arXiv:0908.3101v1 [gr-qc] 21 Aug 2009 † ∗ -85 rmn Germany Bremen, D-28359 ‡ I.Goer fteMVti ansatz McVittie the of Geometry III. V tepst eeaieMVti’ model McVittie’s generalize to Attempts IV. lcrncades [email protected] address: Electronic loa:ZR,Uiest fBee,A Fallturm Am Bremen, of University ZARM, at: Also lcrncades [email protected] address: Electronic I h citemodel McVittie The II. .Pofo rpsto 1 Proposition of Proof A. Conclusion V. .Serfe bevrfilsi spherically in fields observer Shear-free C. .Pofo rpsto 2 Proposition of Proof B. .Introduction I. 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Germany ‡ m 14 2 standard-model is based. garding the motion of the matter and how this, together Given that situation, the task is to combine the virtues with Einstein’s equation, determines one of the two free of both classes of solutions without the corresponding functions in the metric ansatz as a simple function of the deficiencies. This means to find exact solutions for the other. We interpret this condition in terms of an appro- gravitational field of a compact object ‘immersed’ (see priate concept of local mass as saying that the object below) into an otherwise cosmological background. This does not accrete mass from the ambient matter. In Sec- would appear to be an easy task if the field equations tion III we take a second and closer look at the McVittie were linear, for, in that case, one would just add the so- ansatz and note some of its characteristic features which, lution that describes the gravitational field of a compact we feel, have not sufficiently carefully been taken into ac- object in an otherwise empty universe to the cosmological count in [3, 4, 6]. In the light of these observations we solution that corresponds to a homogeneous distribution then discuss in Section IV the attempted generalizations of background matter. Here the mathematical opera- of McVittie’s solution in the references just mentioned. tion of addition appears to be the obvious realization of We find that some of the conclusions drawn are indeed what one might be tempted to call ‘simultaneous physical unwarranted. presence’ and hence, in view of the individual interpre- tations of both solutions, the ‘immersion’ of the compact object into the cosmological background. But this imme- II. THE MCVITTIE MODEL diate interpretation in physical terms of a simple mathe- matical operation is deceptive. This becomes obvious in non-linear theories, like General Relativity (GR), where The characterization of the McVittie model is made no simple mathematical operation exists that produces through two sets of a priori specifications. The first set a new solution out of two old ones and where the very concerns the metric (left side of Einstein’s equations) and same physical question may still be asked. the second set the matter (right side of Einstein’s equa- tions). The former consists in an ansatz for the metric, The proper requirement for a mathematical represen- which can formally be described as follows: Write down tation of the envisaged physical situation must, first of the Schwarzschild metric for the mass parameter m in all, consist in asymptotic conditions which ensure that isotropic coordinates, add a conformal factor a2(t) to the the sought for solution approximates the given (e.g. spatial part, and allow the mass parameter m to depend Schwarzschild) one for small distances and a particu- on time. Hence the metric reads lar cosmological one (e.g. FLRW) for large distances. 2 Second, it must specify somehow the physics in the in- 1 m(t)/2r termediate region. Usually this will include a specifica- g = − dt2 tion of the matter components and their dynamical laws 1+ m(t)/2r 4 (1) together with certain initial and boundary conditions. m(t) 2 2 2 1+ a (t) (dr + r g 2 ) , Needless to say that this will generally result in a complex − 2r S system of partial differential equations. Most analytic 2 2 2 approaches therefore impose further simplifying assump- 2 where gS = dθ + sin θdϕ is the standard metric on tions that automatically guarantee the right asymptotic the unit 2-sphere. Here we restricted attention to the behavior and at the same time reduce the free functions asymptotically spatially flat (i.e. k = 0) FLRW metric, to a manageable number. which is compatible with current cosmological data [11]. In this paper we will discuss a particular such ap- For simplicity we shall refer to (1) simply as McVittie’s proach, which is originally due to McVittie [12] and which ansatz, though this is not quite correct since McVittie has been further analyzed and clarified in a series of care- started from a general spherically symmetric form and fully written papers by Nolan [15, 16, 17]. Our main arrived at (1) with m(t)a(t) = const. after imposing a motivation is that recently McVittie’s solution has been condition that he interpreted as condition for no matter- severely criticized as not being able at all to model the infall. The ansatz (1) is obviously spherically symmetric envisaged situation[3, 6], whereas a family of slightly gen- with the spheres of constant radius r being the orbits of eralized ones [4], in which some restrictions concerning the rotation group.1 In the next section we will discuss the motion of matter and the existence of heat flows is in more detail the geometric implications of this ansatz, lifted, is argued to be free of the alleged problems. The independent of whether Einstein’s equation holds. existence of an exact solution to Einstein’s equation that models local inhomogeneities is clearly of great impor- tance, for example in estimating reliable upper bounds to the possible influence of global cosmological expansion 1 “Spherical symmetry” of a spacetime means the following: There onto the dynamics and kinematics of local systems [1]. exists an action of the group SO(3) on spacetime by isometries, The paper is organized as follows: In Section II we re- which is such that the orbits are either two-dimensional and spacelike or fixed points. The “spheres” implicitly referred to in view what we call the McVittie model. We discuss its this term correspond to the two-dimensional orbits, even though metric ansatz and what its entails regarding the geom- they might in principle also be two-dimensional real projective etry of spacetime. Then we discuss the assumptions re- spaces. In the cases we discuss here they will be 2-spheres. 3
As already discussed in the introduction, the model Equation (4a) can be immediately integrated: here is meant to interpolate between the spherically sym- m0 metric gravitational field of a compact object and the m(t)= , (5) environment. It is not to be taken too seriously in the a(t) region very close to the central object, where the basic where m0 is an integration constant. Below we will show assumptions on the behavior of matter definitely turn that this integration constant is to be interpreted as the unphysical. However, as discussed in [1], at radii much mass of the central body. larger than (in geometric units) the central mass (to be Clearly the system (4) is under-determining. This is defined below) the k = 0 McVittie solution seems to pro- expected since no equation of state has yet been imposed. vide a viable approximation for the envisaged situation. The reason why we did not impose such a condition can The second set of specifications, concerning the matter, now be easily inferred from (4): whereas (4b) implies is as follows: The matter is a perfect fluid with density that ̺ only depends on t, (4c) implies that p depends ̺ and isotropic pressure p. Hence its energy-momentum 2 on t and r iff (˙a/a)˙ = 0. Hence a non-trivial relation tensor is given by p = p(̺) is simply incompatible6 with the assumptions T = ̺ u u + p (u u g) . (2) made so far. The only possible ways to specify p are ⊗ ⊗ − p = 0 or ̺ + p = 0. In the first case (4c) implies that Furthermore, and this is where the two sets of specifi- a/a˙ = 0 if m0 = 0 (since then the second term on the 6 cations make contact, the motion of the matter (i.e. its right-hand side is r dependent, whereas the first is not, four-velocity field) is given by so that both must vanish separately), which corresponds to the exterior Schwarzschild solution, or a(t) t2/3 if ∝ u = e0 , (3) m0 = 0, which leads to the flat FLRW solution with dust. In the second case the fluid just acts like a cos- e ∂ ∂ where 0 is the normalization of / t (compare (10)). mological constant Λ = 8π̺ (using the equation of state Finally, the explicit cosmological constant on the left- ̺ + p = 0 in div T = 0 it implies dp = 0 and this, in hand side of Einstein’s equation is assumed to be zero, turn, using again the equation of state, implies d̺ = 0) which implies no loss of generality, since a non-zero cos- so that this case reduces to the Schwarzschild–de Sitter mological constant can always be regarded as special part solution. To see this explicitly, notice first that (4b,4c) of the matter’s energy-momentum tensor (compare IV). imply the constancy of H =a/a ˙ = Λ/3 and hence No further assumptions are made. In particular, an equa- one has a(t) = a0 exp t Λ/3 . With such a scale- tion of state, like p = p(̺), is not assumed. The reason p factor the McVittie metric (1) with (5) turns into the for this will become clear soon. Later generalizations will p mainly concern (2) and (3). Schwarzschild–de Sitter metric in spatially isotropic coor- The Einstein equation3 now links the specifications dinates. The explicit formulae for the coordinate trans- of geometry with that of matter. It is equivalent to formation which brings the latter in the familiar form can the following three relations between the four functions be found in Section 5 of [18] and also in Section 7 of [10]. m(t),a(t),̺(t, r), and p(t, r): Finally, note from (4a) that constancy of one of the func- tions m and a implies constancy of the other. In this case (am)˙ = 0 , (4a) (4b,4c) imply p = ̺ = 0, so that we are dealing with the 2 exterior Schwarzschild spacetime. a˙ 8π̺ = 3 , (4b) A specific McVittie solution can be obtained by choos- a ing a function a(t), corresponding to the scale function 2 a˙ a˙ ˙ 1+ m/2r of the FLRW spacetime which the McVittie model is 8πp = 3 2 . (4c) required to approach at spatial infinity, and the con- − a − a 1 m/2r − stant m0, corresponding to the ‘central mass’. Rela- Note that here Einstein’s equation has only three inde- tions (4b,4c), and (5) are then used to determine ̺, p, pendent components (as opposed to four for a general and m, respectively. Clearly this ‘poor man’s way’ to spherically symmetric metric), which is a consequence solve Einstein’s equation holds the danger of arriving of the fact that the Einstein tensor for the McVittie at unrealistic spacetime dependent relations between ̺ ansatz (1) is spatially isotropic. This will be discussed in and p. This must be kept in mind when proceeding in more detail in the next section. this fashion. For further discussion of this point we refer to [15, 16]. As will be discussed in more detail in Section III C be- low, in the spherically-symmetric case the concept of lo- 2 Here and in what follows we denote the metric-dual (1-form) cal mass (or energy) is well captured by the Misner–Sharp of a vector u by underlining it, that is, u := g(u, · ) is the 1- (MS) energy [13], whose purely geometric definition in form metric-dual to the vector u. In local coordinates we have µ µ ν terms of Riemannian curvature allows to decompose it u = u ∂µ and u = uµdx , where uµ := gµν u . 3 We speak of “the Einstein equation” in the singular since we into a sum of two terms, one of which comes from the think of it as a single tensor equation, which only upon introduc- Ricci- the other from the Weyl curvature. It is the latter ing a coordinate system decomposes in many scalar equations. which may be identified with the gravitational mass of 4 the central object. Applied to (1), the Weyl contribution which the energy density jumps discontinuously. This to the MS energy can be written in the following form, means that the star’s surface is comoving with the cos- also taking into account (5), mological fluid and hence, in view of (7), that it geomet- rically expands (or contracts). This feature, however, 4π 3 ER = R ̺ , (6a) should be merely seen as an artifact of the McVittie 3 model (in which the relation (7) holds), rather than a EW = m0 . (6b) general property of compact objects in any cosmological spacetimes. Positive pressure within the star seems to The constancy of EW is then interpreted as saying that be only possible if 2aa¨ +a ˙ 2 < 0 (see Eq. (3.27) in [14] no energy is accreted from the ambient matter onto the with a = exp(β/2)), that is, for deceleration parameters central object. q> 1/2. 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 automatically III. GEOMETRY OF THE MCVITTIE ANSATZ vorticity free. The last property is manifest from its hy- persurface orthogonality, which is immediate from (1). In this section we will discuss the geometry of the met- Moreover, u is also shear free. This, too, can be imme- ric (1) independent of the later restriction that it will diately read off (1) once one takes into account the fol- have to satisfy Einstein’s equation for some reasonable lowing result, whose proof we sketch in Appendix C: A energy-momentum tensor. This means that at this point spherically symmetric normalized timelike vector field u we shall not assume any relation between the two func- in a spherically symmetric spacetime ( , g) is shear free tions m(t) and a(t), apart from the first being non neg- iff its corresponding spatial metric, thatM is, the metric g ative and the second being strictly positive. We will restricted to the subbundle u⊥ := v T g(v, u)= discuss the metric’s ‘spatial Ricci-isotropy’ (a term ex- 0 , is conformally flat. The metric{ (1)∈ obviouslyM | is spa- plained below), its singularities and trapped regions, and tially} conformally flat with respect to the choice (3) made also compute its Misner–Sharp energy decomposed into here. Moreover, the expansion (divergence) of u is the Ricci and Weyl parts. We shall start, however, by an- θ =3H, (7) swering the question of what the overlap is between the geometries represented by (1) and the conformal equiva- where H :=a/a ˙ , just as in the FLRW case. In par- lence class of the exterior Schwarzschild geometry. ticular, the expansion of the cosmological fluid is homo- geneous in space. Exactly as in the FLRW case is also the expression for the variation of the areal radius along A. Relation to conformal Schwarzschild class the integral lines of u (that is the velocity of cosmologi- cal matter measured in terms of its proper time and the This question is an obvious one in view of the way areal radius): in which (1) is obtained from the exterior Schwarzschild metric. It is clear that for m = m0 = const. the u(R)= HR, (8) metric (1) is conformally equivalent to the exterior Schwarzschild metric, since upon using a new time co- which is nothing but Hubble’s law. Recall that for a ordinate T with dT = dt/a(t) we can pull out a2(t) as spherically symmetric spacetime the areal radius, de- a common conformal factor. The following proposition, noted here by R, is the function defined by R(p) := whose proof we shall give in Appendix A, states that a (p)/4π, where (p) is the proper area of the 2- constant m is in fact also necessary condition: dimensionalA SO(3)-orbitA through the point p. For the p McVittie spacetime the areal radius is given explicitly Proposition 1. Let McV denote the set of metrics in in (13). The acceleration of u, which in contrast to the the form of the McVittieS ansatz (1) (parametrized by the FLRW case does not vanish here, is given by two positive functions a and m) and cS the set of met- rics conformally equivalent to an exteriorS Schwarzschild ∇ m0 1+ m/2r metric (parametrized by a positive conformal factor and uu = 2 e1 . (9) R 1 m/2r a constant positive Schwarzschild mass M0). Then the − intersection between McV and cS is given by the subset Here e1 is the normalized vector field in radial direction S S of metrics in McV with constant m or, equivalently, by as defined in (10). In leading order in m0/R this corre- S the subset of metrics in cS whose conformal factor has sponds to the acceleration of the observers moving along a gradient proportional toS the Killing field ∂/∂T of the the timelike Killing field in Schwarzschild spacetime. Schwarzschild metric (see (A1b) for notation). It is also important to note that the central gravita- tional mass in McVittie’s spacetime may be modeled by a Note that we excluded the ‘trivial’ cases in which m or shear-free perfect-fluid star of positive homogeneous en- M0 (or both) vanish for the following reason: Comparing ergy density [14]. The matching is performed along a the expressions for the Weyl part of the MS energy of world-tube comoving with the cosmological fluid, across the two types of metrics (see (A10) in Appendix A) it 5
µ follows that m vanishes iff M0 does and this, in turn, where x = t,r,θ,ϕ . Here, and henceforth, we write { } { } leads to a metric conformally related to the Minkowski v := g(v, v) . Note that e0, e1 are orthogonal to k k | | metric where the conformal factor depends only on time, and e2, e3 tangent to the 2-spheres of constant radius r. that is, a FLRW metric. But such a spacetime, being The non-vanishingp independent components of the Ein- homogeneous, is not of interest to us here. stein tensor with respect to the orthonormal basis (10) In particular, Proposition 1 implies that the metric of are: Sultana and Dyer [19] are not of type (1), as suggested 2 Ein(e0, e0)=3F , (11a) in Section IV A of [4] and allegedly shown in Section II of [2] (cf. our footnote 5 at page 8). This immediately 2 A 2 ˙ Ein(e0, e1)= R2 B (am) , (11b) follows from the observation that the conformal factor, Ein(e , e ) = 3F2 +2 A F˙ δ , (11c) expressed as function of the standard Schwarzschild co- i j − B ij ordinates that appear in (A1b), is given by Ω(T, R) = 2 where an overdot denotes differentiation along ∂/∂t. Be- (T + 2M0 ln(R/2M0 1)) (compare Eqs. (8) and (9) of [19]), which also depends− on R and hence does not fore explaining the functions A, B, R, and F , note that satisfy the condition of Proposition 1. We will have to the spatial isotropy of the Einstein tensor follows imme- diately from (11c), since Ein(e , e ) δ . In (11) and say more about this at the beginning of Section IV and i j ∝ ij in Section IV D. in the following we set: A(t, r) := 1 + m(t)/2r, B(t, r) := 1 m(t)/2r , (12) − and B. Spatial Ricci-isotropy 2 m(t) R(t, r)= 1+ a(t) r , (13) 2r An important feature of any metric that is covered by the ansatz (1) is, that its Einstein tensor is spatially where R is the areal radius for the McVittie ansatz (1), isotropic in the following sense: ‘Spatially’ refers to the and also directions orthogonal to ∂/∂t and ‘isotropy’ to the con- · a˙ 1 (am) dition that the spatial restriction of the spacetime’s Ein- F := + . (14) stein tensor is proportional to the spatial restriction of a rB a the metric. Note that, since the spacetime’s metric is In passing we note that both quantities, F and am, that time dependent, the spatial restriction of the spacetime’s appear in the components of the Einstein tensor, have Einstein or Ricci tensor is not the same as the Einstein a geometrical interpretation: the former is one third the or Ricci tensor of the spatial sections with their induced expansion of the vector field e0, that is, F = div(e0)/3, metrics. Hence the notion of spatial isotropy of the Ein- and the latter is the Weyl part of the Misner–Sharp en- stein tensor used here is not the same as saying that the ergy of the metric (1) (see (22), below). Moreover, as we induced metric of the slices is an Einstein metric. already noted in Section II, the observer field e0 is free of Given that the Einstein tensor of (1) is spatially vorticity and shear. Hence, taking into account the rela- isotropic in the sense used here, it is then obvious that tion (C5) between the expansion θ and the shear scalar σ Einstein’s equation will impose a severe restriction upon of an arbitrary spherically-symmetric observer field, the the matter’s energy-momentum tensor, saying that it, expansion of e0 can be simply written as 3dR(e0)/R so too, must be spatially isotropic. The degree of special- that F may be expressed as ization implied by this will be discussed in more detail F = dR(e0)/R. (15) below. Here we only remark that this observation al- ready answers in the negative a question addressed, and In order to estimate the degree of specialization im- left open, in the last paragraph of [3], of whether (1) plied by spatial Ricci-isotropy, we ask for the most gen- is the most general spherically symmetric solution de- eral spherically symmetric metric for which this is the scribing a black hole embedded in a spatially flat FLRW case. To answer this, we first note that any spherically background: It clearly is not. symmetric metric can always be written in the form In passing we make the obvious remark that, since 2 B(t, r) 2 2 4 2 2 2 Ein = Ric (1/2) Scal g, where Ric denotes the Ricci g = dt a (t)A (t, r)(dr + r gS ) . (16) − A(t, r) − tensor, the Einstein tensor is spatially isotropic iff the same holds for the Ricci tensor. For this reason we will This reduces to McVittie’s ansatz (1) if A, B are given from now on refer to spatial Ricci-isotropy to denote the by (12). For the general spherically symmetric metric feature in question. (16), spatial Ricci-isotropy can be shown to be equivalent Now, a way to actually show spatial Ricci-isotropy is to to compute the components of the Einstein tensor with 2 δ (AB) 8(δA)(δB)=0 , (17) respect to the orthonormal tetrad eµ µ∈{0,··· ,3} of (1) − defined by { } where δ := r−1∂/∂r = 2∂/∂r2. It is obvious that there are many more solutions to this differential equation than e := ∂/∂xµ −1 ∂/∂xµ , (10) just (12). µ k k 6
C. Misner–Sharp energy part of the Riemann tensor. The second equality follows then with (11a) and (15). The Weyl part can now be In order to be able to interpret (1) as an ansatz for an obtained as the difference between the full MS energy and inhomogeneity in a FLRW universe, it is useful to com- (21). We use the expression (20) for the former and write 2 2 pute the Misner–Sharp (MS) energy and, in particular, g(∇R, ∇R)= e0(R) e1(R) . The part involving − its Ricci and Weyl parts. This concept of quasi-local e0(R) equals the Ricci part (21) and hence the Weyl part 2 mass, which is defined only for spherically symmetric is given by (R/2) 1 (e1(R)) . From (13) we calculate − spacetimes, and which in this case coincides with Hawk- e1(R) and hence obtain for the Weyl part of the MS ing’s more general definition [8] of quasi-local mass (see energy: e.g. [1]), allows to detect localized sources of gravity. We recall the geometric definition of the MS energy [9, EW = a m . (22) 13]: The Ricci part of the MS energy is that part which, via 3 E := 1 R K, (18) Einstein’s equation, can be locally related to the mat- − 2 ter’s energy-momentum tensor, whereas the gravitational where R denotes the areal radius and K the extrinsic mass of the central object is contained in the Weyl part curvature. More precisely, the equation should be read of the MS energy. Notice that the latter is spatially con- and understood as follows: First of all, the quantities stant (the functions a and m in (22) only depend on time) R and K, and hence also E, are real-valued functions but may depend on time. If the latter is the case we in- on spacetime. In order to determine their values at a terpret this as saying that the central mass exchanges point p, recall that, due to the requirement of spherical energy with the ambient matter. symmetry, there is a unique two-(or zero-) dimensional SO(3) orbit S(p) through p. The value of R at p is as explained below Eq. (8) and the value of K at p is D. Singularities and trapped surfaces
Riem(Xp, Yp, Xp, Yp) K(p) := 2 . (19) Next we comment on the singularity properties of the g(X , X )g(Y , Y ) g(X , Y ) p p p p − p p McVittie ansatz (1). From (11c) one suspect, because of Here Riem is the (totally covariant) Riemannian cur- the term proportional to 1/B, a singularity in the Ricci vature tensor of spacetime and Xp and Yp are any two part of the curvature at r = m/2 (that is at R =2am = linearly independent vectors in the tangent space at p 2EW). In fact, this corresponds to a genuine curvature which are also tangent to the orbit S(p). Note that the singularity, as one can see from looking, for example, at right-hand side only depends of the plane spanned by the following expression for the scalar curvature (i.e. the Xp, Yp and not on the vectors spanning it. Finally we Ricci scalar), note that the minus sign in (18) is just a relict of our Scal = 12F 2 6 A F,˙ (23) signature choice (mostly minus). − − B From the curvature decomposition for a spherically symmetric metric (see [1]) one can rewrite (18) in the which can be quickly computed from (11). In Appendix B form we insert into this expression the definition (14) of F and expand this in powers of 1/(rB). This allows to prove R E = 1+ g(∇R, ∇R) , (20) 2 Proposition 2. The Ricci scalar for a metric of the form (1) becomes singular in the limit r m/2 for any func- where ∇R denotes the gradient vector-field of R. This tions a and m, except for the following→ three special cases: provides a convenient expression for the computation of (i) m =0 and a arbitrary (FLRW), the MS energy. For a self-contained review of the basic (ii) a and m are constant (Schwarzschild), and properties of the MS energy as well as its interpretation · · (iii) (am) =0 and (˙a/a) =0 (Schwarzschild–de Sitter). as the amount of active gravitational energy contained in the interior of the spheres of symmetry (SO(3)-orbits) This means that, as long as we stick to the ansatz (1), at and its relation with the other mass concepts, see [1]. r = m/2 there will always (with the only exceptions listed The decomposition of the Riemann tensor into a Ricci above) be a singularity in the Ricci part of the curvature and a Weyl part leads, together with (18), to a natural and thus, assuming Einstein’s equation is satisfied, also in decomposition of the MS energy into a Ricci and Weyl the energy momentum tensor, irrespectively of the details part (see also [1]). For the Ricci part of the MS energy of the underlying matter model. Hence any attempt to of (1) we get eliminate this singularity by maintaining the ansatz (1) and merely modifying the matter model is doomed to 1 3 R 2 ER = 6 R Ein(e0, e0)= (dR(e0)) . (21) fail. 2 In particular, this is true for the generalizations pre- The first equality in (21) can be derived by merely using sented in [4], contrary to what is claimed in that work the spatial Ricci-isotropy in the expression for the Ricci and its follow ups [3, 6]. We also remark that it makes no 7 sense to absorb the singular factors 1/B in front of the by RS can be written in the form time derivatives by writing (A/B)∂/∂t as e0 and then 2 argue, as was done in [4], that this eliminates the singu- 2E R ε 3 larity. The point is simply that then e0 applied to any − = η + x x +1 . (26) RS x 1+ x(x 1)! − continuously differentiable function diverges as r m/2. − − Below we will argue that this singularity lies within→ a Here we introduced the dimensionlessp radial coordinate trapped region. x := R/RS and the (small) parameters ε := R˙S and Specializing to the McVittie model, recall that in this η := RS/RH , where RH := 1/H denotes the ‘Hubble case it is assumed that the fluid moves along the integral radius’. Recall that since R > RS we have x> 1. curves of ∂/∂t, which become lightlike in the limit as r Consider first the McVittie case, in which ε = R˙ S = 0. tends to m/2. Their acceleration is given by the gradient Then (26) turns into a cubic polynomial in x which is of the pressure, which necessarily diverges in the limit positive for x = 0 and tends to for x . Hence r m/2, as one explicitly sees from (9). For a more ±∞ → ±∞ → it always has a negative zero (which does not interest us) detailed study of the geometric singularity at r = m/2, and two positive zeros iff see [16, 17].
For spherically symmetric spacetimes the Weyl part of RS/RH < 2/3√3 0.38 . (27) the curvature has only a single independent component, ≈ which is 2/R3 times the Weyl part of the MS energy, This clearly corresponds to the physical relevant case − by the very definition of the latter (see [1]). The square where the Schwarzschild radius is much smaller than of the Weyl tensor for the ansatz (1) may then be conve- the Hubble radius. One zero lies in the vicinity of niently expressed as the Schwarzschild radius and one in the vicinity of the Hubble radius, corresponding to two marginally trapped (am)2 spheres. The exact expressions for the zeros can be easily Weyl, Weyl = 48 . (24) h i R6 written down, but are not very illuminating. In leading order in the small parameter η = RS/RH , they are ap- This shows that R = 0 also corresponds to a genuine proximated by curvature singularity, though this is not part of the region 2 4 covered by our original coordinate system, for which r> R1 = RS 1+ η + O(η ) , (28a) 2 m/2 (that is R> 2EW). R2 = R 1 η/2+ O(η ) . (28b) H − It is instructive to also determine the trapped regions of McVittie spacetime. Recall that a spacelike 2-sphere From this one sees that for the McVittie ansatz the radius S is said to be trapped, marginally trapped, or untrapped of the marginally trapped sphere of Schwarzschild space- + if the product θ θ− of the expansions (for the defini- time (RS) increases and that of the FLRW spacetime tions see e.g. [1]) for the ingoing and outgoing future- (RH ) decreases. The first feature can, for the McVit- pointing null vector fields normal to S is positive, zero, tie model, be understood as an effect of the cosmological or negative, respectively. Taking S to be SR, that is, environment, whereas the latter is an effect of the inho- an SO(3) orbit with areal radius R, it immediately fol- mogeneity in form of a central mass abundance. All the lows from the relation 2 θ+θ− = g(∇R, ∇R)/R2 (see [1]) spheres with R < R1 or R > R2 are trapped and those that SR is trapped, marginally trapped, or untrapped iff with R1
2 for R˙S/c > (RS/RH ) /2. However, this also character- ansatz (1), and also outline how to explicitly construct izes the region− of validity of the expansion (29): Given the respective solutions. a positive η, an expansion in ε around zero exists only for ε > η2/2 since there exists no expansion on both (ε, η) around− (0, 0) (this is because the partial derivative of (26) with respect to x does not exist at x = 1). Another exact solution that models an inhomogene- ity in a cosmological spacetime was presented in [19] E. Other global aspects by Sultana and Dyer and was recently analyzed in [2]. Here the metric is conformally equivalent to the exte- Another aspect concerns the global behavior of the rior Schwarzschild metric and the cosmological matter is McVittie ansatz (1). We note that each hypersurface composed of two non-interacting perfect fluids, one being of constant time t is a complete Riemannian manifold, pressureless dust, the other being a null fluid. One might which, besides the rotational symmetry, admits a dis- ask if this solution fits into the class of McVittie models, crete isometry given in (r, θ, ϕ) coordinates by as was suggested in [4]4 and allegedly confirmed explicitly 5 2 −1 in [2] . However, as we already noted at the end of Sec- φ(r, θ, ϕ)= (m/2) r ,θ,ϕ . (30) tion III A above in view of Proposition 1, this is not the case. Two further way to see this are as follows: First, This corresponds to an inversion at the 2-sphere r = 6 m/2, which shows that the hypersurfaces of constant t the Sultana–Dyer metric is not spatially Ricci-isotropic can be thought of as two isometric asymptotically-flat and, second, the McVittie metric is not compatible with pieces joined together at the 2-sphere r = m/2. This 2- the matter model used by Sultana and Dyer, with the sphere is totally geodesic since it is a fixed-point set of an sole exception of trivial or exotic cases, as will be shown isometry; in particular, it is a minimal surface. Except in Section IV D below. for the time-dependent factor m(t), this is just like for the slices of constant Killing time in the Schwarzschild metric (the difference being that (30) does not extend to 4 In Section IV A of [4] it is suggested that the Sultana–Dyer metric an isometry of the spacetime metric unlessm ˙ = 0). Now, 2 3 is equal to the McVittie metric (1) in which a(t) = a0t / and the fact that r 0 corresponds to an asymptotically flat → m(t) = m0, for some constants a0 and m0 (see Eq. (62) in [4]). end of each of the 3-manifolds t = const. implies that Let denote the latter metric by g˜ . Indeed, since m is constant the McVittie metric cannot literally be interpreted as and in view of Proposition 1, g˜ is conformally related to the corresponding to a point particle sitting at r = 0 (r =0 Schwarzschild metric. Moreover, as one may explicitly check via is in infinite metric distance) in an otherwise spatially our Eq. (11c), the Einstein tensor of g˜ has a vanishing spherical part. Despite sharing these two properties, g˜ and the Sultana– flat FLRW universe, just like the Schwarzschild metric Dyer metric are not equal. does not correspond to a point particle sitting at r = 0 5 The problem with the reasoning in Section II of [2] is the following in Minkowski space. Unfortunately, McVittie seems to (numbers refer to equations in [2]): It is true by construction have interpreted his solution in this fashion [12] which that the Sultana–Dyer metric (2.1) is conformally related to the Schwarzschild metric, as expressed in the second line of (2.3) [the even until recently gave rise to some confusion in the first line in (2.3) does not follow], but the conformal function a literature (e.g. [5, 7, 20]). A clarification was given in [16]. depends non-trivially on the Schwarzschild coordinates for time and radius (denoted byη ¯ andr ˜ in [2]: Cf. our discussion in the last paragraph of Section III A). Hence it is not possible IV. ATTEMPTS TO GENERALIZE MCVITTIE’S to introduce a new time coordinate t¯ that satisfies dt¯ = adη¯ MODEL (the right hand side is not a closed 1-form), as pretended in the transition to (2.5). 6 To show this, one has to show that there exists no timelike di- The first obvious generalization consists in allowing for rection with respect to which the Ricci tensor (or, equivalently, a non-vanishing cosmological constant. However, as was the Einstein tensor) is spatially isotropic. This can be shown already indicated before, this is rather trivial since it as follows: First note that the Einstein tensor of the Sultana– Dyer metric has the form Ein = µu ⊗ u + τk ⊗ k (see [19]), merely corresponds to the substitutions ̺ ̺ + ̺Λ and → where u is a normalized future-pointing spherically-symmetric p p + pΛ in (4), where ̺Λ := Λ/8π and pΛ := Λ/8π timelike vector field and k the in-going future-pointing light- are→ the energy-density and pressure associated to the− cos- like vector field orthogonal to the SO(3)-orbits normalized such mological constant Λ. that g(u, k) = 1. In particular, the spherical part of the Ein- stein tensor vanishes: Hence, the Einstein tensor is spatially The attempts to non-trivially generalize the McVittie isotropic iff there exists a non-vanishing spacelike spherically- solution have focused so far on keeping the ansatz (1) and symmetric (i.e. orthogonal to the SO(3)-orbits) vector field s relaxing the conditions on the matter in various ways. In with Ein(s, s) = 0. Without loss of generality one can chose s [4] generalization were presented allowing radial fluid mo- to be normalized: s = sinh χu + cosh χe, where e is the normal- tions relative to the observer vector field ∂/∂t (that is ized vector field orthogonal to u and to the SO(3)-orbits pointing in positive radial direction. Hence one has k = u − e and thus: relaxing condition (3)) as well as including heat conduc- Ein(s, s)= µ sinh2(χ)+τ exp(2χ). Clearly, the latter expression tion. Below we will critically review these attempts, tak- vanishes nowhere in the physically interesting region (cf. Eq. (26) ing due care of the geometric constraints imposed by the in [19]), where both µ and τ are positive. 9
A. Einstein’s equation for the McVittie ansatz What we want to stress here is that the reason for this, as shown in more detail below, lies precisely in the re- In the following we will restrict to those generalizations striction imposed by spatial Ricci-isotropy. of the McVittie model which keep the metric ansatz (1) We take thus the perfect-fluid energy-momentum ten- and thus generalize only the matter model. For this sor (2) for the matter and an arbitrary spherically sym- purpose it is convenient to write down the Einstein’s metric four-velocity u. The latter is given in terms of the equation for an arbitrary spherically symmetric energy- orthonormal basis for the metric (1) by momentum tensor T . Recall that spherical symmetry im- u = cosh χ e0 + sinh χ e1 , (33) plies for the component of T with respect to the orthonor- where χ is the rapidity of u with respect to the observer mal basis (10) that T (ea, eA) = 0 and T (eA, eB) δAB, where a 0, 1 and A, B 2, 3 . Hence, the∝ only field e0 (a positive χ corresponds here to a boost in an independent,∈ { non-vanishing} components∈ { } of T are outward-pointing radial direction). The non-vanishing components of the matter energy-momentum tensor (2) S := T (e0, e0) (31a) with four-velocity (33) are: 2 Q := T (e1, e1) (31b) T (e0, e0)= ̺ + (̺ + p) sinh χ (34a) P := T (e2, e2) (31c) T (e0, e1)= (̺ + p) sinh χ cosh χ (34b) − J := T (e0, e1) , (31d) 2 − T (e1, e1)= p + (̺ + p) sinh χ (34c) and these are functions which do not depend on the an- T (e2, e2)= T (e3, e3)= p . (34d) gular coordinates. Note that S is the energy density, Clearly, the case of vanishing rapidity must lead to Q and P the radial and spherical pressure, and J the the original McVittie model. In this case, in fact, the energy flow—all referred to the observer field e0. The matter energy-momentum tensor (34) is already spatially sing in (31d) is chosen such that a positive J means a isotropic so that (32d) is identically satisfied. Moreover, flow of energy in positive radial direction. Taking (31) (32a) implies (am)˙ = 0 and hence, in view of (14), F = into account, the Einstein equation for the McVittie a/a˙ . Herewith Einstein’s equation reduces to (4) and ansatz (1) and an arbitrary spherically symmetric energy- thus one gets back the original McVittie model. momentum tensor T reduces to the following four equa- In case of non-vanishing rapidity, spatial Ricci- tions: isotropy (32d) implies the following constraint: · 2 B 2 (am) = 4πR A J (32a) ̺ + p =0 . (35) −2 8πS =3F (32b) This means that the energy momentum tensor (2) has the 2 2 div T 8πQ = 3F 2F˙ A (32c) form of a cosmological constant (using (35) in =0 − − B it implies dp = 0 and this, in turn, using again (35), P = Q. (32d) implies d̺ = 0) so that this case reduces to the Schwarzschild–de Sitter solution and hence does not pro- In view of (22), the first equation relates the time varia- vide the physical generalization originally hoped for. tion of the Weyl part of the MS energy contained in the sphere of radius R with the energy flow out of it. The last equation is nothing but spatial Ricci-isotropy. C. Perfect fluid plus heat flow In the following subsections we will consider three models for the cosmological matter which generalize the In a next step one may keep (33) and drop the con- original McVittie model: perfect fluid, perfect fluid plus dition that the fluid be perfect, in the sense of allowing heat flow, and perfect fluid plus null fluid. for radial heat conduction. This is described by a spatial vector field q that represents the current density of heat, B. Perfect fluid which here corresponds to the current density of energy in the rest frame of the fluid. Hence q is everywhere orthogonal to u.7 The fluid’s energy momentum tensor Perhaps the simplest step one can take in trying to generalize 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 to 7 We note that the parametrization of the energy-momentum ten- the ∂/∂t observer field. In this way one could hope to sor given in [4] is manifestly different. Whereas we parametrized avoid a particular singular behavior in the pressure that it in the usual fashion in terms of quantities (energy density, may be due to the ‘no-infall’ condition, though it is clear pressure, current density of heat) that refer to the fluid’s rest that the persisting geometric singularity must show up system, the authors of [4] also write down (36) (their Eq. (79)), but with q orthogonal to e0 (compare their Eq. (93)) rather than somehow in the matter variables as already discussed in u, which affects also the definition of ̺. In fact, marking their Section III D. Unfortunately, as already shown in [4], quantities with a prime, their expression (79) is equivalent to our the relaxation of (3) does not lead to any new solutions. (36) iff p = p′, q = q′ cosh χ, and ̺ = ̺′ − 2q′ sinh χ. 10 then reads These are almost the same as in the case of vanishing rapidity (see (39)), except for the opposite sign on the T = ̺ u u + p (u u g)+ u q + q u . (36) ⊗ ⊗ − ⊗ ⊗ right-hand side of (40b). This simply results from the Taking (33) as fluid velocity and imposing the heat flow- fact that, according to (38b), J = T (e0, e1) = q for vector q to be spherically symmetric, we have vanishing rapidity, whereas, due to the− constraint (40a), J = T (e0, e1) = q for non-vanishing rapidity. This q = q e := q (sinh χ e0 + cosh χ e1) , (37) will be− further interpreted− below. Notice that for the where q is a function of (t, r). Note that a positive q equation of state ̺ + p = 0 (cosmological term) (40a) im- corresponds to heat flowing in an outward-pointing radial plies q = 0, thus leading once more to the Schwarzschild– direction. The independent non-vanishing components of de Sitter solution (see comment below Eq. (35)). Hence- the energy-momentum tensor are now as follows: forth we assume ̺ + p = 0, which implies that one can solve the constraint (40a)6 for the rapidity: T (e0, e0)= ̺ + tanh χ (̺ + p) tanh χ +2 q (38a) 2 T (e0, e1)= q cosh χ (̺ + p) tanh χ +2 q (38b) 2 q − tanh χ = (41) 1 −̺ + p T (e1, e1)= p + 2 sinh(2 χ) (̺ + p) tanh χ +2 q (38c) T (e2, e2)= T (e3, e3)= p . (38d) provided that 2q/(̺ + p) < 1. | | Consider first the case of vanishing rapidity. Then the The Einstein equation gives now four equations for the energy-momentum tensor is already spatially isotropic six functions a,m,̺,p,q, and χ. As in the case of van- and Einstein’s equation (32) reduces to ishing rapidity, this system is under-determining and it is not possible to freely specify any two of these six func- 2 (am)˙ = 4πR2 q B (39a) tions and then determine the the other four. In a similar − A 8π̺ =3F 2 (39b) fashion as before, the easiest way to generate a solution is to specify the two functions a(t) and m(t), to let then 2 8πp = 3F 2F˙ A . (39c) the definitions (12,13,14) determine A,B,R,F , and fi- − − B These are three PDEs (though only time derivatives oc- nally use the Einstein equations (40b,40c,40d) and (41) cur) for the five functions a,m,̺,p, and q so that the to determine q,̺,p, and χ, respectively. Again, choos- system (39) is clearly under-determining. However, it is ing a and m such that their product is constant implies not possible to freely specify any two of these five func- q = 0 and F =a/a ˙ , which leads to the standard McVittie tions and then determine the the other three via (39). For solutions. example, since the left-hand side of (39a) depends only In passing we remark that the condition ̺ + p> 0 can on t, the same must hold for the r.h.s., which implies that be expressed geometrically in terms of the second time- q = f(t)/r2(1 (m/2r)2)2, where f(t) = (am)˙/4πa2. derivative of the areal radius. Indeed, adding either (39b) In particular, the− heat flow must fall-off as− 1/r2. to (39c) or (40c) to (40d) we obtain, taking into account The easiest way to generate a solution in the case of e0 = (A/B)∂/∂t and (15): zero rapidity is to specify the two functions a(t) and e0(R) m(t), then let A,B,R,F be determined by the def- 4π(̺ + p)= e0 , (42) − R initions (12,13,14), and finally let the Einstein equa- tions (39a,39b,39c) determine q,̺, and p, respectively. which is positive iff the rate of change e0(R)/R is a de- Notice that if we happen to specify a and m such that creasing function along the integral lines of the observer am is a constant, this immediately implies q = 0 and e0. In other words, ̺ + p is positive iff ln(R) is a con- F =a/a ˙ , which leads to the standard McVittie solutions. cave function on the worldline of the observer e0, which From (39a) the following is evident: if q> 0 (q < 0), that is implied by, but not equivalent to, the function R being is for outwardly (inwardly) pointing heat flow, the Weyl concave. part of the MS energy decreases (increases), as one would From (41) and (40b), and assuming ̺ + p> 0, one sees expect. the following: If χ> 0 (χ< 0), that is for an outwardly Now we turn to the general case with non-vanishing (inwardly) moving fluid with respect to e0, we have q< 0 rapidity: As it was the case for the perfect fluid in the (q > 0), that is an inwardly (outwardly) pointing heat previous subsection, the condition (32d) of spatial Ricci- flow, and the Weyl part of the MS energy decreases (in- isotropy implies a constraint on the matter: creases). This means that the heat flow’s contribution (̺ + p) tanh χ +2 q =0 . (40a) to the change of EW never compensates that of the fluid Using this, the other components of the Einstein’s equa- motion, quite in accord with naive expectation. Below tion reduces to: we show that for small rapidities the contribution due to the heat flow is minus one-half that of the cosmological ˙ 2 B 2 (am) = +4πR q A (40b) matter. 2 Let us now return to the sign-difference of the right- 8π̺ =3F (40c) hand sides of (39a) and (40b). From (38) one infers that 8πp = 3F 2 2F˙ A . (40d) − − B J is the sum of the two contributions coming from the 11 heat flow 2 1 2 2 T (e0, e0)= ̺ + (̺ + p) sinh χ + λ (48a) Jh := q(1 + 2 sinh χ) (43) 2 1 2 T (e0, e1)= (̺ + p) sinh χ cosh χ λ (48b) − ∓ 2 and from cosmological matter 2 1 2 T (e1, e1)= p + (̺ + p) sinh χ + 2 λ (48c)
Jm := (̺ + p) sinh χ cosh χ , (44) T (e2, e2)= T (e3, e3)= p . (48d) Here and below the upper (lower) sign corresponds to the respectively. The constraint (40a) can be written in the outgoing (ingoing) null field. form In the present case, the condition (32d) of spatial Ricci- 2 2 isotropy is equivalent to the constraint: 2 cosh χJh + (1 + 2 sinh χ)Jm =0, (45) 2 1 2 (̺ + p) sinh χ + 2 λ =0 . (49) which, for small rapidities χ (that is neglecting quadratic terms in χ), implies 2Jh +Jm 0. In this approximation In the physically relevant case in which ̺ + p > 0 the spatial energy-momentum≈ flow due to heat is minus this equation has only the trivial solution χ = 0 and one-half that due to the cosmological matter. For the λ = 0, which leads to the original McVittie model. In total flow this implies J = Jm + Jh Jm/2 Jh. Now the case ̺ + p = 0 (49) implies λ = 0, leading thus the sign difference between (39a) and≈ (40b) is≈− understood to the Schwarzschild–de Sitter spacetime (see comment as follows: In case of vanishing rapidity one has Jm = 0, below (35)). Hence, a new solution is only possible if Jh = q and hence J = q (leading to (39a)), whereas the matter is of an exotic type that satisfies ̺ + p < 0, a short calculation reveals that in case of non-vanishing which either violates the weak energy-condition (̺> 0), rapidity the constraint (45) implies J = Jm + Jh = q, or, less catastrophically, the dominant-energy condition leading thus to (40b). − (̺> p ). In particular, for the matter model considered by Sultana| | and Dyer, one would need to violate the weak energy-condition. D. Perfect fluid plus null fluid
V. CONCLUSION The last tentative generalization we consider is tak- ing for matter the incoherent sum (meaning that the respective energy-momentum tensors adds) of a perfect We conclude by commenting on the the main dif- fluid (possibly with non-vanishing pressure) and a null ferences between these generalizations and the original fluid (eventually representing electromagnetic radiation). McVittie model. First we stress once more that neither This clearly contains as special case the matter model allowing for a nonzero rapidity nor a nonzero heat flow considered by Sultana and Dyer [19] in which the pres- can eliminate the singularity at r = m/2 (R =2am) (as sure vanishes. We already stressed in Section IIIA that erroneously stated in [4]). The only substantial new fea- the metric ansatz of [19] is different from (1). Here we ture of these generalizations is that the Weyl part of the show that the matter model of [19] is essentially incom- MS energy EW = am is not constant anymore. In view patible with (1) except for trivial or exotic cases. of the fact that the combination 2 The matter model consists of an ordinary perfect fluid m/r = A EW/R EW/R (50) and a null fluid (e.g. electromagnetic radiation) with- ≈ out mutual interaction. Hence the matter’s energy- contained in the McVittie ansatz gives the ‘Newtonian’ momentum tensor is just the sum of (2) and part of the potential in the slow-motion and weak-field approximation (see [1]), we deduce that in order to get ± 2 ± ± the geodesic equation for the generalized McVittie model, Tnf = λ l l , (46) ⊗ it suffices to substitute m0 with EW in the equation of where λ is some non-negative function of t and r and motion derived in [1]. This means that the strength of the l+ and l− are, respectively, the outgoing and ingo- central attraction varies in time according to (32a), lead- ing future-pointing null vector fields orthogonal to the ing to an in- or out-spiraling of the orbits if dEW(e0) > 0 spheres of constant radius r partially normalized such or dEW(e0) < 0, respectively. that g(l+, l−) = 1. (It remains a freedom l± α±1l±, We identified the origin of why we could not vary the where α is a positive function). Without loss of7→ generality rapidity and the heat flow independently in the condi- we make use of this freedom and choose: tion (40a) of spatial Ricci-isotropy, which is built into the ansatz (1). We saw that this geometric feature ren- ± l = (e0 e1)/√2 , (47) ders this ansatz special, so that it would be improper to ± call it a general ansatz for spherical inhomogeneities in a where e0 and e1 are the vectors of the orthonormal flat FLRW universe. It remains to be seen whether use- frame (10). The components of the whole energy- ful generalizations exist which are captured by equally momentum tensor with respect to this frame are then: simple ans¨atze. 12
Acknowledgments In order to compute the respective quantities for the met- −2 McV ric Ω ga,m we first give their scaling behavior under D.G. acknowledges support from the Albert-Einstein- conformal transformations. Institute in Golm and the QUEST Excellence Cluster. Clearly, because of their very definitions, for the areal radius and the Weyl part of the MS energy it holds:
2 APPENDIX A: PROOF OF PROPOSITION 1 R(Ψ g)=ΨR(g) (A5) and In this appendix we compute the intersection of the 2 set McV of metrics of type (1), which we denote in the EW(Ψ g)=ΨEW(g) , (A6) S McV following by ga,m , with the set cS of metrics conformally related to an exterior SchwarzschildS metric. Explicitly, respectively. For the whole MS energy it easily follows the latter are of the form from (20) and (A5): cS 2 Schw 2 2 ∇ ∇ gΩ,M0 := Ω gM0 , (A1a) E(Ψ g)=Ψ E(g)+ R g( R, ln Ψ) (A7) where 1 3 ∇ ∇ + 2 R g( ln Ψ, ln Ψ) , −1 Schw 2M0 2 2M0 2 2 g = 1 dT 1 dR R g 2 , where all the quantities on the r.h.s. are referred to the M0 − R − − R − S metric g. Hence, taking the difference between (A7) (A1b) and (A6) one gets that the Ricci part of the MS energy denotes the Schwarzschild metric with mass M0 in ‘stan- scales exactly like the whole MS energy, that is according dard’ coordinates. The question is: for which functions a to (A7). and m and, respectively, for which function Ω and pa- McV 2 Schw Using these scaling properties together with (13) rameter M0 does the equation g = Ω g hold? a,m M0 and (22) we get immediately: Such an equality can be eventually established by finding a coordinate transformation, φ say, between the coordi- R(Ω−2gMcV) =Ω−1(1 + m/2r)2ar (A8) 8 a,m nates (t, r) in (1) and (T, R) in (A1) which brings (1) −2 McV −1 W g in form (A1). This involves solving coupled, non-linear E (Ω a,m )=Ω a m . (A9) partial differential equations for φ, which depend on the The equality between the Weyl part of the MS en- four unknown parameter a, m, Ω, and M0. Needless to ergy (A9) and (A3) implies say that this is not really a thankful task. Alternatively, a better approach would be to compare all the indepen- Ω(t, r) M0 = a(t) m(t) , (A10) dent, algebraic curvature-invariants of the two metrics: This would lead to a system of equations between scalars which gives a condition between the parameter a, m, Ω, which involves the coordinate transformation φ in an al- and M0. Since we assumed that M0 is positive, (A10) gebraic way (i.e. non differentiated). can be read as the expression for the conformal factor in We adopt here an approach which is somewhere in the the (t, r) coordinates. This, together with the equality middle: First, we use just three invariants (the areal ra- between the areal radius (A8) and (A2), implies in turn dius and the Ricci and the Weyl part of the MS energy) M0 to drastically restrict the form of the coordinate transfor- R(t, r)= (1 + m(t)/2r)2r , (A11) mation (see (A14)) and derive thereby constraints on the m(t) free parameters a, m, Ω, and M0 (see (A10) and (A13)). which gives the first component of the coordinate trans- Second, we perform this restricted coordinate transfor- formation φ. Now, using the scaling property (A7) for the mation and determine it completely. To simplify the Ricci part of the MS energy, the expressions (21) and (13) calculation, instead of gMcV = Ω2gSchw, we consider the a,m M0 for the Ricci part of the MS energy and, respectively, the equivalent equation Ω−2gMcV = gSchw. In fact, for the a,m M0 areal radius of the McVittie metric ansatz, and (A10) for Schwarzschild metric (A1b) it is immediate that the the conformal factor, one gets, after some computations, above mentioned quantities are, respectively, given by: 2 Schw −2 McV M0 2 3 m˙ R(g ) = R, (A2) ER(Ω g )= (A ar) . (A12) M0 a,m 2 am m Schw EW(gM0 )= M0 , (A3) Schw The equality between (A12) and (A4) then implies ER(gM0 ) =0 . (A4) m˙ =0 , (A13)
that is m = m0 for some positive constant m0. This, in 8 The transformation between the angular variables is just the turns, implies that the transformation (A11) for R de- identity. pends only on r and not on t. Since the metrics are both 13 in diagonal form, this implies that the transformation for 1, 2, 3 , vanish in this limit, that is iff it holds: T must depend on t only. { } ˙ ˙ Summarizing, so far we have seen that a set of neces- a˙(am) a˙ 4 2 + m =0 (B2a) sary conditions for the equality of the two metrics implies a a 2 the constraints (A10) and (A13) and that the coordinate (am)˙ (am)¨ a˙ (am)˙ transformation between (t, r) and (T, R) is of the form 2 +m = 0 (B2b) a a − a2 ˙ T (t) = f(t) (A14a) m˙ (am) =0 . (B2c)
M0 2 0 These conditions are clearly understood to hold for all R(r)= m0 (1 + m /2r) r , (A14b) times t in which the functions a and m and their deriva- tive exist. In view of (B2c) we have to distinguish be- for some differentiable function f of t. Now, explicitly · expressing the metric Ω2gSchw in the (t, r) coordinates tween two cases:m ˙ = 0 and (am) = 0, respectively. In M0 the first case the system (B2) reduces to the set of con- according to the coordinate transformation (A14) and 2 2 the constraints (A10) and (A13), and putting the result ditions m(¨a/a + 3(˙a/a) ) = 0 and m(¨a/a + (˙a/a) ) = 0, equal to gMcV, the only new condition that one gets is which, in turn, reduces to m = 0 (and a arbitrary), corre- a,m sponding to the FLRW metric, or toa ˙ = 0 (andm ˙ = 0), corresponding to the Schwarzschild metric. In the second M0 1 · · f˙ = . (A15) case, in which (am) = 0, (B2) reduces to m(˙a/a) = 0, ± m0 a which implies either m = 0 (and a arbitrary), correspond- ing again to the FLRW metric, or (˙a/a)· = 0. Together Here, the plus can be chosen in order to exclude a time with (am)· = 0, the latter corresponds to a McVittie inversion. It is important to note that (A15) (together metric with exponentially-growing (or -falling) scale fac- with an initial value) determines f uniquely and do not tor a(t), that is to a Schwarzschild–de Sitter metric. give any constraint on the parameters a, m, Ω, and M0: The only constraints remain thus (A10) and (A13). The proof is concluded noticing that (A10) means that APPENDIX C: SHEAR-FREE OBSERVER the only constraint on Ω is that, expressed in the (t, r) FIELDS IN SPHERICALLY SYMMETRIC coordinates, it depends on t only and hence, in view SPACETIMES of (A14a) and expressed in the (T, R) coordinates, that it depends on T only. More geometrically, this can be Towards the end of Section II we made use of the fol- restated saying that the gradient of Ω must be propor- lowing result: A spherically symmetric normalized time- tional to ∂/∂T , the Killing field of the Schwarzschild like vector field u in a spherically symmetric spacetime metric (see (A1b)). ( , g) is shear free iff the metric hu that g induces on theM subbundle u⊥ := v T g(v, u)=0 by restric- tion is conformally flat.{ ∈ M | } To prove this, we first note that the subbundle u⊥ is APPENDIX B: PROOF OF PROPOSITION 2 integrable, in other words, u is hypersurface orthogonal. This follows from the spherical symmetry of u, which implies that u⊥ contains the vectors tangent to the 2- Inserting the definition (14) of F in the expression (23) dimensional SO(3) orbits. Hence u essentially lives in for the Ricci scalar and organizing the result in powers the 2-dimensional orbit space9, where it is trivially hy- of rB r m/2 we get: ≡ − persurface orthogonal. The hypersurfaces orthogonal to u 2 in 4-dimensional spacetime are then the preimages un- a˙ der the natural projection of the hypersurfaces (curves) Scal = 12 − a in the 2-dimensional orbit space. As a result, we may now locally introduce so-called 6 a˙(am)˙ a˙ ˙ 4 + rA isochronous comoving coordinates, with respect to which − rB a2 a −1 ! u = A(t, r) ∂/∂t and 2 ˙ ¨ ˙ 2 2 2 2 2 6 (am) (am) a˙ (am) g = A (t, r) dt B (t, r) dr R (t, r) g 2 . (C1) 2 +rA − − S − (rB)2 a a − a2 ! 3 m˙ (am)˙ 3 rA . − (rB) a 9 The orbit space is the quotient M/∼, where ∼ is the equivalence (B1) relation whose equivalence classes are the orbits. It is a manifold on the subset corresponding to 2-sphere orbits, to which we re- strict attention here. To say that “u essentially lives in the orbit Hence, the Ricci scalar remains finite in the limit r space” means that u is the pull-back of a 1-form on the quotient −k → m/2 iff all the three coefficient of (rB) , for k via the natural projection. ∈ 14
(Note the different meanings of the functions A and B as radius along u, that we made use of in Section III B: compared to (16)). We now consider the tangent-space endomorphisms ∇u : X ∇X u and their projection σ + θ/3= u(ln(R)) . (C5) into the orthogonal complement7→ of u, i.e., Now, according to (C4), the shear of u vanishes iff the ∇⊥ ⊥ ∇ ⊥ u := Pu u Pu u⊥ , (C2) shear scalar σ does, that is, iff u ln(R/B) vanishes. ◦ ◦ This is equivalent to R/B being independent of t or to ⊥ where Pu := id u u is the projection orthogonal R(t, r) = µ(r)B(t, r) for some function µ, so that the to u (id is the identity− ⊗ endomorphism in the tangent line element (C1) can be rewritten in the spatially con- spaces of ). Note that ∇⊥u is symmetric due to the formally flat form hypersurfaceM orthogonality of u. A direct computation 2 2 2 2 2 ˜ ˜ 2 using (C1) yields g = A (t,ρ) dt C (t,ρ) dρ + ρ gS , (C6a) −
∇⊥ 2 u = u ln(B) Pr + u ln(R) PS , (C3) where A˜(t,ρ) := A(t, r(ρ)), C˜(t,ρ) := C(t, r(ρ)), and
2 where Pr and PS are the projections parallel to ∂/∂r B(t, r)µ(r) and parallel to the tangent 2-planes to the S2-orbits, re- C(t, r)= (C6b) ρ(r) spectively. The trace θ of ∇⊥u, which gives the expan- sion of u, is θ = u ln(B) +2u ln(R) , so that the trace- with free part of ∇⊥u, known as the shear endomorphism σ, r is given by: dr′ ρ(r)= ρ0 exp . (C6c) µ(r′) ⊥ 1 ⊥ r0 σ := u θ id = σ (P 2 2P ) , (C4) Z ∇ − 3 S − r Hence we see that vanishing shear of u implies confor- 1 where σ := 3 u ln(R/B) denotes the shear scalar (only mal flatness of the corresponding spatial metric. For the defined in a spherically-symmetric setting) and id⊥ = converse we first note that, since u and g are spherically ⊥ Pr + PS2 the identity endomorphism in u . In passing, symmetric, the spatial metric hu is itself spherically sym- we note that the defining equations for θ and σ just given metric, so that g can be written in the form (C6a). This immediately lead to the following simple relation between implies that the corresponding R/B depends only on the the shear scalar, expansion, and the variation of the areal radial coordinate and hence that the shear of u vanishes.
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