Nonlinear instability of wormholes supported by exotic dust and a magnetic field
aa r X i v : . [ g r- q c ] F e b Nonlinear instability of wormholes supported by exotic dust and a magnetic field
Olivier Sarbach and Thomas Zannias
Instituto de F´ısica y Matem´aticas, Universidad Michoacana de San Nicol´as de Hidalgo,Edificio C-3, Ciudad Universitaria, 58040 Morelia, Michoac´an, M´exico. (Dated: October 30, 2018)Recently, spherically symmetric, static wormholes supported by exotic dust and a radial mag-netic field have been derived and argued to be stable with respect to linear radial fluctuations. Inthis report we point out that these wormholes are unstable due to the formation of shell-crossingsingularities when the nonlinearities of the theory are taken into account.
PACS numbers: 04.20.-q,04.25.-g, 04.40.-b
I. INTRODUCTION
The most popular wormhole model [1] is described by the spacetime ( M, g ) where M = R × S , g = − dt + dx + ( x + b )( dϑ + sin ϑ dϕ ) , (1)( t, x ) denoting Cartesian coordinates on R and ( ϑ, ϕ ) the standard angular coordinates on S . Here, b > x = 0. In order to satisfy Einstein’s field equations, thissolution needs to be supported by ”exotic” matter, that is, matter fields which violate the null energy condition.Several such matter models have been proposed which admit the solution (1). For example, it has been shown [2–4]that this solution may be supported by a ghost scalar field (a massless scalar field whose kinetic energy has a reversedsign). Also, the wormhole (1) may be sourced by an anisotropic perfect fluid with an appropriate equation of state[5]. More recently, it has been shown [6] that this solution can also be supported by a combination of exotic dust (apressureless fluid with a negative energy density) and a radial magnetic field.Once a suitable matter model which admits the wormhole configuration (1) has been found, a relevant question iswhether or not the solution is stable with respect to fluctuations of the metric and matter fields. Clearly, the stabilityproperties depend on the specific matter model. In the case of black holes, for instance, the magnetically chargedReissner-Nordstr¨om black hole is linearly stable in Einstein-Maxwell theory [7]; however, it is unstable in Einstein-Yang-Mills-Higgs theory [8] if the horizon is sufficiently small. In fact, a similar situation occurs for the Schwarzschildblack hole when it is embedded in a spacetime with one extra compact dimension to form a black string. Whilethe Schwarzschild solution is linearly stable in vacuum general relativity [9] the corresponding black string is linearlyunstable if the circumference of the extra dimension is large enough [10]. Concerning the wormhole model (1), it hasbeen shown to be unstable with respect to linear, radial perturbations [11] when coupled to the ghost scalar field.Numerical simulations in the nonlinear, spherically symmetric case indicate [12–14] that the wormhole either rapidlyexpands or collapses to a Schwarzschild black hole. Attempts have been made to stabilize the wormhole by adding anelectric or magnetic charge [15], resulting in charged generalizations of (1). Although the addition of charge reducesthe time scale of the instability, linearly stable wormholes have not been found in the scalar field model.In a recent article [16] the linear stability of the wormhole (1) has been analyzed in the matter model put forwardin [6] with the conclusion of stability. In this report, we point out that the wormhole (1) is unstable when nonlinear,radial fluctuations of the metric and the exotic dust are taken into account. The reason for the instability is thatfor a large class of initial data shell-crossing singularities form and therefore curvature invariants of the perturbedspacetime blow up in finite time. II. THE MODEL
The model considered in [6] consists of a spherically symmetric, self-gravitating dust cloud coupled to a radialmagnetic field. In terms of a Lagrangian coordinate x which parametrizes the dust shells and proper time τ measuredby an observer comoving with such a shell, the key equation is a master equation for the areal radius r ( τ, x ) which, See the Appendix for a derivation. for each fixed shell x , has the form of a one-dimensional mechanical system,12 ˙ r ( τ, x ) + V ( r ( τ, x ) , x ) = E ( x ) , (2)with potential V ( r, x ) = − µ ( x ) r + Q r . (3)Here, the energy function E ( x ) is determined by the initial data r ( x ) := r (0 , x ) and v ( x ) := ˙ r (0 , x ) for r ( τ, x ) at τ = 0, say, Q is the magnetic charge and µ ( x ) is the mass function for the dust, which, in terms of the initial density ρ ( x ) is µ ( x ) = µ ∞ − πG ∞ Z x ρ ( y ) r ( y ) r ′ ( y ) dy, (4) µ ∞ representing the mass at x → ∞ and G Newton’s constant. Once the function r ( τ, x ) has been determined, themetric g , the electromagnetic field tensor F , the four-velocity u and the density ρ are given by g = − dτ + r ′ ( τ, x ) E ( x ) dx + r ( τ, x ) ( dϑ + sin ϑ dϕ ) , (5) F = Q dϑ ∧ sin ϑ dϕ, (6) u = ∂∂τ , ρ ( τ, x ) = ρ ( x ) (cid:18) r ( x ) r ( τ, x ) (cid:19) r ′ ( x ) r ′ ( τ, x ) . (7)Here and in the following, a dot and a prime denote partial differentiation with respect to τ and x , respectively.Notice that there is a freedom in reparametrizing the dust shells, x f ( x ). In the Tolman-Bondi models (see, forinstance, Ref. [17]) this freedom is usually fixed by labeling each shell by its initial areal radius, r ( x ) = x . Since herewe are interested in describing wormhole spacetimes with one throat, we set r ( x ) = √ x + b instead, where b > x is in equilibrium with respect to the potential V ( r, x ). This ispossible if both Q and µ ( x ) are strictly positive, in which case there is a potential well with global minimum at r = Q /µ ( x ) corresponding to the energy E ( x ) = − Q / (2 r ). This, together with r ( x ) = r ( x ) = √ x + b leads tothe static wormhole solution (1) with b = | Q | , negative density ρ ( x ) = − πG Q ( x + Q ) (8)and zero asymptotic mass µ ∞ = 0. III. PERTURBATION ANALYSIS
Now let us consider a nonlinear perturbation of the static solution described in the previous section for a fixed valueof the magnetic charge Q . Its Cauchy evolution is determined by the initial data for the functions r ( x ), v ( x ) and ρ ( x ) and the value for the asymptotic mass µ ∞ . Keeping the labeling of the dust shells such that r ( x ) = √ x + b ,this amounts in perturbing v ( x ), ρ ( x ) and µ ∞ from their equilibrium values v = 0, ρ given by Eq. (8) and µ ∞ = 0,respectively. The perturbations of ρ and µ ∞ give rise to a new mass function µ ( x ) according to Eq. (4). For thefollowing, we consider a rather large class of nonlinear perturbations which are ”small” in the following sense:(i) The perturbed mass function µ ( x ) is strictly positive for all x ∈ R .(ii) The total energy E ( x ) is strictly negative for all x ∈ R .Condition (i) implies that V ( · , x ) describes a potential well with negative global minimum at r min ( x ) = Q /µ ( x ).Condition (ii) means that the trajectories of the system (2) are bounded; hence the areal radius of each dust shellundergoes periodic oscillations about r min ( x ). If 0 < r ( x ) < r ( x ) denote the zeroes of the function r E ( x ) − V ( r, x )for each fixed x , the corresponding period is T ( x ) = 2 r ( x ) Z r ( x ) dr p E ( x ) − V ( r, x )) = π √ µ ( x ) | E ( x ) | / . (9)Since the static wormhole solution satisfies the conditions (i) and (ii) it is not difficult to construct initial data withthese properties. For example, one could leave ρ and µ ∞ unperturbed which also leaves the potential unperturbedand choose | v | small enough such that v ( x ) < − V ( r ( x ) , x ) = Q /r ( x ) for all x ∈ R .Since small perturbations fulfilling the conditions (i) and (ii) behave like a one-dimensional particle in a potentialwell one might expect the static wormhole (1) to be stable under perturbations. However, a problem arises whendifferent shells of dust touch or cross each other, in which case the density function ρ ( τ, x ) diverges. This occurs atpoints where the function ν ( τ, x ) := r ′ ( τ, x ) /r ′ ( x ) vanishes, see Eq. (7). The time evolution of ν ( τ, x ) is governed bythe following equation which may be obtained by differentiating Eq. (2) twice,¨ ν ( τ, x ) + ω ( τ, x ) ν ( τ, x ) = − πGρ ( x ) (cid:18) r ( x ) r ( τ, x ) (cid:19) , (10)where ω ( τ, x ) = ∂ V∂r ( r ( τ, x ) , x ) = − µ ( x ) r ( τ, x ) + 3 Q r ( τ, x ) . Since ν (0 , x ) = 1, ˙ ν (0 , x ) = v ′ ( x ) /r ′ ( x ), shell-crossing is absent for small enough time provided that the initialfunction ˙ ν (0 , x ) = dv dr ( x )is regular. In particular, this requires the initial velocity profile v ( x ) to have a critical point at the throat. Otherwise,the time evolution of the wormhole is not defined since in this case˙ ρ (0 , x ) = − ρ ( x ) (cid:20) v ( x ) r ( x ) + ˙ ν (0 , x ) (cid:21) diverges. The regularity of ˙ ν (0 , x ) also implies the regularity of the radial part of the metric, − dτ + r ′ ( x ) E ( x ) ν ( τ, x ) dx (11)whose Gaussian curvature is ν − ¨ ν , for small enough time. However, there is a coordinate singularity at x = 0 unless r ′ ( x ) / (1 + 2 E ( x )) is finite and different from zero at x = 0. This yields an additional restriction on the initial energy E ( x ). In order to shed light on this condition, we assume for simplicity that ρ and µ ∞ are unperturbed and set v ( x ) = α ( x ) | Q | /r ( x ) with some smooth function α ( x ) satisfying α ( x ) < x ∈ R . Then,1 + 2 E ( x ) = 1 r ( x ) ( x + Q α ( x ) ) . (12)Since r ′ ( x ) = x/r ( x ), the coordinate singularity is avoided if α ( x ) is proportional to x for small | x | . Together withthe regularity of ˙ ν (0 , x ) this means that the function α ( x ) should satisfy α (0) = 0 , α ′ (0) = 0 , α ( x ) < , x ∈ R . (13)In the following, we assume that the above conditions on α ( x ) are satisfied in order for the coordinate x to beglobally defined on the real axis and in order to avoid shell-crossing singularities for arbitrary small times. To analyzethe occurrence of shell-crossing singularities at later times, we differentiate the equation r ( τ + nT ( x ) , x ) = r ( τ, x ) , n = 0 , , , , ..., τ, x ∈ R , and obtain ν ( τ + nT ( x ) , x ) = ν ( τ, x ) − n ˙ r ( τ, x ) T ′ ( x ) r ′ ( x ) . (14)The evaluation of this equation at a turning point τ = τ ( x ), where ˙ r ( τ ( x ) , x ) = 0, yields ν ( τ ( x ) + nT ( x ) , x ) = ν ( τ ( x ) , x ) , n = 0 , , , , ..., x ∈ R , (15)while for τ = 0 we obtain ν ( nT ( x ) , x ) = 1 − nv ( x ) T ′ ( x ) r ′ ( x ) , n = 0 , , , , ..., x ∈ R . (16)Therefore, for each fixed shell x , the field ν ( τ, x ) oscillates between the two values given in the right-hand side ofEqs. (15) and (16), respectively. If v ( x ) T ′ ( x ) /r ′ ( x ) = 0, this signifies that the oscillation’s amplitude grows withoutbound for τ → ∞ . Furthermore, since ν (0 , x ) = 1, Eq. (16) also implies that for v ( x ) T ′ ( x ) /r ′ ( x ) > ρ and µ ∞ unperturbed and setting v ( x ) = α ( x ) | Q | /r ( x )with the function α ( x ) satisfying the restrictions (13) gives v ( x ) T ′ ( x ) r ′ ( x ) = 4 πα ( x )(1 − α ( x ) ) − / (cid:20) − α ( x ) + 32 r ( x ) α ( x ) α ′ ( x ) x (cid:21) . If α ( x ) is any smooth function which satisfies the conditions (13), is strictly positive for x = 0, and constant outsidea compact interval, then the family of functions α λ ( x ) = λα ( x ) with λ > v ( x ) T ′ ( x ) /r ′ ( x ) > x = 0. We conclude that a large class of nonlinear perturbations leads to the formationof shell-crossing singularities. As a consequence, the wormhole is unstable.We shall finish this report by making a few comments regarding the behavior of linear perturbations on the staticwormhole background defined in the previous section. If we set r ( τ, x ) = r ( x ) + δr ( τ, x ), differentiate Eq. (2) withrespect to τ , we obtain, to linear order in δr ( τ, x ), ∂ δr ( τ, x ) ∂τ + ω ( x ) δr ( τ, x ) = 0 , ω ( x ) = | Q | r ( x ) , (17)and thus, δr ( τ, x ) = δv ( x ) sin( ω ( x ) τ ) ω ( x ) , (18)where we have assumed that r , ρ and µ ∞ are unperturbed. For the following, we also assume that the linearizedinitial velocity δv ( x ) is compactly supported away from the throat at x = 0. The induced linear density perturbation δρ ( x, τ ) is δρ ( τ, x ) = ρ ( x ) " − δr ( τ, x ) r ( x ) − δr ′ ( τ, x ) r ′ ( x ) . (19)Since each shell oscillates with its own frequency ω ( x ), it is seen that δρ ( τ, x ) exhibits linear growth in time (see alsothe discussion in [16]). In terms of the gauge-invariant combination J ( τ, x ) := δρ ( τ, x ) − ρ ′ ( x ) r ′ ( x ) δr ( τ, x ) , this yields J ( τ, x ) = r ( x ) ρ ( x ) x (cid:18) δr ( τ, x ) r ( x ) (cid:19) ′ = r ( x ) ρ ( x ) x (cid:18) sin( ω ( x ) τ ) ω ( x ) δv ( x ) r ( x ) (cid:19) ′ . (20)Therefore, the linear growth in τ cannot be transformed away by an infinitesimal coordinate transformation. As wehave shown above, in the nonlinear case, this growth leads to blowup in finite proper time. IV. CONCLUSIONS
The result of this report is that the wormhole (1), when supported by exotic dust and a radial magnetic field, isunstable due to the formation of shell-crossing singularities. Although shell-crossing singularities have been arguedto be ”mild” in the sense that it is often possible to construct a C -extension of the metric [18], nevertheless thedensity function and the Ricci scalar blow up. Even if one is willing to accept such singularities it is worth pointingout the following: according to the arguments presented below Eq. (16), the extended spacetime possesses the bizarreproperty that for large τ , the norm of the vector field ∂ x , measuring the normal geodesic deviation between neighboringdust shells, undergoes wild oscillations. Acknowledgments
We thank Dar´ıo N´u˜nez for interesting discussions. This work was supported in part by Grants No. CIC 4.7 and4.19 to Universidad Michoacana, PROMEP UMICH-PTC-195 from SEP Mexico, and CONACyT 61173.
Appendix: Derivation of the master equation
The radial components of the Einstein equations for the metric g = − dτ + r ′ ( τ, x ) Γ( τ, x ) dx + r ( τ, x ) ( dϑ + sin ϑ dϕ ) , the four-velocity u = ∂∂τ and the electromagnetic field F = Q dϑ ∧ sin ϑ dϕ read G τ τ = 2 r ˙ΓΓ ˙ r − m ′ r r ′ = − Q r − πGρ, (21) G xx = − r ¨ r − mr = − Q r , (22) G τ x = 2 r ˙ΓΓ r ′ = 0 , (23)where G αβ are the components of the Einstein tensor and m ( τ, R ) is the Misner-Sharp mass function [19] which isdefined by 1 − mr = g ( dr, dr ) = − ˙ r + Γ . (24)These equations are solved as follows: first, Eq. (23) immediately implies that Γ = Γ( x ) is a function of x only.Next, differentiating Eq. (24) with respect to τ and comparing with Eq. (22) yields ˙ m = Q ˙ r/ (2 r ) which may beintegrated to obtain m ( τ, x ) = µ ( x ) − Q / (2 r ( τ, x )) , (25)with µ ( x ) representing the dust mass function. Finally, using this in Eq. (21) gives µ ′ = 4 πGρr r ′ . As a consequence, ρr r ′ must be independent of τ which implies Eq. (7), and Eq. (4) is obtained after integrating in x . The key equation(2) that governs the dynamics of the system is obtained from Eq. (24) after setting Γ( x ) = 1 + 2 E ( x ) and taking intoaccount the relation (25) between the mass functions m and µ . The remaining components of the Einstein equationsand the mass conservation law are automatically satisfied as a consequence of the previous equations and Bianchi’sidentities. [1] M.S. Morris and K.S. Thorne. Wormholes in spacetime and their use for interstellar travel: A tool for teaching generalrelativity. Am. J. Phys. , 56:395–412, 1988.[2] H.G. Ellis. Ether flow through a drainhole: A particle model in general relativity.
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