On the generalization of McVittie's model for an inhomogeneity in a cosmological spacetime
aa r X i v : . [ g r- q c ] A ug On the generalization of McVittie’s modelfor an inhomogeneity in a cosmological spacetime
Matteo Carrera ∗ Institute of Physics, University of Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany
Domenico Giulini † University of Hannover, Appelstrasse 2, D-30167 Hannover, Germany ‡ (Dated: August 21, 2009)McVittie’s spacetime is a spherically symmetric solution to Einstein’s equation with an energy-momentum tensor of a perfect fluid. It describes the external field of a single quasi-isolated objectwith vanishing electric charge and angular momentum in an environment that asymptotically tendsto a Friedmann–Lemaˆıtre–Robertson–Walker universe. We critically discuss some recently proposedgeneralizations of this solution, in which radial matter accretion as well as heat currents are allowed.We clarify the hitherto unexplained constraints between these two generalizing aspects as beingdue to a geometric property, here called spatial Ricci-isotropy, which forces solutions covered bythe McVittie ansatz to be rather special. We also clarify other aspects of these solutions, likewhether they include geometries which are in the same conformal equivalence class as the exteriorSchwarzschild solution, which leads us to contradict some of the statements in the recent literature. PACS numbers: 98.80.Jk, 04.20.Jb
Contents
I. Introduction II. The McVittie model III. Geometry of the McVittie ansatz
IV. Attempts to generalize McVittie’s model
V. Conclusion Acknowledgments A. Proof of Proposition 1 B. Proof of Proposition 2 C. Shear-free observer fields in sphericallysymmetric spacetimes ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Also at: ZARM, University of Bremen, Am FallturmD-28359 Bremen, Germany
References I. INTRODUCTION
Two sets of exact solutions to Einstein’s field equa-tion of General Relativity are of paradigmatic impor-tance: The first set describes the gravitational field ofquasi-isolated objects in an asymptotically flat space-time. Among them is the exterior Schwarzschild solutionthat describes the stationary gravitational field outsidea spherically symmetric star or black hole of mass m with vanishing intrinsic angular momentum (spin) andvanishing electric charge. (The latter two features beingincluded in the three-parameter Kerr–Newman family ofsolutions.) Such asymptotically flat solutions are meantto apply to a region outside the central object which,on one hand, must be sufficiently far from the consideredobject, so as to legitimately neglect small irregularities ofits surface and/or small deviations from perfect spheri-cal symmetry. On the other hand, and more importantly,the region of applicability must also be sufficiently closeto the considered object in order not to include, or comeclose to, other compact sources, or not contain too muchdust-filled space between it and the object which wouldalso act as disturbing source for the gravitational field.In particular, the large-distance asymptotic behavior ofsuch solutions is an idealization and not meant to bestrictly that of any object in the real world.On the other hand, the second set of paradig-matic solutions are the cosmological ones, which aimto model the behavior of space-time at the largestcosmological scales, without trying to be realistic atsmaller scales. Among them is the family of homo-geneous and isotropic Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) cosmologies on which the cosmologicalstandard-model is based.Given that situation, the task is to combine the virtuesof both classes of solutions without the correspondingdeficiencies. This means to find exact solutions for thegravitational field of a compact object ‘immersed’ (seebelow) into an otherwise cosmological background. Thiswould appear to be an easy task if the field equationswere linear, for, in that case, one would just add the so-lution that describes the gravitational field of a compactobject in an otherwise empty universe to the cosmologicalsolution that corresponds to a homogeneous distributionof background matter. Here the mathematical opera-tion of addition appears to be the obvious realization ofwhat one might be tempted to call ‘simultaneous physicalpresence’ and hence, in view of the individual interpre-tations of both solutions, the ‘immersion’ of the compactobject into the cosmological background. But this imme-diate interpretation in physical terms of a simple mathe-matical operation is deceptive. This becomes obvious innon-linear theories, like General Relativity (GR), whereno simple mathematical operation exists that producesa new solution out of two old ones and where the verysame physical question may still be asked.The proper requirement for a mathematical represen-tation of the envisaged physical situation must, first ofall, consist in asymptotic conditions which ensure thatthe sought for solution approximates the given (e.g.Schwarzschild) one for small distances and a particu-lar cosmological one (e.g. FLRW) for large distances.Second, it must specify somehow the physics in the in-termediate region. Usually this will include a specifica-tion of the matter components and their dynamical lawstogether with certain initial and boundary conditions.Needless to say that this will generally result in a complexsystem of partial differential equations. Most analyticapproaches therefore impose further simplifying assump-tions that automatically guarantee the right asymptoticbehavior and at the same time reduce the free functionsto a manageable number.In this paper we will discuss a particular such ap-proach, which is originally due to McVittie [12] and whichhas been further analyzed and clarified in a series of care-fully written papers by Nolan [15, 16, 17]. Our mainmotivation is that recently McVittie’s solution has beenseverely criticized as not being able at all to model theenvisaged situation[3, 6], whereas a family of slightly gen-eralized ones [4], in which some restrictions concerningthe motion of matter and the existence of heat flows islifted, is argued to be free of the alleged problems. Theexistence of an exact solution to Einstein’s equation thatmodels local inhomogeneities is clearly of great impor-tance, for example in estimating reliable upper boundsto the possible influence of global cosmological expansiononto the dynamics and kinematics of local systems [1].The paper is organized as follows: In Section II we re-view what we call the McVittie model . We discuss itsmetric ansatz and what its entails regarding the geom-etry of spacetime. Then we discuss the assumptions re- garding the motion of the matter and how this, togetherwith Einstein’s equation, determines one of the two freefunctions in the metric ansatz as a simple function of theother. We interpret this condition in terms of an appro-priate concept of local mass as saying that the objectdoes not accrete mass from the ambient matter. In Sec-tion III we take a second and closer look at the McVittieansatz and note some of its characteristic features which,we feel, have not sufficiently carefully been taken into ac-count in [3, 4, 6]. In the light of these observations wethen discuss in Section IV the attempted generalizationsof McVittie’s solution in the references just mentioned.We find that some of the conclusions drawn are indeedunwarranted.
II. THE MCVITTIE MODEL
The characterization of the McVittie model is madethrough two sets of a priori specifications. The first setconcerns the metric (left side of Einstein’s equations) andthe second set the matter (right side of Einstein’s equa-tions). The former consists in an ansatz for the metric,which can formally be described as follows: Write downthe Schwarzschild metric for the mass parameter m inisotropic coordinates, add a conformal factor a ( t ) to thespatial part, and allow the mass parameter m to dependon time. Hence the metric reads g = (cid:18) − m ( t ) / r m ( t ) / r (cid:19) d t − (cid:18) m ( t )2 r (cid:19) a ( t ) ( d r + r g S ) , (1)where g S = d θ + sin θ d ϕ is the standard metric onthe unit 2-sphere. Here we restricted attention to theasymptotically spatially flat (i.e. k = 0) FLRW metric,which is compatible with current cosmological data [11].For simplicity we shall refer to (1) simply as McVittie’sansatz , though this is not quite correct since McVittiestarted from a general spherically symmetric form andarrived at (1) with m ( t ) a ( t ) = const . after imposing acondition that he interpreted as condition for no matter-infall. The ansatz (1) is obviously spherically symmetricwith the spheres of constant radius r being the orbits ofthe rotation group. In the next section we will discussin more detail the geometric implications of this ansatz,independent of whether Einstein’s equation holds. “Spherical symmetry” of a spacetime means the following: Thereexists an action of the group SO (3) on spacetime by isometries,which is such that the orbits are either two-dimensional andspacelike or fixed points. The “spheres” implicitly referred to inthis term correspond to the two-dimensional orbits, even thoughthey might in principle also be two-dimensional real projectivespaces. In the cases we discuss here they will be 2-spheres. As already discussed in the introduction, the modelhere is meant to interpolate between the spherically sym-metric gravitational field of a compact object and theenvironment. It is not to be taken too seriously in theregion very close to the central object, where the basicassumptions on the behavior of matter definitely turnunphysical. However, as discussed in [1], at radii muchlarger than (in geometric units) the central mass (to bedefined below) the k = 0 McVittie solution seems to pro-vide a viable approximation for the envisaged situation.The second set of specifications, concerning the matter,is as follows: The matter is a perfect fluid with density ̺ and isotropic pressure p . Hence its energy-momentumtensor is given by T = ̺ u ⊗ u + p ( u ⊗ u − g ) . (2)Furthermore, and this is where the two sets of specifi-cations make contact, the motion of the matter (i.e. itsfour-velocity field) is given by u = e , (3)where e is the normalization of ∂ / ∂ t (compare (10)).Finally, the explicit cosmological constant on the left-hand side of Einstein’s equation is assumed to be zero,which implies no loss of generality, since a non-zero cos-mological constant can always be regarded as special partof the matter’s energy-momentum tensor (compare IV).No further assumptions are made. In particular, an equa-tion of state, like p = p ( ̺ ), is not assumed. The reasonfor this will become clear soon. Later generalizations willmainly concern (2) and (3).The Einstein equation now links the specificationsof geometry with that of matter. It is equivalent tothe following three relations between the four functions m ( t ) , a ( t ) , ̺ ( t, r ), and p ( t, r ):( a m )˙ = 0 , (4a)8 π̺ = 3 (cid:18) ˙ aa (cid:19) , (4b)8 πp = − (cid:18) ˙ aa (cid:19) − (cid:18) ˙ aa (cid:19) ˙ (cid:18) m/ r − m/ r (cid:19) . (4c)Note that here Einstein’s equation has only three inde-pendent components (as opposed to four for a generalspherically symmetric metric), which is a consequenceof the fact that the Einstein tensor for the McVittieansatz (1) is spatially isotropic. This will be discussed inmore detail in the next section. Here and in what follows we denote the metric-dual (1-form)of a vector u by underlining it, that is, u := g ( u , · ) is the 1-form metric-dual to the vector u . In local coordinates we have u = u µ ∂ µ and u = u µ d x µ , where u µ := g µν u ν . We speak of “the Einstein equation” in the singular since wethink of it as a single tensor equation, which only upon introduc-ing a coordinate system decomposes in many scalar equations.
Equation (4a) can be immediately integrated: m ( t ) = m a ( t ) , (5)where m is an integration constant. Below we will showthat this integration constant is to be interpreted as themass of the central body.Clearly the system (4) is under-determining. This isexpected since no equation of state has yet been imposed.The reason why we did not impose such a condition cannow be easily inferred from (4): whereas (4b) impliesthat ̺ only depends on t , (4c) implies that p dependson t and r iff ( ˙ a/a )˙ = 0. Hence a non-trivial relation p = p ( ̺ ) is simply incompatible with the assumptionsmade so far. The only possible ways to specify p are p = 0 or ̺ + p = 0. In the first case (4c) implies that˙ a/a = 0 if m = 0 (since then the second term on theright-hand side is r dependent, whereas the first is not,so that both must vanish separately), which correspondsto the exterior Schwarzschild solution, or a ( t ) ∝ t / if m = 0, which leads to the flat FLRW solution withdust. In the second case the fluid just acts like a cos-mological constant Λ = 8 π̺ (using the equation of state ̺ + p = 0 in div T = 0 it implies d p = 0 and this, inturn, using again the equation of state, implies d ̺ = 0)so that this case reduces to the Schwarzschild–de Sittersolution. To see this explicitly, notice first that (4b,4c)imply the constancy of H = ˙ a/a = p Λ / a ( t ) = a exp (cid:0) t p Λ / (cid:1) . With such a scale-factor the McVittie metric (1) with (5) turns into theSchwarzschild–de Sitter metric in spatially isotropic coor-dinates. The explicit formulae for the coordinate trans-formation which brings the latter in the familiar form canbe found in Section 5 of [18] and also in Section 7 of [10].Finally, note from (4a) that constancy of one of the func-tions m and a implies constancy of the other. In this case(4b,4c) imply p = ̺ = 0, so that we are dealing with theexterior Schwarzschild spacetime.A specific McVittie solution can be obtained by choos-ing a function a ( t ), corresponding to the scale functionof the FLRW spacetime which the McVittie model isrequired to approach at spatial infinity, and the con-stant m , corresponding to the ‘central mass’. Rela-tions (4b,4c), and (5) are then used to determine ̺ , p ,and m , respectively. Clearly this ‘poor man’s way’ tosolve Einstein’s equation holds the danger of arrivingat unrealistic spacetime dependent relations between ̺ and p . This must be kept in mind when proceeding inthis fashion. For further discussion of this point we referto [15, 16].As will be discussed in more detail in Section III C be-low, in the spherically-symmetric case the concept of lo-cal mass (or energy) is well captured by the Misner–Sharp(MS) energy [13], whose purely geometric definition interms of Riemannian curvature allows to decompose itinto a sum of two terms, one of which comes from theRicci- the other from the Weyl curvature. It is the latterwhich may be identified with the gravitational mass ofthe central object. Applied to (1), the Weyl contributionto the MS energy can be written in the following form,also taking into account (5), E R = 4 π R ̺ , (6a) E W = m . (6b)The constancy of E W is then interpreted as saying thatno energy is accreted from the ambient matter onto thecentral object.We now briefly discuss the basic properties of the mo-tion of cosmological matter. Being spherically symmet-ric, the velocity field u specified in (3) is automaticallyvorticity free. The last property is manifest from its hy-persurface orthogonality, which is immediate from (1).Moreover, u is also shear free. This, too, can be imme-diately read off (1) once one takes into account the fol-lowing result, whose proof we sketch in Appendix C: Aspherically symmetric normalized timelike vector field u in a spherically symmetric spacetime ( M , g ) is shear freeiff its corresponding spatial metric, that is, the metric g restricted to the subbundle u ⊥ := { v ∈ T M | g ( v , u ) =0 } , is conformally flat. The metric (1) obviously is spa-tially conformally flat with respect to the choice (3) madehere. Moreover, the expansion (divergence) of u is θ = 3 H , (7)where H := ˙ a/a , just as in the FLRW case. In par-ticular, the expansion of the cosmological fluid is homo-geneous in space . Exactly as in the FLRW case is alsothe expression for the variation of the areal radius alongthe integral lines of u (that is the velocity of cosmologi-cal matter measured in terms of its proper time and theareal radius): u ( R ) = HR , (8)which is nothing but Hubble’s law. Recall that for aspherically symmetric spacetime the areal radius , de-noted here by R , is the function defined by R ( p ) := p A ( p ) / π , where A ( p ) is the proper area of the 2-dimensional SO (3)-orbit through the point p . For theMcVittie spacetime the areal radius is given explicitlyin (13). The acceleration of u , which in contrast to theFLRW case does not vanish here, is given by ∇ u u = m R (cid:18) m/ r − m/ r (cid:19) e . (9)Here e is the normalized vector field in radial directionas defined in (10). In leading order in m /R this corre-sponds to the acceleration of the observers moving alongthe timelike Killing field in Schwarzschild spacetime.It is also important to note that the central gravita-tional mass in McVittie’s spacetime may be modeled by ashear-free perfect-fluid star of positive homogeneous en-ergy density [14]. The matching is performed along aworld-tube comoving with the cosmological fluid, across which the energy density jumps discontinuously. Thismeans that the star’s surface is comoving with the cos-mological fluid and hence, in view of (7), that it geomet-rically expands (or contracts). This feature, however,should be merely seen as an artifact of the McVittiemodel (in which the relation (7) holds), rather than ageneral property of compact objects in any cosmologicalspacetimes. Positive pressure within the star seems tobe only possible if 2 a ¨ a + ˙ a < a = exp( β/ q > / III. GEOMETRY OF THE MCVITTIE ANSATZ
In this section we will discuss the geometry of the met-ric (1) independent of the later restriction that it willhave to satisfy Einstein’s equation for some reasonableenergy-momentum tensor. This means that at this pointwe shall not assume any relation between the two func-tions m ( t ) and a ( t ), apart from the first being non neg-ative and the second being strictly positive. We willdiscuss the metric’s ‘spatial Ricci-isotropy’ (a term ex-plained below), its singularities and trapped regions, andalso compute its Misner–Sharp energy decomposed intothe Ricci and Weyl parts. We shall start, however, by an-swering the question of what the overlap is between thegeometries represented by (1) and the conformal equiva-lence class of the exterior Schwarzschild geometry. A. Relation to conformal Schwarzschild class
This question is an obvious one in view of the wayin which (1) is obtained from the exterior Schwarzschildmetric. It is clear that for m = m = const . themetric (1) is conformally equivalent to the exteriorSchwarzschild metric, since upon using a new time co-ordinate T with dT = dt/a ( t ) we can pull out a ( t ) asa common conformal factor. The following proposition,whose proof we shall give in Appendix A, states that aconstant m is in fact also necessary condition: Proposition 1.
Let S McV denote the set of metrics inthe form of the McVittie ansatz (1) (parametrized by thetwo positive functions a and m ) and S cS the set of met-rics conformally equivalent to an exterior Schwarzschildmetric (parametrized by a positive conformal factor anda constant positive Schwarzschild mass M ). Then theintersection between S McV and S cS is given by the subsetof metrics in S McV with constant m or, equivalently, bythe subset of metrics in S cS whose conformal factor hasa gradient proportional to the Killing field ∂ / ∂ T of theSchwarzschild metric (see (A1b) for notation). Note that we excluded the ‘trivial’ cases in which m or M (or both) vanish for the following reason: Comparingthe expressions for the Weyl part of the MS energy ofthe two types of metrics (see (A10) in Appendix A) itfollows that m vanishes iff M does and this, in turn,leads to a metric conformally related to the Minkowskimetric where the conformal factor depends only on time,that is, a FLRW metric. But such a spacetime, beinghomogeneous, is not of interest to us here.In particular, Proposition 1 implies that the metric ofSultana and Dyer [19] are not of type (1), as suggestedin Section IV A of [4] and allegedly shown in Section IIof [2] (cf. our footnote 5 at page 8). This immediatelyfollows from the observation that the conformal factor,expressed as function of the standard Schwarzschild co-ordinates that appear in (A1b), is given by Ω( T, R ) =( T + 2 M ln( R/ M − (compare Eqs. (8) and (9)of [19]), which also depends on R and hence does notsatisfy the condition of Proposition 1. We will have tosay more about this at the beginning of Section IV andin Section IV D. B. Spatial Ricci-isotropy
An important feature of any metric that is coveredby the ansatz (1) is, that its Einstein tensor is spatiallyisotropic in the following sense: ‘Spatially’ refers to thedirections orthogonal to ∂ / ∂ t and ‘isotropy’ to the con-dition that the spatial restriction of the spacetime’s Ein-stein tensor is proportional to the spatial restriction ofthe metric. Note that, since the spacetime’s metric istime dependent, the spatial restriction of the spacetime’sEinstein or Ricci tensor is not the same as the Einsteinor Ricci tensor of the spatial sections with their inducedmetrics. Hence the notion of spatial isotropy of the Ein-stein tensor used here is not the same as saying that theinduced metric of the slices is an Einstein metric.Given that the Einstein tensor of (1) is spatiallyisotropic in the sense used here, it is then obvious thatEinstein’s equation will impose a severe restriction uponthe matter’s energy-momentum tensor, saying that it,too, must be spatially isotropic. The degree of special-ization implied by this will be discussed in more detailbelow. Here we only remark that this observation al-ready answers in the negative a question addressed, andleft open, in the last paragraph of [3], of whether (1)is the most general spherically symmetric solution de-scribing a black hole embedded in a spatially flat FLRWbackground: It clearly is not.In passing we make the obvious remark that, since Ein = Ric − (1 /
2) Scal g , where Ric denotes the Riccitensor, the Einstein tensor is spatially isotropic iff thesame holds for the Ricci tensor. For this reason we willfrom now on refer to spatial Ricci-isotropy to denote thefeature in question.Now, a way to actually show spatial Ricci-isotropy isto compute the components of the Einstein tensor withrespect to the orthonormal tetrad { e µ } µ ∈{ , ··· , } of (1)defined by e µ := k ∂ / ∂ x µ k − ∂ / ∂ x µ , (10) where { x µ } = { t, r, θ, ϕ } . Here, and henceforth, we write k v k := p | g ( v , v ) | . Note that e , e are orthogonal toand e , e tangent to the 2-spheres of constant radius r .The non-vanishing independent components of the Ein-stein tensor with respect to the orthonormal basis (10)are: Ein ( e , e ) = 3 F , (11a) Ein ( e , e ) = R (cid:0) AB (cid:1) ( a m )˙ , (11b) Ein ( e i , e j ) = − (cid:16) F + 2 AB ˙ F (cid:17) δ ij , (11c)where an overdot denotes differentiation along ∂ / ∂ t . Be-fore explaining the functions A , B , R , and F , note thatthe spatial isotropy of the Einstein tensor follows imme-diately from (11c), since Ein ( e i , e j ) ∝ δ ij . In (11) andin the following we set: A ( t, r ) := 1 + m ( t ) / r , B ( t, r ) := 1 − m ( t ) / r , (12)and R ( t, r ) = (cid:18) m ( t )2 r (cid:19) a ( t ) r , (13)where R is the areal radius for the McVittie ansatz (1),and also F := ˙ aa + 1 rB ( a m ) · a . (14)In passing we note that both quantities, F and a m , thatappear in the components of the Einstein tensor, havea geometrical interpretation: the former is one third theexpansion of the vector field e , that is, F = div ( e ) / e is free ofvorticity and shear. Hence, taking into account the rela-tion (C5) between the expansion θ and the shear scalar σ of an arbitrary spherically-symmetric observer field, theexpansion of e can be simply written as 3 d R ( e ) /R sothat F may be expressed as F = d R ( e ) /R . (15)In order to estimate the degree of specialization im-plied by spatial Ricci-isotropy, we ask for the most gen-eral spherically symmetric metric for which this is thecase. To answer this, we first note that any sphericallysymmetric metric can always be written in the form g = (cid:18) B ( t, r ) A ( t, r ) (cid:19) d t − a ( t ) A ( t, r )( d r + r g S ) . (16)This reduces to McVittie’s ansatz (1) if A, B are givenby (12). For the general spherically symmetric metric(16), spatial Ricci-isotropy can be shown to be equivalentto δ ( AB ) − δA )( δB ) = 0 , (17)where δ := r − ∂/∂r = 2 ∂/∂r . It is obvious that thereare many more solutions to this differential equation thanjust (12). C. Misner–Sharp energy
In order to be able to interpret (1) as an ansatz for aninhomogeneity in a FLRW universe, it is useful to com-pute the Misner–Sharp (MS) energy and, in particular,its Ricci and Weyl parts. This concept of quasi-localmass, which is defined only for spherically symmetricspacetimes, and which in this case coincides with Hawk-ing’s more general definition [8] of quasi-local mass (seee.g. [1]), allows to detect localized sources of gravity.We recall the geometric definition of the MS energy [9,13]: E := − R K , (18)where R denotes the areal radius and K the extrinsiccurvature. More precisely, the equation should be readand understood as follows: First of all, the quantities R and K , and hence also E , are real-valued functionson spacetime. In order to determine their values at apoint p , recall that, due to the requirement of sphericalsymmetry, there is a unique two-(or zero-) dimensional SO (3) orbit S ( p ) through p . The value of R at p is asexplained below Eq. (8) and the value of K at p is K ( p ) := Riem ( X p , Y p , X p , Y p ) g ( X p , X p ) g ( Y p , Y p ) − (cid:0) g ( X p , Y p ) (cid:1) . (19)Here Riem is the (totally covariant) Riemannian cur-vature tensor of spacetime and X p and Y p are any twolinearly independent vectors in the tangent space at p which are also tangent to the orbit S ( p ). Note that theright-hand side only depends of the plane spanned by X p , Y p and not on the vectors spanning it. Finally wenote that the minus sign in (18) is just a relict of oursignature choice (mostly minus).From the curvature decomposition for a sphericallysymmetric metric (see [1]) one can rewrite (18) in theform E = R (cid:0) g ( ∇ R, ∇ R ) (cid:1) , (20)where ∇ R denotes the gradient vector-field of R . Thisprovides a convenient expression for the computation ofthe MS energy. For a self-contained review of the basicproperties of the MS energy as well as its interpretationas the amount of active gravitational energy containedin the interior of the spheres of symmetry ( SO (3)-orbits)and its relation with the other mass concepts, see [1].The decomposition of the Riemann tensor into a Ricciand a Weyl part leads, together with (18), to a naturaldecomposition of the MS energy into a Ricci and Weylpart (see also [1]). For the Ricci part of the MS energyof (1) we get E R = R Ein ( e , e ) = R d R ( e )) . (21)The first equality in (21) can be derived by merely usingthe spatial Ricci-isotropy in the expression for the Ricci part of the Riemann tensor. The second equality followsthen with (11a) and (15). The Weyl part can now beobtained as the difference between the full MS energy and(21). We use the expression (20) for the former and write g ( ∇ R, ∇ R ) = (cid:0) e ( R ) (cid:1) − (cid:0) e ( R ) (cid:1) . The part involving e ( R ) equals the Ricci part (21) and hence the Weyl partis given by ( R/ (cid:0) − ( e ( R )) (cid:1) . From (13) we calculate e ( R ) and hence obtain for the Weyl part of the MSenergy: E W = a m . (22)The Ricci part of the MS energy is that part which, viaEinstein’s equation, can be locally related to the mat-ter’s energy-momentum tensor, whereas the gravitationalmass of the central object is contained in the Weyl partof the MS energy. Notice that the latter is spatially con-stant (the functions a and m in (22) only depend on time)but may depend on time. If the latter is the case we in-terpret this as saying that the central mass exchangesenergy with the ambient matter. D. Singularities and trapped surfaces
Next we comment on the singularity properties of theMcVittie ansatz (1). From (11c) one suspect, because ofthe term proportional to 1 /B , a singularity in the Riccipart of the curvature at r = m/ R = 2 am =2 E W ). In fact, this corresponds to a genuine curvaturesingularity, as one can see from looking, for example, atthe following expression for the scalar curvature (i.e. theRicci scalar), Scal = − F − AB ˙ F , (23)which can be quickly computed from (11). In Appendix Bwe insert into this expression the definition (14) of F andexpand this in powers of 1 / ( rB ). This allows to prove Proposition 2.
The Ricci scalar for a metric of the form(1) becomes singular in the limit r → m/ for any func-tions a and m , except for the following three special cases: ( i ) m = 0 and a arbitrary (FLRW), ( ii ) a and m are constant (Schwarzschild), and ( iii ) ( a m ) · = 0 and ( ˙ a/a ) · = 0 (Schwarzschild–de Sitter). This means that, as long as we stick to the ansatz (1), at r = m/ /B in front of thetime derivatives by writing ( A/B ) ∂ / ∂ t as e and thenargue, as was done in [4], that this eliminates the singu-larity. The point is simply that then e applied to anycontinuously differentiable function diverges as r → m/ ∂ / ∂ t , which become lightlike in the limit as r tends to m/
2. Their acceleration is given by the gradientof the pressure, which necessarily diverges in the limit r → m/
2, as one explicitly sees from (9). For a moredetailed study of the geometric singularity at r = m/ − /R times the Weyl part of the MS energy,by the very definition of the latter (see [1]). The squareof the Weyl tensor for the ansatz (1) may then be conve-niently expressed as h Weyl , Weyl i = 48 ( a m ) R . (24)This shows that R = 0 also corresponds to a genuinecurvature singularity, though this is not part of the regioncovered by our original coordinate system, for which r >m/ R > E W ).It is instructive to also determine the trapped regionsof McVittie spacetime. Recall that a spacelike 2-sphere S is said to be trapped, marginally trapped , or untrapped if the product θ + θ − of the expansions (for the defini-tions see e.g. [1]) for the ingoing and outgoing future-pointing null vector fields normal to S is positive, zero,or negative, respectively. Taking S to be S R , that is,an SO (3) orbit with areal radius R , it immediately fol-lows from the relation 2 θ + θ − = g ( ∇ R, ∇ R ) /R (see [1])that S R is trapped, marginally trapped, or untrapped iff g ( ∇ R, ∇ R ) is positive, zero, or negative, respectively.This corresponds to timelike, lightlike, or spacelike d R ,or equivalently, in view of (20), to 2 E − R being posi-tive, zero, or negative, respectively. Using (21) togetherwith (11a), the MS energy for the McVittie ansatz canbe written as E = E W + R F /
2, so that2 E − R = F R − R + R S . (25)Here we defined the ‘Schwarzschild radius’ as R S := 2 E W ,which generally will depend on time. We wish to de-termine the values of the radial coordinate ( r or R ) atwhich the expression (25) assumes the value zero. Weshall continue to work with R rather than r since R hasthe proper geometric meaning of areal radius. In theregion we are considering (that is r > m/ R > R S ) the inversion of (13) reads r ( R ) = R (cid:0) − R S / R + p − R S /R (cid:1) / a , so that (25) divided by R S can be written in the form2 E − RR S = η + εx − p x ( x − ! x − x + 1 . (26)Here we introduced the dimensionless radial coordinate x := R/R S and the (small) parameters ε := ˙ R S and η := R S /R H , where R H := 1 /H denotes the ‘Hubbleradius’. Recall that since R > R S we have x > ε = ˙ R S = 0.Then (26) turns into a cubic polynomial in x which ispositive for x = 0 and tends to ±∞ for x → ±∞ . Henceit always has a negative zero (which does not interest us)and two positive zeros iff R S /R H < / √ ≈ . . (27)This clearly corresponds to the physical relevant casewhere the Schwarzschild radius is much smaller thanthe Hubble radius. One zero lies in the vicinity ofthe Schwarzschild radius and one in the vicinity of theHubble radius, corresponding to two marginally trappedspheres. The exact expressions for the zeros can be easilywritten down, but are not very illuminating. In leadingorder in the small parameter η = R S /R H , they are ap-proximated by R = R S (cid:0) η + O ( η ) (cid:1) , (28a) R = R H (cid:0) − η/ O ( η ) (cid:1) . (28b)From this one sees that for the McVittie ansatz the radiusof the marginally trapped sphere of Schwarzschild space-time ( R S ) increases and that of the FLRW spacetime( R H ) decreases. The first feature can, for the McVit-tie model, be understood as an effect of the cosmologicalenvironment, whereas the latter is an effect of the inho-mogeneity in form of a central mass abundance. All thespheres with R < R or R > R are trapped and thosewith R < R < R are untrapped. In particular, thesingularity r = m/
2, that is R = 2 E W = R S , lies withinthe inner trapped region.In the case in which ε = ˙ R S is non-zero and ‘small’(see below in which sense), we expect that the zeros (28)vary smoothly in ε so that, in particular, the singularityat R = R S still remains within the inner trapped region.An expansion in ε gives, for the zero in the vicinity ofthe Schwarzschild radius: R ( ε ) = R S (cid:0) η + (2 − η + 13 η ) ε + O ( η , ε ) (cid:1) , (29)which clearly reduces to (28a) for ε = 0. From this ex-pression one sees that, according to the physical expec-tation, in case of accretion ( ε >
0) the inner marginallytrapped sphere becomes larger in area, whereas in the op-posite case ( ε <
0) it shrinks. In our approximation (29),the singularity R = R S continues to lie inside the trappedregion for ‘accretion rates’ ε = ˙ R S > − η / c ,for ˙ R S /c > − ( R S /R H ) /
2. However, this also character-izes the region of validity of the expansion (29): Givena positive η , an expansion in ε around zero exists onlyfor ε > − η / ε, η ) around (0 ,
0) (this is because the partial derivativeof (26) with respect to x does not exist at x = 1). E. Other global aspects
Another aspect concerns the global behavior of theMcVittie ansatz (1). We note that each hypersurfaceof constant time t is a complete Riemannian manifold,which, besides the rotational symmetry, admits a dis-crete isometry given in ( r, θ, ϕ ) coordinates by φ ( r, θ, ϕ ) = (cid:0) ( m/ r − , θ , ϕ (cid:1) . (30)This corresponds to an inversion at the 2-sphere r = m/
2, which shows that the hypersurfaces of constant t can be thought of as two isometric asymptotically-flatpieces joined together at the 2-sphere r = m/
2. This 2-sphere is totally geodesic since it is a fixed-point set of anisometry; in particular, it is a minimal surface. Exceptfor the time-dependent factor m ( t ), this is just like forthe slices of constant Killing time in the Schwarzschildmetric (the difference being that (30) does not extend toan isometry of the spacetime metric unless ˙ m = 0). Now,the fact that r → t = const . implies thatthe McVittie metric cannot literally be interpreted ascorresponding to a point particle sitting at r = 0 ( r = 0is in infinite metric distance) in an otherwise spatiallyflat FLRW universe, just like the Schwarzschild metricdoes not correspond to a point particle sitting at r = 0in Minkowski space. Unfortunately, McVittie seems tohave interpreted his solution in this fashion [12] whicheven until recently gave rise to some confusion in theliterature (e.g. [5, 7, 20]). A clarification was given in [16]. IV. ATTEMPTS TO GENERALIZE MCVITTIE’SMODEL
The first obvious generalization consists in allowing fora non-vanishing cosmological constant. However, as wasalready indicated before, this is rather trivial since itmerely corresponds to the substitutions ̺ → ̺ + ̺ Λ and p → p + p Λ in (4), where ̺ Λ := Λ / π and p Λ := − Λ / π are the energy-density and pressure associated to the cos-mological constant Λ.The attempts to non-trivially generalize the McVittiesolution have focused so far on keeping the ansatz (1) andrelaxing the conditions on the matter in various ways. In[4] generalization were presented allowing radial fluid mo-tions relative to the observer vector field ∂ / ∂ t (that isrelaxing condition (3)) as well as including heat conduc-tion. Below we will critically review these attempts, tak-ing due care of the geometric constraints imposed by the ansatz (1), and also outline how to explicitly constructthe respective solutions.Another exact solution that models an inhomogene-ity in a cosmological spacetime was presented in [19]by Sultana and Dyer and was recently analyzed in [2].Here the metric is conformally equivalent to the exte-rior Schwarzschild metric and the cosmological matter iscomposed of two non-interacting perfect fluids, one beingpressureless dust, the other being a null fluid. One mightask if this solution fits into the class of McVittie models,as was suggested in [4] and allegedly confirmed explicitlyin [2] . However, as we already noted at the end of Sec-tion III A above in view of Proposition 1, this is not thecase. Two further way to see this are as follows: First,the Sultana–Dyer metric is not spatially Ricci-isotropic and, second, the McVittie metric is not compatible withthe matter model used by Sultana and Dyer, with thesole exception of trivial or exotic cases, as will be shownin Section IV D below. In Section IV A of [4] it is suggested that the Sultana–Dyer metricis equal to the McVittie metric (1) in which a ( t ) = a t / and m ( t ) = m , for some constants a and m (see Eq. (62) in [4]).Let denote the latter metric by ˜ g . Indeed, since m is constantand in view of Proposition 1, ˜ g is conformally related to theSchwarzschild metric. Moreover, as one may explicitly check viaour Eq. (11c), the Einstein tensor of ˜ g has a vanishing sphericalpart. Despite sharing these two properties, ˜ g and the Sultana–Dyer metric are not equal. The problem with the reasoning in Section II of [2] is the following(numbers refer to equations in [2]): It is true by constructionthat the Sultana–Dyer metric (2.1) is conformally related to theSchwarzschild metric, as expressed in the second line of (2.3) [thefirst line in (2.3) does not follow], but the conformal function a depends non-trivially on the Schwarzschild coordinates for time and radius (denoted by ¯ η and ˜ r in [2]: Cf. our discussion inthe last paragraph of Section III A). Hence it is not possibleto introduce a new time coordinate ¯ t that satisfies d ¯ t = ad ¯ η (the right hand side is not a closed 1-form), as pretended in thetransition to (2.5). To show this, one has to show that there exists no timelike di-rection with respect to which the Ricci tensor (or, equivalently,the Einstein tensor) is spatially isotropic. This can be shownas follows: First note that the Einstein tensor of the Sultana–Dyer metric has the form
Ein = µ u ⊗ u + τ k ⊗ k (see [19]),where u is a normalized future-pointing spherically-symmetrictimelike vector field and k the in-going future-pointing light-like vector field orthogonal to the SO (3)-orbits normalized suchthat g ( u , k ) = 1. In particular, the spherical part of the Ein-stein tensor vanishes: Hence, the Einstein tensor is spatiallyisotropic iff there exists a non-vanishing spacelike spherically-symmetric (i.e. orthogonal to the SO (3)-orbits) vector field s with Ein ( s , s ) = 0. Without loss of generality one can chose s to be normalized: s = sinh χ u + cosh χ e , where e is the normal-ized vector field orthogonal to u and to the SO (3)-orbits pointingin positive radial direction. Hence one has k = u − e and thus: Ein ( s , s ) = µ sinh ( χ )+ τ exp(2 χ ). Clearly, the latter expressionvanishes nowhere in the physically interesting region (cf. Eq. (26)in [19]), where both µ and τ are positive. A. Einstein’s equation for the McVittie ansatz
In the following we will restrict to those generalizationsof the McVittie model which keep the metric ansatz (1)and thus generalize only the matter model. For thispurpose it is convenient to write down the Einstein’sequation for an arbitrary spherically symmetric energy-momentum tensor T . Recall that spherical symmetry im-plies for the component of T with respect to the orthonor-mal basis (10) that T ( e a , e A ) = 0 and T ( e A , e B ) ∝ δ AB ,where a ∈ { , } and A, B ∈ { , } . Hence, the onlyindependent, non-vanishing components of T are S := T ( e , e ) (31a) Q := T ( e , e ) (31b) P := T ( e , e ) (31c) J := − T ( e , e ) , (31d)and these are functions which do not depend on the an-gular coordinates. Note that S is the energy density, Q and P the radial and spherical pressure, and J theenergy flow—all referred to the observer field e . Thesing in (31d) is chosen such that a positive J means aflow of energy in positive radial direction. Taking (31)into account, the Einstein equation for the McVittieansatz (1) and an arbitrary spherically symmetric energy-momentum tensor T reduces to the following four equa-tions: ( a m ) · = − πR (cid:0) BA (cid:1) J (32a)8 π S = 3 F (32b)8 π Q = − F − F AB (32c) P = Q . (32d)In view of (22), the first equation relates the time varia-tion of the Weyl part of the MS energy contained in thesphere of radius R with the energy flow out of it. Thelast equation is nothing but spatial Ricci-isotropy.In the following subsections we will consider threemodels for the cosmological matter which generalize theoriginal McVittie model: perfect fluid, perfect fluid plusheat flow, and perfect fluid plus null fluid. B. Perfect fluid
Perhaps the simplest step one can take in trying togeneralize the McVittie model is to stick to a single per-fect fluid for the matter, but dropping the condition (3)of ‘no-infall’ by allowing for radial motions relative tothe ∂ / ∂ t observer field. In this way one could hope toavoid a particular singular behavior in the pressure thatmay be due to the ‘no-infall’ condition, though it is clearthat the persisting geometric singularity must show upsomehow in the matter variables as already discussed inSection III D. Unfortunately, as already shown in [4],the relaxation of (3) does not lead to any new solutions. What we want to stress here is that the reason for this,as shown in more detail below, lies precisely in the re-striction imposed by spatial Ricci-isotropy.We take thus the perfect-fluid energy-momentum ten-sor (2) for the matter and an arbitrary spherically sym-metric four-velocity u . The latter is given in terms of theorthonormal basis for the metric (1) by u = cosh χ e + sinh χ e , (33)where χ is the rapidity of u with respect to the observerfield e (a positive χ corresponds here to a boost in anoutward-pointing radial direction). The non-vanishingcomponents of the matter energy-momentum tensor (2)with four-velocity (33) are: T ( e , e ) = ̺ + ( ̺ + p ) sinh χ (34a) T ( e , e ) = − ( ̺ + p ) sinh χ cosh χ (34b) T ( e , e ) = p + ( ̺ + p ) sinh χ (34c) T ( e , e ) = T ( e , e ) = p . (34d)Clearly, the case of vanishing rapidity must lead tothe original McVittie model. In this case, in fact, thematter energy-momentum tensor (34) is already spatiallyisotropic so that (32d) is identically satisfied. Moreover,(32a) implies ( a m )˙ = 0 and hence, in view of (14), F =˙ a/a . Herewith Einstein’s equation reduces to (4) andthus one gets back the original McVittie model.In case of non-vanishing rapidity, spatial Ricci-isotropy (32d) implies the following constraint: ̺ + p = 0 . (35)This means that the energy momentum tensor (2) has theform of a cosmological constant (using (35) in div T = 0it implies d p = 0 and this, in turn, using again (35),implies d ̺ = 0) so that this case reduces to theSchwarzschild–de Sitter solution and hence does not pro-vide the physical generalization originally hoped for. C. Perfect fluid plus heat flow
In a next step one may keep (33) and drop the con-dition that the fluid be perfect, in the sense of allowingfor radial heat conduction. This is described by a spatialvector field q that represents the current density of heat,which here corresponds to the current density of energyin the rest frame of the fluid. Hence q is everywhereorthogonal to u . The fluid’s energy momentum tensor We note that the parametrization of the energy-momentum ten-sor given in [4] is manifestly different. Whereas we parametrizedit in the usual fashion in terms of quantities (energy density,pressure, current density of heat) that refer to the fluid’s restsystem, the authors of [4] also write down (36) (their Eq. (79)),but with q orthogonal to e (compare their Eq. (93)) rather than u , which affects also the definition of ̺ . In fact, marking theirquantities with a prime, their expression (79) is equivalent to our(36) iff p = p ′ , q = q ′ cosh χ , and ̺ = ̺ ′ − q ′ sinh χ . T = ̺ u ⊗ u + p ( u ⊗ u − g ) + u ⊗ q + q ⊗ u . (36)Taking (33) as fluid velocity and imposing the heat flow-vector q to be spherically symmetric, we have q = q e := q (sinh χ e + cosh χ e ) , (37)where q is a function of ( t, r ). Note that a positive q corresponds to heat flowing in an outward-pointing radialdirection. The independent non-vanishing components ofthe energy-momentum tensor are now as follows: T ( e , e ) = ̺ + tanh χ (cid:0) ( ̺ + p ) tanh χ + 2 q (cid:1) (38a) T ( e , e ) = q − cosh χ (cid:0) ( ̺ + p ) tanh χ + 2 q (cid:1) (38b) T ( e , e ) = p + sinh(2 χ ) (cid:0) ( ̺ + p ) tanh χ + 2 q (cid:1) (38c) T ( e , e ) = T ( e , e ) = p . (38d)Consider first the case of vanishing rapidity. Then theenergy-momentum tensor is already spatially isotropicand Einstein’s equation (32) reduces to( a m )˙ = − πR q (cid:0) BA (cid:1) (39a)8 π̺ = 3 F (39b)8 πp = − F − F AB . (39c)These are three PDEs (though only time derivatives oc-cur) for the five functions a, m, ̺, p , and q so that thesystem (39) is clearly under-determining. However, it isnot possible to freely specify any two of these five func-tions and then determine the the other three via (39). Forexample, since the left-hand side of (39a) depends onlyon t , the same must hold for the r.h.s., which implies that q = f ( t ) /r (1 − ( m/ r ) ) , where f ( t ) = − ( a m )˙ / πa .In particular, the heat flow must fall-off as 1 /r .The easiest way to generate a solution in the case ofzero rapidity is to specify the two functions a ( t ) and m ( t ), then let A, B, R, F be determined by the def-initions (12,13,14), and finally let the Einstein equa-tions (39a,39b,39c) determine q, ̺ , and p , respectively.Notice that if we happen to specify a and m such that a m is a constant, this immediately implies q = 0 and F = ˙ a/a , which leads to the standard McVittie solutions.From (39a) the following is evident: if q > q < ̺ + p ) tanh χ + 2 q = 0 . (40a)Using this, the other components of the Einstein’s equa-tion reduces to: ( a m )˙ = +4 πR q (cid:0) BA (cid:1) (40b)8 π̺ = 3 F (40c)8 πp = − F − F AB . (40d) These are almost the same as in the case of vanishingrapidity (see (39)), except for the opposite sign on theright-hand side of (40b). This simply results from thefact that, according to (38b), J = − T ( e , e ) = q forvanishing rapidity, whereas, due to the constraint (40a), J = − T ( e , e ) = − q for non-vanishing rapidity. Thiswill be further interpreted below. Notice that for theequation of state ̺ + p = 0 (cosmological term) (40a) im-plies q = 0, thus leading once more to the Schwarzschild–de Sitter solution (see comment below Eq. (35)). Hence-forth we assume ̺ + p = 0, which implies that one cansolve the constraint (40a) for the rapidity:tanh χ = − q̺ + p (41)provided that | q/ ( ̺ + p ) | < a, m, ̺, p, q , and χ . As in the case of van-ishing rapidity, this system is under-determining and itis not possible to freely specify any two of these six func-tions and then determine the the other four. In a similarfashion as before, the easiest way to generate a solutionis to specify the two functions a ( t ) and m ( t ), to let thenthe definitions (12,13,14) determine A, B, R, F , and fi-nally use the Einstein equations (40b,40c,40d) and (41)to determine q, ̺, p , and χ , respectively. Again, choos-ing a and m such that their product is constant implies q = 0 and F = ˙ a/a , which leads to the standard McVittiesolutions.In passing we remark that the condition ̺ + p > e = ( A/B ) ∂ / ∂ t and (15):4 π ( ̺ + p ) = − e (cid:18) e ( R ) R (cid:19) , (42)which is positive iff the rate of change e ( R ) /R is a de-creasing function along the integral lines of the observer e . In other words, ̺ + p is positive iff ln( R ) is a con-cave function on the worldline of the observer e , whichis implied by, but not equivalent to, the function R beingconcave.From (41) and (40b), and assuming ̺ + p >
0, one seesthe following: If χ > χ < e , we have q < q > E W never compensates that of the fluidmotion, quite in accord with naive expectation. Belowwe show that for small rapidities the contribution due tothe heat flow is minus one-half that of the cosmologicalmatter.Let us now return to the sign-difference of the right-hand sides of (39a) and (40b). From (38) one infers that J is the sum of the two contributions coming from the1heat flow J h := q (1 + 2 sinh χ ) (43)and from cosmological matter J m := ( ̺ + p ) sinh χ cosh χ , (44)respectively. The constraint (40a) can be written in theform 2 cosh χJ h + (1 + 2 sinh χ ) J m = 0 , (45)which, for small rapidities χ (that is neglecting quadraticterms in χ ), implies 2 J h + J m ≈
0. In this approximationthe spatial energy-momentum flow due to heat is minusone-half that due to the cosmological matter. For thetotal flow this implies J = J m + J h ≈ J m / ≈ − J h . Nowthe sign difference between (39a) and (40b) is understoodas follows: In case of vanishing rapidity one has J m = 0, J h = q and hence J = q (leading to (39a)), whereasa short calculation reveals that in case of non-vanishingrapidity the constraint (45) implies J = J m + J h = − q ,leading thus to (40b). D. Perfect fluid plus null fluid
The last tentative generalization we consider is tak-ing for matter the incoherent sum (meaning that therespective energy-momentum tensors adds) of a perfectfluid (possibly with non-vanishing pressure) and a nullfluid (eventually representing electromagnetic radiation).This clearly contains as special case the matter modelconsidered by Sultana and Dyer [19] in which the pres-sure vanishes. We already stressed in Section III A thatthe metric ansatz of [19] is different from (1). Here weshow that the matter model of [19] is essentially incom-patible with (1) except for trivial or exotic cases.The matter model consists of an ordinary perfect fluidand a null fluid (e.g. electromagnetic radiation) with-out mutual interaction. Hence the matter’s energy-momentum tensor is just the sum of (2) and T ± nf = λ l ± ⊗ l ± , (46)where λ is some non-negative function of t and r and l + and l − are, respectively, the outgoing and ingo-ing future-pointing null vector fields orthogonal to thespheres of constant radius r partially normalized suchthat g ( l + , l − ) = 1. (It remains a freedom l ± α ± l ± ,where α is a positive function). Without loss of generalitywe make use of this freedom and choose: l ± = ( e ± e ) / √ , (47)where e and e are the vectors of the orthonormalframe (10). The components of the whole energy-momentum tensor with respect to this frame are then: T ( e , e ) = ̺ + ( ̺ + p ) sinh χ + λ (48a) T ( e , e ) = − ( ̺ + p ) sinh χ cosh χ ∓ λ (48b) T ( e , e ) = p + ( ̺ + p ) sinh χ + λ (48c) T ( e , e ) = T ( e , e ) = p . (48d)Here and below the upper (lower) sign corresponds to theoutgoing (ingoing) null field.In the present case, the condition (32d) of spatial Ricci-isotropy is equivalent to the constraint:( ̺ + p ) sinh χ + λ = 0 . (49)In the physically relevant case in which ̺ + p > χ = 0 and λ = 0, which leads to the original McVittie model. Inthe case ̺ + p = 0 (49) implies λ = 0, leading thusto the Schwarzschild–de Sitter spacetime (see commentbelow (35)). Hence, a new solution is only possible ifthe matter is of an exotic type that satisfies ̺ + p < ̺ > ̺ > | p | ). In particular, for the matter model consideredby Sultana and Dyer, one would need to violate the weakenergy-condition. V. CONCLUSION
We conclude by commenting on the the main dif-ferences between these generalizations and the originalMcVittie model. First we stress once more that neitherallowing for a nonzero rapidity nor a nonzero heat flowcan eliminate the singularity at r = m/ R = 2 am ) (aserroneously stated in [4]). The only substantial new fea-ture of these generalizations is that the Weyl part of theMS energy E W = a m is not constant anymore. In viewof the fact that the combination m/r = A E W /R ≈ E W /R (50)contained in the McVittie ansatz gives the ‘Newtonian’part of the potential in the slow-motion and weak-fieldapproximation (see [1]), we deduce that in order to getthe geodesic equation for the generalized McVittie model,it suffices to substitute m with E W in the equation ofmotion derived in [1]. This means that the strength of thecentral attraction varies in time according to (32a), lead-ing to an in- or out-spiraling of the orbits if d E W ( e ) > d E W ( e ) <
0, respectively.We identified the origin of why we could not vary therapidity and the heat flow independently in the condi-tion (40a) of spatial Ricci-isotropy, which is built intothe ansatz (1). We saw that this geometric feature ren-ders this ansatz special, so that it would be improper tocall it a general ansatz for spherical inhomogeneities in aflat FLRW universe. It remains to be seen whether use-ful generalizations exist which are captured by equallysimple ans¨atze.2
Acknowledgments
D.G. acknowledges support from the Albert-Einstein-Institute in Golm and the QUEST Excellence Cluster.
APPENDIX A: PROOF OF PROPOSITION 1
In this appendix we compute the intersection of theset S McV of metrics of type (1), which we denote in thefollowing by g McV a,m , with the set S cS of metrics conformallyrelated to an exterior Schwarzschild metric. Explicitly,the latter are of the form g cSΩ ,M := Ω g Schw M , (A1a)where g Schw M = (cid:18) − M R (cid:19) d T − (cid:18) − M R (cid:19) − d R − R g S , (A1b)denotes the Schwarzschild metric with mass M in ‘stan-dard’ coordinates. The question is: for which functions a and m and, respectively, for which function Ω and pa-rameter M does the equation g McV a,m = Ω g Schw M hold?Such an equality can be eventually established by findinga coordinate transformation, φ say, between the coordi-nates ( t, r ) in (1) and ( T, R ) in (A1) which brings (1)in form (A1). This involves solving coupled, non-linearpartial differential equations for φ , which depend on thefour unknown parameter a, m, Ω, and M . Needless tosay that this is not really a thankful task. Alternatively,a better approach would be to compare all the indepen-dent, algebraic curvature-invariants of the two metrics:This would lead to a system of equations between scalarswhich involves the coordinate transformation φ in an al-gebraic way (i.e. non differentiated).We adopt here an approach which is somewhere in themiddle: First, we use just three invariants (the areal ra-dius and the Ricci and the Weyl part of the MS energy)to drastically restrict the form of the coordinate transfor-mation (see (A14)) and derive thereby constraints on thefree parameters a, m, Ω, and M (see (A10) and (A13)).Second, we perform this restricted coordinate transfor-mation and determine it completely. To simplify thecalculation, instead of g McV a,m = Ω g Schw M , we consider theequivalent equation Ω − g McV a,m = g Schw M . In fact, for theSchwarzschild metric (A1b) it is immediate that theabove mentioned quantities are, respectively, given by: R ( g Schw M ) = R , (A2) E W ( g Schw M ) = M , (A3) E R ( g Schw M ) = 0 . (A4) The transformation between the angular variables is just theidentity.
In order to compute the respective quantities for the met-ric Ω − g McV a,m we first give their scaling behavior underconformal transformations.Clearly, because of their very definitions, for the arealradius and the Weyl part of the MS energy it holds: R (Ψ g ) = Ψ R ( g ) (A5)and E W (Ψ g ) = Ψ E W ( g ) , (A6)respectively. For the whole MS energy it easily followsfrom (20) and (A5): E (Ψ g ) = Ψ (cid:16) E ( g ) + R g ( ∇ R, ∇ ln Ψ)+ R g ( ∇ ln Ψ , ∇ ln Ψ) (cid:17) , (A7)where all the quantities on the r.h.s. are referred to themetric g . Hence, taking the difference between (A7)and (A6) one gets that the Ricci part of the MS energyscales exactly like the whole MS energy, that is accordingto (A7).Using these scaling properties together with (13)and (22) we get immediately: R (Ω − g McV a,m ) = Ω − (1 + m/ r ) ar (A8) E W (Ω − g McV a,m ) = Ω − a m . (A9)The equality between the Weyl part of the MS en-ergy (A9) and (A3) impliesΩ( t, r ) M = a ( t ) m ( t ) , (A10)which gives a condition between the parameter a, m, Ω,and M . Since we assumed that M is positive, (A10)can be read as the expression for the conformal factor inthe ( t, r ) coordinates. This, together with the equalitybetween the areal radius (A8) and (A2), implies in turn R ( t, r ) = M m ( t ) (1 + m ( t ) / r ) r , (A11)which gives the first component of the coordinate trans-formation φ . Now, using the scaling property (A7) for theRicci part of the MS energy, the expressions (21) and (13)for the Ricci part of the MS energy and, respectively, theareal radius of the McVittie metric ansatz, and (A10) forthe conformal factor, one gets, after some computations, E R (Ω − g McV a,m ) = M a m ( A ar ) (cid:18) ˙ mm (cid:19) . (A12)The equality between (A12) and (A4) then implies˙ m = 0 , (A13)that is m = m for some positive constant m . This, inturns, implies that the transformation (A11) for R de-pends only on r and not on t . Since the metrics are both3in diagonal form, this implies that the transformation for T must depend on t only.Summarizing, so far we have seen that a set of neces-sary conditions for the equality of the two metrics impliesthe constraints (A10) and (A13) and that the coordinatetransformation between ( t, r ) and ( T, R ) is of the form T ( t ) = f ( t ) (A14a) R ( r ) = M m (1 + m / r ) r , (A14b)for some differentiable function f of t . Now, explicitlyexpressing the metric Ω g Schw M in the ( t, r ) coordinatesaccording to the coordinate transformation (A14) andthe constraints (A10) and (A13), and putting the resultequal to g McV a,m , the only new condition that one gets is˙ f = ± M m a . (A15)Here, the plus can be chosen in order to exclude a timeinversion. It is important to note that (A15) (togetherwith an initial value) determines f uniquely and do notgive any constraint on the parameters a, m, Ω, and M :The only constraints remain thus (A10) and (A13).The proof is concluded noticing that (A10) means thatthe only constraint on Ω is that, expressed in the ( t, r )coordinates, it depends on t only and hence, in viewof (A14a) and expressed in the ( T, R ) coordinates, thatit depends on T only. More geometrically, this can berestated saying that the gradient of Ω must be propor-tional to ∂ / ∂ T , the Killing field of the Schwarzschildmetric (see (A1b)). APPENDIX B: PROOF OF PROPOSITION 2
Inserting the definition (14) of F in the expression (23)for the Ricci scalar and organizing the result in powersof rB ≡ r − m/ − (cid:18) ˙ aa (cid:19) − rB a ( a m )˙ a + rA (cid:18) ˙ aa (cid:19) ˙ ! − rB ) (cid:18) ( a m )˙ a (cid:19) + rA (cid:18) ( a m )¨ a − ˙ a ( a m )˙ a (cid:19)! − rB ) rA ˙ m ( a m )˙ a . (B1)Hence, the Ricci scalar remains finite in the limit r → m/ rB ) − k , for k ∈ { , , } , vanish in this limit, that is iff it holds:4 ˙ a ( a m )˙ a + m (cid:18) ˙ aa (cid:19) ˙ = 0 (B2a)2 (cid:18) ( a m )˙ a (cid:19) + m (cid:18) ( a m )¨ a − ˙ a ( a m )˙ a (cid:19) = 0 (B2b)˙ m ( a m )˙ = 0 . (B2c)These conditions are clearly understood to hold for alltimes t in which the functions a and m and their deriva-tive exist. In view of (B2c) we have to distinguish be-tween two cases: ˙ m = 0 and ( a m ) · = 0, respectively. Inthe first case the system (B2) reduces to the set of con-ditions m (¨ a/a + 3( ˙ a/a ) ) = 0 and m (¨ a/a + ( ˙ a/a ) ) = 0,which, in turn, reduces to m = 0 (and a arbitrary), corre-sponding to the FLRW metric, or to ˙ a = 0 (and ˙ m = 0),corresponding to the Schwarzschild metric. In the secondcase, in which ( a m ) · = 0, (B2) reduces to m ( ˙ a/a ) · = 0,which implies either m = 0 (and a arbitrary), correspond-ing again to the FLRW metric, or ( ˙ a/a ) · = 0. Togetherwith ( a m ) · = 0, the latter corresponds to a McVittiemetric with exponentially-growing (or -falling) scale fac-tor a ( t ), that is to a Schwarzschild–de Sitter metric. APPENDIX C: SHEAR-FREE OBSERVERFIELDS IN SPHERICALLY SYMMETRICSPACETIMES
Towards the end of Section II we made use of the fol-lowing result: A spherically symmetric normalized time-like vector field u in a spherically symmetric spacetime( M , g ) is shear free iff the metric h u that g induces onthe subbundle u ⊥ := { v ∈ T M | g ( v , u ) = 0 } by restric-tion is conformally flat.To prove this, we first note that the subbundle u ⊥ isintegrable, in other words, u is hypersurface orthogonal.This follows from the spherical symmetry of u , whichimplies that u ⊥ contains the vectors tangent to the 2-dimensional SO (3) orbits. Hence u essentially lives inthe 2-dimensional orbit space , where it is trivially hy-persurface orthogonal. The hypersurfaces orthogonal to u in 4-dimensional spacetime are then the preimages un-der the natural projection of the hypersurfaces (curves)in the 2-dimensional orbit space.As a result, we may now locally introduce so-calledisochronous comoving coordinates, with respect to which u = A ( t, r ) − ∂ / ∂ t and g = A ( t, r ) d t − B ( t, r ) d r − R ( t, r ) g S . (C1) The orbit space is the quotient M / ∼ , where ∼ is the equivalencerelation whose equivalence classes are the orbits. It is a manifoldon the subset corresponding to 2-sphere orbits, to which we re-strict attention here. To say that “ u essentially lives in the orbitspace” means that u is the pull-back of a 1-form on the quotientvia the natural projection. A and B ascompared to (16)). We now consider the tangent-spaceendomorphisms ∇ u : X ∇ X u and their projectioninto the orthogonal complement of u , i.e., ∇ ⊥ u := (cid:0) P ⊥ u ◦ ∇ u ◦ P ⊥ u (cid:1)(cid:12)(cid:12) u ⊥ , (C2)where P ⊥ u := id − u ⊗ u is the projection orthogonalto u ( id is the identity endomorphism in the tangentspaces of M ). Note that ∇ ⊥ u is symmetric due to thehypersurface orthogonality of u . A direct computationusing (C1) yields ∇ ⊥ u = u (cid:0) ln( B ) (cid:1) P r + u (cid:0) ln( R ) (cid:1) P S , (C3)where P r and P S are the projections parallel to ∂ / ∂ r and parallel to the tangent 2-planes to the S -orbits, re-spectively. The trace θ of ∇ ⊥ u , which gives the expan-sion of u , is θ = u (cid:0) ln( B ) (cid:1) + 2 u (cid:0) ln( R ) (cid:1) , so that the trace-free part of ∇ ⊥ u , known as the shear endomorphism σ ,is given by: σ := ∇ ⊥ u − θ id ⊥ = σ ( P S − P r ) , (C4)where σ := u (cid:0) ln( R/B ) (cid:1) denotes the shear scalar (onlydefined in a spherically-symmetric setting) and id ⊥ = P r + P S the identity endomorphism in u ⊥ . In passing,we note that the defining equations for θ and σ just givenimmediately lead to the following simple relation betweenthe shear scalar, expansion, and the variation of the areal radius along u , that we made use of in Section III B: σ + θ/ u (ln( R )) . (C5)Now, according to (C4), the shear of u vanishes iff theshear scalar σ does, that is, iff u (cid:0) ln( R/B ) (cid:1) vanishes.This is equivalent to R/B being independent of t or to R ( t, r ) = µ ( r ) B ( t, r ) for some function µ , so that theline element (C1) can be rewritten in the spatially con-formally flat form g = ˜ A ( t, ρ ) d t − ˜ C ( t, ρ ) (cid:0) d ρ + ρ g S (cid:1) , (C6a)where ˜ A ( t, ρ ) := A ( t, r ( ρ )), ˜ C ( t, ρ ) := C ( t, r ( ρ )), and C ( t, r ) = B ( t, r ) µ ( r ) ρ ( r ) (C6b)with ρ ( r ) = ρ exp (cid:26)Z rr dr ′ µ ( r ′ ) (cid:27) . (C6c)Hence we see that vanishing shear of u implies confor-mal flatness of the corresponding spatial metric. For theconverse we first note that, since u and g are sphericallysymmetric, the spatial metric h u is itself spherically sym-metric, so that g can be written in the form (C6a). Thisimplies that the corresponding R/B depends only on theradial coordinate and hence that the shear of u vanishes. [1] Carrera, M., and D. Giulini, 2008, “On the influence ofglobal cosmological expansion on the dynamics and kine-matics of local systems,” to appear in Reviews of ModernPhysics, arXiv:0810.2712.[2] Faraoni, V., 2009, “An analysis of the Sultana-Dyer cos-mological black hole solution of the Einstein equations,”arXiv:0907.4473.[3] Faraoni, V., C. Gao, X. Chen, and Y.-G. Shen, 2009,“What is the fate of a black hole embedded in an ex-panding universe?,” Physics Letters B , 7–9.[4] Faraoni, V., and A. Jacques, 2007, “Cosmological Expan-sion and Local Physics,” Physical Review D , 063510(pages 16).[5] Ferraris, M., M. Francaviglia, and A. Spallicci, 1996, “As-sociated radius, energy and pressure of McVittie’s metric,in its astrophysical application,” Nuovo Cimento B111 ,1031–1036.[6] Gao, C., X. Chen, V. Faraoni, and Y.-G. Shen, 2008,“Does the mass of a black hole decrease due to the ac-cretion of phantom energy,” Physical Review D
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