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cosmology and black holes as seen through the eyes of general relativity come together in the investigation of a dynamical black hole embedded in a cosmological background . the interplay between the cosmic dynamics and the black hole gives rise to interesting phenomena and can reveal some unexpected features of the underlying theory of gravity . in this work we restrict our attention , for simplicity , to spherically symmetric systems . the prototypical solution of the einstein equations representing a black hole embedded in a cosmological spacetime is the schwarzschild - de sitter - kottler solution . this metric is special since it admits a timelike killing vector and is , therefore , static in the region between the black hole horizon and the de sitter ( cosmological ) horizon . a less well known solution is the 1933 mcvittie solution @xcite , which is a generalization of the schwarzschild - de sitter - kottler solution . in this case the black hole is embedded in a general friedmann - lematre - robertson - walker ( flrw ) background , so that the region between the black hole horizon and cosmological horizon need not be static . although it has been studied and celebrated by many authors @xcite , it has proved surprisingly difficult to understand ( see the recent work @xcite ) . a simplifying assumption in the study of this solution , explicitly stated in mcvittie s original paper , is the no - accretion condition @xmath0 ( in spherical coordinates , where @xmath1 is the einstein tensor ) . this explicitly forbids any radial flow of material , which should otherwise occur whenever a spherically symmetric local inhomogeneity ( such as a central black hole ) is introduced in the background . when this is modelled however , more general solutions of einstein s theory become possible . these include some generalized mcvittie solutions @xcite ; solutions such as those derived by husain - martinez - nuez @xcite , fonarev @xcite , sultana - dyer @xcite and mcclure - dyer @xcite ; the class of solutions found by szekeres @xcite ; lematre - tolman - bondi black hole solutions @xcite ; and other solutions @xcite . in extended theories of gravity , such as scalar - tensor and @xmath2 gravity , several other solutions of the relevant field equations ( which involve an extra gravitational scalar field or higher derivative terms , respectively ) have been found and sometimes discussed @xcite .    the original motivation for mcvittie s work @xcite was the investigation of the effects of the cosmological expansion on local systems . another approach to this problem later led to the construction of swiss - cheese models by einstein and straus @xcite . however , although this problem has stimulated much discussion over the years @xcite , the scientific community as a whole is yet to arrive at an agreement about the best approach to it ( see the recent review @xcite ) . when solutions representing local inhomogeneities in cosmic backgrounds are considered , the scope of the investigation broadens . for example , a problem of current interest is the possible spatial and temporal variation of the gravitational `` constant '' ( which becomes a scalar field in brans - dicke and scalar - tensor gravity ) @xcite . we now know several solutions of this kind , but before enlarging the catalog further it is important to fully understand the presently known solutions ( for some of them , it is not even known whether the local inhomogeneity is associated with a black hole , a naked singularity , or another kind of object ) . for this reason , we revisit here the no - accretion mcvittie solution , proposing a quick way of locating the associated black hole and cosmological ( apparent ) horizons and studying their evolution . we extend the type of cosmological background to include phantom universes , which have not been considered before in relation to the mcvittie solution .    with the exception of the schwarzschild - de sitter - kottler solution , which incorporates only a static background universe , spherically symmetric black holes in more general cosmological backgrounds are dynamical . this significantly complicates their analyses . since the solutions of einstein s equations corresponding to the mcvittie metric are highly dynamical , it is not convenient for us to study the event horizons ( both black hole and cosmological ) , which may not even exist . it is more instructive to study the dynamical apparent horizons , the importance of which is being increasingly recognized in the literature @xcite . it is known that , for dynamical cosmological black holes , apparent horizons can appear or disappear @xcite , and we would like to shed some light on this bizarre phenomenology . the plan of this paper is as follows . in sec . ii we briefly review the schwarzschild - de sitter - kottler solution ; this ( over-)simplified situation will serve us well when attempting to understand the more complicated phenomenology of dynamical apparent horizons . in sec . iii we locate the apparent horizons of the mcvittie metric for non - phantom cosmological backgrounds and recover the previous results in certain limits . we then continue with the analysis of phantom background universes . finally , sec . iv contains a discussion of our results and our conclusions . throughout this work we use units in which the speed of light @xmath3 and newton s constant @xmath4 are unity , and we mostly follow the notations of ref . in particular , the metric signature is @xmath5 . the schwarzschild - de sitter - kottler solution is the prototypical solution representing a black hole embedded in a cosmological background ( for a certain range of parameter values ) . we will discuss the mcvittie metric by using an analogy with the schwarzschild - de sitter - kottler metric wherever possible , even though the latter corresponds to a very special situation by admitting only a _ static _ black hole in the de sitter background , and its apparent horizons are also event horizons . nonetheless , analogies are made possible by the fact that the schwarzschild - de sitter - kottler solution is contained as a special case in the mcvittie class of solutions . the spherically symmetric schwarzschild - de sitter - kottler solution of the einstein equations has line element    [ 1 ] ds^2=- ( 1 - -h^2r^2 ) dt^2 + ( 1 - -h^2r^2 ) ^-1 dr^2 + r^2 d_(2)^2 , [ eq : sdsk ]    where @xmath6 is the areal radius ( of a sphere with surface area @xmath7 ) , @xmath8 is the metric on the unit 2-sphere , the constant @xmath9 is the hubble parameter of the de sitter background , @xmath10 is the cosmological constant and @xmath11 is a second parameter describing the mass of the central inhomogeneity ( _ e.g. _ , @xcite ) . in general , the locations of the apparent horizons for a spherically - symmetric system can be calculated from the radial element of the inverse metric @xmath12 @xcite . thus the apparent horizons for the schwarzschild - de sitter - kottler solution are defined by the positive roots of the cubic equation [ 3 ] 1- -h^2r^2 = 0.[eq : sdskhor ] following the method outlined by nickalls in @xcite , these roots may be written as @xmath13 where @xmath14 . since @xmath15 and @xmath16 are both necessarily positive ( we only consider expanding universes ) , @xmath17 is negative and therefore unphysical . we thus refer to this spacetime as having only two apparent horizons . we refer to @xmath18 as the black hole apparent horizon , since it reduces simply to the schwarzschild horizon at @xmath19 if there is no background expansion @xmath20 , and we refer to @xmath21 as the cosmological apparent horizon , since it reduces to the static de sitter horizon at @xmath22 if there is no mass present . the metric  ( [ 1 ] ) is static in the region covered by the coordinates @xmath23 , which is comprised between these two horizons . a number of interesting observations can be made . first , both apparent horizons only actually exist if @xmath24 . in this case , since the metric is static between these two horizons , the apparent black hole and cosmological horizons are also event horizons and , therefore , null surfaces . second , if @xmath25 it is easy to show that these horizons then coincide . this case corresponds to the nariai black hole . finally , for @xmath26 both horizons become complex - valued and therefore unphysical , and one is left with a naked singularity . these results can be summarized as follows : @xmath27    the hubble parameter for an idealized de sitter background is a constant , whereas more realistic models incorporate a time - dependent hubble parameter . with a clear understanding of the static horizons in the schwarzschild - de sitter - kottler spacetime , we may now study the dynamical horizons which emerge by considering a more realistic time - dependent metric . we now consider the mcvittie metric for a black hole embedded in an flrw background which is expanding with the hubble flow @xcite . for simplicity , we restrict ourselves to the case in which the background is spatially flat ( curvature index @xmath28 ) . the line element can thus be cast in the form @xcite ds^2=-dt^2 - dtdr + r^2d_(2)^2 . [ 4 ] here @xmath29 , where @xmath30 is the scale factor of the flrw background and an overdot denotes differentiation with respect to the comoving time @xmath31 . note that for the case of a static background in which @xmath32 and @xmath33 , the mcvittie metric actually corresponds to the schwarzschild - de sitter - kottler metric given by  ( [ eq : sdsk ] ) via a simple transformation of the time coordinate @xcite . assuming a perfect fluid stress energy tensor , we may use einstein s equations to calculate forms for the density @xmath34 and pressure @xmath35 of the background fluid in mcvittie s metric . the density turns out to correspond to the known flrw density ( t)= h^2(t ) , one may consider arbitrary flrw backgrounds generated by cosmic fluids satisfying any equation of state ( in fact , in the next section , we study a flrw universe dominated by a phantom fluid ) . for illustrative purposes however , in this section we restrict our attention to a cosmic fluid which reduces to dust at spatial infinity . this corresponds to an equation of state parameter @xmath36 , so the pressure can be shown to be @xcite p(t , r)=(t ) ( -1 ) [ 5 ] . other quantities may be calculated from the inverse metric , given by ( g^ ) =(    cccc - & - & 0 & 0 + & & & + - & ( 1- -h^2r^2 ) & 0 & 0 + & & & + 0 & 0 & & 0 + & & & + 0 & 0 & 0 &    ) . [ 6 ] the misner - sharp - hernandez mass @xmath37 @xcite contained in a sphere of areal radius @xmath6 is defined , in the case of spherical symmetry , by 1- = g^rr . thus , we obtain m_msh= r^3 + m . which is interpreted as the contribution of the energy of the cosmic fluid contained in the ball plus the contribution of the local inhomogeneity . this mass coincides with the hawking - hayward quasi - local mass @xcite .    since for the mcvittie metric @xmath6 is an areal radius and the system is spherically symmetric , the apparent horizons can once again be calculated from @xmath12 , corresponding to [ 8 ] 1 - -h^2(t ) r^2=0 this is clearly equivalent to the schwarzschild - de sitter - kottler horizon condition given by ( [ eq : sdskhor ] ) but with a time - dependent hubble parameter . we denote the resulting time - dependent apparent horizons @xmath38 and @xmath39 , and these correspond to the solutions @xmath18 and @xmath21 given in equation ( [ eq : horizons ] ) but with the replacement @xmath40 . since the apparent horizons for the mcvittie metric are dynamical , rather than static , their relative locations now depend on the cosmic time . analogous to the schwarzschild - de sitter kottler case , the condition for both horizons to exist is @xmath41 , which corresponds to @xmath42 ( and @xmath43 which is always satisfied ) . however , unlike the former case where the hubble parameter is a constant , this inequality will only be satisfied at certain times during the cosmological expansion , and not at others . the time at which @xmath44 is unique for a dust - dominated background with @xmath45 , and we denote it @xmath46 . the three cases may then be characterized as :    * @xmath47 : at early times @xmath48 , so both @xmath38 and @xmath39 are complex and therefore unphysical . _ there are no apparent horizons . _ * @xmath49 : at this time @xmath50 and the horizons @xmath38 and @xmath39 coincide at a real , physical location . _ there is a single apparent horizon at @xmath51_. * @xmath52 : at late times @xmath53 , so both @xmath38 and @xmath39 are real and therefore physical . _ there are two apparent horizons . _    the qualitative dynamical picture which emerges from this analysis is the following and is illustrated in fig . [ figure1 ] .    , hence time @xmath31 and radius @xmath6 are measured in units of @xmath15 ( see text for details).[figure1 ] ]    the lack of apparent horizons for @xmath54 leaves a naked singularity at @xmath55 , where the ricci scalar and pressure also diverge ( see below ) . this is explained by the divergence of the hubble parameter @xmath56 in the early universe , causing the mass @xmath15 to remain supercritical , _ i.e. _ causing @xmath57 to be satisfied . analogous to the schwarzschild - de sitter - kottler solution , a black hole horizon can not be accommodated in such a small universe .    at the critical time @xmath58 a black hole apparent horizon appears and coincides with the cosmological apparent horizon at @xmath59 . for a dust - dominated cosmological background this may be given as @xmath60 . this is the analog of the nariai black hole in the schwarzschild - de sitter - kottler solution , but it is instantaneous . as time progresses , @xmath61 , the single horizon splits into a dynamical black hole apparent horizon surrounded by a time - dependent cosmological horizon . this solution can progressively constitute a better and better toy model for a spherical , non - accreting astrophysical black hole in the late universe with @xmath62 . the black hole apparent horizon shrinks , asymptoting to the spacetime singularity at @xmath19 from above as @xmath63 , while the cosmological apparent horizon expands monotonically , tending to @xmath64 in the same limit . the actual universe is of course not dust - dominated , and is better described by the scale factor for expansion @xmath65^{1/3},\ ] ] consistent with the spatially flat concordance model @xcite . here @xmath66 km s@xmath67 mpc@xmath67 is the current value of the hubble parameter and @xmath68 is the current dark energy density . using this we may calculate actual values for @xmath58 and apparent horizon locations for black holes in our universe . considering , for example , the @xmath69 black hole at the centre of the milky way , we find that a single horizon would have first appeared as early as @xmath70 secs and at a radius very close to the centre @xmath71pc . thereafter , this would have split into two apparent horizons , which would have become increasingly separated . note that a problem with this calculation is that it neglects mass accretion . the results are therefore purely theoretical and would only truly be valid if this black hole had always existed at its current mass . although in reality there were no bound structures in the universe at such an early time , this calculation does at least provide some insight into the scales involved . let us discuss now the well known singularity @xcite . the surface of equation @xmath72 has normal @xmath73 with norm squared n_n^= g^ n_n _ |_r=2 m .=-4m^2h^2(t ) < 0 . @xmath74 is timelike and the surface @xmath75 is spacelike . the ricci scalar r=-8 t^_=8 ( -3p ) = 8 ( t ) ( 4- ) diverges as @xmath76 . this singularity separates spacetime into two disconnected regions @xmath77 and @xmath78 @xcite ; the latter region is described by the metric  ( [ 4 ] ) . at the critical time @xmath79 , when @xmath80 , the normal to the surface of equation @xmath81 is @xmath82 and m^m _ = g^11(r=)= ( -mh(t ) ) = 0 . thus the ( cosmological and black hole ) apparent horizon is instantaneously null .    by differentiating the cubic equation  ( [ 8 ] ) , one may solve for the rate of change in location of the apparent horizons with respect to the comoving time . dropping the @xmath31-dependencies for simplicity , one obtains _ rearranging this , one can compare the expansion rates of the apparent horizons with that of the cosmic substratum , -h= -h ( 1 + ) .[eq : rateofexp ] this equation shows that the apparent horizons are not comoving except for trivial cases . this explains why the black hole can not remain static but is instead forced to expand . in the case of a spatially flat flrw universe ( without the central inhomogeneity ) , it turns out that even the single cosmological horizon at @xmath83 is not comoving , since = - h . it is widely believed that , in the absence of event horizons , an entropy can be meaningfully ascribed to apparent horizons . the thermodynamics of these horizons has been discussed extensively @xcite . therefore , it is interesting to ask whether the total entropy associated with both the black hole and cosmological apparent horizons is a non - decreasing function of time . the area @xmath84 of the black hole apparent horizon is decreasing , but it is bounded from below while this behaviour is more than compensated for by the increase of the area @xmath85 of the cosmological apparent horizon . the total horizon entropy s = s_1+s_2 = ( r_1 ^ 2 + r_2 ^ 2 ) = , where @xmath86 , is plotted in fig . [ figure2 ] .     the total horizon entropy @xmath87 ( in units @xmath88 , where @xmath89 is the boltzmann constant ) associated with the apparent horizons as a function of time . ] since the apparent horizons emerge as a pair at @xmath49 , the horizon entropy @xmath87 exhibits a discontinuous jump from zero value at this time . we now discuss the situation of a cosmological background dominated by a phantom fluid with equation of state satisfying @xmath90 ( @xmath91 ) and violating the weak energy condition . the recent renewed interest in such a field has been motivated by the analysis of data from supernovae ia @xcite and the study of the effects of the accelerating universe @xcite . the consideration of a phantom background has also led to the prediction of a big rip singularity at a finite time in the future @xmath92 @xcite . we now consider a phantom background in the context of the mcvittie solution . surprisingly , this is a situation which has not received much attention in previous studies . one may consider the late time behaviour of the friedmann equation governing a phantom fluid and solve it to obtain a form for the scale factor in terms of @xmath93 and @xmath94 . indeed the solution has been shown to be @xcite : a(t)= , where @xmath95 is a constant . the hubble parameter may therefore be written concisely as h(t)= . note the reverse behaviour of this function compared with the hubble parameter for a dust - dominated universe @xmath45 . the latter diverges at the big bang singularity and gradually decreases over time , tending to zero . the hubble parameter for a phantom fluid , however , takes on a finite value at @xmath96 and slowly increases until the big rip time , at which point it too diverges . this suggests that the horizons around black holes embedded in a phantom fluid might behave in the opposite way to those in a dust - dominated background with @xmath97 . indeed this does turn out to be the case , and the discussion in the previous subsection can be repeated . the result is plotted in fig . [ figure3 ] . the behaviour of the mcvittie apparent horizons versus time in a phantom - dominated background universe for the parameter values @xmath98 and @xmath99 . ]    we may summarize our results in an expanding universe dominated by a phantom fluid as follows . in the early universe , both black hole and cosmological apparent horizons exist , and are approximately located at @xmath19 and @xmath64 , respectively . as time progresses the cosmological horizon shrinks and the black hole horizon expands , until they meet and merge at the critical time @xmath79 . thereafter they disappear , leaving behind a naked singularity . during this evolution the total apparent horizon entropy decreases and has a discontinuous jump to zero value at @xmath79 . this behaviour is yet another manifestation of the `` weirdness '' of the phantom fluid , which seems to violate the second law of thermodynamics in many ways @xcite . the behaviour of the apparent horizons for a phantom cosmic background was derived in ref . @xcite for _ generalized _ mcvittie solutions , which are obtained by relaxing the mcvittie no - accretion condition and allowing for a radial energy flux onto the black hole @xcite . for simplicity of modelling , this radial flux density @xmath100 is necessarily spacelike and violates the energy conditions . the lesson to be learnt by the present discussion of the corresponding mcvittie solution with @xmath101 is that the disappearance of the apparent horizons is not due to the fact that the accreted phantom fluid violates the weak energy condition and the total accreted mass becomes zero : it is due to the phantom character of the fluid which dictates the unusual cosmic expansion leading to the big rip , but not to accretion . in order to understand the bizarre phenomenology of apparent horizons in the mcvittie spacetime , it is useful to first understand the schwarzschild - de sitter - kottler solution of the einstein equations . this is a special case of the mcvittie solution . our study of the simple , static , schwarzschild - de sitter - kottler metric has essentially revealed that a black hole can only fit in a de sitter universe if its horizon size ( determined by its mass ) does not exceed the size of the cosmological horizon . equipped with this clarity , we have then moved on to consider the more complicated mcvittie solution , which accounts for a dynamical background and thus better represents reality . not surprisingly , the condition for the existence of the apparent horizons in this case is analogous to the corresponding one in the static case , with the static hubble constant replaced by a dynamical hubble parameter . this follows from the dynamical nature of the apparent horizons themselves in this case , which we are able to locate throughout their period of existence . the absence of any ( black hole or cosmological ) apparent horizons at early times is now easily understood . at early times the mathematical solutions suggest that the cosmological horizon would be smaller than the black hole horizon , but this is not possible since the universe at this time would be too small to accommodate a black hole apparent horizon at all . one can not then meaningfully distinguish between the `` black hole '' and the `` universe '' in which it is embedded ; rather , the mathematical solutions represent neither and do not possess the properties of a black hole or a universe . thus at early times , not only is there a naked singularity , but the cosmological apparent horizon is also absent . the presence of this naked singularity prevents one from being able to derive the mcvittie solution as the development of regular cauchy data . at some finite time , given by @xmath102 for a dust - dominated background , the cosmological solution is able to catch up with the black hole solution and a single black hole / cosmological apparent horizon appears . these then split and continue to diverge thereafter . the mcvittie metric does not account for accretion onto the central mass . hence the mass parameter @xmath15 is fixed and the horizon dynamics are wholly determined by the expansion of the universe . if the no - accretion assumption is relaxed however , the black hole mass itself is then also determined by the universe s expansion ( possibly with some residual freedom ) and can not be fixed _ a priori_. indeed some generalized mcvittie solutions , for which @xmath15 becomes a function of time , have already been derived @xcite . at late times this class of solutions converges to an attractor with a well - defined mass function @xmath103 @xcite . other solutions for cosmological lumps ( including black holes ) have also been derived and investigated without imposing the no - accretion condition in general relativity and in scalar - tensor and higher derivative gravity @xcite . in some of these studies , the phenomenology of the apparent horizons appears to be even more bizarre than in the mcvittie case and involves the creation or disappearance also of inner black hole apparent horizons @xcite .    locating the apparent horizons and understanding , at least in principle , their behaviour is not the whole story . the recent work @xcite studying the global structure of the mcvittie solution has unveiled a new feature which is believed to be generic : radial ingoing null geodesics do not penetrate the black hole apparent horizon to reach the @xmath55 singularity , but are asymptotic to this horizon . in our opinion , this feature is not too surprising for a solution in which radial flow onto the central black hole is excluded by construction . the property of radial ingoing null geodesics merely reflects the mcvittie no - accretion condition . in fact , the ingoing radial null geodesics can be seen as the test - particle limit of a gravitating null dust ( which however , would be forbidden by the no - accretion condition and could not fit in the mcvittie spacetime ) . future work to fully understand this feature , as well as more general solutions representing black holes embedded in cosmological backgrounds , will be presented elsewhere . thakurta , _ indian j. phys . _ * 55b * , 304 ( 1981 ) ; 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the bizarre behaviour of the apparent ( black hole and cosmological ) horizons of the mcvittie spacetime is discussed using , as an analogy , the schwarzschild - de sitter - kottler spacetime ( which is a special case of mcvittie anyway ) . for a dust - dominated `` background '' universe , 
 a black hole can not exist at early times because its ( apparent ) horizon would be larger than the cosmological ( apparent ) horizon . 
 a phantom - dominated `` background '' universe causes this situation , and the horizon behaviour , to be time - reversed .