A non-geodesic motion in the R^-1 theory of gravity tuned with observations

0.14 MB·English

by  Christian Corda

R 1
−
A non-geodesi
motion in the theory
8 of gravity tuned with observations
0
0
2 Christian Corda

n
a February 2, 2008
J

1
1
INFN - Sezione di Pisa and Università di Pisa, LargoPonte
orvo 3, I - 56127

PISA, Italy; Instituto de Cosmologia,Relatividade e Astrofìsi
a (ICRA-BR),
]
h Centro Brasilero de Pesquisas Fisi
as, Rua Dr. Xavier Sigaud 150, CEP 22290
p -180 Ur
a Rio de Janeiro - RJ Brazil
-
o
r
t
s
a E-mail address:
hristian.
ordaego-gw.it
[

3
v
9 Abstra
t
1
3 R−1In the general pi
ture of high order theories of gravity, re
ently, the
0 theory has been analyzed in two di(cid:27)erent frameworks. Inthis letter
. aR−th1ird
ontext is added,
onsidering an expli
it
oupling between the
1
fun
tion of the Ri

i s
alar and the matter Lagrangian. The result
0
is a non-geodesi
motion of test parti
les whi
h, in prin
iple,
ould be
8

onne
ted with Dark Matter and Pioneer anomaly problems.
0
:
v PACS numbers: 04.50.+h, 04.20.Jb.
i
X
The a

elerated expansion of the Universe, whi
h is today observed, shows
r
a that
osmologi
al dynami
is dominated by the so
alled Dark Energy whi
h
givesalargenegativepressure. Thisisthe standardpi
ture, in whi
hsu
hnew
ingredientis
onsideredasasour
eoftheright hand side ofthe(cid:28)eldequations.
Itshouldbesomeformofun-
lusterednon-zerova
uumenergywhi
h,together
with the
lustered Dark Matter, drives the global dynami
s. This is the so

alled(cid:16)
on
ordan
emodel(cid:17) (ACDM)whi
hgives,inagreementwiththeCosmi
Mi
rowave Ba
kground Radiation (CMBR), Dim Lyman Limit Systems (LLS)
and type la supernovae (SNeIa) data, a good trapestry of the today observed
Universe,butpresentsseveralshort
omingsasthewellknown(cid:16)
oin
iden
e(cid:17) and
(cid:16)
osmologi
al
onstant(cid:17) problems [1℄. An alternative approa
h is
hanging the
left hand side of the (cid:28)eld equations, seeing if observed
osmi
dynami
s
an be
a
hieved extending general relativity [2, 3, 4, 5, 6℄. In this di(cid:27)erent
ontext, it
1
is not required to resear
h
andidates for Dark Energy and Dark Matter, that,
till now, have not been found, but only the (cid:16)observed(cid:17) ingredients, whi
h are

urvatureand baryoni
matter, havetobetakenintoa

ount. Consideringthis
point of view, one
an think that gravity is not s
ale-invariant [7℄ and a room
for alternative theories is present [8, 9, 10, 26, 27, 29℄. In prin
iple, the most
f(R)
popularDarkEnergyandDarkMattermodels
anbea
hieved
onsidering
R
theories of gravity [7, 11℄, where is the Ri

i
urvature s
alar.
In this pi
ture even the sensitive dete
tors for gravitational waves, like bars
and interferometers (i.e. those whi
h are
urrently in operation and the ones
whi
h are in a phase of planning and proposal stages) [12, 13, 31, 33℄,
ould, in
prin
iple,beimportantto
on(cid:28)rmorruleoutthephysi
al
onsisten
yofgeneral
relativity or of any other theory of gravitation. This is be
ause, in the
ontext
of Extended Theories of Gravity, some di(cid:27)eren
es between General Relativity
and the others theories
an be pointed out starting by the linearized theory of
gravity [14, 15, 16, 17, 28, 32℄. R−1
In the general pi
ture of high order theories of gravity, re
ently, the
theory has been analyzed in two di(cid:27)erent frameworks [15, 18℄. In this leRtt−e1r
a third
ontext is added,
onsidering an expli
it
oupling between the
fun
tion of the Ri

i s
alar and the matter Lagrangian. The result is a non-
geodesi
motion of test parti
les whi
h, in prin
iple,
ould be
onne
ted with
Dark Matter and Pioneer anomaly probRle−m1s.
In [15℄ the high order a
tion of the theory of gravity
S = d4x√ gR−1+ ,
Z − Lm (1)
has been analyzed in a
ontext of produ
tion and potential dete
tion of
gravitational waves, while in [18℄ a
osmologi
al appli
ation of su
h a
tion has
been performed.
Equation (1) is aparti
ular
hoi
ewith respe
t to the well known
anoni
al
one of general relativity (the Einstein - Hilbert a
tion [19, 20℄) whi
h is
S = d4x√ gR+ .
Z − Lm (2)
R−1
Now let us
onsider a third a
tion in
luding a
oupling between the
fun
tion of the Ri

i s
alar and the matter Lagrangian:
S = d4x√ g(R−1+R−1 + ).
Z − Lm Lm (3)
Cleary, this feature implies a breakdownof Einstein's equivalen
e prin
iple,
i.e. a non geodesi
motion of test parti
les [21℄.
If the gravitational Lagrangian is nonlinear in the
urvature invariants, the
Einstein(cid:28)eldequationsareanorderhigherthanse
ond[5,6,8,14,15℄. Forthis
reasonsu
htheoriesareoften
alled higher-ordergravitationaltheories. Thisis
exa
tly the
ase of the a
tion (3).
2
g
µν
If one varies this a
tion with respe
t to (see also refs. [8, 14, 15℄ for a
parallel
ompuGta=tio1n)ct=he1(cid:28)eld e~qu=at1ions are obtained (note that in this paper
we work with , and ):
R−2R R−1g + R−2 g (cid:3)R−2 =
− µν − 2 µν ▽µ▽ν − µν
(4)
=2R−2 R 2( g (cid:3))R−2 +(1+R−1)T(m),
m µν µ ν µν m µν
L − ▽ ▽ − L
(cid:3) T(m)
µν
where is the d' Alembertian operator and is the ordinary stress-
energy tensor of the matter.
Followingthe analysisin [22℄, let us
ompute the
ovariantderivativeof the
(cid:28)eld equations (4) , together with the Bian
hi identity [19℄
µG =0
µν
▽ (5)
and with the identity
((cid:3) (cid:3))R−2 =R µR−2,
ν ν µν
▽ −▽ ▽ (6)
obtaining R−2
µT(m) = − (g T(m)) µR.
▽ µν R−1+1 µνLm− µν ▽ (7)
From the last equation it seems that energy
onservation breaks down [21℄.
This apparentshort
oming
an be avoidedin the followingway. Writing down,
expli
itly, the Einstein tensor in equation (4) one gets
G = R2 R−2+R2g (cid:3)R−2+Rg
µν µ ν µν µν
− ▽ ▽
+2 R 2R2( g (cid:3))R−2 +(R2+R)T(m). (8)
m µν µ ν µν m µν
L − ▽ ▽ − L
Then, one
an introdu
e a (cid:16)total(cid:17) stress-energy tensor
T(tot) 1 R2 R−2+R2g (cid:3)R−2+Rg
µν ≡−8π{ ▽µ▽ν µν µν
(9)
+2 R 2R2( g (cid:3))R−2 +(R2+R)T(m) .
m µν µ ν µν m µν
L − ▽ ▽ − L }
In this way the (cid:28)eld equations assume the well known Einsteinian form
G =8πT(tot).
µν µν (10)
In the (cid:16)total(cid:17) stress-energytensor (9) a
urvature
ontribution is addedand
mixedtothematerial one. Thisisbe
ausethehighorderterms
ontribute,like
sour
es,tothe(cid:28)eldequationsand
anbe
onsideredlikee(cid:27)e
tive (cid:28)elds (seeref.
[25℄ for details).
Thus, using equation (10), equation (5) gives
µT(tot),=0
▽ µν (11)
3
i.e. the
onservationofthetotalenergy(material plus
urvature plusmixed
terms). Moreover,using equation(9), equation(11) givesdire
tlyequation(7).
With the goal of testing the motion of test parti
les in the model, one
an
introdu
e the well known stress-energy tensor of a perfe
t (cid:29)uid [19, 20, 22℄
T(m) (ǫ+p)u u pg .
µν ≡ µ ν − µν (12)
Be
ausetwoastrophysi
alexampleswillbe
onsidered(the(cid:28)rstinthegalaxy
andthese
ondintheSolarSystem),thissimplestversionofastress-energyten-
sorforthematter,whi
h
on
erns(cid:16)dust(cid:17) (i.e. starsatthegala
ti
s
ale,planets
andspa
e
raftsattheSolarSystems
ale),
anbeusedinagoodapproximation
ǫ p
[19, 20, 25℄. In equation (12) is the proper energy density, the pressure and
u
µ
the fourth-velo
ity of the parti
les.
Now, following [22℄, one de(cid:28)nes the proje
tor operator
P g u u ,
µα µα µ α
≡ − (13)
gαβP
µβ
the
ontra
tion
an be applied to equation (7), obtaining
d2xα dxµ dxν
+Γ˜α =Fα.
ds2 µν ds ds (14)
The presen
e of the extra for
e
R−2
Fα (ǫ+p)−1Pαν[( − )( +p) R+ p]
≡ R−1+1 Lm ▽ν ▽ν (15)
shows that the motion of test parti
les is non-geodesi
. It is also simple to
see that
dx
Fα α =0,
ds (16)
i.e. the extra for
e is orthogonalto the four-velo
ity of test masses.
TakingtheNewtonianlimitinthreedimensionsofequation(14)oneobtains
a = a + a
−→tot −→n −→ng
(17)
a
−→tot
where the total a

eleration is given by the ordinary Newtonian a
-
a a
−→n −→ng

eleration plus the a

eleration whi
h is due to the extra for
e (non-
geodesi
).
Usingequation(17)and abit ofthree-dimensionalgeometrytheNewtonian
a
−→n
a

eleration
an be written as
1 a
a = (a2 a2 a2 ) −→tot .
−→n 2 tot− n− ng angatot (18)
a a a
−→ng n tot
In the limit in whi
h dominates (i.e. ≪ ) it is
(19)
4
a a a2
tot−→tot ng
a (1 ).
n ≃ 2ang − a2tot (20)
De(cid:28)ning [22, 23, 24, 30℄
1 a2
a−1 (1 ng),
e ≡ 2ang − a2tot (21)
equation (20) be
omes a
tot
a a .
−→n −→tot
≃ ae (22)
From equation (22) one gets
1
atot (aean)2.
≃ (23)
Be
ause the standard Newtonian a

elerationis
M
a = u ,
−→n r2 r (24)
b
the total a

eleration results
a = (aeM)21u = vr2u ,
−→tot r r
r r (25)
b b
where 1
vr =(aeM)4
(26)
istherotationvelo
ityofatestmassunderthein(cid:29)uen
eofthenon-geodesi
for
e.
The extra for
e has environmental nature, thus only phenomenology
an
help us in its identi(cid:28)
ation. a a 10−10m/s
e 0
In a gala
ti

ontext it is natural to identify with ≃ , whi
h
is the a

elerationof Milgrom used in the theoreti
al
ontext of Modi(cid:28)ed New-
tonian Dynami
s to a
hieve Dark Matter into galaxies [22, 24℄.
From another point of view, in the Solar System, if the anomaly in Pioneer
a

eleration is not generated by systemati
e(cid:27)e
ts, but a real e(cid:27)e
t is present
[22, 24℄, one
an in prin
iple put
a =a 8.5 10−10m/s2.
e Pi
≃ × (27)
Thus, the introdu
ed approa
h allows for a uni(cid:28)ed explanation of the two
e(cid:27)e
ts.
Con
lusions
R−1
In this letter the theory of gravity has been analyzed in a third
ontext
whi
h has been added to two di(cid:27)erent frameworks re
ently seen in the genRer−a1l
pi
ture of high order theories of gravity. An expli
it
oupling between the
fun
tion of the Ri

i s
alar and the matter Lagrangian has been performed.
The result is a non-geodesi
motion of test parti
les whi
h, in prin
iple,
ould
be
onne
ted with Dark Matter and Pioneer anomaly problems.
5
A
knowledgements
I would like to thank Salvatore Capozziello, Mauro Fran
aviglia and Maria
Feli
ia De Laurentis for useful dis
ussions on the topi
s of this paper. I thank
the referee for pre
ious advi
es and
omments that allowed to improve this
letter, for the physi
al signi(cid:28)
an
e and for the style too. The EGO
onsortium
has also to be thanked for the use of
omputing fa
ilities. This resear
h has
been partially supported by the ICRA-Brazil.
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7