Riemann Tensor of the Ambient Universe, the Dilaton and the Newton's Constant

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by  Rossen I. Ivanov

Riemann Tensor of the Ambient Universe, the Dilaton and the Newton’s Constant
Rossen I. Ivanov
Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences,
72 Tsarigradsko chaussee, Sofia – 1784, Bulgaria and
School of Electronic Engineering, Dublin City University, Ireland∗
Emil M. Prodanov
School of Physical Sciences, Dublin City University, Ireland†
We investigate a four-dimensional world, embedded into a five-dimensional spacetime, and find
thefive-dimensionalRiemanntensorviageneralisation oftheGauss(–Codacci)equations. Wethen
derive the generalised equations of the four-dimensional world and also show that the square of
the dilaton field is equal to the Newton’s constant. We find plausable constant and non-constant
solutions for thedilaton.
4
0 PACSnumbers: 04.50.+h,04.20.Cv,03.50.De,98.80.-k
0
2
Since the pioneering works of Kaluza [1] and Klein associated with the four-dimensional spacetime M. Let
n
[2], who unified gravitation with electromagnetism, the ψ(xµ) = s = const be the equation of the hypersuface
a
J implications of possible extra dimensions to our world M. One can alternatively express the parametric equa-
5 have been under intense investigation — see [3] for an tions of M as xµ =xµ(yj,s) and treating the parameter
extensive collectionofpapers onhigher-dimensionaluni- sasacoordinate,thisthenrepresentsacoordinatetrans-

1 fication. Jordan [4] and Thiry [5] used the equations formation with inverse:
v
of Kaluza-Klein’s theory to show that the gravitational
7 constant can be expressed as a dynamical field. A con- yj =yj(xµ), s=ψ(xµ). (1)
1
0 stant solutionfor the Newton’s constanthoweveris then
We assume that this transformationis invertible at each
1 allowed only if the square of the Maxwell electromag-
0 netic tensor vanishes. Here we examine a dual set up point. This means that the Jacobimatrices ofthe trans-
4 formation and its inverse have non-vanishing determi-
in which this problem can be avoided. Based on the
0 nants everywhere. Thus, to globally parametrise the fo-
original Kaluza’s model, we are here considering a gen-
/
h eral embedding of a four-dimensional world into a five- liatedfive-dimensionalspacetimeV,itissufficienttouse
t the coordinatesofthe four-dimensionalworldM andthe
- dimensional ambient spacetime. We derive a generalisa-
p foliation parameter s.
tion of Gauss (–Codacci) equations by utilising all de-
e The vector normal to the surface is:
grees of freedom (the entire geometry) of the ambient
h
: spacetime and we also show how they affect the physics ∂ψ
iv of the four-dimensional world. As a result we find a sys- Nµ = ∂xµ . (2)
X tem of equations for the electromagnetic field, gravita-
r tional field and dilaton field. One of these equations is Let us also define:
a
a plausable generalisation of gauge fixed Maxwell equa-
∂xµ ∂xµ ∂yi
tions in the presence of a dilaton field. We also show eµ = , nµ = , Ei = . (3)
j ∂yj ∂s µ ∂xµ
that the square of the dilaton field is equal to (modulo
numerical factors) the Newton’s constant. The gauge
The derivatives are related as follows:
freedom of the electromagnetic fields is transferred to a
freedom in fixing the dilaton field. Apart from the con- ∂ ≡ ∂ = eµ∂ , (4)
stant solution for the dilaton, we give an example for a k ∂yk k µ
non-constant solution describing time-varying Newton’s ∂
∂ ≡ = Ek∂ +N ∂ . (5)
constant in an expanding universe (see also [6]–[11] and ν ∂xν ν k ν s
others). We also give a general formula for generating
different solutions for the dilaton field. Obviously,ifwedenoteeµ5 =nµ andEµ5 =Nµ,then(eµν)
and (Eµ) will be the Jacobi matrices of the transforma-
We consider a four-dimensional world M, embedded ν
tion (xµ)→(yi,s) and its inverse. Therefore:
into a five-dimensional spacetime V (see [12]–[17] and
the references therein for a detailed discussion on em- eµEσ =δµ =Eµeσ. (6)
beddings). Let yi (i = 1,2,3,4) denote the coordi- σ ν ν σ ν
nates on M and xµ (µ = 1,2,3,4,5) denote the coor- This orthogonality condition is equivalent to:
dinates on V. Greek indexes will be related to the five-
dimensional spacetime V, while Latin indexes will be eµEj = δj, (7)
i µ i
2
N nµ = 1, (8) where Q =∇ N =∇ N .
µ µν µ ν ν µ
Eµieνi +Nµnν = δµν, (9) Multiplying (18) across by EαjEβl and applying the or-
thogonality conditions (7)–(11), one easily finds:
N eµ = 0, (10)
µ j
nµEµj = 0. (11) Qαβ =EαiEβjKij +(NαEβi +NβEαi)fi+NαNβχ,(19)
with χ=nµnνQ and
(Note that δµ =dim V=5 and δi =dim M=4.) µν
µ i
Thus the bases (eµj,nµ) and (Eµj,Nµ) are dual. They do f =nµeβQ =AiK + 1 ∂ N, (20)
not depend on the metric of either spacetime, but only j j µβ ij N j
depend on the particular embedding chosen.
where N = N Nµ.
Letusnowintroduceascalarfieldφ(yk,s),avectorfield p µ
The four-dimensional Christoffel symbols
A (yk,s) and the metric tensor g (yk,s) on M. We fur-
i ij
ther define the metric Gµν of the five-dimensionalspace- γi = 1gil(∂ g +∂ g −∂ g ) (21)
time V as an expansion over the basis vectors Ei and jk 2 k lj j lk l jk
µ
N :
µ can then be expressed as:
G =EiEjg +(N Ei +N Ei)A +N N φ. (12) γi =(∂ eµ+eαeβΓµ )Ei −K Ai, (22)
µν µ ν ij µ ν ν µ i µ ν jl j l j l αβ µ jl
Taking xi = yi, x5 = s = const, i.e. eµi = δiµ,nµ = where Γµαβ are the five-dimensional Christoffel symbols.
δµ,Ei = δi ,N = δ5 in (12) corresponds to the orig- Further, the four-dimensional Riemann curvature tensor
5 µ µ µ µ
inal Kaluza’s model [1]. Klein’s modification [2] g →
ij ri =∂ γi −∂ γi +γmγi −γmγi , (23)
g + A A , together with the identification of φ as a jkl k jl l jk jl mk jk ml
ij i j
dilatonisthemodelputforwardbyJordan[4]andThiry using (22) and (7)–(11), becomes:
[5]. We note that the metric (12) has the same form as
ri = EieλeµeνRα +Ei(∇ nα)(eµK −eµK )
the inverse of the metric of Klein’s model, thus the two jkl α j k l λµν α µ k lj l kj
theories aredual: Kaluza’smodel correspondsto slicing, +∇(AiK )−∇ (AiK )
l kj k lj
while Klein’s model correspondsto threadingof the five- +AiAm(K K −K K ), (24)
ml kj mk lj
dimensional spacetime [15]. The case with A = 0 has
i
also been considered (see, for example, [17], [18] and the where Rα is the five-dimensional Riemann curvature
λµν
references therein). tensor. The above is a generalisation of the Gauss equa-
The lack of gauge invariance for the fields A , which we tions. Taking Ai =0 and φ=1, one simply recovers the
i
nevertheless will associate with the electromagnetic po- well known Gauss equations (see, for example, [19]):
tentials, in view of the slicing–threading duality, is com-
ri = EieλeµeνRα +Ki K −KiK . (25)
pensated by the freedom to fix φ. This, as will become jkl α j k l λµν k lj l kj
claer later is the freedom to fix the dilaton field.
Let us now expand the five-dimensional Riemann curva-
Returningto(12),wedefinegij astheinverseofthemet-
ture tensor over our basis. Using its symmetries we can
ric g . Thus Ai =gijA and A2 =gijA A .
ij j i j write:
TheinverseGµν ofthemetricG onVisthengivenby:
µν
R = EiEjEkElU +
λµνσ λ µ ν σ ijkl
Gµν = hijeµeν −θAi(eµnν +eνnµ)+θnµnν, (13)
i j i i + (N Ej −N Ej)EkEl
h λ µ µ λ ν σ
where θ = (φ − A2)−1 and hij = gij + θAiAj. One
+(N Ej −N Ej)EkEl V
can easily check that GµλG = δµ. Using the inverse ν σ σ ν λ µi jkl
λν ν
Gµν (13) of the metric Gµν, we can raise and lower five- +(NλEµj −NµEλj)(NνEσl −NσEνl)Wjl, (26)
dimensional indexes to get:
where the coefficients in this expansion satisfy:
Nµ = GµνN = θ(nµ−Aieµ), (14)
ν i U = U = −U =−U , (27)
N2 = N Nµ = θ, (15) ijkl klij ijlk jikl
µ V = −V , W =W . (28)
n = G nν = A Ei +N φ, (16) jkl jlk jl lj
µ µν i µ µ
n2 = n nµ = φ. (17) Using (24), one can further find:
µ
U = eλeµeνeσR
Note that when Ai = 0 and φ = 1, then Nµ = nµ as in ijkl i j k l λµνσ
= r −(π π −π π ), (29)
the ADM approach [17]. ijkl ik lj il kj
The extrinsic curvature is: Vjkl = nλeµjeνkeσlRλµνσ
1
Kjl = eµjeνlQµν, (18) = AiUijkl − N(∇kπlj −∇lπkj), (30)
3
where π = 1K . Equations(40)(aswewillseebelow)areageneralisation
jl N jl
Finding the remaining tensor W is more complicated. of Maxwell’s equations in a fixed gauge.
jl
One can easily see that One has to make a very important point here. Klein’s
theory corresponds to a threading decomposition of the
Wjl =nµnσeλjeνlRλµνσ =Sjl+AkVlkj, (31) five-dimensional spacetime [15]. Rigorous analysis [16]
shows that the curvature tensor of the hypersurface
where
formed is given by Zelmanov’s curvature tensor, which
1 differs from the ordinary Riemann curvature tensor by
S = nσeλeν(∇ Q −∇ Q ). (32)
jl N2 j l λ σλ σ νλ additional terms containing s-derivatives of the four-
dimensional metric. The cylinder condition forces the
Intheaboveweidentifyderivativesinthedirectionofnσ.
two curvature tensors equal and thus represents a sur-
To handle this type of terms, we will have to explicitly
face forming condition. In Kaluza’s theory, the foliation
invoke the dependence on s.
of the five-dimensional spacetime corresponds to slicing
Firstly, in virtue of (5), we get the following expression
[15]. Then the four-dimensional metric g naturally ap-
ij
for the extrinsic curvature (18):
pears as the slicing metric and imposing a cylinder con-
dition is not at all necessary.
N2 N2
K = − (∇ A +∇ A )+ ∂ g To simplify the analysis of the physics described by the
jl j l l j s jl
2 2
fields A , φ, and N, we will, however, put aside the s-
1 i
− Nλ(g ∂ Ek+g ∂Ek+A ∂ N +A ∂N ) dependentterms. Alsoforsimplicity,wewillassumethat
2 kl j λ kj l λ l j λ j l λ
the basis elements and their duals are constant (thus re-
N2
+ 2 GµνhAk∂k(eµjeνl)−∂s(eµjeνl)i. (33) cwoivllertihnegntvhaeniosrhigfirnoaml K(3a4l)u.za’s theory). The tensor Ωjl
Equation (40) becomes:
Using (20), (31), (32) and (33), it follows that
2
Wjl = N13∇j∇lN − N24(∂jN)(∂lN)+AiAkUijkl ∇kFkl =−2Akrkl + N2(πkl−πjjgkl)∂kN. (41)
+ 1 ∇ (Akπ )+∇ (Akπ ) − 1 Ak∇ π Here Fkl =∇kAl−∇lAk istheMaxwellelectromagnetic
Nh j kl l kj i N k lj tensor with A being the electromagnetic potential.
k
1 1
+ π πk− ∂ π +Ω , (34) The first term on the right-hand-side of (41) describes
N2 kl j N s jl jl aninteractionbetweenelectromagneticandgravitational
fields. We assume that it is much smaller than the re-
where Ω contains only terms which are proportional
jl
maining terms, so that we can neglect it. Note that A
to derivatives of the basis vectors and their duals with k
respect to (yk,s). cannot be “gauged up” to increase the scale of Akrkl.
Furthermore, if N is a constant, then (41) becomes the
Thefive-dimensionalRiccitensorcaneasilybecalculated
usual Maxwell’s equations
from (26):
∇ Fkl =0. (42)
R = EjEl hikU −N2Ak(V +V )+N2W k
µν µ νh ijkl jkl lkj jli
+(N El +N El)(−hjkV +N2AjW ) The remaining two equations are:
µ ν ν µ jkl jl
+N N hjlW . (35) fk
µ ν jl ∇ ( ) = 0, (43)
k
N
Thenthefive-dimensionalEinstein’sequationsinvacuum
1 N2
r − g r = T , (44)
jl jl jl
2 2
R =0 (36)
µν
where r = gikr and r are the four-dimensional scalar
reduce to ik jl
curvatureandfour-dimensionalRiccitensor,respectively.
hikU −N2Ak(V +V )+N2W = 0, (37) The energy-momentum tensor Tjl is therefore given by:
ijkl jkl lkj jl
hjkVjkl−N2AjWjl = 0, (38) T =TMaxwell + gik∇ B + C + D , (45)
jl jl i jlk jl jl
hjlW = 0. (39)
jl
where:
Multiplying (37) by Aj and adding it to (38) allows us
1
to exclude W from equation (38). Then, using the ex- TMaxwell = gikF F − g F Fik,
jl jl ij kl 4 jl ik
pressions (29) and (30) for U and V , (38) becomes:
ijkl jkl B = A ∇ A −A ∇ A −A F +∇ (A A )
jlk k l j l k j j kl j k l
∇ πk−∇ πk =0, (40) +g (Ai∇ A −A ∇ Ai), (46)
k l l k jl k i k i
4
C = g AiAkr −2AiA r −2AiA r , (47) where A = γβa2(t)e2βr, A = A = A = 0, and φ =
jl jl ik l ij j il r t ϕ θ
4 (γ2/s)a2(t)e2βr+α2(1−α)−2a2α(t). We take β to be a
D = (∂ N)(∂ N)
jl N4 j l negativeconstant,sothatthefieldAr willfallofftowards
2 2 infinity. Since a(t) describes the inflation, we note that
−N3∇j∇lN − N2πkk(Al∂jN +Aj∂lN) thefieldAr expandsasa2(t). Thedilaton(whichmodels
2 the Newton’s constant) varies as:
+ −Akπ +Aπk+A πk
N2h jl l j j l N2 =(1−α)2α−2[a(t)]−2α. (54)
−g (Aiπk−Akπi) ∂ N. (48)
jl i i i k Thus:
We will analyse each of these terms separately. The first G˙ a˙
=−2α =−2αH, (55)
one, TjMlaxwell, is the Maxwell energy-momentum tensor. G a
ThetensorC describesinteractionbetweenelectromag-
jl where H is the Hubble’s constant. Observational limits
netic and gravitational fields. From (44) we see that, if [7] put α<10−3.
N2 is very small (as we will confirm later), then r will
jl Oneshouldnote thatthe four-dimensionalmetric is now
be of the order of N2, which justifies the neglection of
s-dependent,butthisdoesnotposeaproblemintheslic-
the interaction terms in (41) and the tensor C .
jl ing formulation. Only the term ∂ g from the extrinsic
s jl
Using (33)in(20)andthen(20)inequation(43),we see
curvature (33) should be recovered.
thataconstantsolutionforN isallowedbyequation(43)
Finally, we note that the generalsolution for the dilaton
if φ satisfies:
field can be written as:
∇k∂kφ= 1FikFik, (49) N2 = (detgik)(detEνµ)2 , (56)
2 detG
µν
whereFik isasolutionof(42). Fortheconstantsolution
for a solution G of (36) and embedding specified with
µν
for N, the tensor Djl vanishes. Moreover: Eµ.
ν
0≡gij∇ T =gij∇ TMaxwell, (50) To recapitulate, we have found plausable generalisations
i jl i jl of Einstein–Maxwell equations and explained the origin
since gmlgnk∇ ∇ B = −2∇ ∇ (fk) = 0 in view of of the constant solutionfor the dilaton (representing the
m n jlk N j k N Newton’s constant G ) as well as the possibilities for
(43). N
modelling non-constant solutions for different cosmolo-
In other words,the conservationlaw (50) is givenby the
usualMaxwellenergy-momentumtensorTMaxwell andN2 gies (representing time-varying GN) in relation to the
jl gauge freedom of our model.
plays the role of the Newton’s constant G :
N
N2 8πG We thank Vesselin Gueorguiev, Brian Dolan, Brien
N
= . (51)
2 c4 Nolan, Siddhartha Sen and the anonymous referee for
useful discussions and comments.
Onehas topointoutherethatinthe set-upofThiry [5],
E.M.P. dedicates this work to his son Emanuel.
andin[14],G ≃φ2,whereφsatisfies(49). Thisimplies
N
that a constant solution for φ and, respectively, G is
N
only possible when the unphysicalconstraintFikF =0
ik
is satisfied.
Incontrast,inthedualset-up,aconstantsolutionispos- ∗ Electronic address: [email protected]
sible. However,N (togetherwithAi andgij)isasolution † Electronic address: [email protected]
to the system of equations (41), (43), and (44) and, in [1] Th. Kaluza, Sitzungsberichte Preussische Akademie der
general, does not need to be a constant. Then it plays Wissenschaften zu Berlin 69, 966–972 (1921).
[2] O. Klein, Zeitschrift Fu¨r Physik 37, 895–906 (1926).
the role of a dilaton field.
[3] T.Appelquist(ed.),A.Chodos(ed.)andP.G.O.Freund
To illustrate this, consider the standard cosmological
(ed.), Modern Kaluza–Klein Theories, Addison–Wesley
metric [11] with Eν =δν:
µ µ Publishing (1987).
[4] P.Jordan andC.Mu¨ller, ZeitschriftFu¨rNaturforschung
ds2(5) = −s2dt2+t2/αs2/(1−α)(dr2+r2dΩ2) 2a, 1–2 (1947);
+α2(1−α)−2t2ds2. (52) [5] Y. Thiry, Compt. Rend. Acad. Sci. Paris 226, 216–218
(1948).
Changing variables by: r → sγeβr with γ = −(1/2)(1+ [6] P.A.M. Dirac, Proc. Roy. Soc. London A165, 199–208
α)/(1−α) and t1/α →a(t), we get: (1938).
[7] Th.Damour,G.W.GibbonsandJ.H.Taylor,Phys.Rev.
ds2 = −s2α2[a(t)]2α−2a˙2(t)dt2+2A drds Lett. 61, 1151–1154 (1988);
(5) r Th. Damour, G.W. Gibbons and C. Gundlach, Phys.
+sa2(t)e2βr(β2dr2+dΩ2)+φds2, (53) Rev. Lett.64, 123–126 (1990).
5
[8] J. Ponce de Leon, gr-qc/0305041. (1995).
[9] E. Garcia-Berro, Yu.A. Kubyshin, P. Loren-Aguilar, gr- [16] A.L.Zelmanov,Dokl.Akad.USSR107,815(1956);A.L.
qc/0302006. Zelmanov, Dokl.Akad.USSR227, 78 (1976).
[10] J.-P. Uzan, Rev.Mod. Phys. 75, 403 (2003). [17] R. Arnowitt, S. Deser and C.W. Misner, in L. Witten
[11] J. Ponce de Leon, Gen. Rel. Grav. 20, 539 (1988). (ed.), Gravitation: an Introduction to Current Research,
[12] S. S. Seahra and Paul S. Wesson, Class. Quant. Grav. 227–265, Wiley (1962).
20,1321–1340 (2003). [18] T.Shiromizu,K.MaedaandM.Sasaki,Phys.Rev.D62,
[13] F. Dahia and C. Romero, J. Math. Phys. 43, 3097–3106 024012 (2000), gr-qc/9910076.
(2002). [19] R.M. Wald, General Relativity, University of Chicago
[14] X.Yang,Y.Ma,J.ShaoandW.Zhou,Phys.Rev.D68, Press (1984).
024006 (2003).
[15] S. Boersma and T. Dray, Gen. Rel. Grav. 27, 319–339