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Accretion Disk Around the Rotating Damour–Solodukhin Wormhole

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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 worm- hole (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 ring- ing 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 prop- erties 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.
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