Table Of ContentMon.Not.R.Astron.Soc.000,1–5(0000) Printed19January2010 (MNLATEXstylefilev2.2)
More on Lensing by a Cosmological Constant
⋆
M. Ishak , W. Rindler†, J. Dossett
Department of Physics, The Universityof Texasat Dallas, Richardson, TX 75083, USA
0 19January2010
1
0
2 ABSTRACT
The question of whether or not the cosmologicalconstant affects the bending of light
n
around a concentrated mass has been the subject of some recent papers. We present
a
J here a simple, specific and transparent example where Λ bending clearly takes place,
and where it is clearly neither a coordinate effect nor an aberration effect. We then
9
1 show that in some recent works using perturbation theory the Λ contribution was
missed because of initial too-stringent smallness assumptions. Namely: Our method
] hasbeentoinsertaKottler(SchwarzschildwithΛ)vacuoleintoaFriedmannuniverse,
h
andtocalculatethetotalbendingwithinthevacuole.Weassumethatnomorebending
p
occurs outside. It is important to observe that while the mass contribution to the
-
o bendingtakesplacemainlyquitenearthelens,theΛbendingcontinuesthroughoutthe
r vacuole.Thusifonedeliberatelyrestrictsone’ssearchforΛbendingtotheimmediate
t
s neighborhoodofthe lens,onewillnotfindit.Lastly,weshowthatthe Λ bendingalso
a follows from standard Weyl focusing, and so again, it cannot be a coordinate effect.
[
Key words: gravitationallensing – gravitation – cosmology:theory.
2
v
6
5
9 1 INTRODUCTION AND EXAMPLE photon orbits because the Λ-independent coordinate equa-
4
tions now refer to a Λ-dependentgeometry.
. Since the publication in 2007 of our first paper on
0 In this first section we begin by exhibiting a specific
this subject (Rindler& Ishak 2007), in which we claimed
1 simple and transparent example, where Λ bending clearly
that Λ contributed to the bending of light, a number
8 takesplace,andwhichinitselfshouldsufficetoanswerthose
0 of authors have supported our findings (Schucker2009; of our critics who categorically deny that Λ contributes to
: Lake2007; Schucker2008; Sereno 2008; Schucker2009;
v thebendingoflight.Thesucceedingsectionsbelowdealwith
Miraghaei et al. 2008) whilst others have questioned them
i moreextensivetopics thathavebeen thesubject of debate.
X (Park 2008; Simpson et al 2008; Khriplovich et al. 2008).
The example in question is a simple consequence
Thoughperhapsnotofimmediatepracticality,thequestion
r of results contained already in our original paper
a of whether Λ does or does not contribute to the bending of
(Rindler& Ishak 2007). There is a well-known generaliza-
light around a concentrated mass is of theoretical interest
tionoftheSchwarzschildmetricthatincludesΛ.Itiscalled
and deserves tobe settled. Inthepresent paperwe hopeto
theKottler(Kottler 1918)metricandrepresentstheunique
contributeto this end.
solution of Einstein’s vacuum field equations with Lambda
Veryearlyon,Eddingtonaddressedtherelatedproblem
for thespacetime around a spherically symmetric mass. By
ofwhetherΛaffectstheorbitsofparticlesaroundaconcen-
an extension of Birkhoff’s theorem, it remains valid in the
tratedmassandfoundthatindeeditdoes;itaddsaminute
vacuumbetween thecentral mass and any spherically sym-
correction to the advance of the perihelion of the planets
metric (not necessarily static) mass distribution around it,
[Eq.(14.25) andEx.14.8 in (Rindler2006)].Eddington ap-
such as a Friedmann universe. It looks as follows:
parently did not investigate the orbits of photons in this
regard. ds2 =α(r)dt2 α(r)−1dr2 r2(dθ2+sin2(θ)dφ2), (1)
− −
Indeed, the usual orbit equations for particles degen-
erate in the case of photons (i.e., when ds2 = 0) in that where relativistic unitsare used (c=G=1) and
theΛtermdropsout,seeEq.(7)in(Rindler& Ishak 2007). 2m Λr2
However,itisourcontentionthatΛneverthelessaffectsthe α(r) 1 . (2)
≡ − r − 3
For small r it approximates Schwarzschild space, and for
large r, if Λ > 0, de Sitter space (in static form). For the
⋆ E-mail:[email protected] sakeof our present simplified example, we assume that this
† F.R.A.S. deSitter space is theactual universe.
(cid:13)c 0000RAS
2 M. Ishak, W. Rindler, J. Dossett
Now consider a ray of light in the coordinate equato- of the volume of the vacuole the radius r is almost of the
rial planeθ=π/2,bendingclosely aroundthecentralmass same magnitude as theproper radius of thevacuole.
(thelens) and connectingtwodiametrically oppositepoints First, as discussed in our previous work
P and Q equidistant from the center (r = r , φ = 0, (Ishak et al. 2008; Ishak 2008), the radius of the vac-
P Q P
φ = π). (See Fig.2 of (Rindler& Ishak 2007)) If Q coin- uole, r , is much larger than the radius R of the lens itself
Q b
cideswithaluminoussource,andPwithaninertialobserver and when using any expression with the radii squared then
momentarilyatrestinthemetric,theobserverwillmeasure the corresponding ratio is (rb)2 103, for example, for the
R ∼
an angle cluster Abel2744 (Smail et al. 1991; Allen et al. 1998), this
2m ΛR3 ratio is rRb 2 ≈2500.
ψ0 (3) Seco(cid:0)nd(cid:1),theratioofthesizeofthevacuolecomparedto
≈ R − 12m
theHubbleradius is about thesame order of magnitude as
between that ray and a ray from the center of the lens the ratio between the Λ-contribution to the bending angle
(R being the radius of the lens). This formula (to first or- compared to the first-order Einstein angle. Indeed, we cal-
der in m/R and ΛR3/m) was established in Eq. (17) of culated in (Ishak et al. 2008), for several clusters, the con-
(Rindler& Ishak 2007),andclearlyshowshowΛaffectsthe tribution of the Λ term to the bending angle and found
bending. A whole cone of such rays will, in fact, reach the it equal to about 10−3 10−4 of the value of the first-
observer from all around the lens, who therefore sees the order Einstein angle. And−as we discussed there, while this
source spread out into an Einstein ring of angular diameter is small, it is as big as the second-order term in the mass
2ψ0. and in some cases even larger. Now, the diameter of the
Of course, our stationary observer is not a cosmologi- vacuole around theclusters is about d 2 10Mpc,
vacuole
cal observer moving with the Hubble flow in the de Sitter (for example 5Mps for Abel2744), so the r≈atio−s between
universe. Such a cosmological observer has a local Hubble d and the angular-diameter distances involved in the
vacuole
velocity lens equation are of the order 10−2 10−3 and finally the
ratio d /R 10−3 10−4−. So by neglecting the
V Hr= Λ/3r (4) vacuole Hubble ≈ −
≈ sizeof vacuole,theeffect ofΛ hasbeenneglected with it as
p
and an aberration factor we show below.
1+V 1/2 Anotherrelevant point here is that the bending dueto
(5) the mass is achieved quickly in the vicinity of the lens and
(cid:18)1 V (cid:19)
− then changes very slightly as we move away from the lens,
relative to the stationary observer (Eq. (4.9) in whereastheeffectofΛinsidethevacuoleaccumulatesupto
(Rindler2006)): his Einstein ring appears that much the boundary of the vacuole. This makes the larger size of
larger. Of course, this factor trivially brings another Λ rb significant.
contribution into play, but this is NOT the one that is We next show how not discarding the diameter of the
underscrutiny here. It does not affect the RATIOin which vacuole compared to the Hubble radius affects the calcula-
m and Λcontribute to thebending. tionof(Simpson et al 2008)andrestoresthecontributionof
In Simpson et al. (Simpson et al 2008), Sec.IV, it is Λ tothe bendingangle.
suggested that our claimed Lambda bending is really an Ref. (Simpson et al 2008) considered the perturbed
aberration effect, and earlier, in Sec II, that it is a gauge Robertson-Walker metric where scalar metric fluctuations
effect: “The potential-like term Λr2/3 in the Kottler f(r) are described byscalar potentials, Φ and Ψ:
[our α(r)] ... does not arise from the true potential Φ, and 2 2 2 2 2 2
ds =(1+2Φ)dt a (t)(1 2Ψ)(dχ +χ dΩ )
thusit should not betaken to cause lensing ... it seems fair − −
to assert that the appearance of a lensing effect from Λ in and where χ is the comoving radius, and dΩ the element
the Ishak-Rindler analysis is purely a gauge artefact.” Our for the unit sphere. The case of a flat universe with no
exampleabovesuggestsotherwise.Itdispenseswiththepo- anisotropicstresseswasconsidered,soΨ=Φ.Andthen,the
tentialformalismoftheusuallensingliterature,andinstead question of how the perturbed FRW metric compares with
dealsdirectly with thespacetimeand itsnullgeodesics in a theexactKottlermetricinsidethevacuolewasexplored.Af-
geometric (coordinate-independent)way.Sothequestion of ter a coordinate transformation and some approximations,
a ”true potential” or of a gauge choice neverarises. theauthorsof(Simpson et al 2008)arrivedatthefollowing
expression for their function f [in theirequation (9)]:
′ 2 2 ∂ln Φ
f =1 2χΦ a˙ χ 1 2Φ | | +2 . (6)
2 THE Λ CONTRIBUTION TO LIGHT − − (cid:20) − (cid:18) ∂lna (cid:19)(cid:21)
BENDING AND APPROXIMATIONS IN
Atthispoint,wedon’tmakethesamesimplification aswas
THE NEWTONIAN GAUGE
madein(Simpson et al 2008)andwedon’tignoretheterms
proportional to Φ in the square brackets since we want to
In this section we discuss how the argument presented in
keep the contribution from the vacuole even if it is small
Simpsonetal. (Simpson et al 2008)isbasedonan assump-
compared with theHubblelength.
tion that eliminates the Λ contribution to the bending of
Next, we recall the 00-equation of the perturbed
light a priori. As the authors acknowledge after equation
Einstein equations (see for example equation (5.27) in
(9) of their section II and at the end of that section, it was
(Dodelson 2003)):
assumed in their work that in a perturbed FLRW universe
the Kottler vacuole around the lens is negligibly small in k2Φ+3a˙ Φ˙ Φa˙ =4πGa2ρ δ . (7)
comparison with the Hubble length and also that, in most a(cid:18) − a(cid:19) m m
(cid:13)c 0000RAS,MNRAS000,1–5
More on Lensing by a Cosmological Constant 3
At the same level of approximation where we want to aswasassumedin(Simpson et al 2008)afterequation(22).
keep the contribution from the vacuole, we don’t drop the Thisinvalidatestheargumentusedafterequation(22)there.
higher order terms on the LHS of (7) and we don’t use
the approximation ∂ln Φ/∂lna = 1 also invoked in
| | −
(Simpson et al 2008). One could for example model small
departuresfrom this relation by 1 ǫ. 3 THE CONTRIBUTION OF Λ TO THE
− ±
Now, using (6), (7), the Friedmann equation, and BENDING OF LIGHT FROM THE WEYL
following the same procedure as in section II of FOCUSING
(Simpson et al 2008), we integrate the potential Φ and ob-
tain thesolution The propagation of light in a given space-time can be
rigorously described using the theory of geometrical op-
m mr2 Λr2
Φ= (8) tics in General Relativity (Sachset al. 1961; Dyer1977).
− r − 2rb3 − 12 The distortions of light bundles are described by the op-
tical scalar equations driven by Ricci and Weyl focusing
plussmaller terms.
(Sachset al. 1961;Dyer1977;Schneideret al. 1992).Were-
Remarkably,ourintegrationofthepotentialgivesthe2
call that the Ricci focusing is given by (Sachset al. 1961;
mass terms of reference (Simpson et al 2008) plus Λ terms
with a leading term that is precisely Λr2 1. Dyer1977; Schneideret al. 1992)
− 12
This Λ term in the potential gives the same Λ con- =R kakb (10)
ab
R
tribution to the light-bending angle (see for example
and theWeylfocusing (upto a phase factor) by
(Ishak 2008)) as the one derived from different approaches
in (Rindler & Ishak 2007; Ishak et al. 2008; Ishak 2008), =C kakbtitj, (11)
aibj
namely, F
whereR istheRiccitensor, C istheWeyltensor, and
ab aibj
αΛ = ΛRrb (9) ka and ti are null vectors. We here follow the notation of
− 3 (Dyer1977; Schneideret al. 1992).
where we assumed R rb 2. IntheKottlervacuumspace(Rab=0),themassandΛ
The result (8) ab≪ovefor thepotential shows that when contributionstolight-bendingcomefromtheWeylfocusing
the size of the vacuole is not neglected in comparison with as covariantly defined in (11). The non-vanishing compo-
the Hubble length then the potential does contain Λ-terms nentsof the Weyltensor read
(as found previously (Ishak et al. 2008; Ishak 2008)) and 2m
C =
confirms that Λ contributes to the bending of light inside trtr − r
thevacuole. m 2m Λr2 2
Therefore, we find that the Λ contribution terms Ctθtθ = r (1− r − 3 )=Ctφtφ/sin (θ)
were missed in (Simpson et al 2008) because of initial too-
m/r 2
stringent smallness assumptions on thesize of the vacuole. Crθrθ = 1 2−m/r Λr2 =Crφrφ/sin (θ)
It was also stated in the same paper by Simpson et al. − − 3
2
(Simpson et al 2008) (section IV) that the Λ bending term Cθφθφ = 2rm sin (θ) (12)
in our equation (3) (see (Rindler& Ishak 2007)) and equa-
Following (Dyer1977), it is easy to verify for the Kottler
tion (9) above (see (Ishak et al. 2008)) appear problematic
metric (1) that
since they do not vanish when the lensing mass is taken to
bezero.Clearlywhenthelensmassiszero(i.e.nolens),the 3mh2
=0 and = (13)
radiusofthelensiszero(i.e.R=0)andtheΛ-contribution R F r5
term vanishes as well (i.e. αΛ = 0). So, as expected, when (equations (27) and (35) in (Dyer1977)). In Schwarzschild
there is no lensing mass inside the de Sitter vacuole, Λ has
space, h is the impact parameter but in Kottler space (not
noeffecton thebendingof light butwhenacentralmass is
asymptoticallyflat)histheconstantofmotionJ/EwhereJ
present as in the Kottler space that is under consideration
andE arerespectivelythemomentumandtheenergyofthe
here,themassbendstherayoflightwhileΛdiminishesthe
photon. The relation between the point of closest approach
deflection inside thevacuole.
r0andtheconstantofthemotionh(orbinothernotations)
Finally, it is worth noting that the structure of the
is given by (e.g (Wald 1984; Kagramanova et al. 2006))
Weyltensor(seeEq.(12)inthenextsection)fortheKottler
spacetime shows that inside the Kottler vacuole, the Weyl r02 2m Λr02
=1 . (14)
tensorisnonvanishingandso thespacetime isnot isotropic h2 − r0 − 3
For Schwarzschild space, the solution r0(h) of this relation
is given on page 145 of (Wald 1984). The solution can be
1 Weverifiedthatwhenourfullsolutionforφ(with3massterms
immediately expanded toinclude Λ and is given by
plus4termswithΛ)isputintoequation(6),ityieldsthefunction
α(r)in(2)asitshouldplushigherorderterms,i.e.O(mΛ). 2 1 3
2inI(nIdsheeadkoeturale.x2p0r0e8ss)iownheforreαwΛe awsassumdeerdivaedsmfraolml aenqguleatΦiobns=(1rR8b). r0 = qh32 +Λcos(cid:16)3arccos(cid:16)−3mrh2 +Λ(cid:17)(cid:17), (15)
Similarly, we justifiably assumed in the step from equation (11)
to (12) in (Ishak2008) that R 2 ≪ 1 thus reducing αΛ = in agreement with footnote 9 in (Finelli et al. 2007). For
rb Λ = 0 it reduces to equation (6.3.37) that was derived for
−ΛR3rb 1− rRb 2 tosimply(cid:0)αΛ(cid:1)=−ΛR3rb. Schwarzschild in (Wald 1984).
q (cid:0) (cid:1)
(cid:13)c 0000RAS,MNRAS000,1–5
4 M. Ishak, W. Rindler, J. Dossett
Now, if one chooses to express the Weyl focusing fully with Kottler in the vacuole and flat FLRW around it,
in terms of the point of closest approach r0 then (14) and and where angles are measured by comoving Friedmann
(13) yield observers. Kantowski et al. take careful account of the fact
that the radius of the Kottler vacuole in the expanding
3mh2 3m 2m Λr02 −1 FLRW universe is larger when the photon exits than when
= = 1 . (16) itentered.Infact,itisthisexpansionthatintheirapproach
F r05 r03 (cid:16) − r0 − 3 (cid:17) givesrisetothemostsignificantpartofthecorrectionterm.
So Λ is present in the final result. Similarly, if one In particular, the authors say, without the expansion there
chooses to express the focusing fully in terms of h, then is no Λ-correction [see after their Eq. (23)]. And later, that
using (15) in order to eliminate r0 from (13) gives as a it arises ”since the cosmological constant contributes to
F
function of m, h, and Λ. Again the final expression for the the extra time ... the Schwarzschild mass has to act on the
Weylfocusing has Λ in it. passing photons.” Their lowest order correction in which Λ
Thisresultisindependentofanglecalculationsandsup- appears is proportional to m√Λ, (their equation (32)) and
ports the conclusion that Λ contributes to the bending of theyfindthatthiscould cause asmuchas a0.02% increase
light the Kottler space in an invariant and thus eminently in the deflection angle of the light that passes through a
coordinate-independent way. rich cluster. It should be noted that a term like (33) can
arise simply from aberration in an expanding universe; it
does for example, when we apply the aberration factor
(1 + V) (mentioned at the end of section 2.1 above)
4 FURTHER REMARKS ON THE ∼
to the mere Einstein bending angle 4m/R in a de Sitter
CONTRIBUTION OF Λ TO LIGHT
universe, where V = Hr, H = Λ/3. Also, Kantowski et
BENDING
al. in their introduction stress tphe desirability to produce
Schucker performed calculations in the Kottler space formulas which, apart from Λ, contain only quantities that
(Schucker2009; Schucker2008) and also used a piece- aremeasurablythesamewithandwithoutΛ.”Toconclude
wise integration of null geodesics (Schucker2009) in whetherΛdoesor doesn’t cause bendingcan easily depend
the Einstein-Strauss model. He confirmed our result on what is held in common and what property is compared
(Rindler& Ishak 2007) that a positive cosmological con- in the two experiments” – namely in a universe with and
stant decreases the bending of light by an isolated spher- one without Λ. Now, their concluding formula (32) for
ical mass. He finds in (Schucker2009) that the decrease the total deflection angle has the factor m/r0 in front of
in the deflection angle due to Λ is attenuated by a ho- it, r0 being the coordinate radial distance of the point of
mogeneous Friedmann background when added around the closest approach. It could perhaps be argued that, in term
spherical mass and by the recession of the observer with- of absolutes, r0 might depend on the geometry of Kottler
out however canceling it. A result consistent with our work space and thus ultimately on Λ. In sum, Kantowski et al.’s
(Ishak et al. 2008; Ishak 2008). work supports Λ-corrections to gravitational lensing in a
Sereno did another calculation (Sereno2008) in the flat ΛCDM model even if they arrive at different Λ-terms.
Kottler space and re-derived our Λ contribution term, Finally, Park (Park 2008) considered a customized
however, he argued that the term can be embedded McVittie metric and did an expansion of the null geodesic
into the angular diameter distances. In a further analysis equationstofirst-orderinthemass.Herealso,second-order
(Sereno2008),hederivedanotherΛcontributionterm that terms including H2 = Λ/3 terms were dropped out in the
hearguedcannotbeincludedintotheangulardiameterdis- calculation in equations(11) and thefinalresult (31), lead-
tances. ingtotheassertionthatΛdoesnotcontributetolensingex-
In the work of Khriplovich & Pomeransky ceptviatheangulardiameterdistances.Infact,evenatthe
(Khriplovich et al. 2008) on this question, the authors same level of approximation used in his paper, Λ contribu-
did an expansion of their equation (16) keeping only the tiontermsweresimplyomittedfrom thefinallensequation
first term in the invariant considered there and dropping (31)includingatermwithH2=Λ/3forreasonofsmallness,
thesecondtermofthatequation(i.e.ωλρintheirnotation) see footnote 3 Indeed, it seems fair to say that the omitted
that contains a Λ contribution. They then claim that part with H2 = Λ/3 in the lens equation in (Park 2008) is
all factors dependent on Λ cancel. Then, the authors do incontradictionwiththestatementfollowingequations(31-
reconsider their position and state in the penultimate 32) there and the conclusion presented in the paper. These
paragraph of their paper the following: “Let us note that terms are perhaps small for current observations but they
somecorrectionsontheorderofλ2ρ2 Λr r tothelensing donot cancel out as stated previously.
r o
∼
effects may exist, as well as other cosmological corrections
in the general case of the FRW Universe. However, such
“short-distance” phenomena are perhaps too small to be
of practical interest.” In fact, using their notation where
λ Λ/3, the neglected term in question is Λr r /3 and 3 We communicated our findings to the author of (Park2008)
g o
rep≡repsents the contribution of Λ to the squared invariant andwearegratefultohimforsendingtousthefollowingpartthat
wasomittedfromhislensequation(31)undertheassumptionof
that was thrown away by assumption.
smallness
Most recently, Kantowski, Chen and Dai
(Kantowski et al. 2009) aimed to reexamine all higher 2mdSL β −H2xSd2L +... . (17)
order corrections to the Einstein deflection angle including dSdL 2 !
Λ-terms. They considered a Swiss-Cheese exact solution
(cid:13)c 0000RAS,MNRAS000,1–5
More on Lensing by a Cosmological Constant 5
5 CONCLUSION RindlerW.,Relativity:Special,General,andCosmological,
Second Edition (Oxford UniversityPress, 2006).
Using a very simple example, we clarified that the effect of RindlerW., M. Ishak,Phys. Rev.D 76 043006 (2007).
Λ on the bending of light is physical and distinct from the Sachs R.K., Proc. Roy.Soc. London.A 264, 309 (1961).
abberation.
SchneiderP.,J.EhlersandE.E.Falco,GravitationalLenses
We showed how the conclusions presented in
(Springer-Verlag, 1992).
(Simpson et al 2008; Khriplovich et al. 2008; Park 2008) Schucker T., General Relativity and Gravitation, 41:67-
about the non-contribution of Λ to the bending of light
75,(2009).
and lensing phenomena are simply due to the fact that Schucker T., N. Zaimen, Astrono. & Astrophys., 484, 103
in the perturbative or expansion approaches used, the Λ
(2008).
contributiontermswereeliminated atsomesteporotherof Schucker T., General Relativity and Gravitation 41:1595-
thecalculation by too-stringent assumptions of smallness.
1610 (2009).
A clarifying point is to realize that the Kottler vac- Sereno M., Phys.Rev. D 77, 043004 (2008).
uole surrounding the lens has a size that is larger than the
Simpson F., J. A. Peacock, A. F. Heavens,
lens and that when compared to the angular-diameter dis-
arXiv:0809.1819v1[astro-ph] (2008)
tances involved in the lens equation provides the derived Smail I. et al., M.N.R.A.S, 252, 19 (1991).
contribution of Λ to lensing. This point was also appreci-
Wald R. General Relativity (University of Chicago Press,
ated using a piecewise integration of the null geodesics in
1984).
an Einstein-Strauss model (Schucker2009; Schucker2008;
Schucker2009).
We also showed that the effect of Λ can be ascribed to
theinvariant Weylfocusing.
While the effect of Λ on the light-bending and
lensing is small as discussed in (Rindler& Ishak 2007;
Ishak et al. 2008;Ishak 2008),theeffectdoesnotcancelout
as has been claimed and is not a coordinate effect.
ACKNOWLEDGMENTS
We thank Fergus Simpson, John Peacock, Alan Heavens,
Minjoon Park, Thomas Schucker, and Mauro Sereno for
stimulating discussions and correspondence. M.I. acknowl-
edges that this material is based upon work supported in
partbyNASAundergrantNNX09AJ55Gandagrantfrom
theTexasSpaceGrantConsortium.Partofthecalculations
for this work havebeen performed on theCosmology Com-
puterCluster funded by theHoblitzelle Foundation.
REFERENCES
Allen S., M.N.R.A.S, 296, 392, (1998).
Dodelson S., Modern Cosmology, Academic Press (2003).
DyerC., Mon. Not. Roy.Astro. Soc. 180, 231 (1977).
Finelli F., M. Galaverni, A. Gruppuso, Phys.Rev. D 75,
043003 (2007).
Ishak M., Phys. Rev.D 78, 103006, (2008).
Ishak M., W. Rindler, J. Dossett, J. Moldenhauer, C. Al-
lison, Monthly Notices of the Royal Astronomical Society,
Volume 388, Issue 3, p1279 (2008).
Kagramanova V., J. Kunz, C. Lammerzahl, Phys.Lett. B
634 465-470 (2006).
Kantowski R., Chen B., Dai X, arXiv:0909.3308 (2009),
ApJ submitted.
Khriplovich I., A. Pomeransky, Int. J. Mod. Phys. D 17,
2255-2259 (2008).
Kottler F., Ann.Phys.361, 401 (1918).
Lake K., arXiv:0711.0673.
Miraghaei H., M. Nouri-Zonoz, arXiv:0810.2006v1 [gr-qc]
(2008).
Park M., Phys.Rev.D.78, 023014 (2008).
(cid:13)c 0000RAS,MNRAS000,1–5
