Gravitomagnetic effects in conformal gravity

0.14 MB·

by  Jackson Levi Said

Gravitomagnetic effects in conformal gravity
Jackson Levi Said∗,1 Joseph Sultana†,2 and Kristian Zarb Adami‡1,3
1Physics Department, University of Malta, Msida, MSD 2080, Malta
2Mathematics Department, University of Malta, Msida, MSD 2080, Malta
3Physics Department, University of Oxford, Oxford, OX1 3RH, United Kingdom
(Dated: 14 January 2014)
Gravitomagnetic effects are characterized by two phenomena: first, the geodetic effect which
describes the precession of the spin of a gyroscope in a free orbit around a massive object, second,
4 theLense-Thirringeffectwhichdescribestheprecessionoftheorbitalplaneaboutarotatingsource
1 mass. We calculate both these effects in the fourth-order theory of conformal Weyl gravity for the
0 test case of circular orbits. We show that for the geodetic effect a linear term arises which may
2 be interesting for high radial orbits, whereas for the Lense-Thirring effect the additional term has
a diminishing effect for most orbits. Circular orbits are also considered in general leading up to a
n
generalization of Kepler’s third law.
a
J
PACSnumbers: 04.20.-q,04.50.Gh
0
1

I. INTRODUCTION sponding field equations are then given by [1]
]
c
q √ gg g δIW = 2αW
- The validity of any alternative theory to Ein- − µα νβδg − µν
r αβ
g stein’s general relativity depends on how well it = 1T , (3)
[ agrees with his theory in the weak field limit as −2 µν

1 well as observational tests. One of the possi- where Tµν is the stress-energy tensor, and
ble alternatives to Einstein’s second order the-
v
8 orywhich has been proposedduring the lasttwo Wµν =2Cαµνβ;βα+CαµνβRαβ. (4)
9 decades is conformal Weyl gravity [1–3]. Instead
8 of choosing the gravitationalaction by requiring Itcanbeseenfrom(3)thatanyvacuumsolution
2 of Einstein’s field equations is also a vacuum so-
thatthetheorybenohigherthansecondorderas
1. in the case of the Einstein-Hilbert action, Weyl lutionofWeylgravity;withtheconversenotnec-
0 essarily being true. Despite being a fourth-order
gravity employs the principle of local conformal
4 theory with highly nonlinear field equations, a
invariance of spacetime to fix the gravitational
1
number of exact solutions [4–9] have been found
: action, meaning that the theory is invariant un-
v which generalize the well-known Kerr-Newmann
der local conformal stretching of the type
i and cylindrical solutions of general relativity.
X
The exact static and spherically symmetric vac-
r g (x) Ω2(x)g (x), (1)
a µν µν uum solution for conformal gravity is given, up
→
to a conformal factor, by the metric [1]
whereΩ(x)isasmoothstrictlypositivefunction.
dr2
Thisleadstoaconformallyinvariantfourthorder ds2 = B(r)dt2+ +r2(dθ2+sin2θdφ2),
− B(r)
theory with a unique action given by
(5)
where
IW = α d4x( g)1/2CλµνκCλµνκ β(2 3βγ)
− − B(r)=1 − 3βγ+γr kr2, (6)
Z − r − −
= 2α d4x( g)1/2[R Rµκ (Rν)2/3]
− − µκ − ν and β, γ, and k are integration constants.
Z
This solution includes as special cases the
+a total derivative, (2)
Schwarzschild solution (γ = 0 = k) and the
Schwarzschild-deSitter (γ =0) solution; the lat-
where C is the conformalWeyl tensorand α
λµνκ ter requiring the presence of a cosmologicalcon-
is a purely dimensionless coefficient. The corre-
stant in Einstein gravity. Moreover the constant
γ has dimensions of acceleration, and so the so-
lutionprovidesacharacteristic,constantacceler-
ation without having introduced one at the La-
∗[email protected]
†[email protected] grangian(such as in the relativistic implementa-
‡[email protected] tion of MOND with TeVeS [10]).
2
The magnitude and the origin (i.e. whether it chosen coordinate system [17], geodetic preces-
is system dependent like β or cosmological like sioncanbeconsideredasduetoaLense-Thirring
k) of the integration constant γ in (6) remains drag. A combined observable phenomenon of
unknown. Whenit is associatedwith the inverse these two effects [18] occurs in the Earth-Moon
Hubble length, i.e. γ 1/R , the effects of the systemaroundtheSunthroughtheprecessionof
H
≃
accelerationdue to the γr terminthe metric are theMoon’sperigee,whichisdetectedbymeasur-
comparablewiththoseduetotheNewtonianpo- ing the lunar orbit using laser ranging between
tentialterm2β/r r /r(r istheSchwarzschild stations on Earth and reflectors on the Moon’s
s s
≡
radius), on length scales given by surface [19–21].
A spaceexperimentto test these twopredictions
r /r2 γ 1/R or r (r R )1/2. (7)
s ≃ ≃ H ≃ s H of Einstein’s theory is Gravity Probe B (GP-B)
[22]whichwaslaunchedon20April2004ina642
As noted in [1], for a galaxy of mass M
1011 M with r 1016 cm and R 1028 cm≃, km polar Earth orbit with data collection last-
⊙ s H
this scale is r ≃1022 cm, i.e. rough≃ly the size ing almost a year. Measurement of the geodetic
∼ and frame-dragging effect was done by means of
of the galaxy, a fact that prompted Mannheim,
cryogenic gyroscopes with one or more of these
O’BrienandKazanastoproducefitstothegalac-
referenced to a remote star by means of an on-
tic rotation curves using the metric of Eq. (6)
board telescope. For the chosen orbit the two
above, without the need to invoke the presence
effects result in a precession along two perpen-
of dark matter as in standard Einstein’s theory
dicularplanessothatGP-Bcouldmeasurethese
(see Refs. [11, 12] for recent work on this issue).
independently. Analysisofthedatafromthefour
However, an issue arises in that in addition to
onboard gyroscopes showed a geodetic drift rate
having flat rotation curves in the region of in-
of 6601.8 18.3 mas/yr and a frame dragging
terest for galaxies, we must also have stability
− ±
drift rate of 37.2 7.2 mas/yr, which are in
for the orbits under consideration [37]. Circular
− ±
accordance with the general relativistic predic-
orbits offer a good toy model for investigating
tions of 6606.1 mas/yr and 39.2 mas/yr re-
different gravitationaleffects in alternativetheo-
− −
spectively.
riesofgravity. Duetotheirimportancewebegin
The classical tests of general relativity including
with a short investigationof circular orbits lead-
the bending of light [23–26], time delay [27] and
ing up to a statement of Kepler’s third law.
perihelionprecession[28]havealreadybeenstud-
The geodetic precession, also known as de Sitter
ied in conformal gravity. In this paper we inves-
precession is a general relativistic phenomenon
tigatecircularorbitsinSec. II,derivingKepler’s
discovered in 1916 by de Sitter [14], who found
third law. In Sec. III the geodetic precession ef-
that the spin of a gyroscope precesses with re-
fect is determined for circular orbits. In Sec. IV
spect to a distant inertial frame when it makes
theLense-Thirringeffectiscalculatedintermsof
free (or “geodetic”) orbits around a nonrotating
the precessionvelocity. Finally in Sec. V we end
massive object. For a circular orbit the amount
with a summary of conclusions and a discussion.
of geodetic precession per orbit is given by
1/2
3GM
∆φgeodetic = 2π"1−(cid:18)1− c2R (cid:19) # II. CIRCULARTOHRIRBDITLSAAWND KEPLER’S
3πGM GM
, for <<1,(8)
≈ c2R c2R
We investigate circular orbits in conformal
where M is the mass of the massive object and gravityusingaLagrangianapproachforthemet-
R is the radius of the orbit. This effect is also ric in Eq.(5). In this case the Lagrangian takes
observed in flat spacetime, where in this case the form
it has a purely kinematical origin and is known 1 ∂xµ∂xν
as Thomas precession [15]. Another closely re- = gµν
L 2 ∂τ ∂τ
latedrelativisticprecessionisthe Lense-Thirring
1 r˙2
effect [16] (or frame dragging effect) discovered = Bt˙2+ +r2θ˙2+r2sin2θφ˙2 , (9)
2 − B
by Lense and Thirring in 1918, and refers to the (cid:20) (cid:21)
precessionofthegyroscopeduetotherotationof where dots denote differentiationwith respectto
the massive object which produces a draggingof proper time, τ. Using the four-velocity normal-
nearby inertial frames. Although these are two ization condition u uµ = 1, where u dxµ,
µ − µ ≡ dτ
independenteffectsinthesensethatthelatterre- it follows that = 1. It must be noted at
L −2
quires rotation of the source of the gravitational this point that this Lagrangian and the associ-
field, it can be shown that in an appropriately ated timelike geodesics representing trajectories
3
of free massive particles are not conformally in- For k <<γ <<β this can be written as
variant, unlike the theory of conformal gravity.
Moreoverconformalinvarianceimpliesthatthere
is no prescribed scale in the theory, which has a
β γ
dimensionless coupling constant α as shown in ω2 + , (14)
Eq. (2). However recently Edery et al. in Ref. ∼ R3 2R
[29] (see also Refs. [30, 31]) considered spon-
taneous symmetry breaking in Weyl gravity and
showedthatthiscanprovideamechanismtogen- which is similar to Kepler’s third law, with the
erate a scale in the theory. second term on the right-hand-side representing
Takingadvantageofthe independence ofthe La- the correction from conformal gravity.
grangianonthet andφ coordinates,andusing
− −
θ = π/2 as the plane of the circular orbits with-
out loss of generality due to sphericalsymmetry,
theEuler-Lagrangeequationsofmotionaregiven
by
Bt˙=E, (10)
III. THE GEODETIC EFFECT
r2φ˙ =L, (11)
where E and L are the energy and angular mo- The precession is calculated for orbits in the
mentum respectively. The third Euler-Lagrange equatorial plane θ = π2 since by symmetry con-
equation is given by siderationsarotationcanalwaysbemadetothis
plane. Rotating coordinates are first introduced
(B−12r˙)˙ [ (β(2−3βγ) +γ 2kr)t˙2 through the transformation
− − r2 −
∂
+ (B−1)r˙2+2rφ˙2]=0. (12)
∂r
φ φ ωt, (15)
For circular orbits r = R, r˙ 0 and therefore → −
≡
the above equation yields
2
dφ 1 β(2 3βγ) where ω is the coordinate angular frequency of
ω2 = = − +γ 2kR .
dt 2R R2 − the rotation. This transforms the metric in
(cid:18) (cid:19) (cid:18) (cid:19)
(13) Eq.(6) to
2
β(2 3βγ) r2ω
ds2 = 1 − 3βγ+γr kr2 r2ω2 dt dφ
−(cid:18) − r − − − (cid:19)" − 1− β(2−r3βγ) −3βγ+γr−kr2−r2ω2 #
dr2 1 β(2−3βγ) 3βγ+γr kr2
+ +r2 − r − − dφ2.
1 β(2−3βγ) 3βγ+γr kr2 1 β(2−3βγ) 3βγ+γr kr2 r2ω2
− r − − − r − − −
(16)
Comparing this with Rindler’s canonical form r2ω
w =
[34] 3 1 β(2−3βγ) 3βγ+γr kr2 r2ω2
− r − − −
(19)
ds2 = e2Φ dt w dxi 2+k dxidxj, (17)
− − i ij k11 =1 β(2−3βγ) 3βγ+γr kr2 (20)
(cid:0) (cid:1) − r − −
the following nonvanishing components arise
1 β(2−3βγ) 3βγ+γr r2 k+ω2
e2Φ =1 β(2−3βγ) 3βγ+γr kr2 r2ω2 k33 = − r − − ,
− r − − − r2(1−3βγ)−βr(2−3βγ)+r(cid:0)3γ−kr(cid:1)4
(18) (21)
4
whereLatinindicesrefertospacelikecoordinates with respect to the inertial frame, where the re-
only. lation uφ =ω = 2π was used.
∆t
We considerfreecircularorbits forwhichthe ac-
celeration [35] vanishes
IV. LENSE-THIRRING EFFECT
a= kijΦ Φ 1/2 =0, (22)
,i ,j
whichimplies tha(cid:0)tΦ,r =0.(cid:1)Hence arelationcan The Lense-Thirringeffect describes the correc-
be established between the angular velocity and tion of a rotating spacetime on the precession
the metric parameters such that of orbits. The effect can be quantified through
a consideration of the Sagnac effect as in [36–
β(2 3βγ) γ
ω2 = − + k. (23) 38],whichrepresentsthedifferenceintraveltime
2r3 2r −
or phase shift of corotating and counterrotating
Substituting this in Eqs.(18, 19, 21) gives light waves in the field of a central massive and
spinning object. In order to calculate this in
3β(2 3βγ) γr
e2Φ =1 − 3βγ+ (24) conformal gravity we must consider the rotating
− 2r − 2
metric found in [4], namely
r2 β(2−3βγ) + γ k
w = 2r3 2r − (25) dr2 dy2
3 1 q3β(2−3βγ) 3βγ+ γr ds2 =(bf ce) +
− 2r − 2 − a d
(cid:18) (cid:19)
1 3β 2−3βγ 3βγ+ rγ 1
k33 = − 2r − 2 . + d(bdφ cdt)2 a(edφ fdt)2 ,
r2(1 3βγ) βr(2 3βγ)+r3γ kr4 bf ce − − −
− − − − (26) − h (31i)
whichis the canonicalCarterformofthe metric,
The proper rotation rate of a gyrocompass with
where
respect to the rotating frame (15) is given in
terms of the canonical form (17) by [35] a=j2+ur+pr2+vr3 kr4, (32)
−
Ω= 1 eΦ kikkjl(w w )(w w ) 1/2 d=1+r′y py2+sy3 j2ky4, (33)
2√2 i,j − j,i k,l− l,k − −
b=j2+r2, (34)
eΦ (cid:2) 1/2 (cid:3)
= k11k33w 2 . (27) e=j 1 y2 , (35)
2 3,1 −
h i c=j, (36)
Substituting Eqs. (24 - 26) in this expressionwe (cid:0) (cid:1)
getΩ=ω,sothatthecoordinaterateω isreally f =1, (37)
the orbital rotational frequency.
and u, p, v, k, r′ and s are constants satisfying
Now for the Mannheim-Kazanas metric (6) the
the constraint uv r′s = 0. The angular mo-
relationbetweenproperandcoordinatetimesfor −
mentum of the source is represented by j. The
a circular orbit (r = R, r˙ = 0) with coordinate
general relativistic result representing the Kerr
angular velocity ω, is given by
solution is recoveredwhen
3β(2 3βγ) γR
∆τ = 1 − 3βγ+ 2R2k ∆t. u= 2MG/c2,
− 2R − 2 − −
r
(28) p=1,
Hence for one complete orbit about the gravi- v =0=k=r′ =s, (38)
tating mass the precessionof the gyroscopewith
while the Kerr-de Sitter solution is obtained
respect to the rotating frame is
when k = Λ/3 and p = 1 kj2; Λ being the
α′ =Ω∆τ cosmologicalconstantandwi−ththeotherparam-
eters taking the same values as in (38). For the
3β(2 3βγ) γR
=2π 1 − 3βγ+ 2R2k. conformal case the additional parameters must
− 2R − 2 −
r be constrained through observation.
(29)
Similar to the general relativistic case we take
This results in a precession angle per orbit of the transformation
α=2π α′ y cosθ, (39)
− →
3β(2 3βγ) 3βγ γR
2π − + +R2k , in order to obtain Boyer-Lindquist-like coordi-
≈ 4R 2 − 4
(cid:18) (cid:19) nates. Noting that dy = sinθdθ the metric
(30) −
turns out to be
5
dr2 sin2θdθ2
ds2 = r2+j2cos2θ +
j2+ur+pr2+vr3 kr4 1+r′cosθ pcos2θ+s cos3θ j2kcos4θ
(cid:18) − − − (cid:19)
+ (cid:0) 1 (cid:1) 1+r′cosθ pcos2θ+scos3θ j2kcos4θ j2+r2 dφ jdt 2
r2+j2cos2θ − − −
j2+ur+pr2h+(cid:0) vr3 kr4 jsin2θdφ dt 2 . (cid:1)(cid:0)(cid:0) (cid:1) (cid:1) (40)
− − −
(cid:0) (cid:1)(cid:0) (cid:1) i
The Sagnac effect is now investigated by con- set r =R so that
sidering corotating and counterrotating circular
ltihgehtcebnetarmalsoinbjtehcet.eqAufatteorriaalcpirlcaunlearθo=rbπ2itatbhoeuset ds2 = dRt22 j2+R2 ω0−j 2
light beams reach a detector (that can also dou- j2+h(cid:0)u(cid:0)R+pR2(cid:1)+vR3(cid:1) kR4 (jω 1)2 .
ble as a source) which is assumed to be rotating 0
− − −
with uniform angular velocity ω0, such that its (cid:0) (cid:1) (42i)
rotation angle is
φ =ω t. (41) Nowgiventhatwearedealingwithraysoflight
0 0
we take ds=0. Eq.(42) then gives two solutions
Since we are considering circular orbits, we also for ω , where
0
1
Ω = 2j uR+(p 1)R2+vR3 kR4
± 2(R4+j2( uR+(2 p)R2 vR3+kR4)) − − −
− − − h
(cid:0) (cid:1)
4j2(uR+(p 1)R2+vR3 kR4)2+4(R4+j2( uR+(2 p)R2 vR3+kR4))(uR+pR2+vR3 kR4) ,
± − − − − − −
q (43) i
these two solutions refer to the rotating and which when substituted back into Eq.(45) gives
counterrotating orbits.
For light the rotation angles are given by
φ =Ω t. (44)
± ±
Ω
±
φ =φ 2π. (48)
0 0
ω ±
In conjunction with Eq.(41) the t coordinate 0
−
can be eliminated
Ω
±
φ = φ . (45)
± 0
ω
0
Solving for φ
0
The first intersection of the rays of light with
the positionofthe observerafter emissionatt=
0, occurs when
2πω
φ+ =φ0+2π, (46) φ0± =±Ω 0ω =±2πω0ξ±, (49)
± 0
φ =φ 2π, (47) −
− 0 where
−
6
ξ = 2 R4+j2 uR+(2 p)R2 vR3+kR4 2j uR+(p 1)R2+vR3 kR4
±
− − − − − −
4(cid:2)j2(cid:0)(uR+(p(cid:0) 1)R2+vR3 kR4)2+4(R4+(cid:1)(cid:1)(cid:3)j2h( uR(cid:0)+(2 p)R2 vR3+kR4))(u(cid:1)R+pR2+vR3 kR4)
± − − − − − −
q −1
2ω R4+j2 uR+(2 p)R2 vR3+kR4 . (50)
0
− − − −
(cid:0) (cid:0) (cid:1)(cid:1)i
CombiningthiswiththemetricinEq.(42),and canthusbedeterminedbyusingthedetectorco-
reverting to physicalunits the proper time delay ordinate relation in Eq.(41)
dφ
dτ = (j2+uR+pR2+vR3 kR4)(jω 1)2 ((j2+R2)ω j)2 (51)
0 0
cRω − − − −
0
q
Integrating between φ and φ gives the Sagnac time delay between the two light beams
0− 0+
φ φ
δτ = 0+ − 0− (j2+uR+pR2+vR3 kR4)(jω 1)2 ((j2+R2)ω j)2
0 0
cRω − − − −
0
q
2π(ξ +ξ )
= + − (j2+uR+pR2+vR3 kR4)(jω 1)2 ((j2+R2)ω j)2. (52)
0 0
cR − − − −
q
In order to calculate the correctiondue solely to obtaining a generalization of Kepler’s third law.
the rotationofthe sourcethe detector rotational The main results of this paper are obtained
parameter, ω , is set to zero, such that in Eq.(30) and Eq.(55), which represent the
0
geodetic and Lense-Thirring effects in conformal
4πjuR+(p 1)R2+vR3 kR4
δτ = − − , (53) gravity respectively.
0
cR uR+pR2+vR3 kR4
Apart from the Einstein term 3πβ/R, Eq.(30)
−
which can bepexpressed in terms of a Lense- also includes terms due to the linear term γr
Thirring precession velocity ω , by in the Mannheim-Kazanas metric as well as
LTCG the cosmological term kr2. We note that as in
ω πR2
δτ =8 LTCG , (54) the case of the other already studied classical
0 c2 p+ u +vR kR2 tests, namely the bending of light [26], time
R −
delay [27] and perihelion precession [28], the
where p
contribution to the geodetic precession from the
u+(p 1)R+vR2 kR3
linear term in the metric has an opposite sign to
ω =jc − − .
LTCG 2R3 that of the Einstein term, so that it reduces the
(cid:18) (cid:19)
(55) amount of precession per orbit. Moreover this
Using the values for the constants given in contribution increases linearly with the radius R
Eq.(38), this gives the general relativistic result of the circular orbit, such that at larger radii its
obtained in Ref. [36]. effect can cancel that of the Einstein term.
The Lense-Thirring precession velocity was
obtained in terms of the Sagnac time delay for a
V. DISCUSSION AND CONCLUSION
stationary observer which occurs solely due to
the drag of spacetime by the rotating central
In this paper, we first considered how circular object. Assuming that the constants u, p and k
orbits behave in conformal gravity, thereby in (55) take the same values as in the Kerr-de
7
Sittersolution,weseethattheconformalgravity of γ, which in our case is substantiated by the
correction through the constant v [which corre- fact that its contribution to the geodetic preces-
sponds to γ when j = 0, θ = π/2 in (40)] has sion in (30) is independent from the mass of the
againadiminishingeffectonthe totalprecession central object, just like the cosmological term.
velocity. However in this case this correction Howeverhaving saidthis, there is nothing in the
is inversely proportional to the radius of the theory that forbids γ from being also system de-
circular orbit. pendent, in which case it may provide the neces-
sary changes in the spacetime geometry to allow
For a gyroscope in the Earth’s orbit, such as theembeddingofasphericallysymmetricmatter
GP-B, the extra contribution to the geodetic distribution in a cosmologicalbackground.
precession from the linear and cosmological To conclude, this paper gives a derivation of
terms inthe metric areinsignificantforpractical Kepler’s third law and the gravitomagnetic ef-
purposes, when the value of γ is taken as the fects in the alternative gravitational theory of
reciprocal of the Hubble radius, as required for conformal gravity. On the solar system scale,
the fitting of galactic rotational curves shown where conformal gravity (like other alternative
in Ref. [1]. In earlier studies such as those in gravitational theories) agrees with general rela-
Refs.[11, 28] constraints were obtained for the tivity, the corrections to the Einstein precession
conformalgravityparameterγ. Howeverinboth areverysmalltobemeasuredbyanyspace-based
cases of geodetic and Lense-Thirring effects the experiment. However these may become signifi-
uncertainties in available GP-B data are too cant on much larger scales, especially when the
large to obtain reasonable constraints. So, for geodetic effect is considered.
example, using the uncertainty in the observed
value for the GP-B geodetic drift mentioned
in the introduction, and the Weyl’s gravity
correction to geodetic effect in (30), one gets ACKNOWLEDGMENTS
γ 1.5 10−20 cm−1, which is eight orders of
≤ ×
magnitude larger than the value obtained from
J. L. S. wishes to thank the Physics Depart-
galactic rotational curves.
ment at the University of Malta for hospitality
during the completion of this work.
It should be remarked, as we did in the intro-
duction, thatthe nature (andthereforethe mag-
nitude) of the constant γ is still unknown. So
whenβ =0orwhenr issufficientlylargesothat
all β dependent terms in (6) can be ignored, the [1] Mannheim, P. D., and Kazanas, D., Astrophys.
metric can be rewritten in the form J. 342, 635 (1989).
[2] Mannheim, P. D., and Kazanas, D., Astrophys.
ds2 = [1−ρ2(γ2/16+k/4)]2 [ dτ2 J. Suppl.76, 431 (1991).
R2(τ)[(1−γρ/4)2+kρ2/4]2 − [3] Riegert, R.J., Phys.Rev.Lett. 53, 315 (1984).
+ R2(τ) (dρ2+ρ2dΩ2)], [4] Mannheim, P. D., and Kazanas, D., Phys. Rev.
[1−ρ2(γ2/16+k/4)]2 2 D 44, 417 (1991).
(56) [5] Mannheim,P.D.,Gen.Relativ.Gravit.22,289
(1990).
where [6] Mannheim,P.D.,Astrophys.J.391,429(1992).
4r [7] Mannheim, P. D., and Kazanas, D., Gen. Rela-
ρ= , (57) tiv. Gravit. 26, 337 (1994).
2(1+γr kr2)1/2+2+γr
− [8] Said, J. L., Sultana, J., and Zarb Adami, K.,
Phys. Rev.D 85, 104054 (2012).
and
[9] Said, J. L., Sultana, J., and Zarb Adami, K.,
Phys. Rev.D 86, 104009 (2012).
τ = R(t)dt. (58) [10] Bekenstein, J. D., Phys. Rev. D 70, 083509
Z (2004).
This is conformally related to the FLRW metric [11] Mannheim,P.D.,andO’Brien,J.G.,Phys.Rev.
with arbitrary scale factor R(τ) and spatial cur- Lett. 106, 121101 (2011).
vatureκ= k γ2/4. Therefore,onemayinter- [12] Mannheim,P.D.,andO’Brien,J.G.,Phys.Rev.
− − D. 85, 124020 (2012).
pret the Mannheim-Kazanas metric as describ-
[13] Nandi, K. K. and Bhadra, A., Phys. Rev. Lett.
ingasphericallysymmetricobjectembeddedina
109, 079001 (2012).
conformallyflatbackgroundspace. Now the fact [14] deSitter,W.,Mon.Not.R.Astron.Soc.77,155
the curvature of this background space depends (1916).
on γ and k, points toward a cosmological origin [15] Thomas, L. H., Nature117, 514 (1926).
8
[16] Lense, J., and Thirring, H., Phys. Z. 19, 156 [28] Sultana, J., Kazanas, D. and Said, J. L., Phys.
(1918). Rev D 86, 084008 (2012).
[17] Ashby,N., and Shadid-Saless, B., Phys. Rev. D [29] Edery, A., Fabbri, L. and Paranjape, M. B.,
42, 1118 (1990). Class. QuantumGrav. 23, 6409 (2006).
[18] Schiff,L. I., Phys.Rev. Lett.4, 215 (1960). [30] Bouchami, J. and Paranjape, M. B., Phys.Rev.
[19] Bertotti, B., Ciufolini, I., and Bender, P. L., D 78, 044022 (2008).
Phys.Rev.Lett. 58, 1062 (1987). [31] Matsuo, N., Gen. Relativ. Gravit. 22, 561
[20] Shapiro, I. I., Reasenberg, R. D., Chandler, J. (1990).
F., and Babcock, R. W., Phys. Rev. Lett. 61, [32] Matsuno, K. and Ishihara, H., Phys. Rev D 80,
2643 (1988). 104037 (2009).
[21] Dickey,J., et. al., Science 265, 482 (1994). [33] Chen,S.andJing,J.,Class.QuantumGrav.27,
[22] Everitt,C.W.F.,Phys.Rev.Lett.106,221101 225006 (2010).
(2011). [34] N. Straumann, General Relativity: With Appli-
[23] Pireaux, S., Class. Quantum Grav. 21, 1897 cations to Astrophysics, (Springer),2004.
(2004). [35] W. Rindler, Relativity: Special, General, and
[24] Pireaux, S., Class. Quantum Grav. 21, 4317 Cosmological (Oxford University Press, second
(2004). edition 2006).
[25] Amore, P., Arceo, S., and Fernandez, F. M., [36] Tartaglia, A., Phys. Rev.D 58, 064009 (1998).
Phys.Rev.D 74, 083004 (2006). [37] Nandi, K. K., Alsing, P. M. Evans, J. C. and
[26] Sultana, J., and Kazanas, D., Phys. Rev.D 81, Nayak,T. B., Phys. Rev.D 63, 084027 (2001).
127502 (2010). [38] Ruggiero, M. L., Gen. Relativ. Gravit., 37, 11
[27] Edery, A., and Paranjape, M. B., Phys. Rev. D (2005)
58, 024011 (1998).