Cosmological black holes from self-gravitating fields
Elcio Abdalla, Niayesh Afshordi, Michele Fontanini, Daniel C. Guariento, Eleftherios Papantonopoulos
aa r X i v : . [ g r- q c ] M a y Cosmological black holes from self-gravitating fields
Elcio Abdalla, ∗ Niayesh Afshordi,
2, 3, † Michele Fontanini, ‡ Daniel C. Guariento,
1, 3, § and Eleftherios Papantonopoulos ¶ Instituto de Física, Universidade de São Paulo,Caixa Postal 66.318, 05315-970, São Paulo, SP, Brazil Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, N2L 2Y5, Canada Department of Physics, National Technical University of Athens, Zografou Campus GR 157 73, Athens, Greece
Both cosmological expansion and black holes are ubiquitous features of our observable Universe,yet exact solutions connecting the two have remained elusive. To this end, we study self-gravitatingclassical fields within dynamical spherically symmetric solutions that can describe black holes in anexpanding universe. After attempting a perturbative approach of a known black-hole solution withscalar hair, we show by exact methods that the unique scalar field action with first-order derivativesthat can source shear-free expansion around a black hole requires noncanonical kinetic terms. Theresulting action is an incompressible limit of k -essence, otherwise known as the cuscuton theory, andthe spacetime it describes is the McVittie metric. We further show that this solution is an exactsolution to the vacuum Hořava-Lifshitz gravity with anisotropic Weyl symmetry. PACS numbers: 04.40.-b, 04.20.Jb, 04.70.-s, 04.70.Bw
I. INTRODUCTION
Scalar fields, being the simplest tool in the hands ofmodel-building physicists, are often used in the contextof gravity and cosmology to build appropriate stress-energy tensors supporting models and to parametrize ef-fects which could be more fundamental, but allow for ascalar description in certain regimes. Besides being easyto use, scalars, and in particular their appearance in fun-damental theories, have recently become more “natural”following the discovery of the first observed scalar parti-cle in nature, the Higgs boson.Self-gravitating scalar fields play an important role inthe formation of primordial black holes from inhomo-geneities in the early Universe [1], and also as effectivedescriptions of fundamental fields in higher-dimensionalblack-hole spacetimes [2]. Another important role playedby scalars in the framework of black-hole physics has beento characterize the matter distribution outside the hori-zon, allowing for theoretical studies of the properties ofblack holes. There exist conjectures and theorems (theno-hair theorem in its many forms) forcing a scalar sur-rounding a black hole to take very specific configurations;for instance, a static black hole cannot have a nontriv-ial scalar hair on its horizon. On the other hand, hairyblack-hole solutions were found [3] in an asymptoticallyflat spacetime with the caveat that the scalar had to di-verge on the horizon. Moreover, such solutions were latershown to be unstable [4]. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected]; [email protected] ¶ [email protected] In general, in order to remedy the pathologies intro-duced by infinities, a regularization procedure has tobe used to make the scalar field finite on the horizon.Roughly speaking, there are two ways to make the scalarfield regular on the horizon, and they both require theintroduction of a new scale in the problem. The firstpossibility is to introduce a scale in the gravity sector ofthe theory through a cosmological constant; then hairyblack-hole solutions were found in which all possible in-finities of the scalar field were hidden behind the horizon[5, 6]. The other way is to introduce a scale in the scalarsector of the theory. This can be done by adding in theEinstein-Hilbert Lagrangian a coupling between a scalarfield and the Einstein tensor. This derivative couplingacts as an effective cosmological constant [7] and regularhairy black-hole solutions were found in asymptoticallyflat spacetime [8, 9].Similarly, in cosmology scalar fields are also widelyused. It would suffice to think of the inflationaryparadigm, following which at early times a semiclassi-cal scalar slowly rolls down its potential dumping energyin the gravitational sector of the theory, thus allowingspacetime to exponentially inflate. But the usefulnessof scalars does not reduce to early-time physics; theyare in fact the main ingredient of many proposals to ex-plain late-time acceleration, among which we recall thequintessence models that employ a canonical scalar fieldcoupled minimally to gravity with a scalar potential [10–12], and the k -essence models in which the explanationof late-time acceleration relies on the nonlinear dynamicsof a scalar field with noncanonical kinetic terms [13].Given that scalars seem to be ubiquitous in gravity andcosmology, it is interesting to investigate what the effectof a cosmological scalar field on a black-hole horizon is. Inthis case, the cosmological scalar field can vary with timeover cosmological scales, and its behavior near the black-hole horizon cannot be inferred from the no-hair theoremswhich strictly apply only to stationary configurations.Considering black holes surrounded by matter, the firstexample to consider is the well known time-dependentsolution described by the Vaidya metric [14]. This is aspherically symmetric solution of the Einstein equationsin the presence of a pure radiation field which propagatesat the speed of light. The Vaidya solution depends on amass function that characterizes the profile of radiation,and therefore any scalar field interacting with the Vaidyablack hole should be massless [15].More realistic solutions can be found for spherical in-homogeneous dust, which are collectively known as theLemaître-Tolman-Bondi (LTB) metric [16]. However, thesolutions generically fail upon shell crossing, which leadsto formation of caustic singularities. Some aspects ofblack-hole dynamics in this context have also been stud-ied [17, 18].A more promising approach to the problem of find-ing black holes embedded in a cosmological back-ground comes from looking for solutions representingblack holes in a Friedmann-Lemaître-Robertson-Walker(FLRW) Universe. There have been many previousattempts to obtain such solutions, among which theEinstein-Straus model represents perhaps the simplestone [19]. In this model the universe is built by patchingSchwarzschild black-hole spheres with a FLRW metricfilling the rest of the volume. However, these black holesare time symmetric, and so they fail to describe an over-all dynamical black-hole configuration in the Universe.Similarly, the “Swiss cheese” models patch FLRW pres-sureless spacetime with spherical holes containing com-pensated LTB solutions. These models have recentlygained popularity as idealized toy models for cosmologi-cal observations in an inhomogeneous universe (see, e.g.,Ref. [20]).Finally, one can consider the McVittie solution [21],a spherically symmetric, time-dependent solution of theEinstein equations that describes a black hole embeddedin a FLRW universe when the expansion asymptotes tode Sitter. The solution approaches a FLRW metric atlarge distances, while it looks like a black hole at shortdistances (at a horizon scale), and, despite the criticismsraised in the literature that the McVittie spacetime con-tains singularities on the horizon [22], recently it wasshown [23] that if a positive cosmological constant is in-troduced, then the metric is regular everywhere on andoutside the black-hole horizon. Similarly, the structure ofthe McVittie solution with a negative cosmological con-stant was discussed in Ref. [24].The usual approach followed to study the effect of acosmological scalar field on a black hole is to use a probeapproximation in which the scalar field does not backre-act onto the metric; i.e., the scalar is assumed to evolvein a static background [25–27]. An improved probe “slow-roll-like” approximation was used in Ref. [28] in which atime-dependent scalar field evolves in a time-dependentblack-hole background.In this work, however, we aim at addressing the fully self-consistent dynamical problem. In Sec. II, we startby considering a perturbative approach to a black-holesolution generated by a scalar field conformally coupledto curvature; we show that for sufficiently smooth devia-tions from such a solution the system of coupled Einsteinand scalar-field equations admits no solution. In Sec. III,we then investigate compatibility of most general scalarfield actions with first-order derivatives (otherwise knownas k -essence) with the McVittie spacetime, which singlesout an incompressible limit of these theories, known asquadratic cuscuton [29]. Finally, Sec. IV concludes thepaper.Throughout the paper, Greek indices run from 0 to 3and we use the ( − , + , + , +) signature. We denote partialderivatives with respect to t by a dot and with respectto r by a prime. II. COSMOLOGICAL BLACK HOLES WITHSTANDARD FIELDS
Consider the McVittie spacetime [21] in isotropic co-ordinates: d s = − (cid:18) − m ar m ar (cid:19) d t + a (cid:16) m ar (cid:17) (cid:0) d r + r dΩ (cid:1) , (1)with a = a ( t ) and m constant. This metric, presentedmany decades ago, is a classical solution of the Ein-stein equations which may describe a black hole (or ablack-hole/white-hole pair) in a shear-free cosmologicalbackground [23, 30–34]. The classical source that gen-erates this geometry is a comoving nonbarotropic per-fect fluid with homogeneous energy density but inhomo-geneous pressure. It is the unique perfect-fluid solutionof the Einstein equations which is spherically symmetric,shear-free, and asymptotically FLRW with a singular-ity at the center [35]. In the limit r ≫ m a , the McVittiespacetime asymptotes to a flat FLRW universe with scalefactor a . If a is set to be constant (say, a ≡ ), we recoverthe Schwarzschild spacetime with Arnowitt-Deser-Misnermass m written in isotropic coordinates. After a lengthydiscussion in the literature, it has been proved that, atleast in some cases (that is, when the scale factor tendsto that of de Sitter space), the line element (1) describesa black hole immersed in a FLRW universe [23, 30].A classical general relativity solution is of course inter-esting on its own, but it can lead to even more interest-ing results when it can be shown to be the gravitationalcounterpart of a field-theoretical model. This, as will beextensively presented later, is the case for the McVittiesolution (1).Inspired by its mathematical simplicity, one might askwhether metric (1) or some generalization thereof is asolution of the system composed of a scalar field cou-pled to gravity. However, a minimally coupled comovingcanonical scalar cannot be the answer, since it has homo-geneous density and pressure, contrary to what is neededto source McVittie. Considering a richer action, say, byadding a vector field, also does not help due to anothercharacteristic of metric (1), namely, spatial Ricci isotropy[32] which is incompatible with a canonical vector fieldand scalar in spherical symmetry.To tackle the problem, one could try a reversed ap-proach, starting from a known scalar-tensor black-holesolution that is close enough to the McVittie metric, or ageneralization of it, and perturb it to try to get at leastin some limit (for instance, at late times) a connectionto metric (1). One such example of a hairy black holein a cosmological background is the so-called Martínez-Troncoso-Zanelli (MTZ) solution [6, 36], which we con-sider as a starting point for our backward search.To better understand the logic behind the perturba-tive approach we develop later, we first retrace the stepstowards the MTZ solution and present a preliminary re-sult. The MTZ metric is sourced by a conformally cou-pled scalar and a vector field, but for simplicity we beginby neglecting the vector and considering the system de-scribed by the action I = Z d x √− g (cid:20) R − π − g αβ φ ; α φ ; β − Rφ − αφ (cid:21) , (2)where α is a dimensionless constant. The correspondingfield equations are G µν + Λ g µν = 8 πT (S) µν , (3) (cid:3) φ = 16 Rφ + 4 αφ , (4)where the energy-momentum tensor of the scalar field is T (S) µν = ∂ µ φ∂ ν φ − g µν g αβ ∂ α φ∂ β φ + 16 [ g µν (cid:3) − ∇ µ ∇ ν + G µν ] φ − αg µν φ , (5)and (cid:3) ≡ g αβ ∇ α ∇ β .Since we look for spherically symmetric solutions witha cosmological constant, a possible heuristic approach isto write a line element of the form d s = − (cid:16) − mr (cid:17) d t + d r (cid:0) − mr (cid:1) + r dΩ (6)in isotropic coordinates, with r = ρ + m : d s = − (cid:18) ρρ + m (cid:19) d t + (cid:18) mρ (cid:19) (cid:0) d ρ + ρ dΩ (cid:1) . (7)If we take (7) and replace ρ ( r ) by a function of the form ρ = ρa ( t ) , we can see that Eqs. (3) and (4) are satisfiedfor d s = − " ρρ + ma ( t ) d t + a ( t ) (cid:20) ma ( t ) ρ (cid:21) (cid:0) d ρ + ρ dΩ (cid:1) , (8a) with φ ( t, ρ ) = r − Λ6 α ma ( t ) ρ , (8b)and the scale factor is a ( t ) = a e √ Λ3 t . (8c)The solution given by (8a) and (8b) does not correspond,however, to a dynamical time-dependent black hole, sinceit can be rewritten as a stationary black hole by a coor-dinate transformation [37].To go further we need to allow more freedom by addinga vector field. Instead of trying to solve the result-ing equations from the action (2) with a time-dependentscalar field and a generic spherically symmetric metric,we modify the action and consider I = Z d x √− g (cid:20) R − π − g µν ∂ µ φ∂ ν φ − Rφ − αφ − π F µν F µν (cid:21) . (9)An exact solution of the coupled Einstein-Maxwell-scalarfield equations resulting from variation of the action (9)is the MTZ solution cited before, and it reads d s = − (cid:20)(cid:16) − mr (cid:17) − Λ3 r (cid:21) d t + (cid:20)(cid:16) − mr (cid:17) − Λ3 r (cid:21) − d r + r dΩ , (10a) φ ( r ) = mr − m r − Λ6 α , (10b) A = − qr d t , (10c)with the extra condition q = m (cid:18) π Λ9 α (cid:19) . (11)This solution describes a hairy static black-hole solutionwith a scalar field which is regular on the horizon and atinfinity.With the above results in mind, one can try to look fora generalization of the MTZ solution (10) which allowsfor more general cosmological histories, while keeping thefield content of the theory unchanged, namely, assuming(9). To introduce a general cosmological behavior we fo-cus on the class of metrics that are obtained by addinga time dependence to what was the cosmological con-stant in (10a) sending Λ → H ( t ) , which corresponds toa generalization of Reissner-Nordström-de Sitter with atime-dependent Hubble factor. Even after making thisspecific choice for the metric, solving the equations di-rectly is quite challenging, so to proceed we restrict ourattention even further to the subclass of spacetimes thatreduce asymptotically in time to a cosmological constant.This allows for a perturbative treatment at large t thatstrongly simplifies the equations. Of course, the validityof the results will be limited to the class of systems thattend “smoothly enough” to the MTZ solution we haveseen above, which was anyway our initial goal when weasked the question of whether or not the McVittie solu-tion or some generalization thereof could be sourced byfields in an analogous way to the MTZ solution.Starting then from the action (9), we look for solutionsof the form d s = − (cid:20)(cid:16) − mr (cid:17) − H ( t ) r (cid:21) d t + (cid:20)(cid:16) − mr (cid:17) − H ( t ) r (cid:21) − d r + r dΩ , (12)requiring in turn that H ( t ) t →∞ → Λ / with some inversepower of t , namely, H ( t ) → H t n + H , ∀ n ≥ , (13)where H is some positive constant and H = Λ / . Onthe field side, the large-time corrections will be generallygiven by φ ( t, r ) = φ MTZ + δφ ( t, r ) , (14) A µ ( t, r ) = A MTZ µ + δA µ ( t, r ) , (15)where the MTZ label refers to the solutions in Eqs. (10)and the corrections die off at time infinity.With these assumptions one can expand both sides ofthe Einstein equations and, comparing the field pertur-bations to the decaying rate of the left-hand side definedby the form of H ( t ) (13), find the large-time behavior of δφ and δA µ . This turns out to be δφ ( t, r ) = Φ( r ) t n (16) δA t ( t, r ) = a t ( r ) t n (17) ∂ t δA r ( t, r ) = a r ( r ) t n (18) δA θ ( t, r ) = O (cid:0) t − n − (cid:1) (19) δA φ ( t, r ) = O (cid:0) t − n − (cid:1) . (20)Although these are the maximal perturbations allowedfor the scalar and the vector, it is easy to see that al-lowing any of the above to be of higher order wouldeliminate from the game either the scalar or the vector,reducing even further the possibility to find solutions.In other words, δφ ( t, r ) could be taken of O (cid:0) t − n − (cid:1) ,but this would just make all the scalar corrections dropfrom the equations at the order at which H t − n correc-tions appear. Similarly, due to the fact that the correc-tions introduced in the metric respect spherical symme-try, of the four components of the vector field the an-gular ones must be negligible, and only the combination δE r ( t, r ) ≡ ∂ r A t ( t, r ) − ∂ t A r ( t, r ) is physical. Moreover,the latter is forced to decay as t − n for the same reasonsdiscussed in the scalar field case.Finally, one can search perturbatively for solutions ofEinstein equations, only to find out that there is none.This result has to be read as the impossibility of pertur-batively building a solution that smoothly tends to MTZat large times [where smoothly is properly defined by thelarge time behavior t − n of H ( t ) , for all n ≥ ] startingwith the ingredients in the action (9): namely, a confor-mally coupled scalar and a vector field, and the ansatzfor the metric in (12). III. MCVITTIE SPACETIME AS A SOLUTIONTO A COSMOLOGICAL SCALAR FIELD
The results from Sec. II imply that canonical scalaror vector fields cannot act as sources to the generalizedMTZ solution with an arbitrary cosmological history. Ifwe consider the McVittie spacetime as the target solu-tion, the restrictions become more severe, due to spatialRicci isotropy of the metric [32], which can be physicallyinterpreted as the fact that a comoving fluid must be de-void of anisotropic stresses in this spacetime. In this case,a canonical scalar (quintessence) field is automaticallyexcluded as the sole source, since Ricci isotropy requiresthe field to be homogeneous. As a consequence, there canbe no radial dependence on the potential, and thereforethe characteristic inhomogeneous pressure which arisesin McVittie cannot be obtained from the field. Similarinconsistencies arise when one also considers vector fieldson the matter action.A possible way out is to relax the assumptions on thescalar field, for instance, by allowing noncanonical kineticterms in the action. In the following sections, we showthat one such k -essence field can provide a consistentsolution to the McVittie metric, both by direct inspectionof the equations and from general requirements over theaction. A. The k -essence field We begin by reviewing a few details of the k -essencefield. Consider the scalar φ introduced [13, 38] via theaction S φ = Z d x √− g L ( X, φ ) , (21)where X = − g µν φ ; µ φ ; ν (22)is the canonical kinetic term of the field φ . Variationof the action (21) with respect to φ gives the k -essenceequation of motion − δSδφ = (cid:0) L ,X g αβ + L ,XX φ ; α φ ; β (cid:1) φ ; αβ + 2 X L ,Xφ − L ,φ = 0 , (23)so when the condition L ,X + 2 X L ,XX > , (24)holds, Eq. (23) is hyperbolic and φ describes a physicaldegree of freedom [39].Variation of the action (21) with respect to g µν givesthe energy-momentum tensor T µν ≡ √− g δS φ δg µν = L ,X φ ; µ φ ; ν + g µν L . (25)Note that the null energy condition also imposes L ,X ≥ .The form of the energy-momentum tensor defined by (25)allows us to define an equivalent fluid velocity [39] u µ ≡ − ∇ µ φ √ X , (26)so we can cast (25) in the form of a perfect fluid: T µν = ρ u µ u ν + p h µν , (27)where h µν ≡ g µν + u µ u ν is the projection tensor in the or-thogonal direction to the flow, and the equivalent densityand pressure are defined as ρ ≡ X L ,X − L , (28) p ≡ L . (29) B. Cuscuton field in spherical symmetry
An interesting particular case of the wide family of k -essence systems is the cuscuton field [29]. This fieldis defined by choosing the following form for the action(21): S φ = Z d x √− g (cid:20) µ q − g αβ φ ; α φ ; β − V ( φ ) (cid:21) , (30)where µ is a constant. The equation of motion (23) thenspecializes to √− g √− gφ ; γ p − g αβ φ ; α φ ; β ! ,γ − µ d V d φ = 0 , (31)and the energy-momentum tensor (25) specializes to T µν = µ φ ; µ φ ; ν p − g αβ φ ; α φ ; β + g µν (cid:20) µ q − g γδ φ ; γ φ ; δ − V ( φ ) (cid:21) . (32) Assuming the most general spherically symmetricansatz for the metric d s = − e ν ( r,t ) d t + e λ ( r,t ) d r + Y ( r, t )dΩ , (33)and a homogeneous cuscuton field φ = ϕ ( t ) , the energy-momentum tensor (32) reduces to T µν = V ( ϕ ) u µ u ν + (cid:2) e − ν µ ˙ ϕ − V ( ϕ ) (cid:3) h µν . (34)Comparison with Eq. (27) and homogeneity of the fieldallow us to associate the potential V ( ϕ ) , which is nowjust a function of time, with the energy density as V ( ϕ ) = ρ ( t ) . (35)The field equation (31) then reduces to YY + ˙ λ ! e − ν = 1 µ d V d ϕ . (36)and using Eq. (35) we can rewrite the right-hand side ofEq. (36) as µ d V d ϕ = ˙ ρµ ˙ ϕ . (37)Moreover, the expansion scalar Θ ≡ u µ ; µ in terms of thecomoving flow with respect to the line element (33) reads Θ = YY + ˙ λ ! e − ν (38)so Eq. (36) really tells us that the expansion scalar is justa function of time. Although we started with the mostgeneral spherically symmetric metric, sourcing it with ahomogeneous field with action given by (30) simplifiesthings, and, in particular, one gets homogeneity in thedensity and expansion. We then write, as is customaryin the case of homogeneous fluids, Θ ≡ H ( t ) . Insertingthis definition for H ( t ) into Eq. (36), we get H ( t ) = 1 µ d V d ϕ . (39)There are several equivalent ways to obtain the above re-sult, be it from the conservation of the energy-momentumtensor (34), whose only nonidentically satisfied compo-nent yields the same result as Eq. (36), or from the traceof the extrinsic curvature. More specifically, since themean extrinsic curvature K αα has the exact same ex-pression as Θ from Eq. (38), Eq. (39) gives K αα = 1 µ d V d ϕ = 3 H ( t ) ; (40)that is, the mean curvature K αα is constant along thesurfaces of constant φ .It has been proven [40] that a regular spherically sym-metric perfect fluid with homogeneous energy density (inits comoving frame) can only drive uniform shear-free ex-pansion (or contraction). We have shown in this sectionthat the cuscuton action (30) satisfies all the conditionsnecessary for this theorem to hold. C. McVittie as a solution to the cuscuton
There are two facts which motivate us to considerMcVittie as a solution to a self-gravitating classical cus-cuton. The first is the fact that the equivalent fluid de-scription of McVittie metric possesses homogeneous den-sity but inhomogeneous pressure, and, unlike a canonicalscalar, a homogeneous cuscuton allows for homogeneousdensity without forcing a homogeneous pressure, as canbe seen by inserting (35) in the first term of Eq. (34). Thesecond comes from considering the extrinsic curvature inMcVittie spacetime. By using the foliation defined by thecomoving flow, the mean extrinsic curvature of metric (1)is K αα = 3 ˙ aa , (41)which is independent of the radial coordinate. Therefore,the line element (1) admits a foliation with constant meancurvature, which coincides with the comoving foliationused to obtain Eq. (40).Based on these similarities, we now assume the McVit-tie metric (1) as an ansatz for the system consisting of aself-gravitating cuscuton, and incidentally by comparisonwith Eq. (40) we make the familiar connection ˙ aa = H .In this metric, the Einstein equations for the cuscutonwith the energy-momentum tensor given by (34) read H = 8 πV ( ϕ ) , (42) H ar + m ar − m + 3 H = 8 π (cid:18) V − ar + m ar − m µ ˙ ϕ (cid:19) , (43)and the equation of motion (39) reduces to − aa + 1 µ d V d ϕ = 0 . (44)Substituting the potential from (42) into (43), we find πµ ˙ ϕ = − ˙ H . (45)Using (42) to eliminate ˙ aa from (44), we find (cid:18) d V d ϕ (cid:19) − πµ V = 0 , (46)which may be solved for V , so V ( ϕ ) = 6 πµ ( ϕ + C ) , (47)where C is an irrelevant integration constant.Inserting (45) into (43) and (35) into (42), we recoverthe familiar forms for the density and pressure in theMcVittie metric, and the system closes with the solu-tion (47). We can conclude therefore that, in fact, theMcVittie metric is a consistent exact solution to thecomplete system consisting of a self-gravitating cuscutonminimally coupled to gravity. D. McVittie for a general k -essence Given the results of the previous section, we mightnow ask what the most general k -essence field that satis-fies Einstein equations for McVittie spacetime (1) is. Inother words, we now address the question of whether ornot a homogeneous cuscuton obtained from the action(30) is the unique k -essence field which is consistent withMcVittie. Starting from the action (21), we may nowcast the energy-momentum tensor (25) under metric (1)by using the equivalent fluid description.So, in McVittie, the kinetic term X from (22) reads X = − a r (2 ar + m ) ( φ ′ ) + 12 (2 ar + m ) (2 ar − m ) ˙ φ . (48)Considering the equivalent four-velocity defined in (26),it becomes clear that, to satisfy the Einstein equations forMcVittie, which requires a comoving perfect fluid, thatis, that the off-diagonal terms of the energy-momentumtensor vanish, one must have either a constant field (thatsimultaneously leads to L ,X = 0 ) or φ = ϕ ( t ) . In otherwords, a homogeneous k -essence field is a necessary con-dition for McVittie to be a solution. A uniform fieldimmediately implies that the spatial Ricci-isotropy con-dition is satisfied, another requirement of the McVittiesolution [32], so the remaining independent equations arethe ( t, t ) and the ( r, r ) components of Einstein’s equa-tions and the equation of motion for the k -essence field.We have seen in Sec. III C that the special case in which L ( X, ϕ ) = µ √ X − V ( ϕ ) is a solution, so now we wantto constrain the functional form of possible L . In orderto do so we perform a simple check: considering the tt component of Einstein’s equations − X L ,X + L = − π (cid:18) ˙ aa (cid:19) , (49)we realize that the right-hand side does not contain anyradial coordinate dependence. We can then take itsderivative with respect to r , to obtain the constraintequation ma ˙ ϕ (2 ar + m )(2 ar − m ) (2 X L ,XX + L ,X ) = 0 , (50)which vanishes only if the last term vanishes. We canthen formally integrate it and find the solution L ( X, ϕ ) = A ( ϕ ) + B ( ϕ ) √ X . (51)This proves that the cuscuton is the only k -essence modelcoming from action (21) that can support the McVit-tie geometry, given that the integration constant A in(51) can be reabsorbed in the potential V ( ϕ ) and that B can be reabsorbed with a trivial field redefinition ϕ → φ = µ − R p B ( ϕ ) dϕ . Note that (51) violates theinequality (24), which means that the field ϕ does notcarry a physical degree of freedom but rather “latcheson” the evolution of the gravitational part, itself fully de-termined by the shape of the potential as per Eq. (42).This is a signature of the parasitic nature of the cuscutonfield [29]. E. Hořava-Lifshitz gravity with anisotropic Weylsymmetry
In this section, we make a surprising observation thatconnects McVittie cosmological black holes to a pro-posal for quantum gravity. Hořava-Lifshitz gravity isa proposal for a power-counting renormalizable theoryof quantum gravity that trades Lorentz symmetry with z = 3 Lifshitz symmetry (or anisotropic space and timescaling symmetry) at high energies [41]. However, effec-tive field theory arguments suggest that relevant Lorentz-violating terms can be generated from quantum correc-tions at low energies and have been the subject of con-troversy and intense scrutiny. This relevant deformationis parametrized by the parameter λ , which is equal to for Einstein gravity, but its value is not protected by anysymmetry in the Lorentz-violating theory.It turns out that the cuscuton action (30) with aquadratic potential, coupled to Einstein gravity, is equiv-alent to the low-energy limit of the (nonprojectable)Hořava-Lifshitz gravity [42]. For a general quadratic po-tential V ( φ ) = M φ , the Lorentz-violating parameter λ in Hořava-Lifshitz gravity is given by λ = 1 − πµ M = 1 − π π = 1 −
23 = 13 , (52)where, in the second line, we used M = 12 πµ fromEq. (47) for the cuscuton theory that satisfies the McVit-tie geometry.The significance of λ = / was already pointed outby Hořava in his original paper [41]. While the z = 3 Lifshitz global anisotropic scaling is the sufficient con-dition for power-counting renormalizability of geometro-dynamics, one may impose the stricter constraint of local anisotropic Weyl symmetry, which significantly limits theallowed terms in the action and fixes λ = / . The otherterm that is allowed by the Weyl symmetry is quadraticin the Cotton tensor for spatial 3-metric and thus van-ishes for McVittie geometry which has a conformally flat(or shear-free) spatial geometry.Therefore, we conclude that McVittie black-hole space-times are also exact solutions to the vacuum Hořava-Lifshitz gravity with anisotropic Weyl symmetry. Furthermore, one can establish a formal (but nonlocal) equiv-alence between anisotropic Weyl+ d diffeomorphisms and d diffeomorphisms, implying that the theory maintains the samenumber of degrees of freedom as general relativity. This formsthe basis for the shape dynamics approach to geometrodynamics[43]. IV. CONCLUSIONS AND FUTUREPROSPECTS
In this work, we have looked at the problem of self-gravitating classical fields, specifically searching for dy-namical spherically symmetric solutions which can de-scribe black holes in an expanding universe, by concen-trating on the particular cases of the MTZ and McVit-tie metrics. We have shown that a generalization of theMTZ metric with an arbitrary cosmological history can-not be obtained perturbatively through the use of canon-ical fields. In McVittie, the situation is even more restric-tive, since the symmetries of the Einstein tensor prohibitcanonical fields from being homogeneous (a required con-dition imposed by Ricci isotropy, meaning that the equiv-alent fluid is free from anisotropic stresses) while retain-ing the characteristic inhomogeneous pressure present inthe McVittie metric. However, by allowing for a non-canonical kinetic term in the scalar action, we have shownthat it is possible to obtain a consistent solution: thecuscuton theory is the unique form of scalar field actionwhich gives the McVittie metric when coupled to gravity.Furthermore, using the equivalence between cuscu-ton and low energy Hořava-Lifshitz gravity, we showedthat McVittie geometry is also an exact vacuum solutionfor the full nonprojectable Hořava-Lifshitz gravity withanisotropic Weyl symmetry.This result opens the possibility of analyzing the prop-erties of self-gravitating fields on richer backgrounds thanMinkowski or Schwarzschild. Moreover, despite the factthat McVittie is unlikely to describe a realistic classicalfluid configuration, the fact that it is a solution to a scalarfield is a decisive step forward in studies of modified grav-ity, since many modified gravity theories can be restatedin the Einstein frame as scalar degrees of freedom coupledto general relativity.Let us make some final observations that may form thebases for directions of future inquiry.1. If we consider the Hořava-Lifshitz theory with Weylsymmetry to be a viable UV completion of gravity,then McVittie will be an appropriate geometry forprimordial black holes that existed at the momentof the big bang, prior to the Planck time. This po-tentially opens an unprecedented window into theinhomogeneous dynamics of the big bang.2. It is interesting that the most general known ex-act solutions for cosmological black holes, LTB andMcVittie spacetimes, are exact opposites, i.e., havefluids with zero and infinite speeds of sound, respec-tively. While LTB can have an arbitrary comovingdensity profile ρ ( r ) , McVittie has an arbitrary ex-pansion history a ( t ) . Could these two spacetimesbe connected via a (yet unknown) duality that re-places space and time?3. Can McVittie black holes teach us something aboutthe physics of black holes that we may encounterin nature? Furthermore, could McVittie providea potential description for the interaction of darkenergy with astrophysical black holes?These are all questions with various degrees of specu-lation and/or physical relevance that continue to extendthe legacy of McVittie into the 21st century. ACKNOWLEDGMENTS
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