Linearization stability of reflection-asymmetric thin-shell wormholes with double shadows
aa r X i v : . [ g r- q c ] J a n Linearization stability of reflection-asymmetric thin-shell wormholes with doublephoton spheres
Naoki Tsukamoto ∗ Department of General Science and Education,National Institute of Technology,Hachinohe College, Aomori 039-1192, Japan
Wormholes are hypothetical objects which can be black hole mimickers with strong gravitationalfields. Recently, Wielgus et al. have discussed the shadows of reflection-asymmetric thin-shellwormholes which are composed of the parts of a Schwarzschild spacetime and a Reissner-Nordstromspacetime with double photon spheres in [M. Wielgus, J. Horak, F. Vincent, and M. Abramowicz,Phys. Rev. D , 084044 (2020)]. In this paper, we study the linearization stability of thereflection-asymmetric thin-shell wormholes with the double photon spheres.
I. INTRODUCTION
Recently, LIGO and VIRGO Collaborations have de-tected gravitational waves from black hole binaries [1]and Event Horizon Telescope Collaboration has reportedthe shadow image of a black hole candidate at the cen-ter of a giant elliptical galaxy M87 [2]. The theoreticaland observational aspects of compact objects in generalrelativity will be more important than before.Wormholes are hypothetical objects with a non-trivialtopology in general relativity [3, 4]. Morris and Thornehave discussed passability of wormholes and they havealso shown that energy conditions violates at least ata throat of static and spherically symmetric wormholesif we assume general relativity without a cosmologicalconstant [3]. Wormholes can be black hole mimickers be-cause they can have strong gravitational fields. For exam-ple, spherically symmetric wormholes with strong grav-itational fields can have unstable (stable) circular lightorbits named photon spheres (antiphoton spheres) [5–13].Thin-shell wormholes whose energy condition is bro-ken only at the throat which is supported by a thinshell [14–17] were considered by Visser [18] with Darmois-Israel matching [16, 17, 19]. Linearization stabilityof a thin shell of a Schwarzschild wormhole was in-vestigated by Poisson and Visser [20] and then stabil-ity of thin-shell wormholes such as Reissner-Nordstromwormholes [21–31], the other static and sphericallysymmetric wormholes [21–23, 32–42], plane symmet-ric wormholes [43], cylindrical symmetric wormhole [44,45] higher-dimensional wormholes [46–51], and lower-dimensional wormholes [52–56] wormholes in an expand-ing spacetime [57] have been discussed. Passability [58]and nonlinear stability [59] of thin-shell wormholes havebeen studied.Usually two copied manifolds are used to construct thethin-shell wormholes while reflection-asymmetric worm-holes also have been investigated in Refs. [32, 60, 61]. ∗ [email protected] Recently, a shadow of a reflection-asymmetric thin-shell wormhole which is composed of the parts of twoSchwarzschild spacetime with different masses were dis-cussed by Wang et al. [62]. Wielgus et al. have investi-gated a shadow by double photon spheres of reflection-asymmetric thin-shell wormholes which are composed ofthe parts of the Schwarzschild manifold and a Reissner-Nordstrom manifold under a constraint on the metrictensors not to jump at the throat [63]. It would be diffi-cult to distinguish a reflection-symmetric wormhole con-structed by the parts of black hole spacetimes from theblack hole by astronomical observations if the wormholehas photon spheres and if there are few light sources inthe other side of the throat. On the other hand, theasymmetric thin-shell wormholes with the asymmetricdouble photon spheres can be distinguished from theblack holes by the observations since light rays can bereflected by a potential wall near the throat [63].In this paper, we investigate the linearization stabilityof the reflection-asymmetric thin-shell wormholes whichare composed of the parts of the Schwarzschild spacetimeand the Reissner-Nordstrom spacetime with the doublephoton spheres constructed in [63].This paper is organized as follows. We review theReissner-Nordstrom spacetime very briefly in Sec. II andwe construct the reflection-asymmetric thin-shell worm-holes with the double photon spheres in Sec. III. Weinvestigate the linearization stability of the reflection-asymmetric thin-shell wormholes in Sec. IV and we con-clude our results in Sec. V. In this paper we use the unitsin which a light speed and Newton’s constant are unity.
II. REISSNER-NORDSTROM SPACETIME
The Reissner-Nordstrom spacetime has a line element ds = − f ( r ) dt + dr f ( r ) + r ( dθ + sin θdφ ) , (2.1)where f ( r ) is given by f ( r ) ≡ − Mr + Q r (2.2)and where M > Q are a mass and a charge, respec-tively. It is a black hole spacetime with an event horizonat r = r EH ≡ M + p M − Q for 0 ≤ Q ≤ M whileit has naked singularity for M < Q . There is a photonsphere at r = r PS ≡ M + p M − Q Q ≤ M / r = r APS ≡ M − p M − Q M ≤ Q ≤ M / III. REFLECTION-ASYMMETRIC THIN-SHELLWORMHOLE
By the Darmois-Israel matching [16, 17, 19], we con-struct a wormhole spacetime without reflection sym-metry or Z2 symmetry. We consider two Reissner-Nordstrom spacetimes with line elements ds ± = − f ± ( r ) dt ± + dr f ± ( r ) + r ( dθ + sin θdφ ) , (3.1)where f ± ( r ) are given by f ± ( r ) ≡ − M ± r + Q ± r . (3.2)Here, we have assumed that we can choose the same co-ordinates r , θ , and φ in the spacetimes. We make twomanifolds M ± ≡ { r > a } , where a is a constant satis-fying a > r EH ± by removing Ω ± ≡ { r ≤ a } from theReissner-Nordstrom spacetimes. The boundaries of themanifolds M ± are timelike hypersurfaces Σ ± ≡ { r = a } and we identify the hypersurfaces Σ ≡ Σ + = Σ − . Asa result, we obtain a manifold M by gluing the mani-folds M ± at a throat located at Σ. The hypersurface Σis filled with a Dirac distribution matter and it is calledthin shell. We permit a = a ( τ ), where τ is the propertime of the thin shell since we are interested in the sta-bility of the thin shell Σ.We assume that we can set the coordinates y i on theboth sides of the hypersurface Σ. The induced metric h ij ≡ g µν e µi e νj on the hypersurface Σ is given by ds = h ij dy i dy j = − dτ + a (cid:0) dθ + sin θdφ (cid:1) . (3.3)Here e µi are basis vectors given by e µi ≡ ∂x µ /∂y i . Weintroduce a bracket [ T ] denoting the jump of a function T across Σ, [ T ] ≡ T + | Σ − T − | Σ , (3.4)where T + and T − are T in M + and M − , respectively.The thin shell satisfies Einstein equations S ij = − π (cid:0)(cid:2) K ij (cid:3) − [ K ] δ ij (cid:1) , (3.5) where S ij is the surface stress-energy tensor of the thinshell given by S ij = ( σ + p ) U i U j + pδ ij , (3.6)where U i is given by U i dy i ≡ u µ e µi dy i = dτ , where u µ isthe four velocity of the thin shell, and where σ = − S ττ and p = S θθ = S φφ are the surface energy density and thesurface pressure of the thin shell, respectively. Here, K ij is the extrinsic curvature given by K ij = ( n µ,ν − Γ ρµν n ρ ) e µi e νj , (3.7)where n ν is the unit normal to the hypersurface andwhere Γ ρµν is Christoffel symbols,The four velocity of the thin shell at t ± = T ± ( τ ) and r = a ( τ ) is given by u µ ∂ µ = ˙ T ± ∂ t ± + ˙ a∂ r , where the over-dot is a differentiation with respect to τ and where ˙ T ± = p f ± + ˙ a /f ± . The unit normals to the hypersurfacein M ± are obtained as n µ ± dx µ ± = ± (cid:16) − ˙ adt ± + ˙ T ± dr (cid:17) and the basis vectors are given by e µτ ∂ µ = ˙ T ± ∂ t ± + ˙ a∂ r , e µθ ∂ µ = ∂ θ , and e µφ ∂ µ = ∂ φ . The extrinsic curvatures ofthe hypersurfaces in M ± are given by K ττ ± = ± p ˙ a + f ± (cid:18) ¨ a + f ′± (cid:19) , (3.8) K θθ ± = K φφ ± = ± p ˙ a + f ± a , (3.9)and the traces are K ± = ± p ˙ a + f ± (cid:18) ¨ a + f ′± (cid:19) ± a p ˙ a + f ± . (3.10)From ( τ, τ ) and ( θ, θ ) components of the Einstein equa-tions (3.5), we obtain σ = − p ˙ a + f + πa − p ˙ a + f − πa (3.11)and p = 18 π p ˙ a + f + (cid:18) ¨ a + ˙ a + f + a + f ′ + (cid:19) + 18 π p ˙ a + f − (cid:18) ¨ a + ˙ a + f − a + f ′− (cid:19) (3.12)and then we obtain, from Eqs. (3.11) and (3.12), d ( σ A ) dτ + p d A dτ = 0 , (3.13)where A ≡ πa is the area of the throat. Equation (3.13)can be expressed by aσ ′ + 2( σ + p ) = 0 , (3.14)where σ ′ = ˙ σ/ ˙ a and the prime denotes the differentiationwith respect to a . We assume that the thin shell is filledwith a barotropic fluid with p = p ( σ ). From Eq. (3.14),we notice that the surface density of the barotropic fluidis expressed as a function of a or σ = σ ( a ). The equationof motion of the thin shell is given by, from Eq. (3.11),˙ a + V ( a ) = 0 , (3.15)where V ( a ) is an effective potential defined by V ( a ) ≡ ¯ f − (cid:18) ∆4 πaσ (cid:19) − (2 πaσ ) , (3.16)where ¯ f and ∆ are given by¯ f ≡ f − + f + , (3.17)and ∆ ≡ f + − f − , (3.18)respectively. The derivative of V with respect to a isobtained as V ′ = ¯ f ′ − ∆ [∆ ′ aσ − ∆( σ + aσ ′ )]8 π a σ − π aσ ( σ + aσ ′ )(3.19)and, from Eq. (3.14), it can be rewritten as V ′ = ¯ f ′ − ∆ [∆ ′ aσ + ∆( σ + 2 p )]8 π a σ +8 π aσ ( σ +2 p ) . (3.20)By using Eq. (3.14) again, the second derivative of V isobtained as V ′′ = ¯ f ′′ − ∆ ′ π a σ − ∆8 π a σ [4∆ ′ ( σ + 2 p ) aσ +∆ ′′ a σ − σ ( σ + p )(1 + 2 β ) + 3∆( σ + 2 p ) (cid:3) − π (cid:2) ( σ + 2 p ) + 2 σ ( σ + p )(1 + 2 β ) (cid:3) (3.21)where β ≡ dp/dσ = p ′ /σ ′ .Here and hereafter, we impose a constraint f + ( a ) = f − ( a ) (3.22)and we concentrate on the case that the manifold M − isthe part of the Schwarzschild black hole spacetime, i.e., Q − = 0, as well as Ref. [63]. The constraint is expressedas Q = 2 aM − ( ξ − , (3.23)where ξ is an asymmetry parameter defined by ξ ≡ M + /M − . The reflection-asymmetric thin-shell worm-holes with double photon spheres must have the throatin domains r EH − < a < r PS − and r EH+ < a < r
PS+ .Permitted parameters ( ξ, a/M − ) for the reflectional-asymmetry thin-shell wormholes with double photonspheres are shown in Fig. 1. Q + = M + (cid:1) aM - FIG. 1. Permitted parameters ( ξ, a/M − ) of the asymme-try wormholes with the double photon spheres are in shadedzones. M + is the part of the Reissner-Nordstrom black hole(BH) spacetime in a deep blue shaded zone while it is thepart of the Reissner-Nordstrom naked singularity (NS) space-time in light blue shaded zones. M − is the subset of theSchwarzschild black hole spacetime. A blue dotted line de-notes a = r PS+ . A red dashed line is explained in Sec. IV.
IV. STABILITY OF THIN-SHELL WORMHOLE
We consider the linearization stability of a static worm-hole with a thin shell at a = a under the constraint(3.22) and Q − = 0. The surface energy density σ andpressure p of the thin shell are given by σ = − √ f πa (4.1)and p = 18 π √ f (cid:18) f a + ¯ f ′ (cid:19) , (4.2)respectively. Here and hereafter a function with sub-script 0 means the function at a = a . Since V = V ′ =0 is satisfied, the effective potential can be expandedaround a = a as V ( a ) = V ′′ a − a ) + O (cid:16) ( a − a ) (cid:17) , (4.3)where V ′′ is given by V ′′ = A − B (cid:0) β (cid:1) , (4.4)where A and B are defined by A ≡ ¯ f ′′ − ∆ ′ f − ¯ f ′ f (4.5)and B ≡ f a − ¯ f ′ a , (4.6)respectively. The thin shell is stable (unstable) when V ′′ > V ′′ < β < (cid:18) A B − (cid:19) (4.7)and B > β > (cid:18) A B − (cid:19) (4.8)and B < ξ, a /M − ) for B > B < B = 0. The boundary of the two zones B = ξ a M - FIG. 2. B < B > a /M − = (7 − ξ ) / √ / ≤ ξ ≤ √ / B = 0. B = 0 is given by a /M − = (7 − ξ ) / √ ≤ ξ ≤ √ . (4.9)On the boundary, the thin shell is unstable for any β .The parameters ( a /M − , β ) for the stable thin shell areshown in Fig. 3. V. CONCLUSION
We have investigated the linearization stability ofthe reflection-asymmetric thin-shell wormhole which iscomposed of parts of the Schwarzschild and Reissner-Nordstrom manifolds with double photon spheres. Wehave imposed the constraint (3.22) as well as Ref. [63].The linearization stability given by Fig. 3 are character-ized by the boundary B = 0 or a /M − = (7 − ξ ) / √ / ≤ ξ ≤ √ / B = 0, the throat with a /M − = 2 .
25 is unstable for any β as shown the leftbottom panel with ξ = 2 . a /M − = r EH − /M − = 2 for 1 ≤ ξ ≤ β as shown the panels in Fig. 3.We comment on the Schwarzschild thin-shell wormholewith reflection symmetry or ξ = 1 .
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