Particle and photon orbits in McVittie spacetimes
McVittie spacetimes represent an embedding of the Schwarzschild field in isotropic cosmological backgrounds. Depending on the scale factor of the background, the resulting spacetime may contain black and white hole horizons, as well as other interesting boundary features. In order to further clarify the nature of these spacetimes, we address this question: do there exist bound particle and photon orbits in McVittie spacetimes? Considering first circular photon orbits, we obtain an explicit characterization of all McVittie spacetimes for which such orbits exist: there is a 2-parameter class of such spacetimes, and so the existence of a circular photon orbit is a highly specialised feature of a McVittie spacetime. However, we prove that in two large classes of McVittie spacetimes, there are bound particle and photon orbits: future-complete non-radial timelike and null geodesics along which the areal radius has a finite upper bound. These geodesics are asymptotic at large times to circular orbits of a corresponding Schwarzschild or Schwarzschild-de Sitter spacetime. The existence of these geodesics lays the foundations for and shows the theoretical possibility of the formation of accretion disks in McVittie spacetimes. We also summarize and extend some previous results on the global structure of McVittie spacetimes. The results on bound orbits are established using centre manifold and other techniques from the theory of dynamical systems.
I Introduction and summary.
Perhaps one of the most striking features of black holes is their ability to create circular photon orbits: by travelling to the vicinity of a Schwarzschild black hole and settling in an orbit at a radius , where is the mass of the black hole, a physicist can exploit the existence of circular photon orbits at this radius to look at the back of their own head. If a Schwarzschild black hole is not available, a charged or rotating black hole can be used for the same purpose. See e.g. Sections 20, 40 and 61 of chandrasekhar1998mathematical (). Indeed circular photon orbits exist in Kerr-Newman-(anti)de Sitter spacetime for all parameter values corresponding to black holes stuchlik2000equatorial (). Thus it seems reasonable to say that the existence of circular photon orbits (CPOs) is a generic feature of black holes.
Similarly, in the case of massive particles, the existence of an innermost stable circular orbit (ISCO) is a feature of black holes that is not generally present in Newtonian gravity (but see amsterdamski2002marginally ()). The existence of the ISCO is central to the formation of thin accrection disks around the black hole which, in turn, encodes useful information about the black hole. See lrr-2013-1 ().
The question arises as to whether or not CPOs and ISCOs arise in black holes in more general, non-vacuum settings. A first step here is to consider how such black holes can be modelled, and this question has arisen in the broader discussion of how the background expansion of the universe affects local systems faraoni2007cosmological () and in the more wide-ranging study of inhomogeneous cosmological models (see e.g. bolejko2011inhomogeneous ()). An early contribution to this discussion was McVittie’s discovery of an intriguing solution of Einstein’s equations that the author himself referred to as a “mass-particle in an expanding universe” mcvittie1933mass (). McVittie’s solution represents an embedding of the Schwarzschild field in an isotropic cosmological background. In the form presented in mcvittie1933mass (), three families of line element were given, corresponding to the curvature index of the isotropic background. There appear to be clear reasons to dispute the interpretation that McVittie’s solutions with represent some form of isolated system in an isotropic background nolan1998point (); nandra2012effect (), and so we will focus on the spatially flat case 111We note that both nolan1998point () and nandra2012effect () identify solutions of the Einstein equations that do for the isotropic spacetimes what the McVittie spacetime does for the class. In comoving coordinates, these solutions are given in terms of elliptic integrals, and so are not readily amenable to the analytic studies that have been done in the case. As far as the author is aware, the possibility that these solutions have an elementary form in area radial coordinates has not been investigated: this may make the solutions more tractable.. In this case, the line element may be written in the form nolan1999point ()
where is a constant, (i.e. we restrict to ) and is the Hubble function of the isotropic background. That is,
where is the scale factor of the spacetime obtained by setting in (1):
The spacetime with this line element will be referred to as the background, and we will use terms such as background metric in the obvious way. We note that is a global time coordinate on the spacetime with line element (1): . We set the time orientation of the spacetime by taking to increase into the future. In (4), is the usual co-moving radial coordinate of the isotropic spacetime. (We note that in (2) and throughout the paper, a prime means derivative with respect to argument.)
The limit of (1) is well-defined, as it corresponds to setting the Weyl tensor of the spacetime to zero. The invariantly defined Newman-Penrose Weyl scalar of the spherically symmetric line element (1) is given by
With , (1) is the line element of the Schwarzschild exterior. With constant, the line element is that of Schwarzschild-de Sitter spacetime with cosmological constant . These limits are likewise well-defined, being respectively the vacuum and Einstein-space limits of (1). In the case , the coordinate transformation with yields the more familiar form
Imposing the Einstein equations with a cosmological constant, and with a perfect fluid energy-momentum tensor, yields expressions for the density and pressure :
Setting yields the background density and pressure:
Assuming that , we see that McVittie spacetimes always have a scalar curvature singularity at .
Comment 1 It should be noted that since and , there is no equation of state of the form in the McVittie spacetime. Thus the term “perfect fluid” is not fully appropriate: we use it in the not uncommon sense of a fluid with isotropic pressure - the radial and tangential pressures are equal. On the other hand, as the background spacetime is homogeneous and isotropic, one can appeal to an equation of state to close the Einstein equations and so govern the evolution. This is the perspective that we take: the insertion of the mass parameter is done post hoc, after the Hubble function has been determined. In theory, one could then consider back reaction effects, redoing CMB and other calculations to take account of the mass parameter . As we will argue below, this mass parameter provides a model of a (highly) localized inhomogeneity in an otherwise isotropic universe (we use the qualification to reflect the fact that the energy density of the universe is unaffected by : - there is no ‘additional’ matter in the universe).
We note that the transformation with and
can be used to write the line element (1) in comoving coordinates, which is the form originally given by McVittie mcvittie1933mass (). In these coordinates, the curvature singularity at arises at . The coordinate transformation is a diffeomorphism only if we restrict to either or . Either interval provides a full cover of . Note also that at fixed , .
With these properties, it is tempting to conclude that the line element (1) corresponds to a spacetime that contains a black hole embedded in an isotropic universe. However, this interpretation is too simplistic. A correct interpretation requires the thorough study of the global properties of the spacetime, based on an analysis of its geodesics. Perhaps the dominant theme that has emerged from the various studies along these lines is that different outcomes emerge depending on the background scale factor and corresponding Hubble function .
As far as the author is aware, the first study of the global structure of McVittie spacetimes was undertaken by Sussman as part of a comprehensive study of the global properties of spherically symmetric, shear-free perfect fluid spacetimes sussman1988spherically (). In this work, a variety of possible global structures was identified. However, the interpretation of some of the results must be questioned as Sussman takes limits and within the same spacetimes. This does not seem possible without the inclusion of the singularity as part of the spacetime (rather than part of its boundary).
For expanding universe models with a big bang at a finite time in the past - that is, when the isotropic background (3) has these features - the singularity forms a past boundary of the spacetime. This was first pointed out in nolan1999point () for backgrounds satisfying a linear equation of state and with , and the result was generalised in kaloper2010mcvittie () and lake2011more (). As we will see below, this feature of McVittie spacetimes is not universal, but does arise if the big bang condition holds. When , the singularity also arises as a future boundary of the spacetime: future-pointing ingoing radial null geodesics run into this singularity in finite affine parameter time. However, as pointed out in kaloper2010mcvittie (), when there is a positive cosmological constant (and when some other technical conditions on hold), ingoing radial null geodesics meet a horizon rather that this singularity. Subsequently, lake2011more () showed how the spacetime can extend through the horizon to a Schwarzschild-de Sitter spacetime. Together, these points indicate that with a background -CDM model with , McVittie spacetimes can indeed model black holes in expanding universes. In fact it was shown in lake2011more () that the boundary includes a black hole horizon and a white hole horizon. However, when , it is very difficult to see how this interpretation can be given: all ingoing radial null geodesics either escape to infinity, or terminate at a scalar curvature singularity rather than reaching a horizon.
Thus, as emphasized in da2013expansion (), McVittie spacetimes can have a variety of global structures depending on the scale factor of the background. In some cases, including the -CDM model, black and white hole horizons arise. Returning to the question of how local systems are affected by cosmological expansion, it is clear that these McVittie spacetimes provide an interesting testing ground for such questions. On the other hand, studying e.g. the existence of CPOs and ISCOs in McVittie spacetimes adds further to our understanding and correct interpretation of these spacetimes. So in this paper, we begin the study of particle and photon orbits in McVittie spacetimes by addressing these questions: do there exist CPOs in McVittie spacetimes? Do there exist bound particle and photon orbits in McVittie spacetimes? We obtain a complete answer to the first of these questions. That is, we explicitly determine all McVittie spacetimes that admit CPOs. Neither the -CDM models of kaloper2010mcvittie () and lake2011more () nor the models of nolan1999point () mentioned above are among these. Such McVittie spacetimes do not possess this characteristic feature of black hole spacetimes. However, both models do have the following feature: both admit bound particle and photon orbits. That is, there are future-complete, non-radial null and timelike geodesics in these spacetimes with the property that , and where is an affine parameter (or proper time) along the geodesic. The constant corresponds to the radius of a circular orbit in either a Schwarzschild or Schwarzschild-de Sitter background. This result is established for two classes of McVittie spacetimes which we define below. These classes include, respectively, the spacetimes of kaloper2010mcvittie () and lake2011more () and those of nolan1999point (). Thus these two classes of McVittie spacetimes both have this characteristic black hole property: particles and photons can be confined to a spatially compact region of spacetime by means of the spacetime geometry.
Before proceeding, we give a brief but detailed summary of the results of this paper, and note their relation to previous work.
i.1 Summary of Section II
In Section II, we write down the relevant geodesic equations and state relevant dynamical systems results. We point out useful bounds relating and along individual geodesics - see (20).
i.2 Summary of Section III
In Section III, we derive and solve an equation that must hold for the Hubble function of a McVittie spacetime that admits a CPO. We briefly discuss global properties of the associated spacetime: a fuller discussion is given in Appendix A. As we see in (1), McVittie spacetimes are determined by a parameter and a function . We show in Section III that there is (only) a 2-parameter family of McVittie spacetimes that admit a CPO. Thus this is a ‘no-go’ result: CPOs are absent from nearly all McVittie spacetimes. We note that the same conclusion holds in relation to circular timelike orbits.
i.3 Summary of Section IV
In Section IV we define a class of McVittie spacetimes that we will refer to as expanding McVittie spacetimes with a big bang background. These are the focus of the remainder of the paper: this class appears to us to be the class of McVittie spacetimes of most physical interest. See Definition 1. We prove that in this class, all causal geodesics (timelike and null, radial and non-radial) originate at the singular boundary at finite affine distance (proper time) in the past. This generalises previous results relating to radial null geodesics nolan1999point (); kaloper2010mcvittie (); lake2011more ().
i.4 Summary of Section V
Here, we deal with the future evolution of causal geodesics. We define two subclasses of expanding McVittie spacetimes with a big bang background that are distinguished by the (future) asymptotic value of the Hubble function . For Class 1, as and for Class 2, . See Definitions 2 and 3. We derive rigorously and generalise results that have been presented previously either in a heuristic manner, or for special cases. In particular, we prove that the global structure (in the sense of a Penrose-Carter conformal diagram) obtained in lake2011more () for a special case - i.e. specified - holds generally for Class 1 (we note that several of the results of lake2011more () were proven in general). Included under this heading are existence proofs relating to important radial null geodesics that delineate the global structure (see Propositions 4 - 7). The crucial role of a non-zero value of was first identified in kaloper2010mcvittie (): this has far-reaching consequences in that it changes radically the interpretation of the corresponding McVittie spacetime, as discussed above. In Section V, our intention, in part, is to put the physical insights of this paper on a sounder mathematical footing. We note also a generalisation and some corrections (Proposition 7 (b) - noted in lake2011more ()) to the results of kaloper2010mcvittie (). For example, the linear equation of state used in Appendix A of that paper is not required. In addition, we generalise some previous results for the case nolan1999point ().
i.5 Summary of Section VI
The main results of the paper are given here. Using various techniques from dynamical systems, we prove the existence of bound photon and particle orbits in both Class 1 and Class 2 spacetimes. That is, we prove the existence of future-complete non-radial timelike and null geodesics with the property that is finite along the whole history of the geodesic. Taking to be the parameter (affine parameter or proper time) along the geodesic, we show that , where is the radius of a circular orbit in the corresponding Schwarzschild-de Sitter (Class 1) or Schwarzschild (Class 2) spacetime. For photon orbits, this forces and for particle orbits, corresponds to a stable circular orbit. See Propositions 10 and 12-14.
We use units with , and we use a to indicate the end of a proof.
Ii General properties of causal geodesics
We begin with the line element (1) and define
Recall that , and note that . The geodesic equations of the spacetime may be written in the following form:
where is the conserved angular momentum of the geodesic, for null geodesics and for timelike geodesics. The overdot represents the derivative with respect to the parameter along the geodesic: an affine parameter for null geodesics and proper time for timelike geodesics. Throughout the remainder of this paper, we use to represent this parameter. We also have the first integral of (13)-(14):
It follows that everywhere on a causal geodesic. Then by a choice of parameter orientation, we have everywhere along all causal geodesics.
We consider the geodesic equations on the region . This region has boundaries and . It will be convenient to consider the geodesic equations as a first order dynamical system. To this end, we define and write the geodesic equations in the form
where , and can be read-off the right hand sides of (13) and (14). We note that , where . In this context, (15) is a zero-order constraint that is propagated by the equations. That is, if (15) holds for , then this equation is valid everywhere along the solution of (16) with the initial condition .
Local existence of solutions of (16) follows from standard results of dynamical systems (see e.g. perkodifferential ()). For any , there is a maximal interval of existence of the initial value problem
The intervals and are respectively the left-maximal and right-maximal intervals of existence. The following theorem plays an important role in the next section:
Roughly speaking, this theorem tells us that solutions of (17) continue to exist while remains bounded. We will exploit this theorem to prove extension results for causal geodesics.
By completing a square, we can write (15) in the form
so that for non-radial geodesics (),
Noting that and are both positive, and that in an expanding () McVittie spacetime, we obtain the following:
At every point on a non-radial causal geodesic in an expanding McVittie spacetime,
Iii Circular photon orbits
We note that if the circular photon orbit has radius , then and we are in Schwarzschild-de Sitter spacetime. So for a non-trivial solution, we exclude . We note then that (equality holds for ). The only remaining condition of (13)-(15) is
This yields the following result.
The function , which is invariantly defined by , plays an important role in spherical symmetry: this leads to an interesting corollary to Proposition 1. As is well known, is proportional to the product of the expansions of the ingoing and outgoing radial null geodesics of a spherically symmetric spacetime. Hence the surface (or surfaces) corresponds to a horizon, where (at least) one of the null expansions vanishes. The regular (or untrapped) region of the spacetime corresponds to , where one null expansion is positive and one is negative, while corresponds to either a trapped region (two negative null expansions) or an anti-trapped region (two positive null expansions). We immediately have the following, which is a particular case of a more general result:
A circular photon orbit of a McVittie spacetime is confined to the regular region of the spacetime.
We can also see from (15) that any turning points (periapsis or apasis, whereat ) of a photon orbit in a McVittie spacetime must lie in the regular region of the spacetime. The same conclusion holds for particle orbits (timelike geodesics), where (15) holds but with on the right hand side. In fact it is readily seen that Corollary 1 holds much more generally in spherically symmetric spacetimes (this is the more general result referred to above). Using double null coordinates , both taken to increase into the future, the null expansion along the future pointing null direction satisfies for some positive function . A corresponding statement holds with replaced by . Then along any future pointing null geodesic,
With increasing into the future, so that and , we see that can vanish only when either the null expansions have opposite sign - that is, when the geodesic is in a regular region of the spacetime - or when both and vanish. This latter situation is non-generic: it occurs for example at the bifurcation 2-sphere of the extended Schwarzschild (Kruskal-Szekeres) spacetime. The former conclusion should hold in general.
Corollary 1 is almost enough to rule out CPO’s in any McVittie spacetime for which the isotropic background has a big bang - i.e. at some time in the past. From (2), we see that will typically diverge as . But then becomes negative at early times. It is very difficult to see how the CPO can exit the trapped region, or terminate in the past at a finite positive value of , without there being some serious pathology of the spacetime. Thus the existence of a CPO is a very strong restriction on McVittie spacetimes with big bang backgrounds.
We turn now to determining which line elements of the form (1) are admitted by Proposition 1. This amounts to solving (21), and ensuring that the resulting is positive. There is a unique non-trivial solution (where non-trivial means that is not constant: recall that this corresponds to Schwarzschild-de Sitter spacetime):
We have used time translation freedom to set at the zero of . The corresponding scale factor is
and we find
There are two inequivalent spacetimes, depending on the sign of . We see from (22) that is positive (respectively negative) if the CPO radius is greater than (respectively less than) . The expansion histories of the background universes corresponding to the two choices are shown in Figures 1 and 2, and their global structure is analysed in the Appendix A.
We find it curious that (21) has a solution which does satisfy the -CDM conditions of lake2011more () (see also Definition 1 below). This solution is . However this yields , and so the second condition (23) for a CPO is violated.
We summarise the main result of this section as follows.
Comment 2 It is straightforward to derive and solve the equation corresponding to (21) in the case of circular particle orbits. This yields a different McVittie spacetime to that of Proposition 2 - but again, only a 2-parameter family of spacetimes arises. We will not pursue this further: the main point is that circular orbits almost never exist in McVittie spacetimes.
Iv Past evolution
We begin the dicsussion by considering the past evolution of causal geodesics in the class of McVittie spacetimes of interest. These correspond to McVittie spacetimes for which the isotropic background is an expanding cosmological model - an expanding isotropic spacetime with a big bang at a finite time in the past.
The spacetime with line element (1) is said to be an expanding McVittie spacetime with a big bang background if the following conditions on the Hubble function hold:
for all ;
for all .
The first condition here includes a technical differentiability condition and the condition for expansion: for all . The second incorporates the big bang condition. This is equivalent to the existence of some - which we set to zero by a translation - for which the scale factor satisfies . The third condition is equivalent to the weak energy condition in both the background and the McVittie spacetime - see (7) and (8).
In this section, we establish the fact that - which, recall, is a curvature singularity - forms the past boundary of the spacetime. That is, all causal geodesics originate in the past at at finite affine distance (null geodesics) or finite proper time (timelike geodesics). All such geodesics extend back to at a positive value of : the background big bang surface is cut off by . These statements are non-trivial and so proofs are given below. We note that it is straightforward to see that forms a spacelike portion of the past boundary of the spacetime kaloper2010mcvittie (): what is not obvious is that all causal geodesics originate here. This is stated formally as Proposition 3 below.
The radial null geodesics of (1) are of particular importance in determining the global structure of the spacetime. These are the geodesics satisfying (13)-(15) with . Noting that is a global time coordinate of (1) that increases into the future by a choice made in Section I above, we have everywhere along a causal geodesic, and so from (15) we can write down
for radial null geodesics (RNG). There are two families: outgoing, corresponding to the upper sign, and ingoing, corresponding to the lower. We introduce the radial coordinate as defined in nolan1999point ():
Notice then that with and as . In terms of , and the RNGs satisfy
Our first step is to establish the following: given any point (so that and ), the ingoing and outgoing radial null geodesics (IRNGs and ORNGs) through both originate on the surface . We then show that the same holds for all causal geodesics passing through , and thereby show that the set does not form part of the past boundary of the spacetime.
In an expanding McVittie spacetime with a big bang background, all outgoing radial null geodesics of the spacetime originate at at finite affine distance in the past. That is, for each ORNG , there exists such that for some .
Proof: Consider the ORNG through with and . This satisfies
for . So for , we have , and hence
where , and we recall from part (i) of Definition 1 that . Integrating this linear differential inequality from to yields
These inequalities hold for . From part (ii) of Definition 1, we see that must reach 0 (and so reaches ) at some positive value of . From (14) with and part (iii) of Definition 1, we see that along the geodesic. This guarantees that reaches at a finite value of the affine parameter .
In an expanding McVittie spacetime with a big bang background, all ingoing radial null geodesics of the spacetime originate at at finite affine distance in the past. That is, for each IRNG , there exists such that for some .
Proof: For IRNGs, we have
We note that for all . Then defining , we find
Integrating from to where eventually yields
Divergence of in the limit (which comes from part (ii) of Definition 1) shows that reaches 0 at a positive value of . The proof that this occurs at a finite value of the affine parameter is identical to the corresponding proof in Lemma 2.
The following results establish that the results of these two lemmas apply to all causal geodesics. We first establish a somewhat obvious ‘radial confinement’ result.
Let and let be a causal geodesic with . Then the past branch of - that is, the set of points
is contained in the interior of the region of bounded by the point , by the paths of the past-directed radial null geodesics through and by the boundary .
Proof: Noting that along , we find from (20) that
The lower and upper bounds correspond to the uniquely defined value at each point of of the slopes of the ingoing and outgoing radial null geodesics. Thus at any given point of , the geodesic crosses the ingoing (respectively outgoing) radial null geodesic from below (respectively above). It follows that to the past of (i.e. for ), remains below (respectively above) the ingoing (respectively outgoing) radial null geodesic through . The conclusion follows.
Comment 3 It is tempting to conclude on the basis of this lemma that any causal must extend back to . However it remains to prove that extends sufficiently far into the past in order that this happens. We now prove this extension result.
Let be a future-pointing causal geodesic of an expanding McVittie spacetime with a big bang background. Then there exists and such that where is an affine parameter (respectively proper time) along a null (respectively timelike) geodesic. That is, all causal geodesics of an expanding McVittie spacetime with a big bang background originate at .
Proof: Let , and as before, let be the left maximal interval of existence for . Applying the radial confinement result of the previous lemma and recalling that along , we have
where is the value of at which the ingoing radial null geodesic through meets . Likewise,
where is the maximum of the value of along the past branches of the radial null geodesics through .
From Lemma 1, and recalling that and that , we have
Then from (14), we can write
where we note that the coefficient of on the right hand side is strictly negative on .
For and , define the compact sets
Then there exists a positive constant such that
Now suppose that the past branch of is confined to - that is, that for all . Then along ,
Using (20), this yields finite upper and lower bounds for :
where the bounding constants depend on the parameter , the initial point and the initial value and derivative of .
This contradicts Theorem 1, and so must exit . The radial confinement result indicates that must exit the set via the lower boundary . That is, for every , there exists so that . It follows that there exists such that . We now show that , completing the proof. (Note that corresponds to .)
Recall that and that with . It follows that for all . Recall that we may write (15) in the form
It follows that
On the right hand side, the terms , and all have finite positive limits as the geodesic approaches . As and both diverge to in the limit, it follows that the first term on the right hand side (rhs) of (52) dominates the other two, so that we may write
Thus there exists such that
It follows by integrating twice that : the geodesic reaches in finite affine (or proper) time.
V Future evolution
The main aim of this section is to build a comprehensive picture of the future evolution of radial null geodesics in expanding McVittie spacetimes with a big bang. We will also, where possible, draw conclusions about non-radial geodesics. The results we obtain both generalise and (attempt to) clarify previous results. The intention is to clarify the connection between the different global structures derived in previous work and the features of the corresponding background spacetimes (as encoded in the Hubble function ). The main results of the paper, which relate to bound photon and particle orbits, are presented in the following section and many of the results of this section are not required for those results. However this section is not wholly an aside: the reader interested in the results of Section VI should review Definitions 2 and 3, Lemmas 5 and 6 and the paragraphs between Propositions 4 and 5.
As is evident from the contrasting results of nolan1999point () and kaloper2010mcvittie (), the asymptotic value as of the Hubble function has a significant influence on the global structure of the spacetime. This is reflected in the results below, and to allow us to present those results in a clear manner, we define two subclasses of expanding McVittie spacetimes with a big bang.
The first class comprises McVittie spacetimes where the scale factor of the isotropic background is that of an expanding, , -CDM isotropic universe. For the purposes of this paper, the defining properties are these:
The spacetime with line element (1) is a Class 1 McVittie spacetime if the Hubble function satisfies (i)-(iii) of Definition 1, plus the following conditions:
Comment 4 We note that the technical condition on the sound speed corresponds to the existence and positivity of the limit and so expresses a physically motivated energy condition, as well as a differentiability condition on .
The second class we consider corresponds to isotropic backgrounds that share features with spatially flat Robertson-Walker universes with equation of state and zero cosmological constant. The defining properties are these:
The spacetime with line element (1) is a Class 2 McVittie spacetime if the Hubble function satisfies (i)-(iii) of Definition 1, plus the following conditions:
Comment 5 The additional differentiability requirement on the equation of state function is a technical condition required for some of the proofs of Section VI below.
Our first result proves the existence of outgoing photon orbits that extend to infinity. This demonstrates the existence of an asymptotic region of every expanding McVittie spacetime with a big bang background where light rays extend to arbitrarily large radii.
Let be an outgoing radial null geodesic of an expanding McVittie spacetime with a big bang background. Then is future complete, and .
Proof: Along an outgoing radial null geodesic, we have
which is positive so that increases with . Since (part (i) of Definition 1), we see that
where and is arbitrary. Integrating shows that (i.e. ) as . From (13) and part (iii) of Definition 1, we see that along the geodesic, and so the affine parameter must extend to as do.
Next, we show that in Class 2 McVittie spacetimes, all photon and particle orbits that originate outside a certain radius extend to infinity and are future complete. This and some subsequent proofs require some details of the horizon structure of McVittie spacetimes nolan1999point (); kaloper2010mcvittie (); lake2011more ().
Recall that the horizon of the McVittie spacetime with line element (1) is the set of points with , the regular region is the set of points with and the anti-trapped region is the set of points with . It is straightforward to show that the expanding condition leads to the region being anti-trapped rather than trapped. We will denote the regular region by , the anti-trapped region by and the horizon by . We note that is the disjoint union of , and and that
There are crucial differences between the horizons of Class 1 and Class 2 McVittie spacetimes. First, we note that in a Class 2 McVittie spacetime, all three sets , and are non-empty. However, in Class 1, this is not necessarily the case. Noting that
We note that this is identically the necessary and sufficient condition for the existence of horizons in the Schwarzschild-de Sitter spacetime with mass parameter and cosmological constant . For Class 1 McVittie spacetimes, we restrict our attention to those cases where a horizon exists:
The horizon existence condition (64) is satisfied.
From (61) and monotonicity of , we can describe the horizon by
where is the inverse of the function : for all . The function has a positive global minimum at , is decreasing on and is increasing on . We denote the global minimum by . Then we may also describe the two branches of the horizon by functions
where by implicit differentiation we find
In a Class 1 McVittie spacetime, as . Then on the horizon, is restricted to the interval where are the larger () and the smaller () of the two positive roots of , and we have as along the horizon. It is straightforward to prove that
From (68), we see that on the inner branch, decreases with from to , with . On the outer branch, increases with from to , and . Note in particular that for all . See Figure 3.
In a Class 2 McVittie spacetime, we have . There is no restriction of - except of course that . From (68), we see that on the inner branch, decreases with from to , and . On the outer branch, increases with from to , and . See Figure 3.