Model for gravity at large distances

0.13 MB·English

by  Daniel Grumiller

Model for gravity at large distances
Daniel Grumiller1
1
Institute for Theoretical Physics, Vienna University of Technology,
Wiedner Hauptstr. 8-10/136, A-1040 Vienna, Austria, Europe
(Dated: January 18, 2011)
Weconstructaneffectivemodelforgravityofacentralobjectatlargescales. Toleadingorderin
thelarge radiusexpansion wefindacosmological constant,aRindleracceleration, atermthat sets
thephysicalscalesandsubleadingterms. Allthesetermsareexpectedfromgeneralrelativity,except
for theRindler term. The latter leads to an anomalous acceleration in geodesics of test-particles.
1
PACSnumbers: 04.60.-m,95.35.+d,96.30.-t,98.52.-b,98.80.-k
1
0
2
Gravityatlargedistancesposessomeofthemostdiffi- EFFECTIVE FIELD THEORY FOR IR GRAVITY

n cult puzzles in contemporary gravitational physics. The
a
J cosmologicalconstantproblem[1]andthenatureofdark Toaddressthe issueofanomalousaccelerationoftest-
matter [2] are the most prominent ones. At a some- particles in the gravitational field of a central object we
7
1 what smaller scale, both in terms of actual size and in first simplify the underlying theory as much as possible.
terms of scientific credibility, there are fly-by anomalies We assume that spacetime is described by a spherically

] [3] and the Pioneer anomaly [4]. The word “anomalous” symmetric metric in four dimensions (see e.g. [8])
O
here refers to the difference between the observed tra-
C ds2 =g dxαdxβ +Φ2 dθ2+sin2θ dφ2 (1)
jectory of a test-particle in the gravitational field of a αβ
h. central object and the calculated trajectory. The pair The 2-dimensional metric gαβ((cid:0)xγ) and the surf(cid:1)ace ra-
p test-particle/centralobjectcanmeane.g.galaxy/cluster, dius Φ(xγ) depend only on the coordinates xγ = t,r .
- star/galaxy,Sun/Pioneerspacecraft,Earth/satellite,etc. Our task is now clear: we should describe the dy{nam}-
o
r Conceptually, there are three ways to resolve the ics of the fields gαβ and Φ. It is possible to do this in
t
s anomalies. Either we modify the matter content of the two dimensions, since the metric g and the scalar field
a Φ are both intrinsically 2-dimensional objects. Each
[ theory(darkmatter),orwemodifythegravitationalthe-
ory itself. The third alternative is that we might not be solution of the equations of motion (EOM) of that 2-
2 dimensionaltheoryleads toa 4-dimensionalline-element
applyingthetheorycorrectlyorbeyonditsrealmofvalid-
v (1),sotheformerfaithfullycapturestheclassicaldynam-
ity. Namely, even though effects that go beyond general
5
ics of the latter. The test-particles are then assumed to
2 relativity (GR) or its Newtonian limit are small locally,
moveinthebackgroundofthe4-dimensionalline-element
6 theymayaccumulateoverlargedistancesand/orthrough
3 averaging,see[5–7]forvariousapproachesthatadvocate (1). The process of “spherical reduction” [9] simplifies
. the 4-dimensional Einstein–Hilbert action to a specific
1 this idea andimplement it in differentways. In this Let-
1 ter we advocate a new approach to describe gravity at 2-dimensional dilaton gravity model, see e.g. [10]. The
0 novelty of our approach is that we allow for IR modifi-
large distances that is agnostic about the issue which of
1 cations of the dilaton gravity model. We explain now in
these three alternatives is realized in Nature. The main
:
v strengthofourmethodliesinitsrigidity—thereisonly detail how this works.
Xi one new free parameter — and in its ab initio nature. We construct the most general 2-dimensional theory
with the field content g and Φ compatible with the fol-
r One well-establishedkey ingredientto our approachis lowing assumptions. First of all, we require the theory
a
toconstructaneffectivefieldtheorybywritingdownthe tobepower-countingrenormalizable,assumingthatnon-
mostgeneralactionconsistentwithalltherequiredsym- renormalizabletermsaresuppressed. Thisleadsuniquely
metriesandwithadditionalassumptions(powercounting to the action [11, 12]
renormalizability,analyticity,...). Acrucialpointis that
1
we impose spherical symmetry in addition to diffeomor- S = d2x√ g f(Φ)R+2(∂Φ)2 2V(Φ) (2)
phisminvariance,whichoftenisagoodapproximationin −κ2 Z − h − i
the IR [27]. Spherical symmetry effectively reduces the We have exploited field redefinitions to bring the kinetic
theory to two dimensions. The main result of this Let- term for the dilaton field Φ into a convenient form. The
ter is that the most general action consistent with our gravitational coupling constant κ does not play any es-
assumptions leads to an additional term in the gravita- sential role in our discussions. We assume that the free
tionalpotentialascomparedtoGR.Thisnoveltermgen- functionsf,V areanalyticinΦinthelimitoflargeΦ,like
erates accelerations similar to the ones observed in vari- in spherically reduced GR [10]. An analysis of the EOM
ous“anomalous”systemsinNatureandhasabeautifully (seebelow)revealsthatthecouplingfunctionf thatmul-
simple geometric interpretation as Rindler acceleration. tiplies the Ricci scalar R must be given by f = Φ2 [28]
2
to reproduce the Newton potential M/r. If we consid- derived from the action (4). The result is [30]
2 −
eredinsteadf =cΦ thenthepotentialwouldchangeto
M/r1/c. Anexperimentalboundoncis c 1 <10−10 g dxαdxβ = K2dt2+ dr2 Φ=r (7)
−[13]. Changing the power of Φ in the fun|ct−ion|f would αβ − K2
be even more drastic, leading to exponential growth or 2 2M 2
K =1 Λr +2ar (8)
decay of the potential. In this workwe make the conser- − r −
vativeassumptionthatf =Φ2 isnotrenormalizedinthe
The line-element (7) exhibits a Killing vector ∂ with
t
IR, in excellent agreement with the experimental data.
norm K given by (8). The 2-dimensional Ricci scalar
We still have to choose the potential V in the action R= d2K2/dr2 =2Λ+4M/r3 only depends on Λ and
(2). Inthe4-dimensionallanguageweconsiderlargesur- on M−, but not on a. The quantity M is a constant of
faceareasaroundsomecentralobject. Aftersphericalre- motion. If a=Λ=0 we recover the Schwarzschild solu-
duction the limit of large surface areas leads to the limit tionwithM being theblackholemass. IfΛ=0wehave
of large dilaton field Φ. We assume that the potential V additionally a cosmological constant. If a=60 we obtain
behaves as follows: a contributionto the Killing normK that6does notarise
in vacuum Einstein gravity. The geometric meaning of
V(Φ)=Λ˜Φ2+a˜Φ+˜b+ (1/Φ) (3) this new term is clear [10]: it generates Rindler acceler-
O
ation. Indeed, for M = Λ = 0 the line-element (7) is
the 2-dimensional Rindler metric [8]. Thus, our effective
The Ansatz (3) for the potential V is pivotal for the dis-
fieldtheory (4)that describesgravityin the IRdiffers in
cussion, so let us pause andexplain the rationale behind
one aspect from GR: it allows for an arbitrary Rindler
it. That the leading term in the large Φ-expansion is
acceleration term. This is our second key result [31].
quadratic follows from physics: if we did allow for pow-
ershigher thanΦ2 thenthe ensuing metricwouldhavea
curvaturesingularityforlargeΦ. Thus,(3)isthegeneric
PHYSICAL CONSEQUENCES
asymptoticresultprovidedwerequiretheabsenceofcur-
vaturesingularitiesandanalyticityinΦ inthe IR,which
WetakenowthepresenceoftheRindlertermin(8)for
is what we do. The term (1/Φ) leads to a contribu-
O granted and discuss some of its physical consequences.
tionto the gravitationalpotentialproportionalto lnr/r,
Order of magnitude of the parameters Thecosmolog-
whose effects are subleading in the deep IR as compared
ical constant is given by Λ 10−123 [14, 15], and this is
toeffectscomingfromthefirstthreetermsin(3). There- ≈
thevalueweshallbeusing,thoughinseveralapplications
fore,we ignorethis term(andfurther subleadingterms).
we set Λ=0 for simplicity. Note that we are employing
[29]
natural units c=~=G =1. The Rindler acceleration
By rescaling simultaneously Φ and κ we can fix one N
a in principle may depend on the scales of the system
ofthe subleading coefficientsin the asymptoticpotential
under consideration. We motivate now an order of mag-
(3), which amounts to fixing a specific physical length nitude estimate for a in the deep IR. The product Λr2
scale. Without loss of generality we choose˜b b= 2.
→ − that enters the Killing norm (8) becomes order of unity
Droppingallasymptoticallysubleadingtermsandchoos-
if r is of the order of the radius of the visible Universe.
ing convenient normalizations of the coupling constants
If we suppose that the same happens for the 2ar-term
as well as κ=1 the action (2) simplifies to then we obtain the estimate a 10−62 10−61, which
≈ −
is approximately the scale were anomalous accelerations
S = d2x√ g Φ2R+2(∂Φ)2 6ΛΦ2+8aΦ+2 (4) areobserved: theMONDaccelerationisabout10−62[16]
−Z − h − i and the Pioneer acceleration is about 10−61 [4].
Geodesics of test-particles Westudynowgeodesicsof
The action (4) is our first key result. It provides the timelike test-particles (of unit mass) propagating on the
generic effective theory of gravity in the IR consistent 4-dimensionalbackgrounddetermined by the spherically
with the assumptions spelled out above. The theory de- symmetric line element (1) with (7)-(8). Consider for
fined by the action (4) depends on two coupling con- simplicity motion in the plane θ = π/2. Using standard
stants, Λ and a. Solutions of the EOM descending from methods [8] we obtain the well-known result φ˙ = ℓ/r2,
the action (4) describe spherically symmetric line ele- where ℓ is the conservedangular momentum, and
ments (1) that model gravity in the IR.
r˙2
It is straightforwardto find all solutions to the EOM +Veff =E (9)
2
2 4a with E =const. The effective potential reads
R= gαβ ∂ Φ+6Λ (5)
α β
Φ ∇ − Φ
2Φ ∇µ∂ν −gµν∇α∂α Φ−gµν ∂Φ 2 =gµνV(Φ) (6) Veff =−Mr + 2ℓr22 − Mr3ℓ2 − Λ2r2 +ar 1+ rℓ22 (10)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
3
v v
0.0005
0.001
0.0004
0.0008
0.0003
0.0006
0.0002
0.0004
0.0001
0.0002
r r
5·1054 1·1055 1.5·1055 2·1055 2.5·1055 1·1055 2·1055 3·1055 4·1055 5·1055
FIG. 1: Toy model rotation curvefor dwarf galaxy FIG. 2: Toy model rotation curvefor large spiral galaxy
The first term in Veff is the Newton potential, the sec- (spiral)galaxies,see e.g.[18]. The scale wherethe veloc-
ond the centrifugal barrier, the third the GR correction, ityprofileflattensisv 10−3,whichroughlycorresponds
the fourth a cosmic acceleration term and the last term to 300km/s, in reason≈able agreement with the data.
proportional to the Rindler acceleration a is novel. For
It is worthwhile mentioning that the velocity profile
vanishing cosmological constant and vanishing angular
(12) was studied on purely phenomenological grounds,
momentum the force Feff derived from the effective po-
see Eq. (125) in [19] in section 6.3 (“Clues from data”).
tential contains only two terms.
It is gratifying that the effective field theory (4) predicts
a velocity profile (12) that was arguedto be a goodphe-
∂Veff M
Feff = = a (11) nomenological fit to the data [32]. We note, however,
− ∂r (cid:12)ℓ=Λ=0 −r2 − that so far there is no empirical hint that generically ro-
(cid:12)
(cid:12) tation curves of galaxies rise linearly at large distances,
Thus, for very small Newton potentials the Rindler ac-
seee.g.[20],soonehastotakethisroughmodelofgalax-
celerationtermbecomesrelevantandprovidesaconstant
ies with a few grains of salt. Of course, allowing for an
accelerationtowards the source (if a is positive) or away
r-dependent function a in (12) one can fit any given ro-
from the source (if a is negative). This effect becomes
tationcurve. Itcouldbe arewardingexercisetoperform
important at sufficiently large distances only.
such fits to the data, thereby determining the Rindler
Galactic rotation curves In the Newtonian limit the
acceleration a as a function of r.
velocity profile is determined from the mass by v(r)
M(r)/r, where we assume some radial mass profi≈le Solar system precision tests and Pioneer anomaly
Mp(r) and set to zero the cosmological constant and the Physics in the solar system appears to be well-
Rindler acceleration. If we take into account also the understood. Nevertheless, there are a few tentative
Rindler accelerationwe obtain a new formula for the ra- anomalieslikefly-byanomaliesandthePioneeranomaly,
dial velocity profile in a galaxy: for a review cf. [21]. The force law (11) leads to an
anomalous constant acceleration towards the Sun (if a
M(r) is positive). Interestingly, this effect has been observed
v(r)≈r r +ar (12) by the Pioneer spacecraft [4]. Taking the data at face
value requires to fix a 10−61, which is not that differ-
≈
Let us now construct a rough model of galaxies, start- entfromthevalueofausedinthetoymodelforgalaxies
ing with dwarf galaxies. We assume that the density is above. EventhoughthereisstillroomfordoubtifthePi-
constant in the center up to r 1054 (less than a kilo- oneer anomaly is a genuine effect [22], it is striking that
≈
parsec) and drops to zero thereafter. For a total mass of our effective field theory (4) is capable to account for
8
M 10 solar masses we obtain from (12) the velocity it. It is notoriously difficult to reconcile modifications
≈
profiledepictedinFig.1. ThedashedlinegivestheKep- of gravitational theories that explain, for instance, the
lerianvelocityprofileandthesolidline givesthevelocity Pioneer anomaly without spoiling solar system precision
profile deduced from (12) with a 10−62. Strikingly, tests [23]. Our Rindler acceleration term does not nec-
≈
Fig. 1 resembles the rotation curves for dwarf galaxies, essarily spoil the solar-system precision tests, since the
see e.g. [17]. To describe large galaxies we assume that value of a depends on the system that we describe. For
thedensityisconstantinthecenteruptor 1055(about instance, in the system Galaxy/Sun a 10−62, in the
≈ ≈
3 kiloparsecs) and drops to zero thereafter. For a total systemSun/Mercury a may be muchsmaller,in the sys-
11 −61
mass of M 10 solar masses we obtain from (12) the temSun/Pioneerspacecrafta 10 ,andinthesystem
≈ ≈
velocity profile depicted in Fig. 2. The dashed line gives Earth/satelliteamaybemuchbigger. Itisachallenging
the Keplerianvelocity profile and the solid line gives the open issue to understand precisely what determines the
velocityprofilededucedfrom(12),againwitha 10−62. scale of a. It is unlikely that such an understanding can
≈
Strikingly, Fig. 2 resembles the rotation curves for large be achieved within our 2-dimensional effective model.
4
4-dimensional interpretation The viewpoint takenso and therefore naturally depends on the specific system
far was that the effective theory for gravity in the IR is under consideration.
a2-dimensionalscalar-tensortheory(4),whosesolutions
I thank Alan Guth, Roman Jackiw, Wolfgang Kum-
of the (vacuum) EOM lead to 4-dimensional spherically
mer, Philip Mannheim, Stacy McGaugh, Florian Preis,
symmetric metrics (1) with (7)-(8). Following this phi-
DimitriVassilevichandRichardWoodardfordiscussions.
losophywederivedabovesomeremarkableconsequences.
I am grateful to Herbert Balasin and Tonguc Rador for
Oneissue,however,isnotentirelysatisfactory: wehadto
independently pointing outanincorrectsigninEq.(13).
take a detour via spherical reduction to two dimensions
DG is supported by the START project Y435-N16 of
to establishour results. Itwouldbe interesting to derive
the Austrian Science Foundation (FWF).
— or at least to interpret — the same results directly
in four dimensions. Such a top-downderivation is acces-
sible only by reintroducing model assumptions that lift
the agnosticism of the current approach. In the present
work we confine ourselves to possible 4-dimensional in- [1] S. M. Carroll, Living Rev. Rel. 4, 1 (2001), astro-
terpretations of the 2-dimensional Rindler acceleration. ph/0004075.
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equations are obeyed in four dimensions by metrics of (2005), hep-ph/0404175.
[3] J. D. Anderson, J. K. Campbell, J. E. Ekelund, J. Ellis,
type(1)with(7)-(8). Obviously,anymetriccanformally
and J. F. Jordan, Phys.Rev.Lett. 100, 091102 (2008).
bewrittenasasolutiontoEinstein’sequationswithsome
[4] J. D. Anderson et al., Phys. Rev. Lett. 81, 2858 (1998),
suitable energy momentum tensor.
gr-qc/9808081.
[5] T.BuchertandM.Carfora, Phys.Rev.Lett.90,031101
1
Rµ δµR+3δµΛ=8πTµ (13) (2003), gr-qc/0210045.
ν − 2 ν ν ν [6] M. Reuter and H. Weyer, JCAP 0412, 001 (2004), hep-
th/0410119.
In our case this effective energy momentum tensor turns
[7] J. F. Donoghue, PoS EFT09, 001 (2009), 0909.0021.
out to be like the one of an anisotropic fluid
[8] R. M. Wald, General Relativity (The University of
Chicago Press, 1984).
Tνµ =diag(−ρ, pr,p⊥,p⊥)µν (14) [9] B. K. Berger, D. M. Chitre, V. E. Moncrief, and
Y. Nutku,Phys. Rev.D5, 2467 (1972).
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a 1
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[12] S.D.OdintsovandI.L.Shapiro,Phys.Lett.B263, 183
Note that our effective fluid does not behave at alllike a
(1991).
cold dark matter fluid modelled by dust (pr = p⊥ = 0). [13] C. Talmadge, J. P. Berthias, R. W. Hellings, and E. M.
We stress that there is only one free parameter in our Standish, Phys.Rev. Lett.61, 1159 (1988).
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[15] S.Perlmutteretal.(SupernovaCosmology Project), As-
a is negative. We found above that this is not the sign
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neededtomodelthegalacticrotationcurvesandthePio-
[16] M. Milgrom, Astrophys.J. 270, 365 (1983).
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[17] G. Rhee, O. Valenzuela, A. Klypin, J. Holtzman, and
predictanunusualequationofstate(15)fordarkmatter B. Moorthy, Astrophys. J. 617, 1059 (2004), astro-
that could be tested by combining gravitational lensing ph/0311020.
and rotation curve data, as explained in Ref. [24]. This [18] Y. Sofue and V. Rubin, Ann. Rev. Astron. Astrophys.
could provide the simplest route to falsify our model. 39, 137 (2001), astro-ph/0010594.
[19] P.D.Mannheim,Prog.Part.Nucl.Phys.56,340(2006),
Outlook Whether the anisotropic fluid (14), (15) is
astro-ph/0505266.
“real”inanysenseorjustaneffectivewaytoincorporate
[20] W. J. G. de Blok and S. S. McGaugh, Mon. Not. Roy.
IReffects (that either accumulatewithin GRorthat fol-
Astron. Soc. 290, 533 (1997), astro-ph/9704274.
lowfromsomemodificationofGR)isofconsiderablethe- [21] C. Lammerzahl, O. Preuss, and H. Dittus (2006), gr-
oreticalinterest,butgoesbeyondthescopeofthecurrent qc/0604052.
Letter. Still,wementionanintriguingfactoid: Themost [22] H. Dittus et al. (PIONEER), ESA Spec. Publ. 588, 3
generalsphericallysymmetricsolutiontogαβ ∂ R=0 (2005), gr-qc/0506139.
α β
ortogαβ R =0ortotheEOMarisin∇ginconfor- [23] K. Tangen, Phys. Rev. D76, 042005 (2007), gr-
∇α∇β µν qc/0602089.
malWeylgravity[25]isgivenbyoursolution(1),(7)-(8)
[24] T. Faber and M. Visser, Mon. Not. Roy. Astron. Soc.
[in the first case supplemented by a Reissner–Nordstro¨m
372, 136 (2006), astro-ph/0512213.
termQ2/r2intheKillingnorm(8)]. Inallthreecasesthe [25] P.D.MannheimandD.Kazanas,Astrophys.J.342,635
Rindler acceleration a emerges as a constant of motion (1989).
5
[26] D. Cangemi and R. Jackiw, Phys. Rev. Lett. 69, 233 isolateddegeneratesolutionsthatwedonotdiscusshere.
(1992), hep-th/9203056. [31] There isasimpleway toeliminate theRindlertermand
[27] We use the particle physicists’ jargon and refer to the recover GR with a cosmological constant Λ, namely to
large distance limit as ‘infrared’, abbreviated by IR. require the action (4) to be symmetric under Φ → −Φ.
[28] Actually,evenf =Φ2+AΦ+B+... isallowed.However, However, there is no a priori reason for this symmetry.
theAΦterm can beeliminated bya constant shift of Φ, [32] There is a slight difference to the phenomenological fit
the B term does not contribute to the EOM and the for the velocity profile studied in [19]: while ourRindler
subleading terms can beneglected in theIR. acceleration a is a prescribed constant, in Mannheim’s
[29] The lnr/r-term in the gravitational potential has a fall- work it consists of two terms, one of which is also a pre-
off behaviorcomparable to the Newton term, but is log- scribed constant,whiletheotherscaleswith thenumber
arithmically larger. Thus, for comparison with precision of stars in the galaxy. The latter contribution does not
data it may be worthwhile to take it into account, even arise in our scenario since we have assumed for simplic-
thoughitissuppressedbyafactorr2 ascomparedtothe ity that a takes the same value in all galaxies. We can
Rindler term. The next subleading term in the gravita- drop this assumption and recover precisely Mannheim’s
tionalpotentialisaReissner–Nordstro¨mterm,whichcan fit if we assume that a depends on the scales of the sys-
safely be neglected in the IR. Further subleading terms tem under consideration, a=a0+a1N, where N is the
correspond not to IR but rather toUV modifications. numberofstarsinthegalaxy.Fordwarfgalaxiesthecon-
[30] This calculation is done most conveniently in the gauge tribution coming from a1 does not play any role, while
theoretic formulation analog to [26], see [10] for details. for large galaxies it plays only a modest role.
In addition to the generic solutions (7)-(8) there can be