Grav. Cosmol. No. 2, 2013

Friedmann-Robertson-Walker metric in curvature coordinates

and its applications

Abhas Mitra1

Theoretical Astrophysics Section, Bhabha Atomic Research Centre, Mumbai, India

Received September 13, 2012Received in final form February 20, 2013

For the first time, we express the general Friedmann-Robertson-Walker (FRW) metric (k = +1, 0,−1) intoexplicit “Schwarzschild” or “Curvature” form, which is important from the viewpoint of cosmology. Withthis form of the FRW metric, we reconsider the old problem of embedding a Schwarzschild mass (SM) in apre-existing FRW background from the viewpoints of both (1) the enigmatic McVittie metric, obtained in1933 and (2) the Einstein-Straus approach (1945) of scooping out a spherical cavity in the same background.Since the exterior of the SM is, by definition, described in the Schwarzschild coordinates, for a definitivestudy of the Einstein-Straus approach we employ this form of the FRW metric. We find that a necessarycondition for a SM to participate in the cosmic expansion is that the background fluid is dust.

1 Introduction

It is believed that our universe is expanding andis described by the Friedmann-Robertson-Walker(FRW) metric

ds2 = dt2 − a2(t)(

dr2

1− kr2+ r2dΩ2

)(1)

where r is the comoving radial coordinate, t is thecomoving cosmic time, a(t) is the scale factor, dΩ2

is the metric on a unit 2-sphere, and k = +1, 0,−1.On the other hand, what we actually see around usare lumpy distributions of matter in the form ofstars, galaxies, galaxy clusters and so on. The lo-cal spacetime structure around such objects is cer-tainly different from what is given by the abovemetric. In fact, it is not known whether the space-time within or around such objects is participatingwith the supposed cosmic expansion or not. As atest case, if we focus attention on a single isolatedastrophysical object, under the simplest assump-tions, the spacetime exterior to it is given by thevacuum Schwarzschild metric

ds2 =

(1−2M

R

)dT 2−

(1−2M

R

)−1dR2−R2dΩ2,

(2)

1e-mail: amitra@barc.gov.in

where M is the gravitational mass of the object andR is the area/curvature coordinate. Clearly (2) isan example of a metric in curvature coordinatesR, T . A fundamental question here is whetherone can embed this simplest ‘Schwarzschild Mass”(SM) into the supposed background cosmologicalmetric in a successful manner. Formally, this ques-tion was first investigated by McVittie as long agoas in 1933 [1]. He obtained an exact solution of theEinstein equations which attempted to accommo-date both these pictures:

ds2 =(1− f

1 + f

)2dt2 − [1 + f(r, t)]4eβ(t)

(1 + r2/4R20)2

× (dr2 + r2dΩ2), (3)

where

f(r, t) = Me−β(t)/22r(1 + r2/4R20)1/2. (4)

Here M ≥ 0 is an integration constant and“1/R2

0 gives the curvature of space as a whole” [1].Obviously, it is tempting to consider M as thegravitational mass of the embedded SM. It mustbe noted here that the symbol r used in the fore-going metric is different from the correspondingsymbol appearing in Eq. (1); a clarification on thispoint will follow later. Though the McVittie met-ric (MM) has often been considered as an exam-ple of a Schwarzschild mass embedded in the FRW

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background, it actually remained controversial andat best enigmatic. There have been innumerablestudies to understand the MM and then arrive ata successful scheme for embedding a Schwarzschildmass into a FRW background — one of the lat-est being [2]. However, to the best of our knowl-edge, none has been fully successful so far, andthe problem continues to be unsolved. Further, itwas pointed out that the MM suffers from inher-ent limitations [3], at least for an arbitrary signof R2

0 . In particular, the pressure is infinite atthe Schwarzschild radius, while the energy is nega-tive for particular values of negative universe curva-ture [3]. We can briefly note that the MM reducesto the FRW metric for M = 0 and for an arbitraryβ(t). And to this extent, McVittie’s attempt wassuccessful. But for a finite M it does not reduce tothe Schwarzschild form even with β(t) = 0. Thisis obviously a failure of the scheme. Hence in orderto attain the exact Schwarzschild form, for an arbi-trary sign of R2

0 , one should have both (i) β(t) = 0and (ii) R0 =∞ .

Since McVittie’s approach to the problem hasnot resulted in a general solution even now, we maylook back at a different line of attack, first contem-plated by Einstein and Straus [4]. Here, one nolonger aims at an idealistic picture of embeddinga SM over and above a pre-existing FRW back-ground. Instead, one cuts out a spherical vacuumregion (cavity) from the background uniform fluid.Now, a condensed SM having a mass equal to themass of the cut-out region

Mb = Mv =4π

3ρ(t)R3

v(t) (5)

is symmetrically placed within the cavity. HereRv(t) is the radius of the cavity or vacuole. Sincethe mean density of the SM is ρm ρ(t), theradius of the SM Rb Rv(t). In this picture,the intervening region Rb ≤ R ≤ Rv(t) contin-ues to be vacuum unlike the McVittie metric sce-nario. In the absence of the SM mass, by Birkhoff’stheorem, the spherical cavity would represent flatMinkowski spacetime. As long as the SM is notemitting any radiation, the intervening spacetimeRb ≤ R ≤ Rv(t) continues to be static by Birkhoff’stheorem. In fact, there is no harm if the SM itselfwould be pulsating: Rb = Rb(t), without piercingthe vacuole interface: Rv(t).

The best way to ensure that two different solu-tions can indeed match smoothly will be to express

both of them in the same set of coordinates andthen study the junction conditions. If the two solu-tions indeed glue together physically, they must doso when expressed in the same coordinates. Andwe would like to revisit the Einstein-Straus exer-cise by expressing the FRW metric in the samecurvature coordinates (R, T ) in which the staticvacuole field is expressed. The problem of express-ing the FRW metric in the Schwarzschild form isin its own right an outstanding problem. It maybe reminded again that the FRW spacetime willlose explicit isotropy in non-comoving coordinates.Note that earlier Gautreau attempted this problemonly for the simplified k = 0 case [5]. In this paperwe consider the general case (k = +1, 0,−1).

Very interestingly, it will be found that, inSchwarzschild coordinate system, the FRW metrichas a nice similarity with the vacuum Schwarzschildmetric. As a consequence, it will be found that onemight embed a “Schwarzschild Mass” onto a FRWbackground in a smooth manner provided the FRWmetric corresponds to zero pressure.

2 The FRW Metric in curvaturecoordinates

A direct comparison of Eqs. (1) and (2) suggeststhat, at the very beginning, we must leave the an-gular part unaltered and put

R = R(r, t) = ra(t), (6)

so that

dR = radt+ adr, (7)

where an overdot denotes ∂/∂t . If we define

H(t) = a/a, (8)

Eq. (7) becomes

adr = dR−HRdt. (9)

We also define two auxiliary parameters:

Γ2 = (1− kr2) = 1−KR2, (10)

ξ2 = 1− kr2 − a2r2 = 1−KR2 −H2R2

= Γ2 −H2R2, (11)

where K = k/a2 . In terms of Γ and ξ , the FRWmetric (1) becomes

ds2 =(Γ2dt+ aardr)2

ξ2Γ2− dR2

ξ2−R2dΩ2 (12)

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Since, in general, t = t(T,R), one can write

dt = αdT + βdR, (13)

where

α =∂t

∂T; β =

∂t

∂R. (14)

Then it turns out that

Γ2dt+ aardr

ξΓ=αξ

ΓdT +

(βξ

Γ+HR

ξΓ

)dR. (15)

And in order that the FRW metric (12) be di-agonal in the curvature coordinates, it is seen thatwe must set here

β = −HR/ξ2, (16)

so that the Schwarzschild-FRW (S-FRW) metric(12) eventually becomes

ds2 =α2ξ2

Γ2dT 2 − dR2

ξ2−R2dΩ2. (17)

In the k = 0 case, (17) indeed reduces to theform considered by Gautreau [5]. It is seen thatwhile the form of gRR is unique, in view of thepresence of the α term in gTT , the latter is not.Then one may proceed by demanding that dT mustbe a perfect differential in the expression

gTTdT2 =

(Γ2dt+ aardr)2

ξ2Γ2, (18)

which is free from α . And since there are es-sentially two known cases of the S-FRW Metric,namely, the de-Sitter and Milne metrics, we testthe freshly obtained S-FRW Metric in the light ofthese two cases.

2.1 The de Sitter metric

The de-Sitter metric corresponds to k = 0 anda(t) = eHt , H = const, so that in this case wehave

Γ2 = 1; a = Ha;

ξ2 = 1−H2a2r2 = 1−H2R2 (19)

Therefore from Eq. (17) we obtain

gRR = −ξ2 = −(1−H2R2)−2 (20)

and from (18)

gTTdT2 = (1−H2R2)

[dt− HRdR

1−H2R2

]2. (21)

In fact the quantity within the square brackets canbe expressed as a perfect differential

dt− HRdR

1−H2R2= d

(t− 1

Hlog√

1−H2R2

).

(22)

This means that in this case we can identify

T = t− 1

Hlog√

1−H2R2, (23)

dT = dt− HRdR

1−H2R2, (24)

and

gTT = (1−H2R2) = ξ2 (25)

Thus we obtain

ds2 = (1−H2R2)dT 2−(1−H2R2)−1dR2−R2dΩ2,

(26)

and this is indeed the correct Schwarzschild formof the de-Sitter metric.

2.2 The Milne metric

The Milne metric

ds2 = dt2 − t2[dr2

1 + r2+ r2dΩ2

](27)

is a very special case of FRW metric with

a(t) = t, k = −1. (28)

In this case, one finds

a = 1, H = 1/t, (29)

Γ2 = 1 + r2, ξ2 = 1. (30)

Clearly, here gRR = −1 and

gTTdT2 =

(Γ2dt+ aardr)2

Γ2=

(Γdt+

aardr

Γ

)2

(31)

Now by noting that

d√

1 + r2 =rdr√1 + r2

(32)

4

we may further write (33) as

gTTdT2 =

(Γdt+

1

2

tdΓ

Γ

)2

= [d(t√

1 + r2)]2

(33)

This suggests that, for dT to be a perfect differen-tial, we should have

T = t√

1 + r2 = tΓ; gTT = 1. (34)

Therefore, the Schwarzschild form of the Milne casebecomes

ds2 = dT 2 − dR2 −R2dΩ2, (35)

i.e., the Minkowski metric in spherical coordinates,as is well known.

It is of interest that both these known S-FRWmetrics are static.

2.3 A simplified form including a cosmo-logical constant

In the presence of a cosmological constant, oneshould modify the density as

ρ→ ρe = ρ+Λ

8π(36)

and then we can define the modified mass function

Me(r, t) =4π

3R3ρe = M +

Λ

6R3 (37)

eqNow let us recall the Friedmann equation con-necting k and ρ :

k =8π

3ρea

2 − a2. (38)

Then we see that by using the two foregoing equa-tions, we can rewrite ξ2 in an interesting form:

ξ2 = 1− ka2r2 − a2r2

= 1− 8π

3ρer

2a2 = 1− 2Me

R. (39)

Accordingly, we rewrite the S-FRW metric (17)in a highly physically significant form:

ds2 = α2 1− 2Me/R

1−KR2dT 2− dR2

1− 2Me/R−R2dΩ2.

(40)

3 Back to the Einstein-Straus prob-lem

Obviously, the S-FRW metric (42) has a consider-able similarity with the vacuum Schwarzschild met-ric (2) or the Kottler metric [6]

ds2 = (1− 2Me/R)dT 2 − dR2

1− 2Me/R−R2dΩ2,

(41)

and which may be necessary if the FRW spacetimeshould self-consistently accommodate an isolatedSchwarzschild mass. This is so because, after all,the exterior spacetime of the Schwarzschild massis to be described by the vacuum Schwarzschildmetric or the Kottler metric. However, there is alatent important difference between the two cases:while in the Schwarzschild case one has

Me 6= Me(T ), (42)

in the FRW case Me is apparently a function oftime because [7]

M = −4πR2Rp 6= 0; if p 6= 0, (43)

where p is the isotropic pressure of the cosmic fluid.But if the cosmic fluid is assumed to be dust, withpressure p = 0, then M = const. In this case theS-FRW metric looks very similar to the black holemetric. Therefore any “Schwarzschild Mass” Mv

might be embedded in a vacuole of radius Rv cre-ated from a background isotropic and homogeneousuniverse filled with dust, such that

Mb =

∫ Rb(t)

04πρm(r, t)R2R′dr

=4π

3ρ(t)R3

0(t) = Mv = const. (44)

4 Conclusions

Here, for the first time, we have expressed the gen-eral FRW metric (k = +1, 0,−1) in an explicit“Schwarzschild” form. Having done this, we recon-sidered the problem of embedding a SchwarzschildMass (SM) in a pre-existing FRW background fromthe viewpoints of (1) the enigmatic McVittie met-ric obtained in 1933 and (2) the Einstein-Strausapproach of scooping out a spherical cavity in thesame background. Since the exterior of the SM is,

5

by definition, described in the Schwarzschild/curvaturecoordinates, for a decisive study of the Einstein-Straus approach, it is necessary to express the FRWmetric in similar coordinates. And one of the mainapplications of the FRW curvature metric was toseek an objective answer as to the feasibility of em-bedding a SM into a preexisting FRW background.By means of such parallel studies of this prob-lem, we found that one of the necessary conditionsfor the spacetime associated with a Schwarzschildmass to participate in the cosmic expansion is thatthe background fluid is dust. And indeed, Einsteinand Stauss considered the background universe tobe filled with dust [4].

Acknowledgements

The author thanks two anonymous referees formaking several suggestions which led to an im-proved version of this manuscript.

References

[1] M. C. McVittie, Mon. Not. R. Astron. Soc. 93, 325(1933).

[2] N. Kaloper, M. Kleban, and D. Martin, Phys. Rev.D 81, 104044 (2010).

[3] M. Ferris, M. Francaviglia and A. Spallicci, NuovoCim. IIIB, 1031 (1996)

[4] A. Einstein and E. G. Straus, Rev. Mod. Phys. 17,120 (1945).

[5] R. Gautreau, Phys. Rev. D 29, 186 (1984).

[6] F. Kottler, Ann. Phys. 56, 410 (1918).

[7] L. Landau and E.M. Lifshitz, Classical Theory ofFields, (Pergamon, Oxford, 1962).