a r X i v :g r -q c /0403115v 2 27 S e p 2004Arnowitt–Der–Misner gravity with variable G and Λand fixed
point cosmologies from the renormalization group
Alfio Bonanno,1,2Giampiero Esposito 3,4and Claudio Rubano 4,3
1Osrvatorio Astrofisico,Via S.Sofia 78,95123Catania,Italy
2Istituto Nazionale di Fisica Nucleare,Sezione di Catania,
Corso Italia 57,95129Catania,Italy
3Istituto Nazionale di Fisica Nucleare,Sezione di Napoli,Complesso Universitario di Monte S.Angelo,Via Cintia,Edificio N’,80126Napoli,Italy 4Dipartimento di Scienze Fisiche,Complesso Universitario di Monte S.Angelo,Via Cintia,Edificio N’,80126Napoli,Italy Abstract Models of gravity with variable G and Λhave acquired greater relevance after the recent evidence in favour of the Einstein theory being non-perturbatively renormalizable in the Weinberg n.The prent paper applies the Arnowitt–Der–Misner (ADM)formalism to such a class of gravitational models.A modified action functional is then built which reduces to the Einstein–Hilbert action when G is consta
nt,and leads to a power-law growth of the scale factor for pure gravity and for a massless φ4theory in a Univer with Robertson–Walker symmetry,in agreement with the recently developed fixed-point cosmology.Interestingly,the renormalization-group flow at the fixed point is
found to be compatible with a Lagrangian description of the running quantities G and Λ.PACS:04.20.Fy,04.60.-m,11.10.Hi,98.80.Cq
I.INTRODUCTION
Recent studies support the idea that the Newton constant G and the cosmological con-stantΛare actually spacetime functions by virtue of quantumfluctuations of the background metric[1,2,3,4].Their behaviour is ruled by the renormalization group(hereafter RG) equations for a Wilson-type Einstein-Hilbert action where√g become relevant
operators in the neighbourhood of a non-perturbative ultravioletfixed point in four dimen-sions[5].The theory is thus asymptotically safe in the Weinberg n[6]becau the continuum limit is recovered at this new ultravioletfixed point.In other words,the theory is non-perturbatively renormalizable[7,8,9,10].
Within this framework,the basic ingredient to promote G andΛto the role of spacetime functions is the renormalization group improvement,a standard device in particle physics in order to add,for instance,the dominant quantum corrections to the Born approximation of a scattering cross ction.The basic idea of this approach is similar to the renormalization group bad derivation of the Uehling correction to the Coulomb potential in massless QED [11].The“RG improved”Einstein equations can thus be obtained by replacing G→G(k),Λ→Λ(k),where k is the running mass scale which should be identified with the inver of cosmological time in a homogeneous and isotropic Univer,k∝1/t as discusd in Refs. [3,4],or with inver of the proper distance k∝1/d(P)of a freely falling obrver in a Schwarzschild background[1,2].
In this way the RG running gives ri to a dynamically evolving,spacetime dependent G andΛ.The improvement of Einstein’s equations can then be bad upon any RG trajectory k→(G(k),Λ(k))obtained as an(approximate)solution to the exact RG equation of(quan-tum)Einstein-Gravity.Within this framework it has been shown that the renormalization group derived cosmologies provide a solution to the horizon andflatness problem of standard cosmology without any inflationary mechanism.They reprent also a promising model of dark energy in the late Univer[12].A similar approach has also been discusd in Ref.护照过期怎么办
[13],where the RG equation aris from the matterfieldfluctuations.
In comparison to earlier work[14,15,16,17,18]on cosmologies with a time dependent G,Λand possiblyfine structure constantαthe new feature of the RG derived cosmologies is that the time dependence of G andΛis a condary effect which results from a more fundamental scale dependence.In a typical Brans-Dicke type theory the dynamics of the
Brans-Dickefieldω=1/G is governed by a standard local Lagrangian with a kinetic term ∝(∂µω)2.In this approach there is no simple Lagrangian description of the G-dynamics a
眼霜是什么priori.It rather aris from an RG equation for G(k)and a cutoffidentification k=k(xµ). From the point of view of the gravitationalfield equations,G(xµ)has the status of an external scalarfield who evolution is engendered by the RG equations.It is nevertheless interesting to notice that a dynamically evolving cosmological constant and asymptotically free gravitational interaction also appear in very general scalar-tensor cosmologies[19,20].
In general,the RG equations do not admit a gradientflow reprentation and it is not clear how to embed the RG behaviour into a Lagrangian formalism.This problem has been widely discusd in Ref.[21]where a consistent RG improvement of the Einstein-Hilbert action has been propod at the
level of the four-dimensional Lagrangian.
Instead,the relevant question we would like to study in this paper is how to achieve a modification of the standard ADM Lagrangian where G andΛare dynamical variables according to a prescribed renormalized trajectory(e Ref.[22]for afirst attempt in this direction).The simplest non-trivial renormalization group trajectory is reprented by the scaling law near afixed point for which
G(xµ)Λ(xµ)=const,(1) and we shall u the simple relation(1)to constrain the possible dynamics.
Indeed,a theory with an independent dynamical G is known to be equivalent to metric-scalar gravity already at classical level.In particular,it can be reduced to canonical form with the standard expression of the kinetic term for the scalar by a conformal rescaling of the metric(e a discussion of this point in Ref.[23]).At this stage,a theory where Lambda is an independent dynamical variable meets rious problems,in agreement with what we find below.
Following thefixed-point relation as implemented in Eq.(1),in this paper we thus discuss a dynamics according to whichΛdepends on G which is,in turn,a function of position and time.IfΛand G were instead taken to be independent functions of position and time, the primary constraint of vanishing conjugate momentum toΛwould lead to a condary constraint which is very pathological,and details
will be given later in Sec.II to avoid logical jumps.
In Sec.II we introduce and motivate a modified action functional for theories of grav-
惠州游玩itation with variable G andΛ.Such a result is applied,in Sec.III,to pure gravity and to gravity coupled to a massless lf-interacting scalarfield in a Univer with Robertson–Walker(hereafter RW)symmetry.Concluding remarks and open problems are prented in Sec.IV,while the full Hamiltonian analysis is performed in the appendix.
II.MODIFIED ACTION FUNCTIONAL
According to the ADM treatment of space-time geometry,we now assume that the space-time manifold(M,g)is topologicallyΣ×R and is foliated by a family of spacelike hyper-surfacesΣt all diffeomorphic toΣ.The metric is then locally cast in the ADM form
g=−(N2−N i N i)dt⊗dt+N i(dx i⊗dt+dt⊗dx i)+h ij dx i⊗dx j,(2) where N is the lap function and N i are the components of the shift vector[24].To obtain the ADM form of the action,one has to consider the induced Riemannian metric h ij dx i⊗dx j
onΣ,the extrinsic-curvature tensor K ij=1
∂t
, and similarly for N,N i and G),and add a suitable boundary term to the Einstein–Hilbert action,which is necessary to ensure stationarity of the full action functional in the Hamilton variational problem[25].More precily,the fundamental identity in Ref.[26](hereafter h≡det h ij,K≡K i i,and(3)R is the scalar curvature ofΣ)
√h(K
ij
K ij−K2+(3)R)−2(K√
h KN i−h ij N,j ,(4) suggests using the Leibniz rule to express
1
h),0=G,0
h+ K√G ,0,(5)
体势
1
什么菜最有营养∂x i =
G,i
∂x i f i
16π
N√G(K ij K ij−K2+(3)R−2Λ)−2G,0h+2G,i f i
after adding to the action functional the boundary term (cf.Ref.
[25])
I Σ=1h
G resulting
from (3)and (6)yields vanishing contribution,having taken ∂Σ=∅).
The Lagrangian (7),however,suffers from a rious drawback,becau the resulting momentum conjugate to the three-metric reads as
p ij ≡δL h h物业租赁
女生暗恋你的表现NG (G ,0−G ,k N k ).(8)
This would yield,in turn,a Hamiltonian containing a term quadratic in G ,0(since K ij depends,among the others,on p ij and on G ,0h ij ),despite the fact that (7)is only linear in G ,0when expresd in terms of first and cond fundamental forms.There is therefore a worrying lack of equivalence between K ij and p ij .
We thus decide to include a “bulk”contribution in order to cancel the effect of G ,0and G ,i in Eq.(7)by writing
I ≡1
读书手抄报图片大全
G ( x ,t )√8π M (K √G ( x ,t )d 4x −1G ( x ,t )
d 4x,(9)as a starting action defining a gravitational theory with variabl
e G and Λ,where the added terms in Eq.(9)have integrands which are not four-dimensional total divergences.More precily,upon division by 16πG in the integrand o
f the Einstein–Hilbert term (first integral in Eq.(9)),the cond and third integral in Eq.(9)cancel the effect of −2(K
√8πG Σ
K √
8πG dt ∂Σ
f i n i d 2x,
which vanishes if Σis the smooth boundary of M (since then ∂Σ=∂∂M =0).
In other words,on renormalization-group improving the gravitational Lagrangian in the ADM approac
h,one might think that G and Λhave the status of given external field,who evolution is in principle dictated by the RG flow equation.However,it is also interesting to understand whether one can generalize the standard ADM Lagrangian in order to consider G as a dynamical field and investigate which dynamics is consistent with the RG approach.In this spirit we eventually consider the following general ADM Lagrangian:
L =1h
16π g ρσG ;ρG ;σ−g d 3x
=µh N 2G ,0G ,i − h ij −N i N j
h/(8πG ),
which would vanish on the constraint manifold.This is very pathological,becau it implies that either the lap vanishes or the induced three-metric on the surfaces of constant time has vanishing determinant.Neither of the alternatives ems acceptable in a viable space-time model.
