Gravitational perturbations in the Newman-Penrose formalism: Applications to wormholes
aa r X i v : . [ g r- q c ] F e b Spherically symmetric wormholes can be linearly stable
Juan Carlos Del ´Aguila ∗ and Tonatiuh Matos † Departamento de F´ısica, Centro de Investigaci´on y de EstudiosAvanzados del IPN, A.P. 14-740, 07000 Ciudad de M´exico, M´exico. (Dated: Received: date / Accepted: date)In this work we study the problem of linear stability of gravitational perturbations in stationaryand spherically symmetric wormholes. For this purpose, we employ the Newman-Penrose formalismwhich is well-suited for treating gravitational radiation in General Relativity, as well as the geo-metrical aspect of this theory. With this method we obtain a “master equation” that describes thebehavior of perturbations that are “vacuum-like” and of odd-parity in the Regge-Wheeler gauge.This equation is later applied to a specific class of Morris-Thorne wormholes and also to the metricof an asymptotically flat scalar field wormhole. The analysis of the equations that these space-timesyield reveals that they are stable with respect to the type of perturbations here studied.
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I. INTRODUCTION
Ever since the concept of “wormhole” was first introduced in the literature by Misner and Wheeler in 1957 [1], therehas been general interest in the fascinating physical and geometrical properties that the non-trivial topology of theseobjects could possess. Previously, Einstein and Rosen had already proposed an interpretation of the Schwarzschildspace-time consisting of two identical “sheets” connected through a “bridge”, this is nowadays known as an Einstein-Rosen bridge [2]. While the Scwarzschild metric can indeed admit the interpretation of wormhole, Fuller and Wheelerwould later prove that it is not a traversable one since, in order for a signal to cross it, causality must be violated [3].Nevertheless, these ideas can be considered as the precursors of the modern version of a wormhole, an hypotheticalcompact object that would allow communication between distant regions of the same universe, or even between twodifferent universes. Several years later, Morris and Thorne established the characteristics that a traversable wormholemust feature, along with a general metric that should describe them [4]. The metric was assumed to be stationary andspherically symmetric. In their work they arrived to the unfortunate conclusion that a traversable wormhole requiresof “exotic” matter, this is, matter that violates the energy conditions, as its gravitational source. Until now there isno observational evidence on the existence of wormholes, and thus, they have remained within the speculative realmof the theory of General Relativity. Moreover, there are additional concerns regarding the realistic existence of suchobjects in the universe, one of them is that of their stability.It can be argued that, in order for a stellar object to be of any astrophysical interest, it has to stable. Otherwise, aslight perturbation or deviation from its initial state would result in the collapse of the space-time itself. Remarkablepapers that solve this problem, at least in a linear theory and using analytical methods, are well-known. The stabilityof the Schwarzschild metric was first studied by Regge and Wheeler by adding a perturbation term h µν to thebackground metric and keeping terms up to first order of the perturbation [5]. Based on their results, later worksthen confirmed that said space-time is indeed stable against small perturbations [6, 7]. With a less straightforwardapproach, Teukolsky managed to find a “master equation” that describes gravitational, electromagnetic, and neutrinofield perturbations of a spinning black hole [8]. Analyzing then the mentioned equation with numerical techniques,the stability of the first modes of vibration of the Kerr black hole was concluded [9]. From a practical point of view,this result was of great relevance to the existence and possible observation of a realistic black hole. However, froma theoretical standpoint, maybe the most interesting aspect of this series of papers is that of the derivation of themaster equation. For this purpose, Teukoslky exploited the full potential of the Newman-Penrose formalism [10] andthe underlying geometric properties of the Kerr metric, in particular, the fact that it is of type D in the algebraicclassification of space-times. As impressive as the master equation is, unfortunately, it is only valid for vacuumspace-times of type D. This rules out the possibility of applying it to wormholes.It is clear that the problem of stability for black holes has been thoroughly studied. Over the years, these develop-ments have contributed to the physical relevance of this outstanding prediction of General Relativity. On the otherhand, wormholes are still only theoretical entities. They are commonly, but not uniquely, proposed as stationary andspherically symmetric space-times supported by a phantom scalar field, i.e., a scalar field whose kinetic energy hasa reversed sign (sometimes referred too as ghost scalar field). Maybe the most simple model of such a wormhole isthat of Ellis [11]. In recent years, many works have been written about the question of the linear stability of thesetype of wormholes and a handful of them report that they are generally unstable. Thus adding another problematicissue to their set of particular properties. In [12], scalar field wormholes consisting of one asymptotically flat end,while the other end being asymptotically AdS, were proven to be unstable against axial perturbations. Similar resultsregarding the instability of particular cases of wormholes with phantom fields were obtained in papers [13–15] whenperturbing the space-time metric. Said perturbations were assumed to be only radial and, of course, time-dependent.Particularly in [14], the case of the Ellis wormhole was studied exhaustively. On the contrary, there have been a fewother works [16, 17] studying certain wormholes that are supposedly stable. The perturbation analysis of [16] can beconsidered to be somewhat more general in the sense that it includes angular-dependent terms.In this paper we will first develop a framework for treating linear gravitational perturbations using the Newman-Penrose formalism. This will benefit us with some previously used tools and results that have been found over the yearsfor the problem of gravitational radiation in General Relativity. We will focus on what we shall call “vacuum-like”perturbations, that is, perturbations such that the first order change of the Ricci tensor is δR µν = 0. Additionally, inthis first approach, we will specialize to the odd-parity perturbations, named so by Regge and Wheeler in their paper[5]. The results we obtain will later on be applied to wormhole metrics, some of them supported by phantom scalarfields. The work is organized as follows. In section II we will study the general problem of gravitational perturbationswithin the tetrad formalism. This treatment will be particularized to stationary and spherically symmetric space-timesin section III, here, the master equation will also be presented. In section IV the meaning of physical regularity thatmust be imposed to the mentioned perturbations will be discussed. Examples of the application of our master equationwill be given in sections V and VI, first on the Morris-Thorne wormholes, and then on a specific phantom scalar fieldwormhole. Finally, conclusions and final comments will be given. We also included two appendixes containing mostlylaborious calculations that were carried out throughout the paper. II. GRAVITATIONAL PERTURBATIONS IN THE TETRAD FORMALISM
We shall follow the notation utilized by Newman and Penrose in their seminal paper [10] and hereafter may referto this work as the NP paper. In this formalism a null tetrad ( l µ , n µ , m µ , ¯ m µ ) is introduced into every point of afour-dimensional pseudo-Riemannian manifold of signature (+1 , − , − , −
1) and metric g µν . The vectors l µ and n µ are real, while m µ and ¯ m µ are complex. In this paper we will use a bar over any given quantity to denote its complexconjugate. The vectors of the tetrad must also satisfy the orthogonal property l µ n µ = − m µ ¯ m µ = 1, with the rest ofthe vector combinations being zero. The space-time metric can then be expressed as g µν = l µ n ν + n µ l ν − m µ ¯ m ν − ¯ m µ m ν . (1)This relation can be rewritten in a more compact way as g µν = z mµ z nν γ mn if one conveniently defines z mµ = ( l µ , n µ , m µ , ¯ m µ ) ,z µm = ( l µ , n µ , m µ , ¯ m µ ) ,γ mn = γ mn = −
10 0 − , (2)where γ mp γ pn = δ mn . Using (2) we can also write the orthogonality properties simply as z µm z nµ = γ mn . The metric γ will be used to raise or lower tetrad indices. Newman and Penrose define 12 complex spin coefficients that dependon linear combinations of the quantities Z mnp = z µm z νn ∇ µ z pν , which are anti-symmetrical in their last two indices ,i.e., Z mnp = −Z mpn . Additionally, four differential operators are introduced D = l µ ∇ µ , ∆ = n µ ∇ µ , δ = m µ ∇ µ , δ ∗ = ¯ m µ ∇ µ , (3) We will use Greek indices ( µ, ν = 0 , , ,
3) to denote tensor indices and lower-case Latin indices ( a, b, m, n = 0 , , ,
3) to denote tetradindices. In the case of γ and Z mnp our notation will slightly vary from that of the NP paper. They use η instead of γ for the tetrad metric, and γ pnm instead of Z mnp . Note that the order of the indices for these last quantities is also different. or more compactly D m = z µm ∇ µ with D m = ( D, ∆ , δ, δ ∗ ). Using the 12 spin coefficients, along with the operators(3), Newman and Penrose obtained a set of numerous equations that are the equivalent of the Bianchi identities andthe components of the Ricci and Weyl tensors in tetrad form, this is now known as the Newman-Penrose formalism.Since the Einstein field equations make use of the curvature tensors yielded by a given space-time metric, one candiscuss any problem in General Relativity (at least its geometrical aspects) within this formalism.Here we will develop a general framework for perturbation theory using the Newman-Penrose formalism. Thescheme we follow is the typical one for linear gravitational perturbations, that is, we add a perturbation term h µν to a certain background metric g µν , and then compute the components of the Ricci tensor keeping terms up to firstorder of the perturbation. The perturbation term is assumed to be small compared to its background counterpart.In this formalism, the perturbation term of the metric will be represented by a perturbation in the null tetrad, forexample, l µ = e l µ + ˆ l µ , where we will establish the convention that a tilde denotes any given background quantity andthe hat denotes the perturbation term of said quantity. To proceed, we expand the perturbation terms of the tetradin the basis of the background tetrad, hence, z µm = e z µm + ˆ z µm = e z µm + ˆΣ nm e z µn ,z mµ = e z mµ + ˆ z mµ = e z mµ + ˆΩ nm e z nµ . (4)To maintain the vectors l µ and n µ real, the ˆΣ nm matrix has to satisfy ˆΣ nm ∈ R and ˆΣ m = ˆΣ ∗ m for m, n = 0 , = ˆΣ ∗ , ˆΣ = ˆΣ ∗ , and that ˆΣ m = ˆΣ ∗ m for m = 0 , m µ and ¯ m µ remainas complex conjugates of each other. To simplify notation we drop the hat off the perturbation terms ˆΣ nm , ˆΩ nm andkeep in mind through the rest of this section that now Σ nm and Ω nm carry exclusively perturbed quantities. Then, themetric can be written as g µν = γ mn z mµ z nν = e g µν + Ω mn ( e z mµ e z nν + e z mν e z nµ ) + O (Ω ) , (5)where Ω mn = γ mp Ω np . Our first task will be to find a relation between Σ nm and Ω nm such that the orthogonalproperties of the tetrad formalism hold to first order of Σ , Ω. Of course, these properties are assumed to be satisfiedfor the background tetrad. It is not difficult to prove that the relation we are looking for is Ω nm = − γ mp Σ pq γ qn ,or alternatively, Ω mn = − Σ nm . Using this result one can next verify that g µρ g ρν = δ µν , and so, the fundamentalequations of the formalism are consistent.With the tetrad given by equation (4) the quantities Z abc related to the spin coefficients may be computed. However,note that the connection Γ associated to the operator ∇ appearing in these quantities is compatible with the metric g ,not with the background metric e g . Naturally, the components of the connection Γ can be expressed as Γ ρµν = e Γ ρµν + ˆΓ ρµν .We obtain, thus, Z abc = e Z abc − ˆΓ cab + e D a Ω cb + e Z abp Ω pc + e Z apc Σ pb + e Z pbc Σ pa , (6)where we have defined ˆΓ cab = e z cα ˆΓ αµν e z µa e z νb . The components of the perturbed connection may be found by usingthe compatibility condition ∇ α g µν = 0 and the torsion free symmetry ˆΓ abc = ˆΓ acb . A straightforward, but somewhatlong, calculation yields ˆΓ abc = e D ( b Ω c ) a + e D [ b Ω a ] c + e D [ c Ω a ] b + e Z ( bc ) p Ξ pa + e Z [ ba ] p Ξ pc + e Z [ ca ] p Ξ pb , (7)with Ξ nm = Ω nm − Σ nm , and also Π nm = Ω nm + Σ nm , which will be used in the next equation. Substituting (7) in (6),and after some algebraic simplifications, we get Z abc in terms only of background quantities and metric perturbations, Z abc = e Z abc + e D [ b Σ a ] c − e D [ c Σ a ] b + e D [ b Σ c ] a + Ξ m [ b e Z c ] am + e Z am [ c Π mb ] + Ξ ma e Z [ cb ] m + e Z mbc Σ ma . (8) In equation (5) we have explicitly indicated there are second order terms of Ω nm . From this point forward we will omit the second orderdependency in every equation for compactness and, unless otherwise noted, every equal sign should be understood as such only to firstorder of Σ or Ω. Round brackets will be used to denote symmetrization of the indices enclosed, while square brackets to denote anti-symmetrization.
This equation is manifestly anti-symmetric in its last two indices as the quantity Z abc should be. Though lengthy,equation (8) describes how the spin coefficients, which are necessary for the Newman-Penrose formalism, change tofirst order for any given perturbation Σ nm . Perturbed Tetrad Rotations
Consider a transformation of the perturbation terms Ω mn → Ω mn + Ω ′ mn . From (5) it can be seen that g µν → g µν + Ω ′ mn ( e z mµ e z nν + e z mν e z nµ ) . Since the expression in parenthesis is symmetric in its tetrad indices, the metric will then be invariant under thesetype of transformations if we demand that Ω ′ mn = − Ω ′ nm . Not only the metric will be invariant, but naturally,also any other scalar or tensor derived from it, so long as the tensor does not possess tetrad indices. Therefore,there exists liberty in choosing the perturbation tetrad ˆ z µm = Σ nm e z µn since the Ω mn that corresponds to a certainperturbed metric is not unique (recall that Ω nm = − γ mp Σ pq γ qn ). This of course is related to the group of Lorentztransformations that leave invariant the orthogonality properties of the formalism (see reference [18]). However, forthis case, the parameters of the Lorentz group should be taken as infinitesimal.Under the transformation Ω mn → Ω mn + Ω ′ mn , the previously defined Ξ nm is invariant, whileΠ nm → Π nm + 2Ω ′ nm , with Ω ′ nm = γ mp Ω ′ pn . Note that Ω ′ mn = − Ω ′ nm implies that Ω ′ mn = Σ ′ mn . Using these relations, we have that thequantities Z abc transform as Z abc → Z abc + 2 e Z am [ c Ω ′ mb ] + e Z mbc Ω ′ ma . Perhaps the most important benefit that the perturbed tetrad rotations provide lies in the differential operators D m . They evidently change as D m → D m + Σ ′ nm e D n , but because there is some freedom in choosing the perturbationtetrad vectors, we may then conveniently pick them so that, for instance, D m = e D m + χ e D n for some fixed m = n ,and a scalar field χ . In the following section we take advantage of this particular property, simplifying thus ourcalculations.It is important to notice that, when performing any rotation through Ω mn → Ω mn +Ω ′ mn , one has to be careful thatthe rotated vectors l µ and n µ end up being real, and that m µ and ¯ m µ remain as complex conjugates. This restrictsthe possible valid rotations that can be done. Taking into account these restrictions, one can be convinced that thereis a total of six degrees of freedom, which is consistent with the fact that the group of Lorentz transformations is asix parameter group. III. GRAVITATIONAL PERTURBATIONS IN SPHERICALLY SYMMETRIC SPACE-TIMES
For the remainder of this work we focus on four-dimensional stationary and spherically symmetric space-times(
M, g µν ) whose line element, without loss of generality, can be written in the form ds = g ( r ) dt − g ( r ) dr − g ( r ) d Ω , (9)where we have introduced a radial coordinate r and the metric elements g , , ( r ), which are arbitrary functions ofsaid coordinate. Also, d Ω is the standard metric on the two-sphere. An orthonormal frame for this metric is simplygiven by X = 1 √ g ∂ t , X = 1 √ g ∂ r , X = 1 √ g ∂ θ , X = 1 √ g sin θ ∂ ϕ . (10)From frame (10), a null tetrad can be constructed by taking appropriate linear combinations of the X vectors. Inthis paper we will take advantage of the symmetries of the space-time, namely the fact that ∂ t and ∂ ϕ are Killingvectors, and choose e l µ and e n µ so that they lie in the subspace spanned by said Killing vectors. This can also beextended to axially symmetric space-times. Hence, the vectors of the null tetrad will be e l µ = 1 √ X µ + X µ ) , e n µ = 1 √ X µ − X µ ) , e m µ = 1 √ X µ + iX µ ) . (11)A direct evaluation of the spin coefficients of the Newman-Penrose formalism with metric (9) and tetrad (11) yieldsthat the only non-vanishing coefficients are e κ , e ν , e τ , e π , e α and e β . Additionally, the following properties hold e κ + e ν ∗ = e τ + e π ∗ = e α + e β = 0 , e κ + e ν = − e τ − e π, e α = 14 ( e ν + e ν ∗ + e τ + e τ ∗ ) , (12)with e α, e β ∈ R . Notice that as a consequence of our choice of vectors e l µ and e n µ we will have that e D e φ = e ∆ e φ = 0 forany background scalar quantity e φ , including these spin coefficients.We now add a perturbation term h µν to the background metric introduced in this section. Following the pioneeringwork of Regge and Wheeler [5] we consider a perturbation of the form h µν = h h h h , (13)with h µν expressed in the coordinate basis { t, r, θ, ϕ } . Regge and Wheeler obtained this particular (and simple)expression for h µν through a gauge transformation of the most general perturbation whose angular part consists ofproducts of scalar, vector, and tensor spherical harmonics Y l,m ( θ, ϕ ) on the 2-sphere. This gauge is sometimes calledthe Regge-Wheeler gauge. Furthermore, (13) represents a perturbation of ( − l +1 parity, which Regge and Wheelernamed as odd, due to its negative symmetry under reflections about the origin. The ϕ -dependence of the perturbationcan be eliminated without significant loss of information by setting m = 0. This is possible since the backgroundspace-time is spherically symmetric. Therefore, we have that h , = h , ( t, r, θ ).Using the one-forms of the background tetrad ne l µ , e n µ , e m µ , e ¯ m µ o as a basis, we can write h µν = f ( e n µ e n ν − e l µ e l ν ) + 2 f [ e l ( µ e m ν ) + e l ( µ e ¯ m ν ) − e n ( µ e m ν ) − e n ( µ e ¯ m ν ) ] , where f = h / √ g g sin θ and f = h / √ g g sin θ . It can be verified that an acceptable tetrad for the perturbedmetric g µν = e g µν + h µν is given by l µ = e l µ + 12 f e n µ − f ( e m µ + e ¯ m µ ) , n µ = e n µ − f e l µ + f ( e m µ + e ¯ m µ ) , m µ = e m µ , (14)from which the elements of Ω nm can be easily read offΩ nm = f / − f − f − f / f f . It will more helpful, though, to represent the perturbation in terms of Σ nm = − γ mp Ω pq γ qn , obtaining thusΣ nm = − f / f / f − f f − f . (15)As it was already stated in the previous section, any given metric perturbation does not uniquely define Ω nm , andconsequently, Σ nm . We will be interested in a perturbed tetrad such that D e φ = ( χ e D + χ e ∆) e φ = 0 , ∆ e φ = ( ξ e D + ξ e ∆) e φ = 0 , (16)where again, e φ is any background scalar quantity and χ , , ξ , are elements of Σ nm . It turns out that preciselythe matrix given by (15) describes the perturbation tetrad with this desired property. Nonetheless, it is important tomention that one can always find, through an adequate tetrad rotation, a perturbation tetrad such that (16) holds ina spherically symmetric (even in an axially symmetric, for that matter) background space-time. This is possible toodue to our previous election of background vectors e l µ and e n µ , namely, the fact that they lie in the subspace spannedby Killing vectors. Another advantage that this Σ nm possess is that δ e φ = e δ e φ . However, this will not always be thecase for an arbitrary metric perturbation, even performing a perturbed tetrad rotation. Additionally it can be seenthat e Df , = e ∆ f , because of the ϕ independence of those functions.With finally an explicit expression for the perturbation matrix Σ nm , we can proceed to compute the perturbed spincoefficients using equation (8). Our objective then will be to write the components of the Ricci tensor R µν in termsof these spin coefficients using the equations of the Newman-Penrose formalism. It can already be foreseen that wewill obtain second-order partial differential equations for the perturbation functions f and f due to the fact that theformalism provides first-order partial differential equations for the spin coefficients, and these in turn, have first-orderderivatives of said functions. Since the calculation of the NP quantities is pretty much straightforward and the resultsare numerous, they will be shown separately in Appendix A, and we should cite them in the following as needed.In the Newman-Penrose formalism, 10 curvature related quantities Φ AB ( A, B = 0 , , R/
24, are defined.These are merely the projection of the tetrad vectors into the Ricci tensor, i.e., R µν z µm z νn , and a rescaling of the Ricciscalar R , respectively. See equations (NP 4.3b) for their explicit expressions. By either a direct calculation of thesequantities, or by the vanishing of the background spin coefficients, the following holds for the background metric e Φ = e Φ = e Φ = e Φ = 0 , e Φ = e Φ . (17)This also follows from the fact that the background Ricci tensor admits the form e R µν = diag h e R , e R , e R , e R i in the { t, r, θ, ϕ } basis. Using (NP 4.2a), (NP 4.2n) and (12), it can be seen that the last equality in (17) implies that e δ e ν + e δ ∗ e κ = 2 e α ( e ν + e κ ) . (18)It is important to realize that, apart from Λ, the Φ AB quantities depend manifestly on the tetrad choice. Evenupon fixing the background tetrad, Φ AB will vary with perturbed rotations such as the ones described in the previoussection. Since the Ricci tensor itself is invariant to these type of transformations, we look for expressions of itscomponents in the coordinate basis and constructed from the quantities Φ AB and Λ. More precisely, we will look forthe components of R µν in the orthonormal frame (10).In terms of the background tetrad basis, the orthonormal basis can be written as X µα = e Γ mα e z µn . From (11) it canbe easily seen that, e Γ mα = 1 √ − i i − . Similarly, in terms of the perturbed tetrad we have that X µα = Γ mα z µm . Using the fact that z µm = ( δ nm + Σ nm ) e z µn ,we obtain Γ nα = e Γ mα ( δ nm − Σ nm ) to first order in Σ, or explicitly,Γ nα = 1 √ − f / f / − f f − i i f / − f / . (19)We can then write, X µ = 1 √ (cid:20)(cid:18) − f (cid:19) l µ + (cid:18) f (cid:19) n µ (cid:21) , X µ = 1 √ m µ + ¯ m µ + 2 f ( n µ − l µ )] ,X µ = i √ m µ − m µ ] , X µ = 1 √ (cid:20)(cid:18) f (cid:19) l µ − (cid:18) − f (cid:19) n µ (cid:21) . (20)The X vectors of equation (20) can be shown to be invariant under transformations Σ nm → Σ nm + Σ ′ nm , and to firstorder in Σ, by noting that Γ nα → Γ nα − e Γ mα Σ ′ nm , and z µm → z µm + Σ ′ nm e z µn . Thus, we may find the desired invariantequations for the Ricci components by contracting these vectors with the Ricci tensor field. Unfortunately, this hasto be done for the 10 independent components of said tensor, yielding the following relationsˆ R = − ˆΦ − ˆΦ + 2(3 ˆΛ − ˆΦ ) , ˆ R = − ˆΦ − ˆΦ − ˆΦ − ˆΦ , ˆ R = i ( ˆΦ − ˆΦ + ˆΦ − ˆΦ ) , ˆ R = − ˆΦ + ˆΦ + 2(3 e Λ − e Φ ) f , ˆ R = − ˆΦ − ˆΦ − ) , ˆ R = i ( ˆΦ − ˆΦ ) , ˆ R = − ˆΦ + ˆΦ − ˆΦ + ˆΦ + 4(3 e Λ − e Φ + e Φ ) f , ˆ R = ˆΦ + ˆΦ − ) , ˆ R = i ( ˆΦ − ˆΦ + ˆΦ − ˆΦ ) , ˆ R = − ˆΦ − ˆΦ − − ˆΦ ) , (21)where we have defined ˆ R αβ = ˆ R µν X µα X νβ . In equations (21) we have written only the perturbation terms (denotedby a hat), that is, the terms of first order in f , . Naturally, the background terms that should appear on both sidesof the equations, which are of order zero, cancel each other out. Hereafter, we drop the tilde off the backgroundquantities and so, any quantity or operator without a hat should be understood to be of the background space-time,except for the perturbation functions f and f (same convention as Appendix A).Taking the results (A.6) from Appendix A, one can realize that the only non-vanishing components of ˆ R αβ areˆ R = 2( δ + − α ) Df + (cid:2) ( δ − + 2 κ + ) δ − − ( δ + + κ − + 3 π − ) δ + + 4( κ − − κ ) + 2(3Λ − Φ ) (cid:3) f , ˆ R = 2 (cid:2) D + ( δ − + 4 κ + )( δ − − κ + ) + 2(3Λ − Φ + Φ ) (cid:3) f − ( δ + − κ − + π − ) Df , ˆ R = i ( δ − − κ + ) Df − i ( δ + + κ − + 3 π − )( δ − − κ + ) f . (22)With the help of the commutator [ δ − − κ + , δ + ] = ( κ − + π − )( δ − + 2 κ + ), the ˆ R component of the past equationscan be rewritten as ˆ R = i ( δ − − κ + ) [ Df − δ + + 2 π − ) f ] . In this paper we are interested in what we will call ”vacuum-like linear perturbations”, this is, perturbations suchthat the first order term ˆ R µν vanishes, which implies that ˆ R µν = 0. Hence, we can attempt to solve the system ofequations (22) by making said assumption. Consider the ˆ R component, we have already factored it in a way thatcan be easily solved. Notice that the expression in parentheses cannot vanish since δ − is a differential operator and κ + is a scalar quantity. Thus, ˆ R will vanish if the expression in square brackets also does, or by the application ofthe operator in parentheses to the quantity in square brackets. We will examine the first possibility, that is, Df = 2( δ + + 2 π − ) f . (23)By inserting (23) in the ˆ R component of (22), an equation for the perturbation function f can finally be found, (cid:2) D + ( δ − + 4 κ + )( δ − − κ + ) − ( δ + − κ − + π − )( δ + + 2 π − ) + 2(3Λ − Φ + Φ ) (cid:3) f = 0 . (24)We are left, however, with the ˆ R = 0 equation yet to solve with the inconvenient that the perturbation functions f and f have already been used to solve the other two equations in system (22). By applying the operator D in theˆ R component, then using (23) and (24), this equation can be shown to vanish only if (see Appendix B),( δ + − α )(3Λ − Φ + Φ ) = 0 . (25)Unfortunately at this point, we are forced to abandon the generality that has been conserved until now regardingthe spherically symmetric space-times here considered, and restrict ourselves to those for which equation (25) holds.By considering the explicit form of the spin coefficients and operators shown in (A.2), this past condition can berewritten as 1 g √ g ddr [ g (3Λ − Φ + Φ )] = 0 . If this equation is true everywhere in space-time the implication is that3Λ − Φ + Φ = c/ g , (26)where c is an integration constant chosen so for future convenience. We will be able to associate certain physicalsignificance to a particular value of this constant in Section V. Examples that fulfill condition (26), besides vacuumspace-times, are the solutions of the Einstein-scalar field equations R µν = ±∇ µ φ ∇ ν φ , for this particular case, c = 0(see Appendix B for details). Many wormhole space-times arise as solutions to this class of field equations, hence, ourresults can be applied to them.For reasons explained in the next section, we shall opt to replace the perturbation function f with Q = 2 √ g sin θf .To do so, the following helpful relations can be verified to be true by examining the spin coefficients and operators of(A.2), δ − (cid:18) θ (cid:19) = − κ + sin θ , δ + (cid:18) √ g (cid:19) = κ − − π − √ g . Substituting f = Q/ √ g sin θ in (24), and employing these two equalities, we at last arrive to our master equationfor odd-parity perturbations, (cid:2) D + ( δ − + 2 κ + )( δ − − κ + ) − ( δ + − κ − )( δ + + π − + κ − ) + 2(3Λ − Φ + Φ ) (cid:3) Q = 0 . (27)The notation introduced throughout the paper allows us to easily identify the terms appearing in the masterequation. The D operator is associated to the time dependence of the perturbation, the second term is associatedwith the angular part due to it containing the δ − operators, and the third term is related to the radial part becauseof the δ + operators. In (27) there also appears a background matter term which is purely radial. It is natural thento propose a separable ansatz of the form Q = T ( t ) R ( r )Θ( θ ). With such a proposed solution, the angular part ofthe master equation will yield the following differential equation when inserting the explicit expressions for the spincoefficients and operators, d Θ dθ − θ d Θ dθ = − l ( l + 1)Θ . (28)Equation (28) has for solution Θ = sin θdP l (cos θ ) /dθ , where P l (cos θ ) are the well-known Legendre polynomials.This result was of course, expected, owing to the spherical symmetry of the line element (9) and to the decompositionin tensor spherical harmonics of the perturbation that Regge and Wheeler previously used. In fact, this part of thesolution is, obviously, the same that appears in their paper. It also can be verified that the radial equation yielded by(27), reduces to that of Regge-Wheeler when inserting the corresponding spin coefficients for the Schwarzschild metric.During the rest of this work we will analyze the radial part of the master equation (27), along with its properties. IV. PHYSICAL REGULARITY OF THE PERTURBATION
In order for the gravitational perturbation to be of any physical relevance, it has to display an ”acceptable”behavior throughout space-time, or at least asymptotically. One might naturally impose the condition that the metricperturbation functions of h µν do not grow without bound as r → ∞ and deem that as physical regularity. Nevertheless,due to the gauge freedom that exists in General Relativity, this condition is not quite precise. Fortunately, theNewman-Penrose formalism can also be used to describe more accurately what this acceptable behavior is expectedto be by means of the so-called “peeling theorem” [10].Consider the following vectors tangent to ingoing and outgoing radial null geodesics of the background metric, k µ ± = X µ ± X µ = 1 √ (cid:16)e l µ + e n µ ± e m µ ± e ¯ m µ (cid:17) . The next null rotations of our initial tetrad yield a new one such that the unperturbed part of l ′′ µ and n ′′ µ is alignedto the k µ + and k µ − vectors, respectively, l ′ µ = l µ + a ¯ m µ + a ∗ m µ + k a k n µ , m ′ µ = m µ + a n µ , n ′ µ = n µ ,n ′′ µ = n ′ µ + a ¯ m ′ µ + a ∗ m ′ µ + k a k l ′ µ , l ′′ µ = l ′ µ , (29)with a = 1 and a = − /
2. In equations (29) and (30), and only in those equations, we temporarily restore theconvention of section II in which any given quantity ξ of the space-time is written as the sum of a background termand a perturbation term, i.e., ξ = e ξ + ˆ ξ . Under transformations (29), the Weyl scalars we need change as (see reference[18]) ψ ′ = ψ + 4 a ψ + 6 a ψ + 4 a ψ + a ψ , ψ ′ = ψ + 3 a ψ + 3 a ψ + a ψ , ψ ′ = ψ + 2 a ψ + a ψ ,ψ ′′ = ψ ′ + 2 a ∗ ψ ′ + a ∗ ψ ′ . (30)If a = 1 and a = − /
2, then ψ ′′ = ψ / − ψ / ψ /
4. After substituting the expressions found in (A.7), theperturbed part of this Weyl scalar reduces toˆ ψ ′′ = 12 ( δ − + 2 κ + ) [( δ + − κ − ) f − Df ] . (31)The physical significance of ˆ ψ ′′ can be revealed by applying the operator D to (31), and then reducing it accordinglywith some of the relations of the formalism here derived along with the master equation, thus obtaining D ˆ ψ ′′ = 12 √ g sin θ δ − [( δ − + 2 κ + )( δ − − κ + ) + 2(3Λ − Φ + Φ )] Q. (32)We have already solved the angular part of the master equation whose terms appear again in (32). By making useof said solution and some properties of the Legendre equation, the past expression can be rewritten as ∂ ˆ ψ ′′ ∂t = − il ( l + 1) g / [( l − l + 2) + c ] T ( t ) R ( r ) P l (cos θ ) , (33)where we have made use of restriction (26) too. In the case of space-times that solve the Einstein-scalar fieldequations (and vacuum space-times too) we have that c = 0, and the meaning of ∂ ˆ ψ ′′ ∂t = − i ( l + 2)! g / ( l − T ( t ) R ( r ) P l (cos θ ) (34)becomes clearer, as well as the reason behind the use of the perturbation function Q . The peeling theorem establishesthat the Weyl scalar ψ asymptotically decays at null infinity as 1 /λ ′ , where λ ′ is the affine parameter of a null geodesicthat reaches said infinity. Since r is an appropriate radial coordinate, this affine parameter can be chosen as λ ′ = r for the case of the background radial null geodesics to which the unperturbed part of l ′′ µ and n ′′ µ are tangent to.Furthermore, the metric component appearing in (34) goes asymptotically as g ( r ) ∼ r . The perturbation function0 Q = T ( t ) R ( r )Θ( θ ), hence, manifestly describes the peeling property that the ψ scalar should display at null infinity.From this analysis we can state that a regular behavior of Q is one that does not alter the 1 /r decay of the Weylscalar ˆ ψ ′′ when r → ∞ . Also in this case, and from the reduced form of ∂ t ˆ ψ ′′ , it can be seen that the l = 0 and l = 1solutions will not yield a physical perturbation due to the vanishing of this Weyl scalar, i.e., the lowest multipole ofgravitational radiation is the quadrupole ( l = 2) [19]. The relation shown in equation (34) was previously found inthe case of perturbations of the Schwarzschild black hole in [20]. There, it was also shown that ˆ ψ ′′ is invariant underinfinitesimal null tetrad rotations and under gauge transformations as well, making this quantity measurable by anyobserver. Such properties are also valid for the ˆ ψ ′′ of the gravitational perturbations discussed in this paper. V. THE MORRIS-THORNE WORMHOLES
In this section we will apply the master equation found in section III to the wormhole space-times introduced in[4]. The general line element is the following, ds = e r ) dt − dr − b ( r ) /r − r d Ω , (35)where Φ( r ) is known as the redshift function and b ( r ) as the shape function. Both of these metric componentsfulfill certain conditions in order for the geometry of the space-time to be that of a wormhole. In particular, thereexists a minimum radius r = b > b ( b ) = b . This value defines the throat of the wormhole, and hence,the domain of the radial coordinate is r ∈ [ b , ∞ ). It should be clarified that this coordinate decreases from positiveinfinity to b as the throat is approached from one of the two universes it connects, and then increases back to infinitywhen emerging on the other universe. An additional requirement of the shape function is 1 − b ( r ) /r ≥
0, along withΦ( r ) being everywhere finite. This last condition on the redshift function is related to the non-existence of eventhorizons in the space-time so that hypothetical travelers may move from one universe to the other in both directions.If the wormhole is to be asymptotically flat, then the limits Φ( r ) → b ( r ) /r → r → ∞ must also be imposed.After this brief presentation on the features of the Morris-Thorne wormholes our intention next is to apply themaster equation (27) to them. Nevertheless, it should be reminded that this equation is not valid for the entire familyof Morris-Thorne metrics, but only for those that satisfy the condition (26). In fact, this condition determines aconstraint on the redshift and shape functions, namely, rb ′ ( r ) + b ( r )2 r + ( b ( r ) − r ) Φ ′ ( r ) = c. (36)We will now show that the class of Morris-Thorne metrics defined by (36), satisfies the conditions that a wormholemust possess. The most compelling way to accomplish this is to rearrange the defining constraint of the class so thatthe shape function, without its first derivative, is in terms only of the redshift function. This will allow us to picka suitable Φ( r ), specifically an everywhere finite function, and find the corresponding expression for b ( r ). Using thebasic theory of first-order differential equations one can show that the desired relation between these functions is b ( r ) = r + 2 e − r ) r [( c − F ( r ) + c ] , where c is an integration constant and F ( r ) = Z re r ) dr. The integration constant can be chosen so that the condition b ( b ) = b on the minimum radius r = b is fulfilled.Obtaining thus, b ( r ) = r + 2( c − e − r ) r Z rb r ′ e r ′ ) dr ′ . (37)1From (37) and the fact that the integrand there is strictly positive in the domain of integration, it can be seen thatthe condition 1 − b ( r ) /r ≥ c <
1. This also implies that the vector ∂/∂r remains everywhere space-like.Furthermore, by examining the limit r → ∞ for which Φ( r ) →
0, one can realize that b ( r ) /r → c . Then, in order forthe wormhole to be asymptotically flat, the constant c has to be set as c = 0. Recall that, for a vanishing value of c ,we concluded from the analysis in the previous section of the Weyl scalar ψ that the lowest radiative multipole is thequadrupole. This is in full agreement with the expected behavior of gravitational perturbations in an asymptoticallyflat space-time. It can be argued, therefore, that only those space-times with c = 0 are physically meaningful.We have obtained an upper bound for the constant c (and a fixed value of physical relevance) for which, given anappropriate redshift function, the metrics studied here possess indeed the geometry of a wormhole. With this, weshall examine the perturbation equation yielded by the line element (35). The relevant background spin coefficientsand operators are κ + = − π + = i cot θ √ r , π − + κ − = − α = 1 r s (cid:18) − b ( r ) r (cid:19) , π − − κ − = Φ ′ ( r ) s (cid:18) − b ( r ) r (cid:19) ,δ + = s (cid:18) − b ( r ) r (cid:19) ∂∂r , δ − = i √ r ∂∂θ , D = 1 √ (cid:18) e − Φ( r ) ∂∂t + 1 r sin θ ∂∂ϕ (cid:19) , which are computed using a tetrad of the form (11). With these expressions we obtain the following second-orderpartial differential equation from (27), e − ∂ Q∂t − r (cid:18) ∂ Q∂θ − θ ∂Q∂θ (cid:19) − r − br ∂∂r r − br ∂Q∂r ! − Φ ′ (cid:18) − br (cid:19) ∂Q∂r + (cid:18) Φ ′ ( b − r ) + rb ′ − b r (cid:19) Qr + 4(3Λ − Φ + Φ ) Q = 0 . The equation obtained for the perturbation can be further simplified by considering the previously introducedansatz for Q , whose angular part has already been solved in section III, and with the additional assumption of anharmonic dependence on time, i.e., Q = e iωt R ( r ) sin θdP l (cos θ ) /dθ . Furthermore, through the following coordinatechange for r , ddr ∗ = ± e Φ r − br ddr , and substituting equations (26, 28), the master equation for this restricted class of Morris-Thorne wormholes canfinally be rewritten in a very compact form as, d Rdr ∗ − (cid:0) V ( r ) − ω (cid:1) R = 0 , (38)with V ( r ) = e r (cid:20) l ( l + 1) + 3 (cid:18) c − br (cid:19)(cid:21) . In the coordinate transformation performed, one can always choose an adequate integration constant so that thethroat of the wormhole is located at r ∗ = 0. Moreover, the r ∗ coordinate takes the positive sign for one side of thethroat, and the negative sign for the other side. When r → ∞ , one has that r ∗ → ± r/ √ − c , where both coordinatescoincide (up to the sign) if the wormhole joins asymptotically flat sides. The coordinate r ∗ thus takes values on thewhole real line, i.e., r ∗ ∈ ( −∞ , ∞ ).With the domain of this new coordinate established, equation (38) then defines an eigenvalue problem for ω andthe operator H = − d /dr ∗ + V ( r ) which is linear and self-adjoint in a L ( R , dr ∗ ) space. An operator of this type issometimes called a Schr¨odinger operator. The stability analysis lies now in determining if there exist eigenvalues ofthe equation H R = ω R which represent perturbations that grow without bound as t → ∞ , but are physically regular2 TABLE I: Metric components of a few examples from the stable class of Morris-Thorne wormholes (1 > c > / c = 0 for asymptotically flatspace-times). e r ) − b ( r ) /r − c )(1 − b /r )1 + e − ( r/b ) (1 − c ) e − r ) [1 − b ( e r ) − e − ) /r ]1 + b / ( x + b ) (1 − c ) e − r ) (cid:2) b (ln (cid:2) / r / b (cid:3) − /r (cid:3) / r/b − /π (1 − c ) e − r ) h e r ) − b /πr − b (cid:0) π/ − − r/b ) ] (cid:1) /πr i otherwise. By equation (33) and the peeling theorem, any eigenfunction R ∈ L ( R , dr ∗ ) will describe physicallyregular perturbations due to it being square-integrable. Since the operator H is self-adjoint, the eigenvalues ω mustbe real. Hence, considering the time dependent part of the proposed ansatz, any instability will appear as a purelyimaginary ω , this is, as a negative eigenvalue.The discussion of the eigenvalue spectrum of (38) follows in a fairly simple manner based on the properties of thepotential V ( r ) of the Schr¨odinger operator H . First, we shall focus on the wormholes with vanishing c since theyrepresent asymptotically flat space-times. For this case it is readily seen that V ( r ) ≥ r ∈ [ b , ∞ ], due to the1 − b ( r ) /r ≥ l takes positive integer values starting from l = 2. For a strictly positivepotential there cannot exist negative eigenvalues (energy bound states) and thus, all of the vibrational modes of thisclass of wormholes are linearly stable under vacuum-like perturbations of odd-parity.For the sake of completeness, we may also consider the cases of space-times with c = 0. Unlike the asymptoticallyflat metrics, and looking at equation (33) for the Weyl scalar ψ , it must be noticed that the l = 1 modes can actuallyyield a gravitational perturbation. In order for the potential V ( r ) to be everywhere positive, and thus implyingstability for these wormholes too, the following lower bound has to be imposed for the discussed constant 1 > c > / − b ( r ) /r ≥ c < /
3, then V ( r ) becomes negative in the region near the throat of the wormhole, potentially leading to the instability of one ofthe l = 1 modes.To finalize this section we provide some examples of this class of stable Morris-Thorne wormholes in table 1. Theyare easily obtained utilizing equation (37) for the shape function. This process requires only of a well-behaved andbounded redshift function as input and so, can be used to yield as many space-times as functions that exist of this type.Note that the asymptotically flat with Φ( r ) = 0 case reduces to the well-known Ellis wormhole , which additionallyis a solution of the Einstein-scalar field equations with a negative sign. Unfortunately, since all of these wormholesbelong to the family of Morris-Thorne metrics, they violate the energy conditions at least near their throats.Interestingly enough, and though the l = 0 modes do not generate a gravitational perturbation, the potential V ( r )we deduce here reduces to that studied in [14, 15] for the Ellis metric when inserting the l = 0 value. In those worksthe instability of that wormhole follows due to their corresponding potential being negative. This indicates that theangular dependance of the solution proposed here is crucial to deduce stability. Of course, the reason why we obtaina different result lies in the type of perturbation we have analyzed during this work. VI. A STABLE PHANTOM SCALAR FIELD WORMHOLE
In section III we mentioned that the master equation derived there is valid for solutions of the Einstein-scalar fieldequations. In fact, one of the examples of stable Morris-Thorne wormholes shown in table 1 is indeed a solution ofthis type, namely the Ellis metric. In what follows we will present one last example of a stable wormhole supportedby a phantom scalar field, i.e., a solution to R µν = −∇ µ φ ∇ ν φ . This space-time was found in [21] and interpretedas a rotating scalar field wormhole. Here, we will focus on its static version since our master equation can only beapplied to that reduced form of the metric. Its line element in Boyer-Lindquist coordinates is To obtain its more familiar line element, the transformation from the radial coordinate r to r ∗ = ± q r − b is needed. In this case the r ∗ coordinate is the proper radial distance. ds = f dt − f (cid:2) dr + ( r − rr + r ) d Ω (cid:3) , with f = e − φ ( λ − π/ and λ = arctan h ( r − r ) / p r − r i . In this coordinate system we have for the Boyer-Lindquist radius that −∞ < r < ∞ , covering this way both universes. The quantities r and r are constantparameters whose units are that of length, and for which r > r . The scalar field is given by φ = p φ / λ − π/ φ a constant without units. In this wormhole the throat joins two asymptotically flat sides, nevertheless, thesesides are not symmetrical. This can be seen when taking the asymptotic limits of the f function,lim r →∞ f = 1 , lim r →−∞ f = e φ π . By rescaling the t and r coordinates to t − = e φ π/ t and r − = e − φ π/ r , it can be realized that indeed the otherside of the throat is asymptotically flat as well. The wormhole becomes symmetric only if φ = 0, in which case, theline element reduces to that of the Ellis metric. It results convenient to replace the coordinate r with x = ( r − r ) /L ,where L = r − r . Thus, ds = f dt − L f (cid:2) dx + ( x + 1) d Ω (cid:3) , (39)and λ = arctan x . In these coordinates the throat of the wormhole is located at x = 0, while the upper and loweruniverses are described by x > x <
0, respectively.To obtain the equation that governs the gravitational perturbations of this space-time we proceed with the samescheme as in the previous section. The spin coefficients and operators are κ + = − π + = i cot θ L s f x + 1) , π − + κ − = 2 x + Φ L ( x + 1) r f , π − − κ − = − Φ L ( x + 1) r f ,δ + = 1 L r f ∂∂x , δ − = iL s f x + 1) ∂∂θ , D = 1 √ √ f ∂∂t + 1 L sin θ r fx + 1 ∂∂ϕ ! . (40)They yield the following expression when inserted into the master equation (27), along with the assumption of asimilar ansatz as the one used throughout this paper Q = e iωt X ( x ) sin θdP l (cos θ ) /dθ , − ω f Q + l ( l + 1) fL ( x + 1) Q − √ fL ∂∂x (cid:18)p f ∂Q∂x (cid:19) + Φ f L ( x + 1) ∂Q∂x + 3 fL ( x + 1) (cid:18) φ x + φ − (cid:19) Q = 0 . Once again, a change of radial coordinate is needed to arrive at an eigenvalue equation as (38), this is, ddx ∗ = fL ddx . Since f is regular for all x ∈ R and because of the asymptotic form of said function at both infinities, the newcoordinate ranges over the values −∞ < x ∗ < ∞ . A suitable integration constant can also be picked so that thethroat is described by x ∗ = 0. As a result of this transformation, we have that d Xdx ∗ − (cid:0) V ( x ) − ω (cid:1) X = 0 , (41)where now V ( x ) = f L ( x + 1) (cid:20) l ( l + 1) + 3 x + 1 (cid:18) φ x + φ − (cid:19)(cid:21) . H = − d /dx ∗ + V ( x ) in the L ( R , dx ∗ )space of square-integrable functions X ( x ). Its properties are the same as those of the previous case in section V.Additionally, it can be easily verified that the second term that appears inside brackets in the expression of V ( x )has a global minimum u min = − x = − φ /
2. Hence, appealing to the fact that the l = 2vibrational modes are the lowest possible, the potential V ( x ) is strictly positive for all x ∈ R . By the same argumentsas those mentioned for the former class of Morris-Thorne wormholes, we can conclude that this scalar field wormholeis stable when perturbed by vacuum-like gravitational perturbations of odd-parity.As mentioned before, and just like in the class of Morris-Thorne metrics previously discussed, the Ellis space-timeis again a particular case of this phantom scalar field wormhole when the parameter φ vanishes . The wormholepresented here also violates the energy conditions as a result of it being a solution of the Einstein-scalar field equationswith a negative sign. CONCLUSIONS AND SOME ADDITIONAL COMMENTS
We have utilized the Newman-Penrose formalism to obtain a so-called master equation that describes the linearbehavior of gravitational perturbations in stationary and spherically symmetric space-times. The perturbations wereassumed to be vacuum-like and of odd-parity in the Regge-Wheeler gauge. This framework allowed us to write thederived master equation in a compact (and may we dare say elegant) manner through the use of the spin coefficientsand operators that characterize the formalism. Our master equation is not applicable, though, to the whole generalityof space-times with spherical symmetry, this is due to a constraint on certain components of the Ricci tensor thathas to be obeyed. Despite this, we showed that it is well-suited to analyze some interesting examples of metrics thatdescribe wormholes, for instance, the solutions of the Einstein-scalar field equations. Other space-times that werefound to be within the range of validity of our master equation belong to the family of Morris-Thorne wormholes.In fact, they define a particular class of wormholes that were concluded to be stable after we applied to them theaforementioned master equation. The explicit metric components of some of this type of space-times were presentedtoo. Finally, we gave one last example of a static scalar field wormhole that, according to the properties of itscorresponding master equation, is stable against the perturbations here studied.It should be borne in mind that, while our results indicate stability for some wormholes, it is only with respectto perturbations of odd-parity. Future developments of this work include the study of their even-parity counterpartswithin the Newman-Penrose formalism. However, the complexity of the calculations involved for this purpose increasescompared to the odd case. Another interesting aspect to determine is the possibility to generalize the scheme presentedhere for gravitational perturbations in the context of the tetrad formalism to axially symmetric space-times. This inturn implies a generalization of the Regge-Wheeler gauge to this kind of metrics. Yet again, the whole process mayrequire of lengthy calculations that hopefully are still manageable from an analytical approach.
APPENDIX A
Here we show all the relevant quantities of the Newman-Penrose formalism calculated for metric (9) with backgroundtetrad (11) and perturbation matrix (15). We will reference the equations of the NP paper from which our resultsare derived. To simplify notation, the use of the tilde for background quantities will be dropped and the hat will bekept for the perturbation terms. Thus, any quantity or operator without a hat should be understood to be of thebackground space-time, except for the perturbations functions f and f .From (NP 4.1a) and equation (8) of our text, the perturbation term of the spin coefficients is given byˆ κ = ˆ Z = Df − δf , ˆ π = − ˆ Z = 0 , ˆ ν = − ˆ Z = ∆ f − δ ∗ f , ˆ τ = ˆ Z = 0 , ˆ ρ = ˆ Z = ( − δ − + κ + − π + ) f , ˆ λ = − ˆ Z = 0 , For this particular case the relation between the proper radial length r ∗ and the coordinates x = x ∗ is Lx = r ∗ , with L = b . µ = − ˆ Z = ( δ − − κ + + π + ) f , ˆ σ = ˆ Z = 0 , ˆ α = 12 ( ˆ Z − ˆ Z ) = 12 ( νf − Df ) , ˆ β = 12 ( ˆ Z − ˆ Z ) = −
12 ( κf − Df ) , ˆ ε = 12 ( ˆ Z − ˆ Z ) = 12 (( − δ − + κ + − π + ) f − ∆ f ) , ˆ γ = 12 ( ˆ Z − ˆ Z ) = 12 (( δ − − κ + + π + ) f − Df ) , (A.1)with the definitions δ ± = ( δ ± δ ∗ ) / κ ± = ( κ ± ν ) /
2, and π ± = ( π ± τ ) /
2. In terms of the metric components thesenewly defined coefficients and operators take the explicit form κ + = − π + = i cot θ √ g , π − + κ − = − α = g ′ g √ g , π − − κ − = g ′ g √ g ,δ + = 1 √ g ∂∂r , δ − = i √ g ∂∂θ , (A.2)where a prime in this set of equations denotes derivation with respect to the radial coordinate r . Note that δ ∗ + = δ + , δ ∗− = − δ − , and that κ − , π − ∈ R while κ + , π + are purely imaginary. With this notation, identity (18) can be expressedas δ + κ + = 2 ακ + . (A.3)The linearized perturbed components of the Ricci tensor in tetrad form, sometimes called the Ricci identities, canbe computed by the Newman-Penrose equations (NP 4.2). Thereby, we obtain D ˆ ρ − δ ∗ ˆ κ = − κ ∗ ˆ τ − ˆ κ ∗ τ − κ (3 ˆ α + ˆ β ∗ − ˆ π ) − ˆ κ (3 α + β ∗ − π ) + ˆΦ , (NP 4.2a) D ˆ σ − δ ˆ κ = − ( τ − π ∗ + α ∗ + 3 β )ˆ κ − (ˆ τ − ˆ π ∗ + ˆ α ∗ + 3 ˆ β ) κ + ˆ ψ , (NP 4.2b) D ˆ α − δ ∗ ˆ ε = (ˆ ρ + ˆ ε ∗ − ε ) α + β ˆ σ ∗ − β ∗ ˆ ε − κ ˆ λ − κ ∗ ˆ γ + (ˆ ε + ˆ ρ ) π + ˆΦ , (NP 4.2d) D ˆ γ − ∆ˆ ε = (ˆ τ + ˆ π ∗ ) α + (ˆ τ ∗ + ˆ π ) β + τ ˆ π + ˆ τ π − ν ˆ κ − ˆ νκ + ˆΨ − ˆΛ + ˆΦ , (NP 4.2f) D ˆ λ − δ ∗ ˆ π = 2 π ˆ π + ( α − β ∗ )ˆ π + (ˆ α − ˆ β ∗ ) π − ν ˆ κ ∗ − ˆ νκ ∗ + ˆΦ , (NP 4.2g) D ˆ µ − δ ˆ π = π ˆ π ∗ + ˆ ππ ∗ − π (ˆ α ∗ − ˆ β ) + ˆ π ( α ∗ − β ) − ν ˆ κ − ˆ νκ + ˆΨ + 2 ˆΛ , (NP 4.2h)∆ˆ λ − δ ∗ ˆ ν = (3 α + β ∗ + π − τ ∗ )ˆ ν + (3 ˆ α + ˆ β ∗ + ˆ π − ˆ τ ∗ ) ν + ˆ ψ , (NP 4.2j) δ ˆ α − δ ∗ ˆ β = α ˆ α ∗ + ˆ αα ∗ + β ˆ β ∗ + ˆ ββ ∗ − α ˆ β − αβ − ˆΨ + ˆΛ + ˆΦ , (NP 4.2l) δ ˆ ν − ∆ˆ µ = − ν ∗ ˆ π − ˆ ν ∗ π + ( τ − β − α ∗ )ˆ ν + (ˆ τ − β − ˆ α ∗ ) ν + ˆΦ , (NP 4.2n) δ ˆ γ − ∆ ˆ β = ( τ − β − α ∗ )ˆ γ + ˆ µτ − ˆ σν − ˆ εν ∗ − β (ˆ γ − ˆ γ ∗ − ˆ µ ) + α ˆ λ ∗ + ˆΦ , (NP 4.2o) δ ˆ τ − ∆ˆ σ = ( τ + β − α ∗ )ˆ τ + (ˆ τ + ˆ β − ˆ α ∗ ) τ − κ ˆ ν ∗ − ˆ κν ∗ + ˆΦ , (NP 4.2p)where we have taken advantage of the property ˆ D m φ = 0 that our particular choice of tetrad gives us for arbitrarybackground scalars φ . We have also omitted the background terms that should appear on both sides of these equationssince they cancel each other out.The commutators (NP 4.4) of the background differential operators of the formalism are[∆ , D ] = 0 , [ δ, D ] = − π ∗ D + κ ∆ , [ δ, ∆] = − ν ∗ D + τ ∆ , [ δ ∗ , δ ] = 2 α ( δ − δ ∗ ) , (A.4)which can be utilized to derived the commutation relations for our previously introduced operators δ ± ,[ δ ± , D ] = ∓ π ∓ D + κ ∓ ∆ , [ δ ± , ∆] = κ ∓ D ∓ π ∓ ∆ , [ δ − , δ + ] = − αδ − . (A.5)When applying these commutators to ϕ -independent scalar quantities φ , as will always be the case in this work,there is a further simplification [ δ − , D ] φ = [ δ − , ∆] φ = 0, since Dφ = ∆ φ and κ + + π + = 0.6After some considerable algebraic steps, reduced equations for the linearized Ricci identities can be obtained byinserting the perturbed spin coefficients (A.1) into the (NP 4.2) equations presented above, along with the further aidof the commutators in (A.5) and the spin coefficient properties (12, 17). Doing so yieldsˆΦ = − ˆΦ = 12 (cid:2) ( δ + + κ − + 3 π − ) δ + − ( δ − + 2 κ + ) δ − + 4( κ − κ − ) (cid:3) f − ( δ + − α ) Df , ˆΦ = ˆΦ ∗ = − ˆΦ = − ˆΦ ∗ = 12 (cid:2) D + ( δ + + δ − + κ − + 3 π − + 4 κ + )( δ − − κ + ) (cid:3) f −
14 ( δ + + δ − + π − − κ − − κ + ) Df , ˆΦ = ˆΦ = ˆΦ ∗ = ˆΛ =0 . (A.6)For the Weyl scalars of interest we obtainˆ ψ = − ˆ ψ ∗ = 12 ( δ + + δ − − κ − + π − + 2 κ + ) [( δ + + δ − ) f − Df ] − κ + + κ − ) [ Df + ( κ + + κ − ) f ] , ˆ ψ = δ − Df + ( κ − δ − − κ + δ + ) f . (A.7) APPENDIX B
A more detailed proof of the consistency condition (25) of the linearized Einstein field equations under vacuum-likeperturbations is presented in this appendix. We also show that a background space-time that obeys the Einstein-scalarfield equations satisfies this condition.When applying the operator D to it, and using the commutators (A.4), the component ˆ R of the system (22)reduces to D ˆ R =2( δ + − α + 2 π − ) D f + (cid:2) ( δ − + 2 κ + ) δ − − ( δ + + 4 π − )( δ + + π − − κ − ) + 4( κ − − κ ) + 2(3Λ − Φ ) (cid:3) Df . (B.1)Expression (23) for the perturbation function f can now be substituted in (B.1). The resulting terms can berearranged as D ˆ R =2( δ + − α + 2 π − ) (cid:0) [ D + ( δ − + 4 κ + )( δ − − κ + ) − ( δ + − κ − + π − )( δ + + 2 π − )] f + 2 f [ δ − + 4 κ + ] κ + (cid:1) − Df (cid:2) ( δ + + κ − + 3 π − ) κ − + 2 κ −
3Λ + Φ (cid:3) . Careful attention must be paid on the order in which the operators are being applied. The previous equationcan be simplified by using (24) and defining the quantities A = 2( δ − + 4 κ + ) κ + − − Φ + Φ ), as well as B = ( δ + + κ − + 3 π − ) κ − + 2 κ −
3Λ + Φ . Despite the appearance of differential operators in these quantities, A and B should not be understood as such. They are merely scalar quantities, the operators δ ± in them are to be appliedonly to the spin coefficients κ ± . Hence, we can write D ˆ R = 2 [( δ + − α + 2 π − ) A − B ( δ + + 2 π − )] f . (B.2)Expanding the first term of (B.2) results in D ˆ R = 2 f [( δ + − α + 2 π − ) A − π − B ] + 2( A − B ) δ + f . (B.3)Using the background Ricci identities of the Newman-Penrose formalism (NP 4.2a) and (NP 4.2b), it can be proventhat A = B = 4 κ − ( ψ + ψ ∗ ) / − Φ − − Φ ). Another helpful identity, consequence of (A.3) and (A.5), is( δ + − α )( δ − + 4 κ + ) κ + = 0. Equation (B.3) thereby simplifies to7 D ˆ R = − f ( δ + − α )(3Λ − Φ + Φ ) . From the fact that on the previous equation only f is time-dependent, and since D contains a time derivative, itfollows that in order for the ˆ R component to vanish, the next condition has to hold( δ + − α )(3Λ − Φ + Φ ) = 0 . (B.4)This is the result that was anticipated in section III of the main text.Next we demonstrate that if R µν = ±∇ µ φ ∇ ν φ for a given static and spherically symmetric (even axially symmetric)space-time, which is true for the Einstein field equations minimally coupled to a massless scalar field φ without aself-interaction potential, then (B.4) is satisfied when the null tetrad is given by (11).Let { x µ } be a suitable coordinate system chosen so that ∂/∂x and ∂/∂x are the Killing vector fields associatedto the symmetry of a static space-time, be it spherical or axial. This implies that the metric g µν , as well as itscorresponding Ricci tensor, depend at most on the x and x coordinates. If R µν = ±∇ µ φ ∇ ν φ , then consequently φ = φ ( x , x ) and so, the components R ij vanish ( i, j = 0 , { l µ , n µ , m µ , ¯ m µ } in which thevectors l µ and n µ are appropriate combination of the Killing vectors ∂/∂x and ∂/∂x , we have that R µν l µ n ν =2(3Λ − Φ ) = 0. In the first equality sign we have used the (NP 4.3b) equations, the second equality is due tothe components R ij being zero . Furthermore, Φ = − R µν l µ l ν / − Φ + Φ = 0, and (B.4) holds.One last comment may be in order regarding the perturbations discussed throughout this paper in the context ofthis kind of space-times. Their field equations are of the type R µν = ±∇ µ φ ∇ ν φ, ∇ µ ∇ µ φ = 0 . (B.5)In the case of spherically symmetric space-times, both of this equations can be shown to hold up to first order ofthe metric perturbation (13), with the additional condition that one does not add a perturbation term to the scalarfield φ , i.e., ˆ φ = 0. We assume, of course, that the background metric satisfies its particular set of field equations.Hence, the equal sign holds for the zero order terms of the linearized field equations (B.5). Since we are consideringthe vacuum-like property ˆ R µν = 0 too, the linearized version of the first equation is evidently satisfied when ˆ φ = 0.For the second field equation this is not so easily seen, but is equally trivial. Its first order term is given by[ ∇ µ ∇ µ φ ] (1) = h µν ( ∂ µν − e Γ αµν ∇ α ) φ − e g µν ˆΓ αµν ∇ α φ, (B.6)where again a tilde denotes background quantities and a hat their respective perturbations, also φ = e φ . All ofthe terms in the past equation vanish when performing their corresponding index contractions, this is mainly due tothe exclusive r -dependency of the scalar field φ , combined with the zero entries of the background and perturbationinverse metrics, along with the vanishing of the following connection coefficientsˆΓ rµµ = e Γ rtϕ = e Γ rϕt = e Γ rrϕ = e Γ rϕr = 0 . (B.7)Thus, [ ∇ µ ∇ µ φ ] (1) = 0 and the vacuum-like perturbations of odd-parity studied in this paper are compatible withthe linearized field equations (B.5). Acknowledgments.
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