damour–Solodukhin wormhole

R. Kh. Karimov1,R.N.Izmailov1,a , K. K. Nandi1,2 1 Zel’dovich International Center for Astrophysics, Bashkir State Pedagogical University, 3A, October Revolution Street, Ufa 450008, RB, Russia 2 High Energy Cosmic Ray Research Center, University of North Bengal, Darjeeling, WB 734 013, India

Received: 5 October 2019 / Accepted: 12 November 2019 / Published online: 21 November 2019 © The Author(s) 2019

Abstract A new rotating generalization of the Damour– more commonly known as the Damour–Solodukhin worm- Solodukhin wormhole (RDSWH), called Kerr-like wormhole (DSWH). The DSWH differs from the Schwarzschild hole, has recently been proposed and investigated by Bueno black hole (SBH) by a dimensionless real deviation parame- et al. for echoes in the gravitational wave signal. We show a ter λ and represents a twice-asymptotically flat regular space- novel feature of the RDSWH, viz., that the kinematic prop- time connected by a throat. The authors showed that if λ is erties such as the ISCO or marginally stable radius rms,effi- so small that the time scale Δt = 2GMln 1 is longer λ λ2 ciency and the disk potential Veff are independent of than the observational time scale, the signals emitted by a (which means they are identical to their KBH counterparts source falling into a wormhole will contain the usual QNM for any given spin). Differences however appear in the emis- ringing signature of a black hole, in spite of the absence of . <λ≤ sivity properties for higher values 0 1 1 (say) and for a true horizon [1]. Völkel and Kokkotas [2] have recently = . the extreme spin a 0 998. The kinematic and emissivity shown, using an inverse method, that the knowledge of the are generic properties as variations of the wormhole mass observed QNM spectrum can allow one to also accurately and the rate of accretion within the model preserve these construct the double hump Pöschl–Teller potential approxi- properties. Specifically, the behavior of the luminosity peak mating that of DSWH. Bueno et al. [3] suitably redefined the is quite opposite to each other for the two objects, which static DSWH and further generalized it the into a Kerr-like could be useful from the viewpoint of observations. Apart wormhole, called here the rotating DSWH or RDSWH for Δ from this, an estimate of the difference λ in the maxima of brevity. They studied its gravitational wave echo properties. ( ) flux of radiation F r shows non-zero values but is too tiny Gravitational deflection of relativistic massive particles by λ< −3 to be observable at present for 10 permitted by the RDSWH has been recently studied by Jusufi et al. [4]. strong lensing bound. The broad conclusion is that RDSWH QNM ringing by BHs have been well reviewed, see, e.g. are experimentally indistinguishable from KBH by accretion Bertietal.[5] but ringing by WHs still remained an open characteristics. question. A first step towards answering the question was initiated by Cardoso et al. [6,7], who studied the QNM ringing using a wormhole assembled by means of Visser’s cut- 1 Introduction and-paste surgery of two copies of Schwarzschild black holes (SBH) at a radius close to the horizon [8] (For future perspec- There is a renewed interest in wormholes after it has been tives and new directions of research on ultracompact objects realized that they can mimic post-merger ring-down ini- (UCO) including wormholes, see [9]). The authors of [8] tial quasi-normal mode (QNM) spectrum of gravitational showed that, while the time evolution of the early QNMs waves. To our knowledge, this possibility of “wormhole accurately mimic those from SBH horizon, the differences (if QNM mode” was first put forward by Damour and Solo- any) would appear only at later times. This work inspired an dukhin [1], where resonances were trapped in a double- investigation in [10], where it has been shown that the mass- hump potential associated with what they termed “black less Ellis–Bronnikov wormhole (EBWH) [11–13], made of hole foil”. This is an example of a horizonless wormhole the minimally coupled exotic scalar field, can also reproduce the black hole QNM spectrum in the eikonal limit (large ). Ringing by massive EBWH was studied in [14]. These developments prompt a natural inquiry as to whether static a e-mail: [email protected] DSWH can exhibit mimicking of SBH in other phenomenon 123 952 Page 2 of 9 Eur. Phys. J. C (2019) 79 :952 as well. A recent work [15] shows that it indeed can, e.g., 2 Thin accretion disk it can very accurately mimic SBH strong field lensing properties for λ ≤ 10−3. However, this is only an upper bound, We assume geometrically thin accretion disk, which means which obviously does not rule out values of λ arbitrarily that the disk height H above the equator is much smaller than close to zero. Strong field lensing is an excellent diagnostic the characteristic radius R of the disk, H R. The disk is for studying the signatures of WHs versus BHs, see, e.g., assumed to be in hydrodynamical equilibrium stabilizing its [16]. vertical size, with the pressure and vertical entropy gradient There is another important diagnostic, namely, the accre- being negligible. An efficient cooling mechanism via heat tion phenomenon around compact objects that has already loss by radiation over the disk surface is assumed to be func- been a very active field of research (see, e.g., [17–27]). The tioning in the disk, which prevents the disk from collecting first comprehensive study of accretion disks using a Newto- the heat generated by stresses and dynamical friction. The nian approach was made in [28]. Later a general relativistic thin disk has an inner edge defined by the marginally stable model of thin accretion disk was developed in three seminal circular radius rms, while the orbits at higher radii are Kep- papers by Novikov and Thorne [29], Page and Thorne [30] lerian. In the steady-state approximation, the mass accretion ˙ and Thorne [31] under the assumption that the disk is in a rate M0 is assumed to be a constant and the physical quantities steady-state, that is, the mass accretion rate M˙ is constant in describing the accreting matter are averaged over a charac- time and does not depend of the radius of the disk. The disk teristic time scale Δt, over the azimuthal angle Δφ = 2π for is further supposed to be in hydrodynamic and thermody- a total period of the orbits, and over the height H [29,30]. namic equilibrium, which ensure a black body electromag- The orbiting particles with the four-velocity uμ form a netic spectrum and properties of emitted radiation. The thin disk of an averaged surface density Σ, where the rest mass μ accretion disk model further assumes that individual parti- density ρ0, the energy flow vector q and the stress tensor cles are moving on Keplerian orbits, but for this to be true tμν are measured in the averaged rest-frame. Then the central object is assumed to have weak magnetic field, H otherwise the orbits in the inner edge of the disk will be Σ(r) = ρ0dz, (1) deformed. −H Accretion around wormholes has also been an active area where ρ0 rest mass density averaged over Δt and 2π and of research [32,33], more so because wormholes are special the torque density types of objects sourced by materials that violate at least H the Null Energy Condition (hence exotic). Research included r r Wφ = tφ dz, (2) also other types of objects, e.g., quark, boson and fermion −H stars, brane-world black holes, gravastars, naked singularities with the component tr averaged over Δt and 2π. The time (NS) [34–50], f (R)-modified gravity models of black holes φ and orbital average of the energy flow vector qμ gives the [51–53] and so on. One of the most promising method to radiation flux F(r) over the disk surface as distinguish different types of astrophysical objects through α their accretion disk properties is the profile analysis of K F(r) =qz. (3) iron line [54–58]. μ In this paper, we shall study the kinematic as well as emis- For the stress energy tensor Tν of the disk, the energy μ sivity properties such as the luminosity spectra, flux of radi- and angular momentum four-vectors are defined by −E ≡ μ ν μ μ ν ation, temperature profile, efficiency of a thin accretion disk Tν (∂/∂t) and J ≡ Tν (∂/∂φ) respectively. The struc- around a stellar sized RDSWH using the Page–Thorne model ture equations of the thin disk can be derived by integrating [28–31]. As a numerical example, we shall assume the accre- the conservation laws of the rest mass, of the energy, and of tion to take place only in the attractive positive mass mouth the angular momentum [29,30]. From the rest mass conser- ˙ 19 −1 μ with mass 15M and an accretion rate M0 ∼ 10 gs . vation, ∇μ(ρ0u ) = 0, it follows that the average rate of the The motive is to analyze how far the accretion profiles are accretion is independent of the disk radius, influenced by λ and the dimensionless spin parameter a∗. ˙ ≡− π Σ r = . The paper is organized as follows: In Sect. 2, we out- M0 2 r u constant (4) line the main formulas relating to the thin accretion disk to In the steady-state approximation, specific energy E and be used in the paper. The kinematic accretion formulas for specific angular momentum L of accreting particles have generic spinning spacetime are presented in Sect. 3. These are depend only on the radius of the orbits. Defining black hole applied to RDSWH in Sect. 4. Numerical estimates are made rotational velocity Ω = dφ/dt, the energy conservation law μ in Sect. 5, while Sect. 6 concludes the paper. We take units ∇μ E = 0 yields the integral such that G = c = 1 unless restored and metric signature r −, +, +, + ˙ − π Ω φ = π F . ( ). M0 E 2 r W ,r 4 r E (5) 123 Eur. Phys. J. C (2019) 79 :952 Page 3 of 9 952

This is a balance equation, which states that the energy trans- 3 Generic rotating spacetime ˙ ported by the rest mass flow, M0 E, and the energy transported r by the torque in the disk, 2πrΩWφ, is balanced by the energy The accretion disk is formed by particles moving in circular radiated away from the surface of the disk, 4πrF E. orbits around a compact object, with the geodesics deter- μ The angular momentum conservation law, ∇μ J = 0, mined by the space-time geometry around the object, be it a states the balance of three forms of angular momentum trans- WH, BH or NS. For a rotating geometry the metric is gener- port, viz., ically given by r 2 2 2 ˙ − π φ = π F. =− + φ φ + M0 L 2 rW ,r 4 r L (6) ds gtt dt 2gt dtd grr dr 2 2 r +gθθ dθ + gφφ dφ . (12) By eliminating Wφ from Eqs. (5) and (6), and applying the energy–angular momentum relation for circular geodesic |θ − π/ | , At and around the equator, i.e., when 2 1 we orbits in the form E,r = Ω L,r , the flux F of the radiant assume, with Harko et al. [43], that the metric functions gtt, energy, or power, over the disk can be expressed as [29,30], g φ, g , gθθ and gφφ depend only on the radial coordinate r. t rr ˙ Ω r Ω M0 ,r The angular velocity , of the specific energy E, and of the F (r) =− √ E − Ω L L, dr. (7) π − 2 r specific angular momentum L of accreting particles in the 4 g E − Ω L rms above geometry are given by The disk is supposed to be in thermodynamical equilibrium, dt Egφφ + Lg φ as explained, so the radiation flux emitted by the disk surface = t , (13) τ 2 + will follow Stefan–Boltzmann law: d gtφ gttgφφ ( ) = σ 4 ( ) , dφ Eg φ − Lg F r T r (8) =− t tt , (14) τ 2 + d gtφ gttgφφ where σ is the Stefan–Boltzmann constant. The observed 2 2 2 luminosity L (ν) has a redshifted black body spectrum [18] dr E gφφ + 2E Lgtφ − L gtt grr =−1 + . (15) π τ 2 + π rout 2 ν3 ϕ d gtφ gttgφφ 2 8 h cos i e rdrd Lν = 4πd I (ν) = , 2 ν c r 0 h e − One may hence define an effective potential term defined in exp κ T 1 B as (9) 2 2 E gφφ + 2E Lgtφ − L gtt i d V (r) =−1 + . (16) where is the disk inclination angle to the vertical, is the eff 2 + gtφ gttgφφ distance between the observer and the center of the disk, rin and rout are the inner and outer radii of the disc, h is the Existence of circular orbits at any arbitrary radius r in the ν (ν) Planck constant, e is the emission frequency, I is the equatorial plane demands that Veff (r) = 0 and dVeff /dr = 0. Planck distribution, and kB is the Boltzmann constant. The These conditions allow us to write the kinematic parameters observed photons are redshifted and received frequency ν is as related to the emitted ones by νe = (1 + z)ν. The redshift gtt − gtφΩ factor (1 + z) has the form: E = , (17) 2 gtt − 2gtφΩ − gφφΩ 1 + Ωr sin φ sin i 1 + z = , (10) gtφ + gφφΩ 2 = , −gtt − 2Ωgtφ − Ω gφφ L (18) g − 2g φΩ − gφφΩ2 tt t where the light bending effect is neglected [59,60]. 2 dφ −gtφ,r + (gtφ,r ) + gtt,r gφφ,r Another important characteristic of the thin accretion disk Ω = = . (19) dt gφφ, is its efficiency , which quantifies the ability with which the r central body converts the accreting mass into radiation. The where, throughout the paper, X,r ≡ dX/dr. Stability of 2 2 efficiency is measured at infinity and it is defined as the ratio orbits depends on the signs of d Veff /dr , while the condi- 2 2 of two rates: the rate of energy of the photons emitted from tion d Veff /dr = 0 gives the inflection point or marginally the disk surface and the rate with which the mass-energy is stable (ms) orbit or innermost stable circular orbit (ISCO) transported to the central body. If all photons reach infinity, at r = rms. The thin disk is assumed to have an inner edge the Page–Thorne accretion efficiency is given by the specific defined by the marginally stable circular radius rms, while energy of the accreting particles measured at the marginally the orbits at higher radii than r = rms are Keplerian. Note stable orbit [30]: that marginally stable circular orbits play a crucial role when an accreting BH is surrounded, e.g., by quintessence matter = 1 − E(r ) , (11) ms [16,62]. The generic Eqs. (7–19) are valid for any rotating As the definition indicates, should be non-negative. spacetime that will be explicitly calculated in what follows. 123 952 Page 4 of 9 Eur. Phys. J. C (2019) 79 :952

4 Accretion disk properties of RDSWH

We start with a Kerr-like wormhole spacetime, recently considered by Bueno et al. [3] that generalized the static Damour–Solodukhin wormhole [1], which we call here RDSWH. The spacetime metric in Boyer–Lindquist coor- dinates is given by 2Mr 4Mar sin2 θ Σ ds2 =− 1 − dt2 − dtdφ + dr2 Σ Σ Δˆ 2Ma2r sin2 θ +Σdθ 2 + r 2 + a2 + 2 θdφ2, Σ sin (20) Fig. 1 The value of the radial coordinate for the throat r+ (dashed ˆ where Σ and Δ are expressed by line) and marginally stable rms (solid line) orbits as functions of the − spin parameter a for the λ = 10 3 Σ ≡ r 2 + a2 cos2 θ, Δˆ ≡ r 2 − 2M(1 + λ2)r + a2. (21)

It contains a family of parameters where a and M corre- Here we see that expression for effective potential of sponds to the spin and the mass of wormhole, and λ2 is the RDSWH solution does not depend on λ and therefore coin- deformation parameter.1 A non vanishing λ2 differs this met- cides with expression for effective potential of Kerr BH and ric from Kerr metric, we can recover the Kerr metric when so the ISCO radius or rms is the same as that of Kerr BH. λ2 = 0. Although the throat of the wormhole can be easily The remaining expressions determining kinematic proper- obtained by equating the Δˆ to zero, ties of RDSWH are specific energy, specific angular momentum, angular velocity and radius of the marginally stable orbit 2 r+ = (1 + λ )M + M2(1 + λ2)2 − a2, (22) which follow from Eqs. (17)–(19). They also coincide with those of Kerr BH: which represents a special region that connects two different √ asymptotically flat regions. For any values of λ = 0the 2 r (r − 2M) + aM(2 Mr − a) metric is regular everywhere and with small values of λ2 ∼ 0, E = √ , 3( − ) + ( )3/2 − 2 ( + ) + 3 it is practically indistinguishable from a Kerr BH. r r r 3M 6a Mr 3a M r M 2a Mr (24) Special feature of RDSWH metric is that kinematic prop- √ √ erties of accretion disk coincide with those of Kerr BH. As an Mr7 − 3aMr2 + a2 Mr(r + 2M) − a3 M L = , example let us concider the effective potential, given by Eq. √ r r 3(r − 3M) + 6a(Mr)3/2 − 3a2 M(r + M) + 2a3 Mr (16), which determines the geodesic motion of the test par- (25) ticles in the equatorial plane of RDSWH solution (20)–(21). √ Mr3 − aM This is given by Ω = , (26) r 3 − a2 M

1 2M(L − aE)2 + r(aE − L)(aE + L) + E2r 3 r = 3M + 3M2 + a2 + P − 72M2 − 8(6M2 − a2) V =−1 + . ms 2 eff 2 r a + r(r − 2M) 1/2 2 2 2 − 1 −4P + 64a M(3M + a + P) 2 , (27) (23) where 1 We clarify that M here is the Keplerian mass as considered by Bueno 4 2 2 4 2M 9M − 10a M + a 1 et al. [3], viz., gtt ∼ 1 − in their Eq. (36) and explicitly so stated = + 3 , r P 1 K after their RDSWH metric (46). Keplerian mass is the one measured by K 3 the orbiting particles in the accretion disk or the one responsible for the K = 27M6 − 45a2 M4 − 8a3 M3 + 17a4 M2 + 8a5 M + a6. echoes [3], both being strong ﬁeld phenomena. On the other hand, the ADM mass is measured by an asymptotic static observer and is given by The reason of coincidence of kinematic properties is that M = M + λ2 [61]: ADM 1 . While the ADM mass is not relevant for λ studies in accretion or echoes, the existence of two distinct masses here parameter appears only in the grr component of metric reveal that the RDSWH possesses characteristics of Machian Brans– which is not used in Eqs. (16)–(19). Dicke solutions, where the Keplerian mass is distinct from the ADM We shall consider a stellar sized wormhole of mass 15M λ mass. It would thus be rewarding to relate to some kind of scalar ﬁeld as a central compact object with an accretion rate M˙ ∼ and develop a scalar-tensor theory for which RDSWH could be an exact 0 19 −1 λ solution. We thank the anonymous reviewer whose query has led us to 10 gs and study the effect of different values of in think of this interesting possibility to be pursued in the future. the spacetime described by RDSWH solution (20)–(21). We 123 Eur. Phys. J. C (2019) 79 :952 Page 5 of 9 952

Table 1 The maximum values of the time averaged radiation ﬂux F(r), The critical values of radius rcrit and of frequency νcrit where the corre- temperature distribution T (r) with corresponding critical radii and the sponding maxima occur is shown in the columns 5 and 7 respectively emission spectra with corresponding critical frequencies for RDSWH.

−1 −2 14 7 30 15 a λ Fmax(r) [erg s cm ]×10 Tmax(r) [K] rcrit [cm]×10 νL(ν)max [erg]×10 νcrit [Hz]×10

006.170 57435 1.893 4.888 2.928 0.16.161 57414 1.893 4.883 2.927 0.55.936 56881 1.901 4.788 2.902 15.180 54976 1.934 4.469 2.813 0.75 0 1.014 × 102 115630 0.968 1.791 × 101 5.046 0.11.009 × 102 115517 0.968 1.788 × 101 5.042 0.59.108 × 101 112577 0.982 1.703 × 101 4.942 15.568 × 101 99548 1.100 1.364 × 101 4.542 0.95 0 1.012 × 103 205566 0.569 1.791 × 101 5.046 0.11.002 × 103 205025 0.570 1.788 × 101 5.042 0.57.264 × 102 189184 0.607 1.703 × 101 4.942 11.657 × 102 99548 0.946 1.364 × 101 4.542 0.998 0 1.265 × 104 386466 0.331 1.108 × 102 8.527 0.11.192 × 104 380761 0.334 1.093 × 102 8.513 0.52.692 × 103 262502 0.494 7.059 × 101 8.165 12.518 × 102 145149 0.919 2.813 × 101 6.187

Table 2 The rms and the λ = 0. Choice a = 0.75 is expected from the collapse a rms [M] efﬁciency for RDSWH. They of a maximally rotating polytropic star [63]. Another highly are the same as those of Kerr . . 0600 0 057 probable value of spin parameter is a = 0.95, which comes BH. The general relativistic 0.34.98 0.069 = Schwarzschild black hole from different observational methods [64,65]. Case a corresponds to a = 0. Here 0.73.39 0.104 0.998 is a maximally allowed value coming from the Page– a = a/M 0.92 2.18 0.205 Thorne limit for Kerr BH [30]. 0.95 1.94 0.228 0.998 1.24 0.321 5 Numerical estimates

Table 3 Difference of the − − λΔλ[erg s 1 cm 2] maxima of fluxes of radiation We present below three tables showing observable charac- between RDSWH and Kerr BH, 10−5 2.54 × 105 teristics of the thin disk (Table 1), the minimum stable radius using a = 0.7 10−6 2.42 × 103 and the accretion efficiency (Table 2) and the difference of − the maxima of flux of radiation between RDSWH and Kerr 10 7 28 BH (Table 3). 10−8 0 Figure 2a–d display the flux of radiation F(r) emitted by 10−9 0 the disk between r and received at an arbitrary radius r −10 ms 10 0 away from the center of the disk (Eq. (7)). Figure 3a–d show variation of temperature over the disk from rms to an arbitrary radius and Fig. 4a–d show observed luminosity variations should ensure that rms > r+, which is a necessary condition over different frequency ranges. The peaks values of emis- so that the accreting particles do not reach the throat and sivity properties of accretion disk are given in the Table 1. disappear into the other universe even for extreme limit a = In Table 2, we show the variation of marginally stable orbits 1. A special feature of the RDSWH is that it does satisfy this rms and with the spin parameter a in the range used in condition for λ<10−3, the strong field lensing constraint the previous plots. The conversion efficiency of RDSWH is for mimicking BHs [15](seeFig.1). independent of λ, so is identical with that of Kerr BH. For illustration, we will consider different values of spin From Fig. 2a–d we see that the flux emitted from the disk parameter a. Choice a = 0 corresponds to static case of around Kerr BH always bigger than RDSWH depending on DSWH, which in turn goes to Schwarzschild solution when a. To show the difference of profiles we will use their max- 123 952 Page 6 of 9 Eur. Phys. J. C (2019) 79 :952

Fig. 2 ThetimeaveragedfluxF(r) as a function of the radial coordinate r (in cm) radiated by the disk for a RDSWH plotted for different values of λ andcomparedwiththeKerrBH ima given in Table 1. With λ ≤ 0.1 flux of the accretion disk Recently, by Bueno et al. [3] have considered small values − − is practically indistinguishable from Kerr BH. Only limit- λ = 10 5, 10 10 and a = 0.7, to analyze how the quasi- ing case a = 0.998 shows the difference of fluxes up to normal modes of the RDSWH differ from those of the Kerr 6 %. With the increase of the spin parameter the difference BH. If we consider the accretion disk properties, we can see, of fluxes of Kerr BH and RDSWH is also increasing. The that only flux of radiation of wormhole is differ from BH. most significant difference appears when λ = 1, here we see Others, temperature and emission spectra indistinguishable that maximum value of flux of the wormhole is only 2 % then λ = 10−5, 10−10. of maximum value of Kerr BH flux. As r →∞the fluxes In Table 3 below is shown the difference Δλ in the maxima of RDSWH and Kerr BH become indistinguishable. Another of flux of radiation between RDSWH and Kerr BH: interesting feature of RDSWH is the shift of maxima of fluxes Δλ = − . with the increase of λ and spin parameter a. Fmax λ=0 Fmax λ =0 (28) Similar characteristics appear in the disk temperature profiles, depicted in Fig. 3a–d. For all values of the spin parameter the disks rotating around the Kerr black holes are hot- 6 Conclusions ter than those around the wormhole. With the increase of parameter λ the temperature of the disk around rotating WH Accretion around spinning wormholes is an active area of decreases. current research. Along this line, gravitational wave echoes In Fig. 4a–d, we display the disk spectra for the spinning of RDSWH have been recently studied by Bueno et al. [3]. wormholes compared to Kerr BH. The cut-off frequencies for We studied in this work the thin accretion disk profiles of the the Kerr black hole are systematically higher than those for same wormhole using the steady-state Page–Thorne model. the wormholes. The biggest deviations in radiation emitted Our aim was to examine how far the attractive positive mass by the accretion disk appears in ultraviolet range. Same as mouth of RDSWH mimicks profiles of the Kerr BH for small flux and temperature profiles the emissivity profile displays values of λ inspired by strong field lensing, which in general the most significant difference between RDSWH and Kerr is an excellent diagnostic for comparison between two cat- BH when a = 0.998 and λ = 1. egories of objects (BH and WH) with differing topology, 123 Eur. Phys. J. C (2019) 79 :952 Page 7 of 9 952

Fig. 3 Temperature distribution T (r) as a function of the radial coordinate r (in cm) of the accretion disk for a RDSWH plotted for different values of λ andcomparedwiththeKerrBH

see. e.g., [15,16]. We chose for illustration a rotating stellar increment of λ leads to an increasing shift in r = rcrit at sized wormhole as the accreting object and obtained certain which the peaks of Fmax(r), Tmax(r) decrease, while there is non-trivial generic characteristics of the disk. Specifically, a decreasing shift in the critical frequency νcrit at which the we show that RDSWH parameter λ has no influence on the peak of νL(ν)max decreases. On the other hand, for a fixed kinematic profiles of the disc. That is, the accretion efficiency λ, an increment of a leads to a decreasing shift in r = rcrit , radius of the marginally stable orbit rms (Table 2), the spe- at which the peaks of Fmax(r), Tmax(r) increase, while there cific angular momentum L, the specific energy E, angular is an increasing shift in the critical frequency νcrit at which velocity Ω for a test particle in a circular equatorial orbit the peak of νL(ν)max increases. These are generic features around RDSWH are independent of λ. as variations of the wormhole mass and the rate of accre- However, differences appear in the emissivity properties tion within the RDSWH model preserve these characteristics. for higher values 0.1 <λ≤ 1 (say) and for the extreme Thus the behavior of the luminosity peak is generically quite spin a = 0.998. We numerically computed the time aver- opposite to each other in the indicated variations, which in aged flux F(r), temperature distribution T (r) and emission principle could be useful from the viewpoint of observations. spectra νL(ν) of the accretion disk for different λ for the Apart from luminosity, it follows that an estimate of same dimensionless spin parameter a. The relevant results the difference Δλ in the maxima of flux of radiation F(r) are displayed in Figs. 2a–d, 3a–d, and 4a–d for an accret- between RDSWH and Kerr BH shows non-zero values ing wormhole with assumed mass 15M and accretion rate (Table 3) but is unobservable at present for small values ˙ 19 −1 −3 M0 ∼ 10 gm.sec for λ = 0 (Kerr BH), λ = 0.1, λ<10 permitted by the strong lensing bound. Given the λ = 0.5 and λ = 1 (RDSWH) respectively. Specifically, fact that the accretion profiles are not known with great pre- we observe certain characteristics of the emissivity profile, cision, the overall conclusion is that RDSWH are experimen- viz., the maxima (peak) of the F(r), temperature distribu- tally indistinguishable from KBH by accretion characteristics tion T (r) and emission spectra νL(ν) shift in the following as of now. Nevertheless, microlensing perturbations to the manner as can be read off from Table 1:Forafixeda,an flux ratios of gravitationally lensed quasar images can con-

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Fig. 4 The emission spectra νL(ν) of the accretion disk for a RDSWH plotted for different values of λ and compared with the Kerr BH strain the temperature proﬁles of their accretion disks (con- 5. E. Berti, V. Cardoso, A.O. Starinets, Class. Quantum Gravit. 26, nected by Stefan-Boltzmann law), which could be a more 163001 (2009) accurate test for the accretion process [66] and may effec- 6. V.Cardoso, E. Franzin, P.Pani, Phys. Rev. Lett. 116, 171101 (2016) 117 λ 7. V. Cardoso, E. Franzin, P. Pani, Phys. Rev. Lett. (E), 089902 tively lead to a tighter constraint on . Work is underway. (2016) 8. M. Visser, Lorentzian wormholes-from Einstein To Hawking (AIP, Acknowledgements The reported study was funded by RFBR accord- New York, 1995) ing to the research project No. 18-32-00377. 9. V. Cardoso, P. Pani, Nat. Astron. 1, 586 (2017) 10. R.A. Konoplya, A. Zhidenko, JCAP 12, 043 (2016) Data Availability Statement This manuscript has no associated data or 11. H.G. Ellis, J. Math. Phys. 14, 104 (1973) the data will not be deposited. [Authors’ comment: This is a theoretical 12. H.G. Ellis, J. Math. Phys. 15, 520 (1974). (E) study and no experimental data has been listed.] 13. K.A. Bronnikov, Acta Phys. Polon. B 4, 251 (1973) 14. K.K. Nandi, R.N. Izmailov, A.A. Yanbekov, A.A. Shayakhmetov, Open Access This article is distributed under the terms of the Creative Phys.Rev.D95, 104011 (2017) Commons Attribution 4.0 International License (http://creativecomm 15. K.K. Nandi, R.N. Izmailov, E.R. Zhdanov, A. Bhattacharya, JCAP ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, 07, 027 (2018) and reproduction in any medium, provided you give appropriate credit 16. R.N. Izmailov, A. Bhattacharya, E.R. Zhdanov, A.A. Potapov, K.K. to the original author(s) and the source, provide a link to the Creative Nandi, Eur. Phys. J. Plus 134, 384 (2019) Commons license, and indicate if changes were made. 17. A.E. Broderick, R. Narayan, Class. Quantum Gravit. 24, 659 (2007) Funded by SCOAP3. 18. D. Torres, Nucl. Phys. B 626, 377 (2002) 19. H. Zhang, M. Zhou, C. Bambi, B. Kleihaus, J. Kunz, E. Radu, Phys. Rev. D 95, 104043 (2017) 20. N. Lin, Z. Li, J. Arthur, R. Asquith, C. Bambi, JCAP 09, 038 (2015) References 21. C. Bambi, E. Barausse, Phys. Rev. D 84, 084034 (2011) 22. C. Bambi, Astrophys. J. 761, 174 (2012) 1. T. Damour, S.N. Solodukhin, Phys. Rev. D 76, 024016 (2007) 23. C. Bambi, Phys. Lett. B 730, 59 (2014) 2. S.H. Völkel, K.D. Kokkotas, Class. Quantum Gravit. 35, 105018 24. E. Babichev, V. Dokuchaev, Yu. Eroshenko, Phys. Rev. Lett. 93, (2018) 021102 (2004) 3. P. Bueno, P.A. Cano, F. Goelen, T. Hertog, B. Vercnocke, Phys. 25. C.S.J. Pun, Z. Kovács, T. Harko, Phys. Rev. D 78, 084015 (2008) Rev. D 97, 024040 (2018) 26. C.S.J. Pun, Z. Kovács, T. Harko, Phys. Rev. D 78, 024043 (2008) 4. K. Jusuﬁ, A. Banerjee, G. Gyulchev, M. Amir, Eur. Phys. J. C 79, 28 (2019) 123 Eur. Phys. J. C (2019) 79 :952 Page 9 of 9 952

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