The effect of a massive object on an expanding universe
A tetrad-based procedure is presented for solving Einstein’s field equations for spherically-symmetric systems; this approach was first discussed by Lasenby, Doran & Gull in the language of geometric algebra. The method is used to derive metrics describing a point mass in a spatially-flat, open and closed expanding universe respectively. In the spatially-flat case, a simple coordinate transformation relates the metric to the corresponding one derived by McVittie. Nonetheless, our use of non-comoving (‘physical’) coordinates greatly facilitates physical interpretation. For the open and closed universes, our metrics describe different spacetimes to the corresponding McVittie metrics and we believe the latter to be incorrect. In the closed case, our metric possesses an image mass at the antipodal point of the universe. We calculate the geodesic equations for the spatially-flat metric and interpret them. For radial motion in the Newtonian limit, the force acting on a test particle consists of the usual inwards component due to the central mass and a cosmological component proportional to that is directed outwards (inwards) when the expansion of the universe is accelerating (decelerating). For the standard CDM concordance cosmology, the cosmological force reverses direction at about . We also derive an invariant fully general-relativistic expression, valid for arbitrary spherically-symmetric systems, for the force required to hold a test particle at rest relative to the central point mass.
keywords:gravitation – cosmology: theory – black hole physics
Among the known exact solutions of Einstein’s field equations in general relativity there are two commonly studied metrics that describe spacetime in very different regimes. First, the Friedmann–Robertson–Walker (FRW) metric describes the expansion of a homogeneous and isotropic universe in terms of the scale factor . The FRW metric makes no reference to any particular mass points in the universe but, rather, describes a continuous, homogeneous and isotropic fluid on cosmological scales. Instead of using a ‘physical’ (non-comoving) radial coordinate , it is usually written in terms of a comoving radial coordinate , where , such that the spatial coordinates of points moving with the Hubble flow do not depend on the cosmic time . Here the comoving coordinate is dimensionless, whereas the scale factor has units of length. For a spatially-flat FRW universe, for example, using physical coordinates, the metric is
in comoving coordinates, where , is the Hubble parameter and primes denote differentiation with respect to the cosmic time (we will adopt natural units thoroughout, so that ).
Second, the Schwarzschild metric describes the spherically symmetric static gravitational field outside a non-rotating spherical mass and can be used to model spacetime outside a star, planet or black hole. Normally the Schwarzschild metric is given in ‘physical’ coordinates and reads
but another common representation of this spacetime uses the isotropic radial coordinate , where , such that
The main problem with the Schwarzschild metric in a cosmological context is that it ignores the dynamical expanding background in which the mass resides.
McVittie (1933, 1956) combined the Schwarzschild and FRW metrics to produce a new spherically-symmetric metric that describes a point mass embedded in an expanding spatially-flat universe. McVittie demanded that:
at large distances from the mass the metric is given approximately by the FRW metric (2);
when expansion is ignored, so that , one obtains the Schwarzschild metric in isotropic coordinates (4) (whereby , with defined below);
the metric is a consistent solution to Einstein’s field equations with a perfect fluid energy-momentum tensor;
there is no radial matter infall.
McVittie derived a metric satisfying these criteria for a spatially-flat background universe:
where has been used to indicate McVittie’s dimensionless radial coordinate, rather than our ‘physical’ coordinate. In Section 3.1 we will see how the two are related and point out some of the problems with (5). One sees that (5) is a natural combination of (2) and (4); nonetheless, there has been a long debate about its physical interpretation. This uncertainty has recently been resolved by Kaloper, Kleban & Martin (2010) and Lake & Abdelqader (2011), who have shown that McVittie’s metric does indeed describe a point-mass in an otherwise spatially-flat FRW universe. In this paper we have also independently arrived at the same conclusion, as discussed in Section 3.1. McVittie also generalised his solution to accommodate spatially-curved cosmologies, which are discussed further in Section 3.
For a given matter energy-momentum tensor, Einstein’s field equations for the metric constitute a set of non-linear differential equations that are notoriously difficult to solve. Moreover, the freedom to use different coordinate systems, as illustrated above, can obscure the interpretation of the physical quantities. In a previous paper, Lasenby et al. (1998) presented a new approach to solving the field equations. In this method, one begins by postulating a tetrad (or frame) field consistent with spherical symmetry but with unknown coefficients, and the field equations are instead solved for these coefficients. In this paper, we follow the approach of Lasenby et al. (1998) to derive afresh the metric for a point mass embedded in an expanding universe, both for spatially-flat and curved cosmologies, and compare our results with McVittie’s metrics and with work conducted by other authors on similar models. We also discuss the physical consequences of our derived metrics, focussing in particular on particle dynamics and the force required to keep a test particle at rest relative to the point mass.
The outline of this paper is as follows. In Section 2, we introduce the tetrad-based method for solving the Einstein equations for spherically-symmetric systems, and derive the metrics for a point mass embedded in an expanding universe for spatially-flat, open and closed cosmologies. In Section 3, we compare our metrics with those derived by McVittie. The geodesic equations for our spatially-flat metric are derived and interpreted in Section 4. In Section 5, we derive a general invariant expression, valid for arbitrary spherically-symmetric spacetimes, for the force required to keep a test particle at rest relative to the central point mass, and consider the form of this force for our derived metrics. Our conclusions are presented in Section 6.
We note that this paper is the first in a set of two. In our second paper (Nandra, Lasenby & Hobson 2011; hereinafter NLH2), we focus on some of the astrophysical consequences of this work. In particular, we investigate and interpret the zeros in our derived force expression for the constitutents of galaxies and galaxy clusters.
2 Metric for a point mass in an expanding universe
We derive the metric for a point mass in an expanding universe using a tetrad-based approach in general relativity (see e.g. Carroll, 2003); our method is essentially a translation of that originally presented by Lasenby et al. (1998) in the language of geometric algebra. First consider a Riemannian spacetime in which events are labelled with a set of coordinates , such that each point in spacetime has corresponding coordinate basis vectors , related to the metric via . At each point we may also define a local Lorentz frame by another set of orthogonal basis vectors (Roman indices). These are not derived from any coordinate system and are related to the Minkowski metric via . One can describe a vector at any point in terms of its components in either basis: for example and . The relationship between the two sets of basis vectors is defined in terms of tetrads, or vierbeins , where the inverse is denoted :
It is not difficult to show that the metric elements are given in terms of the tetrads by .
We now consider a spherically-symmetric system, in which case the tetrads may be defined in terms of four unknown functions , , and . Note that dependencies on both and will often be suppressed in the equations presented below, whereas we will usually make explicit dependency on either and alone. We may take the non-zero tetrad components and their inverses to be
In so doing, we have made use of the invariance of general relativity under local rotations of the Lorentz frames to align and with the coordinate basis vectors and at each point. It has been shown in Lasenby et al. (1998) that a natural gauge choice is one in which , which we will assume from now on. This is called the ‘Newtonian gauge’ because it allows simple Newtonian interpretations, as we shall see. Using the tetrads to calculate the metric coefficients leads to the line element
We now define the two linear differential operators
and additionally define the functions , and by
where is a constant. Assuming the matter is a perfect fluid with density and pressure , Einstein’s field equations and the Bianchi identities can be used to yield relationships between the unknown quantities, as listed below (Lasenby et al., 1998):
From the equation we now see that plays the role of an intrinsic mass (or energy) interior to , and from the equation it also becomes clear that is interpreted as a radial acceleration. In such a physical set up is the cosmological constant.
In order to determine specific forms for the above functions it is sensible to start with a definition of the mass . For a static matter distribution the density is a function of alone, , and . Setting equal to a constant leads specifically to the exterior Schwarzschild metric in ‘physical’ coordinates (3). For a homogeneous background cosmology, and , leading to the FRW metric in ‘physical’ coordinates (1). In this work we choose to describe a point object with constant mass , embedded in a background fluid with uniform but time-dependent spatial density:
which is easily shown to be consistent with the equation above.
We point out that the background fluid is in fact a total ‘effective’ fluid, made up of two components: baryonic matter with ordinary gas pressure, and dark matter with an effective pressure that arises from the motions of dark matter particles having undergone phase-mixing and relaxation (see Lynden-Bell (1967) and Binney & Tremaine (2008)). The degree of pressure support that the dark matter provides depends on the degree of phase-mixing and relaxation that the dark matter particles have undergone, which (in the non-static, non-virialised case) will be a variable function of space and time. The properties of this single ‘phenomenological’ fluid, with an overall density , are studied in more detail in our companion paper NLH2. Here we simply calculate the total pressure of the background fluid required, in the presence of a point mass , to solve the Einstein field equations in the spherically-symmetric case. The ‘boundary condition’ on this pressure (at least for a flat or open universe, where spatial infinity can be reached) is that the pressure tends at infinity to the value appropriate for the type of cosmological fluid assumed. This is in the present case, since we are matching to a dust cosmology. The total pressure at finite may be thought of as a sum of the baryonic gas pressure and dark matter pressure, but without an explicit non-linear multi-fluid treatment we do not break the fluid up into its components in the strong-field analysis presented below.
Note that the central point mass in our model is inevitably surrounded by an event horizon. The fluid contained within this region remains trapped and cannot take part in the universal background expansion, and so our expression for in (12) is only valid outside the Schwarzschild radius. We thus expect the metric describing the spacetime to break down at this point. However, since one is usually most interested in the region (roughly equivalent to ), it is appropriate to continue using this definition for to study particle dynamics far away from the central point mass.
This is easily solved for , and hence , to yield
where we have defined the new function
It should be noted that, by interpreting as the Hubble parameter, the three terms on the right-hand-side of (17) correspond to via the Friedmann equation for a homogeneous and isotropic universe, where is the curvature parameter and is the scale factor. Calculating the function is now straightforward from its definition in (10):
Finally, the function can then be calculated from the equation in (11). We thus have expressions for all the required functions , , , and .
We conclude our general discussion by noting the relationship between the fluid pressure and the function . Combining the and equations in (11), one quickly finds
This first-order linear differential equation can be easily solved for by finding the appropriate integrating factor, and one obtains
Using the Friedmann acceleration equation for a homogeneous and isotropic universe, one then finds that , where is the equation-of-state parameter of the cosmological fluid. Hence the relationship (20) between the fluid pressure and becomes simply
For an FRW universe without a point mass (), in which , we recover the relationship .
2.1 Spatially-flat universe
A number of observational studies, such as WMAP (Larson et al., 2011), indicate that the universe is spatially flat, or at least very close to being so. In this case and the resulting expressions for the quantities , , , and are easily obtained; these are listed in the left-hand column of Table 1. We note that the given expression for is obtained by imposing the boundary condition as ; from (22) this follows from the physically reasonable boundary condition that the fluid pressure as .
From (8) this leads to the metric (in ‘physical’, i.e. non-comoving coordinates):
In this case, the general expression (22) for the fluid pressure becomes
This can be checked directly by substituting our form for in the equation from (11), from which it follows that and are related by
Imposing the boundary condition that the pressure tends to zero as , this leads to (24), as expected.
We note that the metric (23) is singular at . This is, however, unlike the coordinate singularity of the standard Schwarzschild metric. The latter arises due to a poor choice of coordinates and by converting to another more suitable coordinate system, such as Eddington–Finkelstein coordinates, it can be shown that the Schwarzschild metric is actually globally valid. On the contrary, for our derived metric, we see from (24) that the fluid pressure becomes infinite at , which is thus a real physical singularity. Hence our metric is only valid in the region . In reality, this region is usually deeply embedded within the object. In an attempt to make our solution globally valid, we shall present an extension of this ‘exterior’ work to the interior of the object in a subsequent paper, where we will take its spatial extent into proper consideration and follow a similar approach to that used in this work. We point out that Nolan (1999b) has suggested considering a different type of fluid altogether, such as a tachyon fluid, to define an equivalent metric inside this region, but we leave this type of approach for future research.
2.2 Open universe
For an open universe , one has and the resulting expressions for the quantities , and are easily obtained and are listed in the middle column of Table 1. In this case, however, the expression for (and hence ) is less straightfoward to obtain. Combining (18) with the equation from (11), one finds that may be written analytically in terms of an elliptic integral:
where the constant of integration, or equivalently the limits of integration, must be found by imposing an appropriate boundary condition.
To avoid the complexity of elliptic functions, we instead expand as a power series in , since astrophysically one is most interested in the region , i.e. values of lying between (but far away from) the central point mass and the curvature scale of the universe. Recasting in its differential form gives
The series solution to this equation is
where the arbitrary function can only be determined by the imposition of a boundary condition. For an open universe we expect and hence as . Therefore, from (22) we expect as . This gives , and hence
As shown in Fig. 1, expanding to first order in is sufficient to represent the solution to high accuracy for our region of interest.
An approximate form for the metric in the case of an open cosmology is therefore
It can be verified that in the limit (or ) this reduces to the standard FRW metric. Also, in the limit and working to first-order in , the metric coefficients in (29) reduce to those in the spatially-flat case (23).
We also note that the metric is not singular at , but instead becomes singular where
Indeed, is singular there. Multiplying through by , the resulting cubic equation has a positive discriminant and hence only one real root, which occurs at a radial coordinate inside the standard Schwarzschild radius . Since and hence the fluid pressure are singular there, then, as in the spatially-flat case, this is a true physical singularity rather than merely a coordinate singularity. We further point out that, in contrast to the spatially-flat case, the radial coordinate at which this singularity occurs is a function of cosmic time .
2.3 Closed universe
For a closed universe (), one has and the resulting expressions for , and are listed in the right-hand column of Table 1. As in the open case, the expression for (and hence ) requires more work. One finds that can similarly be given analytically in terms of an elliptic integral:
where, once again, the constant of integration or limits of integration, must be found by imposing an appropriate boundary condition. As we will see below, however, the imposition of such a boundary condition requires considerable care in this case, since the limit is not defined for a closed cosmology. Recasting in its differential form gives
It is again convenient to expand as a power series in , which reads
where the arbitrary function can only be determined by the imposition of a boundary condition, to which we now turn.
2.3.1 Boundary condition on
The main problem in defining an appropriate boundary condition for a closed universe is that our ‘physical’ radial coordinate only covers part of each spatial hypersurface at constant cosmic time . One can see from Table 1 that and hence the metric becomes singular when
This also corresponds to where becomes singular, from equation (31), and hence where the pressure diverges, from equation (22). Multiplying through by , the resulting cubic equation has a negative discriminant and hence three real roots, provided . It is easily shown that one of these roots lies at negative , and is hence unphysical, and the remaining two roots lie at
It is straightforward to show that corresponds to the ‘black-hole’ radius, and lies outside the Schwarzschild radius . At this point , and thus the fluid pressure, are also singular, and so this corresponds to a true physical singularity, as in the spatially-flat and open cases.
The other root, , is easily shown to correspond to the ‘cosmological’ radius, and lies inside the curvature radius . This radius should be merely a coordinate singularity, which we verify in Section 2.3.2. Thus one would not expect the fluid pressure, and hence , to be singular there. Moreover, one would also expect to be non-singular there. From the expression (32), one quickly finds that for the latter condition to hold, one requires
Thus, the series expansion of about the cosmological radius takes the form
where the coefficients may be determined by substitution into (32), and we have momentarily dropped the explicit dependence of and on for brevity. One finds that the first coefficient, which is the only one of interest, reads
Thus, we have determined the (finite) values of both and at the cosmological radius . Since the differential equation (32) for is first-order in its radial derivative, one can thus, in principle (or numerically), ‘propagate’ out of the cosmological radius, towards smaller values. The boundary conditions (36) and (38) therefore uniquely determine .
We point out that, in addition to being singular at the ‘black-hole’ radius , the function (and hence the fluid pressure) will also be singular at the zeros of the integral given in equation (31). If the inner-most zero occurs at , which is some (unique) function only of and , we may thus represent in the integral form
It is not clear how to find an analytical expression for , but numerical results show that lies slightly outside the radius . Moreover, as , both the absolute and fractional radial coordinate distance between the two radii decreases. Indeed, in any practical case, the two will be indistinguishable.
Turning to the power series approximation (33) of , valid for our region of interest , the identification of the appropriate boundary conditions at the cosmological radius now allows us to fix the arbitrary function straightforwardly. From the small approximation of (34), it is clear that the limit is equivalent to approaching from below. If were non-zero, then would be singular at the cosmological radius, owing to the term. We thus deduce that we require , so that
As shown in Fig. 2, this expansion is sufficient to represent the solution to high accuracy for our region of interest. Thus, an approximate form for the metric in the case of a closed cosmology is
It can be verified that in the limit , this reduces to the standard FRW metric. Also, in the limit and working to first-order in , the metric coefficients in (41) reduce to those in the spatially-flat case (23).
2.3.2 Alternative radial coordinate
We have mentioned that, for a closed universe, our ‘physical’ radial coordinate only covers part of each spatial hypersurface at constant cosmic time . The metric has a singularity at the ‘cosmological’ radius , which we now verify is merely a coordinate singularity.
Even without the presence of a point mass, a similar problem arises when using the ‘physical’ -coordinate in the case of a pure closed FRW metric, which has a coordinate singularity at . This issue is discussed in Lasenby et al. (1998), where an alternative radial coordinate was introduced, which removes the singularity at the cosmological radius. The solution presented there amounts to a two-stage coordinate transformation: one first transforms to a comoving radial coordinate and then performs a stereographic transformation. The final form of the metric is ‘isotropic’ in the sense that its spatial part is in conformal form. Thus, an obvious approach in our case (with ) is to seek an isotropic form for the metric, which reduces to the form found by Lasenby et al. (1998) when .
We note that only the radial coordinate is transformed, so the coordinate keeps its meaning as cosmic time. We thus consider a new radial coordinate of the general form . In fact, it is more convenient in what follows to consider the inverse transformation . It should be understood here that is a new function to be determined, the value of which is equal to the old radial coordinate.
We begin by considering the metric in the form (8), where the functions and are given by the analytical expressions given in the right-hand column of Table 1. For the moment, we will not assume a form for . Performing the transformation , we will obtain a new metric in and (and the standard angular coordinates and ). By analogy with the approach of Lasenby et al. (1998), we require this new metric to be in isotropic form, i.e. the coefficient of must equal that of and the cross-term must disappear. The first condition leads to
while the second condition yields the following direct formula for :
When these conditions are satisfied, the resulting metric is given by
which depends only on the single function . Substituting this metric into the Einstein field equations, one can verify that it is consistent with a satisfactory perfect fluid energy-momentum tensor, in which the fluid density depends only on and the radial and transverse pressures are equal and satisfy the relation (22).
One can solve (42) to obtain an expression in integral form for in terms of . There are two solutions, one of which reads
Here we have used as the lower limit of integration, so that we can more easily make a connection with our integral expression for given in (39). This might not be the appropriate integration limit in this case, however, and so we include the arbitrary function to absorb any discrepancy.
The arrangement of integration limits in (45) enables us to consider the -range from near the point mass to the cosmological radius. One can now proceed beyond the cosmological radius, however, by writing the second solution to (42) as
The maximum value of occurs at the cosmological radius and, thereafter, continues to increase as decreases again. The geometrical interpretation of this result is illustrated in Fig. 4, with one spatial dimension suppressed.
In essence, as increases, one is following a great circle on the surface of a sphere, starting at the original point mass and ending at an image mass at the antipodal point of the universe.
We note that there is an interesting relationship between any two values that correspond to the same value of . If two such values are and , then
Since the right-hand side is a function only of , then at any given cosmic time the values and are reciprocally related. This behaviour also occurs in the case of a pure closed FRW model, as discussed in Lasenby et al. (1998).
So far, we have left undetermined, but one can in fact obtain an expression for by combining the standard partial derivative reciprocity relation
with the expression (43) for , which gives
Although we do not have an explicit expression for , we have shown that there is an operational method for determining it (i.e. where becomes singular when numerically propagating it inwards from the cosmological radius). If we know and at each time slice (see below for the latter), we can evaluate the above integral for . The only remaining ambiguity is an arbitrary additive constant, which corresponds to an arbitrary multiplicative constant in the definition of in terms of , i.e. it is not possible to identify a unique overall scale for , which seems reasonable.
We note that our solution involving an ‘image mass’ at the antipodal point of the universe ties in with the scenario recently investigated by Uzan, Ellis & Larena (2011), who also considered a closed universe with masses embedded symmetrically at opposite poles. They showed a static solution was not possible for this case, which fits in well with the fact that here we have an explicit exact solution for the masses embedded in an expanding universe. The exact nature of the correspondence with their work will be the subject of future investigation, however.
2.3.3 Cosmological evolution
When considered as a function of the new radial coordinate , the radial derivative of is given by