Trapped ghost wormholes and regular black holes. The stability problem

Transcription

1 Trapped ghost wormholes and regular black holes. The stability problem Kirill Bronnikov in collab. with Sergei Bolokhov, Arislan Makhmudov, Milena Skvortsova (VNIIMS, Moscow; RUDN University, Moscow; MEPhI, Moscow) Modern Physics of Compact Stars and Relativistic Gravity 2017 Yerevan, K. Bronnikov, S. Bolokhov, A. Makhmudov, M. Skvortsova Trapped ghost wormholes and regular black holes. The stability problem

2 Motivation and plan Singularity problem in GR and other classical theories of gravity. Black hole singularities, their avoidance: a regular center or no center at all. The notions of a wormhole and a black universe. The necessity of exotic matter for both wormholes and black universes if considered in GR. In particular, such a source is a phantom (ghost) scalar field (more or less favored by cosmol. observations). Ghosts are predicted by some theories (e.g. multidimensional) but not observed around us. Possible explanation: they exist only in strong field regions trapped ghosts, like a jenie in a bottle [KB, S. Sushkov, 2010]. Explicit examples of trapped-ghost wormholes and black universes. Stability under radial perturbations: gauge-invariant effective potentials, generic poles at throats and transition surfaces from a usual scalar field to a ghost. Poles at throats: regularization and known instability results. Poles at transition surfaces to ghosts stabilizing properties due to boundary conditions.

3 What is a wormhole?

4 Kip Thorne on wormholes

5 Wormholes 2

6 Wormholes 3

7 Wormholes 4

8 Wormholes 5

9 Wormholes 6

10 Wormholes 7

11 Wormholes 8

12 Wormholes 9

13 What is a black universe? A black universe (BU) is a regular black hole where, beyond the horizon, instead of a singularity there is an expanding, asymptotically isotropic, e.g., ds space-time (K.B., J. Fabris, 2005; K.B. et al., 2006). For a spherically symmetric metric ds 2 = A(x)dt 2 dx 2 /A(x) r 2 (x)(dθ 2 + sin 2 θdϕ 2 ), A 1 and r as x +, A r 2, so that B(x) = A/r 2 B 0 < 0 as x.

14 Black universe 2 A black universe combines the properties of the following objects: A black hole (BH) a Killing horizon separating static and non-static spacetime regions; A wormhole (WH) no center and a regular minimum of the area of coordinate spheres A nonsingular cosmological model at large times the nonstatic region reaches a de Sitter (ds) mode of isotropic expansion Black universes, having no center at all, make an alternative to widely discussed BH models with a regular center, for example, with a de Sitter core, and sources like nonlinear electrodynamics (e.g., Dymnikova, 1992; K.B., 2001)

15 Trapped ghosts: possible origin Example from multidimensional gravity [KB, R. Konoplich, S. Rubin, CQG 24, 1261 (2007)]: S = d D x g D [F (R) + c 1R AB R AB + c 2R ABCD R ABCD ] converted after reduction to 4D theory in Einstein s picture to S d 4 x g 4[R 4 + K Ein( φ) 2 2V Ein]. With F (R) = R + cr 2 2Λ D and some particular parameter values:

16 Basic equations for trapped ghosts in 4D Action: S = m2 4 2 g [ ] R + 2h(φ)g µν µ φ ν φ 2V (φ) ; h(φ) > 0 for a canonical scalar field, h(φ) < 0 for a phantom one. Metric: ds 2 = A(x)dt 2 dx 2 A(x) r 2 (x)(dθ 2 + sin 2 θ ϕ 2 ). Equations: Scalar : 2(Ar 2 hφ ) Ar 2 h φ = r 2 dv /dφ, (1) R t t = : (A r 2 ) = 2r 2 V ; (2) R t t R x x = : r /r = h(φ)φ 2 ; (3) R t t R θ θ = : A(r 2 ) r 2 A = 2, (4) G x x = : 1 + A rr + Ar 2 = r 2 (haφ 2 V ), (5) Eq. (4) is once immediately integrated: B (x) (A/r 2 ) = 2(3m x)/r 4, B A/r 2. m = const. (6)

17 Wormholes and black universes with trapped ghosts: Conditions and choice of r(x) Conditions: < x <. Asympt. flatness or (A)dS: r(x) x, r(x) must have a minimum x ±. h(φ) < 0 near it. Trapped ghost: h(φ) > 0 far from that minimum. Wormhole: A > 0; A 1 (flat asympt.), A x 2 (AdS asympt.) Black universe: A 1 (x + ), A x 2, x. Example: choose r(x) = a (x/a)2 + 1, n = const > 2. For n = 3, a = 1: (x/a)2 + n 5 r(x), n=3 0.4 r^2 r''(x), n= r < 0 h < 0 for x <

18 Example with n = 3: solution for B(x) and V (x) m = -0,02 m = 0 1 [ B = B x 2 + 6x 4 6(1 + x 2 ) 3 ] + 3mx( x x 4 ) + 39m arctan x, B0 = const, 2 1 { V (x) = 32(6 x 2 3x 4 ) 12(1 + x 2 ) 2 (3 + x 2 ) 3 6mx( x x x x 8 ) } + 117m(1 + x 2 ) 2 ( x x 4 + 3x 6 )(π 2 arctan x). B(x), n= m = 0, m = m = 0 m = V(x) Plots of B(x) for different values of m: wormholes for m 0, black universes for m > 0 (with B 0 chosen to provide asymptotic flatness as x )

19 The stability problem: monopole perturbations Action: S gd ] x[ 4 R + 2h(φ)g αβ φ ;αφ ;β 2V (φ), Equations: 2h µ µφ + dv /dφ = 0, G ν µ = T ν µ [φ] = h(φ)[2φ µφ ν δ ν µφ α φ α] + δ ν µv (φ). Metric: ds 2 = e 2γ dt 2 e 2α du 2 e 2β dω 2, where γ, α, β, φ are now functions of both u (radial coordinate) and time t. Here u is an arbitrary coordinate [we previously used the coordinate condition α + γ = 0 and denoted u = x, e 2γ = A(x), e β = r(x).] Consider linear spherically symmetric perturbations of static solutions to the field equations, so that φ(u, t) = φ(u) + δφ(u, t), γ(u, t) = γ(u) + δγ and similarly for other quantities, with small deltas.

20 Linear perturbation equations 2 e 2α 2γ hδ φ 2h[δφ + δφ (γ + 2β α ) + φ (δγ + 2δβ δα )] 2δh[φ + φ (2β + γ α )] h δφ φ δh + δ( e 2α V φ ) = 0, (7) δ β + β δ β β δ α γ δ β = hφ δ φ, (8) δ( e 2α 2β ) + e 2α 2γ δ β δβ δβ (γ + 2β α ) β (δγ + 2δβ δα ) = δ( e 2α V ), (9) In this problem, we have two independent forms of arbitrariness: one consists in the freedom of choosing a radial coordinate u in the static configuration, the other is a perturbation gauge, connected with the freedom to choose a reference frame in perturbed space-time. This can be fixed in imposing a certain relation for δα, δβ, δγ, δφ. In what follows we employ both kinds of freedom. The above equations have been written in the most universal form, without fixing the u coordinate or the perturbation gauge.

21 Master equation for perturbations The only dynamic degree of freedom for radial perturbations is δφ. Thus: Choose the gauge δβ = 0 for convenience and the tortoise (wave) coordinate u z such that α = γ. Exclude δα, δγ from Eq. (7) using (8) and (9). Reduce the resulting wave equation for δφ to its canonical form using the substitution δφ = ψ(z, t) e η, η = β + h /(2h). The result is ψ ψ + V eff (z)ψ = 0, (10) [ V eff (z) = e 2α 2hφ 2 (V e 2β ) + 2φ β 2 β V φ h φ 4h V 2 φ + V ] φφ + β + β 2. 2h One can show that the final wave equation is gauge-invariant. A further substitution, possible since the background is static, ψ(x, t) = Y (x) e iωt, ω = const, leads to the Schrödinger-like equation Y + [ω 2 V eff (x)]y = 0. (11)

22 Physical conditions for perturbations Now, if there is a nontrivial solution to (11) with Im ω 0 satisfying some physically reasonable conditions at the ends of the range of z (finiteness of δφ, absence of ingoing waves), then the static system is unstable since δφ can exponentially grow with time (or linearly if Im ω = 0). Otherwise the static system is linearly stable. Thus, as usual in such studies, the stability problem is reduced to a boundary-value problem for the Schrödinger-like equation (11). Conditions: Regularity of δφ in the whole static region (the whole space for a wormhole, the domain of outer communication for a black universe). δφ 0 as z ±. z + : spatial infinity, z : another spatial infinity for a wormhole, the event horizon for a black universe.

23 Perturbations near the throat If the throat z = z 0 has a generic shape, that is, r(z) = r throat + O((z z 0) 2 ), then independently of model details V eff (z) = 2 + O(1). (12) (z z 0) 2 In a more general case, such that r(z) = r throat + O((z z 0) 2n ), V eff (z) = 2(2n 1) + O(1). (13) (z z 0) 2 It can seem that such a potential wall completely separates perturbations on different sides of the throat. But with such potentials we are unable to consider the most important mode: changes of the throat radius itself. Way out: Regularization of V eff (z) using the so-called S-transformation [C. Gundlach et al., 1995; J. Gonzalez at al., 2009]. However, it turns out that this method really regularizes V eff (z) only in the case n = 1, that is for the behavior (12) but does not work for other n, e.g., for long throats (n > 1).

24 S-transformation Consider a wave equation of the type (10) ψ ψ + W (z)ψ = 0, (14) with an arbitrary potential W (z) (its specific example is the above potential V eff ). Let us present W (z) in the form W (z) = S 2 (z) + S, a Riccati equation for S(z), so this is always possible. Then Eq. (14) can be rewritten as ψ + ( z + S)( z + S)ψ = 0. (15) Now, introduce the new function χ = ( z + S)ψ then, applying the operator z + S to Eq. (15), we obtain a wave equation for χ: χ χ + W 1(z)χ = 0, (16) with the new effective potential W 1(x) = S + S 2 = W (z) + 2S 2. (17) One can verify that if W = 2/z 2 + O(1), then W 1(z) is regular at z = 0, and all perturbation modes can be studied.

25 Some known stability results-1 Regularized effective potential for perturbations of some anti-fisher wormhole solution (with a massless phantom scalar) and the time-domain profile showing the instability. [J. Gonzalez et al., 2009; KB, J. Fabris, A. Zhidenko, 2011]

26 Some known stability results-2 Regularized effective potentials for perturbations of some black universe solutions with phantom scalars and the time-domain profiles. One of the models is stable: it is a black universe where the throat coincides with the horizon. [KB, R. Konoplya, A. Zhidenko, 2012]

27 Perturbations near a transition surface h = 0 Near a surface h = 0, say, z = 0, independently of model details, the effective potential behaves as V eff 1/(4z 2 ). (18) It is an infinitely deep potential well. In QM such a potential would lead to unlimitedly deep energy levels (like falling to a center ). Here: other physical conditions: instead of quadratic integrability, we require δφ < everywhere. With Eq. (18), the Schrödinger-like equation (11) gives, independently of ω, Y (z) = z (C 1 + C 2 ln z ) + O(z 3/2 ), C 1, C 2 = const. (19) Since δφ Y / h Y / x, we have δφ C 1 + C 2 ln z + O(z). Physical requirements C 2 = 0. It is a new constraint to be added to other boundary conditions the whole space splits into regions where the boundary-value problem should be solved separately. There is an instability if there are coinciding eigenvalues ω 2 0. Thus a canonical phantom transition plays a strong stabilizing role. But for each particular trapped-ghost solution a special study is necessary.

28 Example B(x), n=3 2 h = 0 h = 0 1 V eff(x) 3 m = -0,02 m = m = 0,02-1 (a) B(x) for three values of m, (b) The effective potential for a symmetric wormhole with m = 0. There are 4 regions in which the boundary-value problem should be considered (though only two different ones). Moreover, due to the pole at the throat x = 0 the potential V eff needs regularization before seeking ω; then its whole shape will change, including the behavior at the surfaces h = 0. -2

29 Example 2 2 B(x) Veff(x) m= m= m= (a) B(x) for three larger values of m (black universe solutions); at m there is no throat outside the horizon (B = 0). (b) The effective potentials for black universes with no throat outside the horizon. In such cases a regularization of V eff is not needed. A tentative result is that ω = 0 is a common eigenvalue in the regions to the left and to the right of h = 15 at m = , hence there is instability with a linear growth of δφ(t).

30 Conclusion A possible explanation of the absence of phantom matter around us: ghosts are trapped in strong field regions like jenies in bottles. There are many examples of explicit solutions for trapped-ghost wormholes and black universes. The effective potentials for radial perturbations of wormhole and black universe configurations contain universal forms of poles at throats and transition surfaces from a usual scalar field to a ghost. Generic poles at throats can be regularized, and instabilities have been revealed for many known wormhole and black universe solutions. Poles at transition surfaces to ghosts exhibit stabilizing properties due to the perturbation regularity condition. A tentative result for some trapped-ghost solutions is that they are unstable with at least linear growth of perturbations. It is desirable to verify these results and to try to build stable solutions.

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