Journal of the Korean Physical Society, Vol. 61, No. 8, October 2012, pp. 1181∼1186
Wormhole Accretion onto a Black Hole Sung-Won Kim∗ Department of Science Education, Ewha Womans University, Seoul 120-750, Korea (Received 23 August 2012, in final form 14 September 2012) In this paper, the process of wormhole accretion onto a black hole is studied. We adopt the thin-shell wormhole for the wormhole model, as well as spherical collapse around the black hole and outer axial accretion for the accretion process model. The critical distance of wormhole breaking was roughly estimated by balancing the black-hole tidal force and the force caused by the wormhole tension. PACS numbers: 04.70.Bw, 97.10.Gz Keywords: Wormhole accretion, Thin-shell wormhole, Spherical collapse, Axial accretion DOI: 10.3938/jkps.61.1181
I. MOTIVATION
1. Morris-Thorne Wormhole
The black hole and the wormhole have very close common points and quite different opposite points. They are astrophysical objects that satisfy the Einstein equation. One can travel through the wormhole both ways while one can travel only one way through the black hole. The interaction issues of a wormhole and a black hole with the stability problem are very interesting and arouse great interest. We wondered what would happen to a wormhole when it meets a black hole. In this respect, information about problems of stability and accretion is needed. Recently, meaningful results on dark-energy accretion onto a black hole have been obtained [1]. The mass change rate was founded to be determined by the equation-of-state parameter, and phantom energy accretion cause a decrease in the black-hole mass. Thus, if wormhole pieces are accreted onto a black hole, then these consequences are useful of the process and the final state of the system are to be considered.
II. WORMHOLE MODELS Generally, a wormhole is defined as a bridge connecting two asymptotically-flat spacetimes (cosmological spacetimes are possibly allowed instead of flat spacetimes in a wormhole model in the Friedmann-Robertson-Walker (FRW) universe [2]). Here, two simple examples of the wormhole model are introduced: the Morris-Thorne type and thin-shell delta-function type.
The Morris-Thorne type wormhole [3] is static and spherically symmetric, and its spacetime metric is given by ds2 = −e2Φ(r) dt2 +
dr2 + r2 dΩ2 . 1 − b(r)/r
(1)
Here, Φ is the lapse function, and b is the wormhole shape function. The function Φ is everywhere finite so that there is no horizon. If the wormhole structure is to be maintained, the flare-out condition d2 r b − b r = >0 dz 2 2b2
(2)
is required at or near the throat, r = b. The exoticity function ζ should be positive at or near the throat by the flare-out condition ζ≡
τ −ρ = |ρ|
b r
− b − 2(r − b)Φ > 0, |b |
(3)
where τ is the tension and ρ is the energy density. The flare-out condition implies sustainability of the wormhole’s shape and the exoticity of the wormhole-consisting matter. The matter is very exotic in that tension is large compared to the energy density.
2. Thin-shell Wormhole
The second example of a wormhole is a delta-function– type wormhole. The exotic matter is assumed to be distributed at the throat only, such that ∗ E-mail:
ρ = ρ0 δ(r − b0 ),
[email protected]
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τ = τ0 δ(r − b0 ),
(4)
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Journal of the Korean Physical Society, Vol. 61, No. 8, October 2012
and the shape function is constant as ∞ b= 8πρr2 dr = 8πb20 ρ0 = const = b0 .
The conservation of stress-energy is (5)
b0
The thin-shell wormhole consists of the matter at the throat, and two flat spaces so that the energy-momentum tensor is given as + − Tµν = δ(η)Sµν + Θ(η)Tµν + Θ(−η)Tµν ,
(6) ±
where Sµν is the surface stress energy and T are the matters on both sides of the throat. If we take the covariant derivative of Eq. (6), we can get the energymomentum conservation law. By adopting an orthonormal frame specially adapted to the surface in the spherically symmetric case, one may put the normal in the form nµˆ = (0, 0, 0, 1), ⎛
Sµˆνˆ
⎞ σ 0 0 0 ⎜ 0 −ϑ 0 0 ⎟ =⎝ , 0 0 −ϑ 0 ⎠ 0 0 0 0
⎛
hµˆνˆ
−1 ⎜ 0 =⎝ 0 0
0 +1 0 0
0 0 +1 0
⎞ 0 0⎟ , 0⎠ 0 (7)
where σ is the surface energy density and ϑ is the principal surface tension (negative pressure). Visser suggested the thin-shell wormhole by taking two Schwarzschild spacetimes instead of two flat spaces in a delta-function–type wormhole [4]. It is the model given by the cut of two copies of Schwarzschild black holes just outside the horizon that are then pasted as in a surgical procedure. The surface energy density and the surface tension are given as 1 2M 2M 1 , ϑ=− . (8) 1− 1− σ=− 2πb b 4πb b For the dynamical case, the unique variable b should be b(τ ), where τ is the proper time as measured by a comoving observer on the wormhole’s throat. The position of the throat is xµ (τ, θ, φ) ≡ (t(τ ), b(τ ), θ, φ). The four-velocity is ⎛ ⎞
˙2 1 − 2M/b + b dt db ˙ 0, 0⎠ , (9) , , 0, 0 = ⎝ , b, uµ = dτ dτ 1 − 2M/b with unit normal to the throat
˙ b 2M , 1− + b˙ 2 , 0, 0 . ξµ = 1 − 2M/b b Thus, the Einstein field equation becomes 2M 1 + b˙ 2 , 1− σ = − 2πb b 1 1 − M/b + b˙ 2 + b¨b ϑ = − . 4πb 1 − 2M/b + b˙ 2
(10)
(11)
σ˙ = −2(σ − ϑ)
b˙ b
or
D d (σb2 ) = ϑ (b2 ). Dτ dτ
(12)
III. CLOSE ENCOUNTER SCENARIO We want to know the process and the result of the wormhole and the black hole getting close to each other. First, the relative sizes of the two objects are very important before a detailed analysis can be performed.
1. Sizes of Two Objects
When one of the two objects is very large and the other is extremely small, the problem is rather easy. By intuitive analysis, the smaller one would be treated as the perturbed one not to give any significant effect on the larger one, and it is treated as the test particle for the gravitational field of the larger one. When the large wormhole approaches and meets the black hole, the tiny black hole travels through the wormhole’s throat without any serious perturbation to the wormhole. When a very small wormhole meets a large black hole, the wormhole will be absorbed into the black hole. What happens to the wormhole if a comparable-size wormhole meets the black hole? Comparable size is defined from the wormhole’s throat size and the black hole’s horizon size. Because the thin-shell wormhole mass is calculated from σ as M ∼ σdA, (13) the relation that makes a wormhole comparable to a black hole is just b ∼ RS , where b is the wormhole’s throat size and RS is the Schwarzschild radius of the black hole.
2. Stability
The geometry will significantly change when the wormhole approaches the black hole, which may modify both the tension of the wormhole and the tidal force of the black hole. Here, we assume that the issue of the instability of the wormhole caused by the tidal force of the black hole will be dominant. Intuitively, the instability of the wormhole triggers its being torn in pieces. Even though the wormhole’s shape is maintained by the tension, in the near region the tidal force dominates the tension force and tears the wormhole in pieces, and these pieces will accrete onto the black hole.
Wormhole Accretion onto a Black Hole – Sung-Won Kim
Many issues are related to the stability problem of the wormhole. A very important one is the close approach of the wormhole and the black hole. Because a wormhole has no guarantee of stability, it can be easily broken and torn by any external tidal forces. For the case of the Schwarzschild-type thin-shell wormhole, the stability of the dynamic wormhole is decided with the potential problem from Einstein’s equation, Eq. (11) by using 2M − (2πσb)2 = −1 b˙ − b
∆aj = −Rˆiˆjˆipˆξ pˆ.
thin-shell model of delta-type–distribution, static, spherical shell with σ = −1/2πb, ϑ = −1/4πb,
(18)
which are from Eq. (8) with the limit of vanishing M . The throat position is xµ (τ, θ, φ) ≡ (t, b(τ ), θ, φ). We need to estimate the distance where the tension force is the same as the tidal force in the azimuthal direction. The matters are represented as σ=−
1 c2 , 2πa G
ϑ=−
1 c4 , 4πb G
d = 2b
Rtˆrˆtˆrˆ and Rtˆϕˆtˆϕˆ .
4πb2 σ ·
MG 1 c4 · · 2πb or · 2b ∼ r3 4πb G
∆aϕ = −
GM d, r3
(17)
where and d are the separations for the radial and the azimuthal directions, respectively. They all increase as the distances diminish. We can treat this problem for two cases: collapse and disruption (see Fig.˜ 1). The azimuthal tidal force is useful for collapse, and the radial tidal force is useful for disruption.
r03 =
G 16b2 M, c2 (20)
where the wormhole’s effect on the black hole is neglected. For the special case of comparable sizes such 2 as 2M ∼ b cG , the critical position should be
(16)
There are two kinds of tidal accelerations, radial and azimuthal tidal accelerations:
(19)
with fundamental constants by using dimensional adjustments. In a simple setting, the tidal force is roughly the same as the force caused by the surface tension:
(15)
In the case of a Schwarzschild black hole, the nonvanishing components of the Riemann tensor are
2GM , r3
Fig. 1. (Color online) (a) Collapsing model of a wormhole around a black hole. (b) Wormhole accretion onto a black hole
(14)
with the energy E = −1. The Schwarzschild-type thinshell wormhole is everywhere unstable. Thus, the deltafunction–type thin-shell wormhole is also unstable in the limit of vanishing M . Next, we can think further about the close-encounter scenario. Before they meet, the wormhole structure is still maintained, because the tension of the wormhole is very strong. When they get closer, the tidal forces caused by the black hole will affect the wormhole’s structure. At the critical distance r0 , the tidal force caused by the black hole is similar to the tension of the wormhole. We can estimate the critical distance r0 from a simple model comparison. Because the delta-function–type thin-shell wormhole is unstable, the tidal force perturbs the wormhole, and the wormhole will start being torn into pieces; finally, those pieces will accrete onto the black hole. The tidal acceleration by the black hole is
∆ar =
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r0 = 2M = b,
(21)
which means that the wormhole collapses and will be torn at r0 = b. Here, the fundamental constants are omitted. Just before the wormhole collapses, the four-velocity of the throat is u0 = 1, u = b˙ = 0.
(22)
At the onset of collapse, b˙ is decided by the gravitational attraction of the black hole as ˙ u0 = 1 + b˙ 2 , u = b. (23) From the geodesic equation,
3. Collapse
For the case of radial collapse, we assume that the black hole is in the center of the wormhole and that the system is in quasi-equilibrium state before collapse so that there is no effect on the wormhole. We consider the
α β d2 xµ µ dx dx = −Γ αβ dτ 2 dτ dτ
or 2 ¨b = d b = −Γ1 dt dt − Γ1 db db . 00 11 dτ 2 dτ dτ dτ dτ
(24)
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as 2M ∼ b cG , the critical position in natural units becomes r0 = 2b.
Fig. 2. Graph of b˙ in terms of τ when r > M and r < M . τ0 is the time of the velocity transition, and τ1 is the divergent time. The b in r < M collapses with more acceleration than it does in r > M .
After integration, the velocity of the wormhole throat is ⎧ ⎪ β ⎪ ⎨ α tan( |αβ|τ + C), r < M, (25) b˙ = ⎪ β ⎪ ⎩ α tanh( |αβ|τ + C), r > M, −1
where α = 4M 2 /r3 (M/r − 1) (1 − 2M /r) and β = M/r2 (1 − 2M /r) for the Schwarzschild black-hole case. The radial velocity of the wormhole throat is shown in Fig. 2. If we connect two stages of the throat velocity at r = M , the type of the velocity increase changes. When it collapses, the velocity of throat shows a hypertangent–type increasing nature in r > M , and it changes into a tangent type at τ = τ0 . This result means that the collapse time rate increase faster than before in r < M . Finally, b˙ is divergent when τ1 = π/(2 |αβ|).
(27)
The closer they are, the larger the tidal forces are. Because the tidal forces surmount the tension, the wormhole tension cannot maintain the wormhole’s shape any more. Finally, the wormhole will be torn into pieces, and the pieces form small wormholes because each piece still has enough exotic matter properties to form a wormhole. The small wormholes are absorbed onto the black hole without special restrictions. In these processes, the gravitational effect of the wormhole on the black hole is neglected. We will review the recent result of the sphericallysymmetric accretion model for a black hole to see the final state of the black hole. The accretion model of the static spherically symmetric case was suggested by Michel [5] and was recently developed by Babichev et al [1]: ds2 = eν dt2 − eλ dr2 − r2 dΩ2 .
For the static spherically-symmetric case, the radial flow onto the black hole adopts the four-velocity as uµ = (u0 , u, 0, 0),
When the wormhole approaches the black hole from the outside, the radial tidal force will dominate the force caused by wormhole tension. Thus, the critical distance, with fundamental constants, will be given as 2M G 1 c4 · · 2πb · 2b ∼ 3 r 4πb G
Conservation of the energy-momentum tensor is given by T0 µ :µ = 0 or
√ d (T0 1 −g) = 0 or dr
√ T0 1 −g = C1 . (31)
(ρ + P )uν :ν + ρ,ν uν = 0.
(32)
A T0 1 = (ρ + P )u0 u = − √ (ρ∞ + P (ρ∞ )). −g
(33)
The mass flow term is
Thus, the black-hole’s mass change rate is A M˙ = −4πr2 T0 1 = 4πr2 √ (ρ∞ + P (ρ∞ )) −g
or G 16a2 M, c2
(29)
In case of cosmology, the co-moving coordinate uµ = δ0µ is used in general while u0 here is given by (30) u0 = (1 − u2 g11 )/g 00 .
uµ T µν :ν = 0 or
r03 =
uµ uµ = 1.
Here, C1 is considered as a constant independent of r. The projection of the energy-momentum conservation law on uµ is also given by
4. Accretion
4πa2 σ ·
(28)
(26)
where the wormhole’s effect on the black hole is neglected. For the special case of comparable sizes such
= 4πAM 2 (1 + ω)ρ, where ω is the equation-of-state parameter. If the accreted matter is a phantom energy, (1+ω) < −1, M˙ < 0; i.e., the black-hole mass decreases.
Wormhole Accretion onto a Black Hole – Sung-Won Kim
In the case of a disruption process, the wormhole is disrupted, it will be torn to pieces, and the pieces will accrete onto the black hole. The energy-momentum tensor of the thin-shell wormhole is Tµν = (ρ+P )uµ uν −P gµν = [(σ−ϑ)uµ uν +ϑhµν ]δ(r−b(τ )). (34) For the delta-function–type thin-shell wormhole, the matters are given as σ = 2ϑ = −2P < 0
or
ρ + P < 0,
(35)
which are the same as those in the case of phantom energy accretion onto a black hole. This result is the same in the case of collapse-accretion. Therefore, when a wormhole is accreted onto the black hole, the mass of the black hole decreases, as in the phantom energy accretion case. We can extend the accretion problem to the slowlyrotating Kerr black-hole case. The slowly-rotating Kerr spacetime is given as
−1 2M 2M 2 2 ds = 1 − dt − 1 − dr2 r r
4J 2 2 2 2 2 −r dθ − r sin θ dφ − 3 dtdφ r with the small angular momentum J M, a = J/M 1. The four-velocity becomes uµ = (u0 , ur , uθ , uφ ),
uθ = 0, uφ 1
IV. AFTER ACCRETION As we studied before, the black-hole’s mass will decrease after the accretion of the wormhole. This is similar to the case of phantom energy accretion, Eq. (34). When the accreted dark energy is the quintessence, the pressure and the energy density are P =
1 2 ϕ˙ − V (ϕ), 2
ρ=
1 2 ϕ˙ + V (ϕ). 2
(40)
Thus, P + ρ = ϕ˙ 2 and the rate of change of the mass is M˙ ∝ ϕ˙ 2 . In the ghost field case, the negative kinetic part (ϕ˙ 2 → −ϕ˙ 2 ) shows a negative rate of change for the black-hole’s mass as M˙ < 0, which is consistent with the phantom energy case. In related research on the interaction between the black hole and the wormhole, Frolov and Novikov [6] introduced the wormhole straw to see the internal structure of the black hole. It is a kind of the extreme case because they used a very small wormhole. We should stress that, nevertheless, no fundamental theorems concerning black-hole physics are violated. The study of the black-hole’s interior practically does not change its gravitational field. Another example is the conversion between the black hole and the wormhole by putting a ghost field onto a black hole [7]. The final state of the black hole is the wormhole or Minkowski space. Therefore, we will conclude that the black hole will evaporate after the full accretion of a comparable size wormhole.
(36)
and u0 = g 00 u0 + g 03 u3 ,
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u3 = g 33 u3 + g 30 u0 .
(37)
Thus, from the relation uµ uµ = 1, 1 = u0 u0 + u1 u1 + u3 (g33 u3 + g30 u0 ) (g 00 u0 + g 03 u3 )u0 + g11 u2 + g30 u3 u0 or 1 g 00 u20 + 2g 03 u0 u3 + g11 u2 .
(38)
The mass change rate M˙ will be M˙ = −4πr2 T0 1 = −4πr2 (ρ + P )u0 u A = 4πr2 √ (ρ∞ + P (ρ∞ )), −g
V. SUMMARY We studied wormhole models, stability problems, and accretion problems for dealing with the interaction between a black hole and a wormhole. Thus the scenario is as follows: wormhole approach, breaking and tearing into pieces due to instability of the wormhole, accretion of its fragments, and black-hole mass decrease. Finally, the black hole will evaporate when the exotic matter of the wormhole is accreted fully. Here, we only considered a thin-shell–type wormhole. For further research, we will study other models of the wormhole and extend the analysis to more general cases, such as the axisymmetric case, for black holes’ numerical values.
where √
−g =
(r − 2M )r4 + 4J 2 r sin2 θ sin2 θ (r − 2M )
ACKNOWLEDGMENTS
1/2 . (39)
Therefore, the (ρ + P ) property does not change M˙ like √ the Schwarzschild case, though the form of −g is rather complicated. That is, even when the phantom energy or the wormhole is accreted onto a slowly-rotating Kerr black hole, the mass of the black hole will decrease.
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0013054).
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REFERENCES [1] E. Babichev, V. Dokuchaev and Yu Eroshenko, Phys. Rev. Lett. 93, 021102 (2004). [2] S.-W. Kim, Phys. Rev. D 53, 6889 (1996). [3] M. Morris and K. S. Thorne, Am. J. Phys. 56, 395 (1988).
[4] [5] [6] [7]
M. Visser, Nucl. Phys. B 328, 203 (1989). F. C. Michel, Astrophys. Space Sci. 15, 153 (1972). V. Frolov and I. Novikov, Phys. Rev. D 48, 1607 (1993). S. A. Hayward, S.-W. Kim and H. Lee, Phys. Rev. D 65, 064003 (2002).
