arXiv:1909.07959v3 [gr-qc] 20 Feb 2020 aeycnetr,wudb ipie yuigtefntoa reno functional the using ultima [ by the simplified is be which would surface, conjecture, critical safety dimensional finite the on couplings unu ed tteU ii hc ol n oapeitv renormaliza predictive a to [ end would which limit UV the at successfu fields quantum a proposed conjecture safety asymptotic Weinberg’s Introduction I hswudsmlf h a ootiigtedsrdqatte.TeΓ The quantities. desired the obtaining to way the simplify would This vrg cinΓ action average etmflcutosi h ucinlitga r upesdby suppressed are integral at functional achieve the To in modes. fluctuations mentum fluctuation momentum high the over averaged 2 1 .Teeoe nta ffntoa nerto vralspacetime all over integration functional of instead Therefore, ]. traje the for point fixed UV non–Gaussian a finding of process The ]. edsccnrec nqatmipoe spacetimes improved quantum in congruence Geodesic nacnrec nteqatmipoe spacetime. improved quantum the in congruence a lea in necessarily not does condition energy derived strong is classical equation Raychaudhuri improved The spacetimes. eivsiaetegoei eito qaini h conte the in equation deviation geodesic the investigate We k aifigteeatrnraiaingopeuto (ERGE), equation group renormalization exact the satisfying eateto hsc,Uiest fTha,Tha,Iran Tehran, Tehran, of University Physics, of Department k∂ k Γ k .Mt n .Shojai A. and Moti R. = 2 1 Tr Abstract Γ k (2) 1 + R k − 1 k∂ otecnegneo geodesics of convergence the to d k R k eomlzblt odto for condition renormalizibility l n ti hw htthe that shown is it and , . to unu improved quantum of xt nabtaymass–squared arbitrary an n a s h effective the use can one , mlzto ru method group rmalization hsga,telwmo- low the goal, this k ega fasymptotic of goal te sa ffcieaction effective an is l hoyo gravity of theory ble tre fessential of ctories (1) dimension IR–cutoff function k2R(0)(p2/k2), where the dimensionless function R(0)(p2/k2) Rk ∝ is a kind of smeared step function [3]. Therefore, without any impression by low–momentum fluctuations, the high–momentum fluctuations are integrated out such as a kind of Wilsonian loop. Truncation of the theory space would decrease the complexity of the infinite dimensional dynamical system of ERGE, and enables one to solve it nonperturbatively although approxi- mately. It has been shown [4, 5] that the Einstein–Hilbert truncation which spanned the theory space on the √g and √gR basis, leads to an anti–screening gravitational running coupling
G(k ) G(k)= 0 (2) 1+ ωG(k )(k2 k2) 0 − 0
2 where ω = 4 (1 π ) and the k is a reference scale. The Newton’s constant G seems to be π − 144 0 N a sutiable choice as the reference, hence G G(k 0) = G [4, 5]. N ≡ 0 → 0
Usually people assume the decoupling idea [3, 6] and improve the coupling constant G0 of the classical theory to the running one. It seems that this could be a proper approximate shortcut from UV to IR limit in a process of searching for the effects of this quantization method. The cutoff momentum k is a common choice for the scaling parameter of decoupling. On the flat background, this renomalization group parameter can be identified with the inverse of
1 a function of coordinate with dimension of length, k D− (x). But this coordinate dependent ∝ identification is not useful for the curved spacetimes. Introducing a function of all independent curvature invariants as χ(χi), in which χi=1,... are the independent curvature invariants, seems to be a suitable coordinate independent choice for the scaling parameter in curved spacetimes [7].
1 Hence, a proper identification in curved spacetime is k = ζχ− (χi) where ζ is some dimensionless constant. By this coordinate independence property, improving the coupling constant is possible at the level of the action functional as
1 4 √ g 4 SI = d x − R + d x √ g m , (3) 16π Z G(χ) Z − L without any concerns about breaking the general covariance [7].
2 As a result, the quantum corrections would emerge dynamically in the gravitational field equation
Gαβ =8πG(χ)T˜αβ + G(χ)Xαβ(χ) , (4)
2 δ(√ g m) ˜ − L where the Tαβ = √− g δgαβ is the classical energy–momentum tensor. The dynamical effects − of the running coupling are contained in
1 δχ 1 Xαβ(χ)= α β gαβ G(χ)− R (χ) αβ + ∇ ∇ − − 2 K δg
δχ δχ ∂µ R (χ) αβ + ∂µ∂ν R (χ) αβ , (5) K δ(∂µg ) K δ(∂ν∂µg ) which explicitly depends on the identification function χ, its derivatives, the running coupling
2 ∂G(χ)/∂χ G(χ) and its derivative through (χ) 2 . K ≡ G(χ) It has to be noted that, in general χ could be a function of all the independent curvature invariants, but an important question is that what is this function? It should be determined via some physical conditions and/or selection rules. This could be energy conditions and the behavior of the geodesic congruence as discussed in this paper, or other physical conditions to be addressed in a forthcoming work. For simple models like the ones given by χ = R,
αβ 1/2 αβγσ 1/2 χ = (RαβR ) and χ = (RαβγσR ) the Xαβ(χ) term is derived in [7]. Note that it is quite possible that some curvature invariants like R be identically zero, even though the curvature of the manifold is not zero. For this case, such simple models are ruled out and invariants constructed out of untraced curvature tensors seem to be a better choice (such as the Kretschmann scalar). Other choices like invariants obtained from derivatives of curvature
αβγδ;ǫσ tensor (e.g. Rαβγδ;ǫσR ) are also possible and maybe useful. The metric solution of this equation is the one describes the spacetime which fulfills the asymptotic safety conjecture. The effect of such quantum corrections on the classical solutions are studied for some specific problems like cosmological solutions [6, 8] and black holes and their thermodynamics [4, 9, 10]. Here we investigate the effects on the geometrical concepts such as geodesic deviation and the
3 Raychaudhuri equation. To do so, we first evaluate the quantum corrections to the connection in section II. Then, the geodesic deviation and Raychaudhuri equations are derived in the next two sections.
II Improved metric connection
Determining the eligible connection is an unavoidable issue in the process of investigating the
2 2 behavior of neighboring geodesics. For this purpose, considering G(χ)= G0/(1+ ωG0ζ χ ), we expand the improved equation of motion (4) up to first order of quantum corrections
G =8πG T˜ + ω (χ)+ (2) , (6) αβ 0 αβ Gαβ O
2 8πζ 1 where (χ) 2 T˜ + X (χ). Gαβ ≡ χ αβ ωG0 αβ Neglecting the higher orders, the term (χ) behaves as a source of perturbation to the Gαβ classical (non–improved) Einstein equation. Thus, the approximate solution gαβ =g ˜αβ + qαβ could be a proper solution for this equation where the quantum correction qαβ generally depends on the chosen symmetries of the spacetime and also on the classical metricg ˜αβ. Using this decomposition into the quantum and classical effects, the Levi–Civita connection Γγ = 1 gγκ(∂ g + ∂ g ∂ g ) can be written as αβ 2 α κβ β ακ − κ αβ
Γγ = Γ˜γ + γ (QC) + γ (QQ) , (7) αβ αβ Cαβ Cαβ where the Γ˜γ = 1 g˜γκ(∂ g˜ +∂ g˜ ∂ g˜ ) is the classical connection. The quantum–classical αβ 2 α κβ β ακ − κ αβ part γ (QC) = q g˜γσΓ˜κ +˜gκγ (8) Cαβ − κσ αβ Qαβκ is the contribution of both the classical connection and the quantum part qκσ to the full connec- tion. On the other hand the part γ (QQ) qκγ , where = 1 (∂ q + ∂ q ∂ q ), Cαβ ≡ Qαβκ Qαβκ 2 α βκ β κα − κ αβ describes the pure quantum corrections and since it is of the second order in qαβ, we neglect it in what follows. Also, because γ (QC) is defined as the difference between two connections, it Cαβ 4 behaves like a tensor.
III Improved geodesic deviation
The behavior of neighboring geodesics is described by the tidal forces which deflect the Eu- clidean parallel geodesics in curved spacetime. As in general relativity, this behavior can be investigated by considering a 2-surface which is covered by a congruence of timelike geodesics. S If τ be an affine parameter along the specified geodesic and distinct geodesics are labeled by a parameter λ, then xµ xµ(τ,λ) is a parametric equation of the surface . Any point on ≡ S this surface is specified by two vector fields, uµ = ∂xµ/∂τ as a tangent vector of the geodesic and ξµ = ∂xµ/∂λ which connects two nearby curves. Therefore, on using the Lie derivative relations ξµ = uµ = 0 besides symmetric properties of a connection, one gets Lu Lξ
uβ ξα = ξβ uα . (9) ∇β ∇β
The geodesic deviation, which describes the relative acceleration of neighboring geodesics is interpreted as the parallel transportation of Dξα/Dτ = uµ ξα. From eq. (9), this covariant ∇µ derivative becomes D2ξα = Rα uβξγuσ + ξβ (uγ uα) . (10) Dτ 2 − βγσ ∇β ∇γ
Note that, conservation of stress tensor ( T˜µν = 0) for a congruence of particles forces the ∇µ second term to vanish. But in the approximation gµν =g ˜µν + qµν , the conservation relation reduces to ( T˜µν) + other terms linear in q 0. Therefore, particles does not ∇µ evaluated ong ˜µν µν ≃ move on the geodesics ofg ˜µν, and thus the second term is nonzero up to the first order in qµν . On using this approximation and equation (6), and by virtue of Bianchi identity, αG = 0, ∇ αβ this term becomes
β 1 β β u βuα = Gβα (χ)+ Xβα(χ) , (11) ∇ −8π ∇ J ∇
5 1 where G− (χ). Hence, the geodesic deviation equation in the asymptotic safety context is J ≡
D2ξα 1 = Rα uβξγuσ ξβ γ [ (χ)gακG + gακX (χ)] . (12) Dτ 2 − βγσ − 8π ∇β∇ J γκ γκ
This shows that, in addition to the tidal forces, the deviation vector would be affected by the variations of the improvement terms. Clearly, this variation depends on the chosen scaling parameter χ.
The relation can be written more clearly on substituting gαβ =g ˜αβ + qαβ, and expanding up to the first order as
D2ξα = R˜α u˜βξγu˜σ 2R˜α u˜βξγuσ (QC) ∆α u˜βξγu˜σ+ Dτ 2 − βγσ − βγσ − βγσ
1 ξβ γ [ (χ)gακG + gακX (χ)] , (13) 8π ∇β∇ J γκ γκ where ∆α Rα R˜α is βγσ ≡ βγσ − βγσ
∆α = ∂ α (QC) ∂ α (QC) + Γ˜α λ (QC) + α (QC)Γ˜λ Γ˜α λ (QC) α (QC)Γ˜λ . (14) βγσ γ Cσβ − σCγβ γλCσβ Cγλ σβ − σλCγβ − Cσλ γβ
Note that any quantity with a tilde is a classical quantity.
While the term R˜α u˜βξγu˜σ in (13) is the classical one, the others are the effects of the − βγσ quantum improvement. All the correction terms are proportional to the relative distance and could affect geodesics like the classical case. The second term on the r.h.s, 2R˜α u˜βξγuσ (QC), − βγσ describes the tidal force between the classical geometry and its quantum fluctuations. The third one, ∆α u˜βξγu˜σ, determines how two classical parallel geodesics deviate because of − βγσ α the quantum fluctuations of the classical background. It is notable that the ∆ βγσ is nothing but the derivatives of quantized metric, which is compatible with the geometrical definition of the tidal force. And, the last term at the r.h.s is caused by dynamical self–coupling of geometry and the matter terms. This additional quantum effect may be attractive or repulsive depending on the structure and symmetries of the spacetime and the quantum aspects. This symmetry dependence is the
6 result of the dependence of the cutoff function χ(χi) on the metric. It has to be noted that any singular curvature invariant, should be avoided in χ, at least in the vicinity of the singularity. This property would be clarified more by studying the evolution of the cross–sectional volume of a congruence of geodesics or Raychaudhuri equation. In the next section we would see that the improved Raychaudhuri equation is so that the strong energy condition could not guarantee the attractive property of quantum improved gravity.
IV Improved Raychaudhuri equation
The behavior of gravity can be investigated more precisely by studying the evolution of the expansion parameter, which describes the cross–sectional volume of geodesics congruence in the Raychaudhuri equation framework. Although the strong energy condition in the classi- cal Raychaudhuri equation assures the attractive behavior of gravity, we would see that the improvement may change the situation in certain conditions. As in the classical case, to obtain the Raychaudhuri equation, we first investigate the kine- matics of a congruence of geodesics.
A Kinematics of a timelike geodesic congruence
The purely transverse tensor B u measures the parallel transport failure of ξα along αβ ≡ ∇β α the congruence. For the linear decomposition (4), we would have
Bα = B˜α + α (QC) (15) β β Bβ where the B˜α ˜ uα is the classical form of Bα and the α (QC) α (QC)uγ is the quantum β ≡ ∇β β Bβ ≡ Cβγ correction on it. The expansion, 1 B = θh + σ + ω (16) αβ 3 αβ αβ αβ where hαβ = gαβ + uαuβ is the spacelike hypersurface induced metric, would facilitate under-
7 α α β standing the evolution of deviation vector Dξ /Dτ = Bβ ξ .
The scalar expansion θ which is the trace of Bαβ becomes
θ Bα = θ˜ + ϑα (QC) (17) ≡ α α
α (QC) αβ α (QC) where for the linear solution g =g ˜ + q , the term ϑα = q B˜ + α is the first αβ αβ αβ − αβ B order correction to the classical expansion parameter θ˜ B˜α. ≡ α
The shear tensor σαβ is the traceless symmetric component of the decomposition (16) and can be expanded as 1 σ B h θ =σ ˜ + ς (QC) (18) αβ ≡ (αβ) − 3 αβ αβ αβ
(QC) (QC) γ (QC) where theσ ˜ is the classical shear tensor and the ς = (h˜ ϑγ + q θ˜)/3 is αβ αβ B(αβ) − αβ αβ the first order quantum correction.
Finally, the last term of Bαβ is the antisymmetric rotation tensor ωαβ which up to the first order correction becomes ω B =ω ˜ + w (QC) (19) αβ ≡ [αβ] αβ αβ with the correction term w (QC) = (QC). αβ B[αβ] α α β Since Dξ /Dτ = Bβ ξ , any small displacement from a spacelike hypersurface to another
α α β one would be described by ∆ξ = Bβ ξ (t0)∆t. Three distinct cases can portray each element of this deviation from parallel transportation, best:
(i) The dynamical spacetime without any rotation and shearing, or σαβ = ωαβ = 0. In this case, only the parameter θ would be the source of displacement such as
1 ∆ξα = θξα(t )∆t , (20) 3 0
which is just the pure expansion. This expansion may results from either the deviation
of the classical metric field ( α (QC)uγ), or from the interaction of the classical metricg ˜ Cβγ αβ α (QC) with the quantum correction qαβ which has footprints in ϑα .
It is important to note that the zero value of any σαβ or ωαβ quantity, does not mean the
8 vanishing of their classical counterparts. Instead, the classical spacetime may have non
zero rotating or shearing properties, canceled by quantum terms.
(ii) The dynamical spacetime which does not experience any shearing or expansion, in other
words, σαβ = 0 and θ = 0. Hence,
α α β ∆ξ = ωβ ξ (t0)∆t . (21)
(QC) Since the quantum corrected term w0i may have nonzero value, this case, unlike the previous one, could end up to asynchronized effects like what happens in Palatini f(R)
theories [11].
Same as for the first case, the vanishing values of the elements σαβ and θ does not mean that their classical values are zero.
(iii) The last case, is the spacetime which does not make any rotation or expansion between
the geodesics, so ωαβ = 0 and θ = 0. This assumption leads to
α α β ∆ξ = σβ ξ (t0)∆t . (22)
α Again, the quantum correction of transverse vector ξ to the classical shearingσ ˜αβ is not
limited to extra shearing ςij. And asynchronization which is caused by the element ς0i could be considerable for this kind of spacetimes, too.
B Improved Raychaudhuri equation
To derive the improved Raychaudhuri equation, which describes the evolution of expansion
α scalar, we begin by studying the time derivation of Bβ . We have
DB αβ = uγ B = uγ ˜ B˜ + uγ ˜ (QC) uγ κ (QC)B˜ uγ κ (QC)B˜ . (23) Dτ ∇γ αβ ∇γ αβ ∇γBαβ − Cγβ ακ − Cαγ κβ
9 The first correction terms, uγ ˜ (QC), is the contribution of quantum semi–connection evolu- ∇γ Bαβ tion. On the other hand, the last two terms describe the interaction of the classical deviation source and the quantum one along the geodesic passage. Now, the evolution of expansion scalar is obtained by taking the trace of the above equation in the improved spacetime:
D D˜ θ θ γ αβ ˜ κα β (QC) ˜ γ αβ ˜ (QC) = u q γ +2˜g κγ Bαβ + u g˜ γ αβ . (24) Dτ Dτ − ∇ C ∇ B
Since Dθ˜ 1 = θ˜2 σ˜αβσ˜ +˜ωαβω˜ R˜ uαuβ (25) Dτ −2 − αβ αβ − αβ and ˜ γ Rαβ = Rαβ + ∆αγβ (26) and by considering equations (17)–(19) the improved Raychaudhuri equation is derived as:
Dθ 1 = θ2 σαβσ + ωαβω R uαuβ (27) Dτ − 2 − αβ αβ − αβ +2θϑα (QC) +2σαβς(QC) 2σ βσ qαγ 2ωαβς(QC) +2ω βω qαγ α αβ − α γβ − αβ α γβ γ α β γ αβ ˜ κα β (QC) ˜ γ αβ ˜ (QC) ∆αγβu u u q γ +2˜g κγ Bαβ + u g˜ γ αβ . − − h ∇ C i ∇ B
Although in the improved spacetime the transversity of σαβ and Bαβ are saved, other terms of this equation can change the standard attractive behavior of gravity and should be investigated separately. It is interesting to notice that since the first order corrections are considered, all the non– classical terms of the improved Raychaudhuri equation describe the behavior of the quantum fluctuations of the geodesic on the classical background.
Indeed, since the Einstein equation is improved in this context, the strong energy condition
α β does not guarantee the non–negativity of Rαβu u for the quantum improved spacetime.
10 Rewriting the improved field equation (4) as
1 1 1 T Tg = (χ)R X (χ)+ X(χ)g , (28) αβ − 2 αβ 8π J αβ − αβ 2 αβ the strong energy condition (T 1 Tg )uαuβ > 0 results in αβ − 2 αβ
1 1 R uαuβ > X (χ)uαuβ X(χ) (29) αβ (χ) αβ − 2 J
α β where X(χ) is the trace of Xαβ(χ). Therefore, to get the condition Rαβu u > 0, the non– negativity of the quantum term (X (χ)uαuβ 1 X(χ))/ (χ), is necessary. This is not what αβ − 2 J would happen always for any spacetime and any choice of χ.
V Concluding remarks
The asymptotic safety conjecture predicts a renormalizable field theory when there is a non– Gaussian fixed point at the UV limit of the theory. By using the functional renormalization group methods, with some consideration, one would end to an antiscreening running gravita- tional coupling which has a non–Gaussian fixed point needed by this conjecture. It is shown that a general function of curvature invariants, χ(χi), seems to be a suitable scaling parameter for gravity theory as a renormalizable quantum field theory [7]. Considering the running gravi- tational coupling G(χ), and improving the Einstein–Hilbert action would result in the improved equation of motions which contain this kind of quantum correction effects. In this paper, we investigated the properties of a geodesics congruence in this context. First the geodesic deviation equation is derived. It is shown that the deviation contains improvement terms. The improved Raychaudhuri equation shows that the strong energy condition, unlike the classical general relativity, does not necessarily end to a converging quantum corrected geodesic congruence, because of the presence of correction terms.
To make this more clear, consider a congruence of timelike geodesics which are hypersurface
11 orthogonal. Since the improved equation of motion (4) results in
8π 1 1 1 R = (T Tg )+ (X Xg ) , (30) αβ (χ) αβ − 2 αβ (χ) αβ − 2 αβ J J the convergence of this congruence depends on the condition
8π 1 1 1 (T Tg )uαuβ + (X Xg )uαuβ (31) (χ) αβ − 2 αβ (χ) αβ − 2 αβ J J +2θϑα (QC) +2σαβς(QC) 2σ βσ qαγ 2ωαβς(QC) +2ω βω qαγ α αβ − α γβ − αβ α γβ γ α β γ αβ ˜ κα β (QC) ˜ γ αβ ˜ (QC) ∆αγβu u u q γ +2˜g κγ Bαβ + u g˜ γ αβ > 0 . − − h ∇ C i ∇ B
αβ As an example, the Xαβ(χ) for a special case χ = RαβR would be [7]
X (χ)= g (χ)+ (χ)RR R gκγ αβ ∇α∇β − αβ J K ακ βγ 1 1 γ ( (χ)RR )+ ( (χ)RR )+ g ( (χ)RRγκ) (32) −∇ ∇β K αγ 2 K αβ 2 αβ∇γ∇κ K
2 ∂G(χ)/∂χ where (χ) 2 . Up to the first order, only the first term should be considered. Hence, K ≡ G(χ) X = (χ) and equation (31) reduces to − J
8π 1 1 1 (T Tg )uαuβ + ( g ) (χ)uαuβ (χ) αβ − 2 αβ (χ) ∇α∇β − 2 αβ J J J +2θϑα (QC) +2σαβς(QC) 2σ βσ qαγ 2ωαβς(QC) +2ω βω qαγ α αβ − α γβ − αβ α γβ γ α β γ αβ ˜ κα β (QC) ˜ γ αβ ˜ (QC) ∆αγβ u u u q γ +2˜g κγ Bαβ + u g˜ γ αβ > 0 . (33) − − h ∇ C i ∇ B
This shows that it is quite possible that for a general improved spacetime, the correspondence of attractiveness and strong energy condition, breaks down.
Acknowledgment: This work is supported by a grant from Iran National Science Foun- dation (INSF).
12 References
[1] S. Weinberg, in Understanding the Fundamental Constituents of Matter, edited by A. Zichichi, New York: Plenum Press (1978),
S. Weinberg, in General Relativity, edited by S.W. Hawking and W. Isreal, Cambridge: Cambridge University Press (1979).
[2] M. Reuter, Phys. Rev. D. 57, 971 (1998),
M. Reuter, arXiv: hep-th/9605030 (1996), A. Codello, R. Percacci and C. Rahemde, Int. J. Mod. Phys. A 23, 143 (2008), D. Benedetti, P.F. Machado and F. Saueressig, Nucl. Phys. B 824, 168 (2010).
[3] M. Reuter and F. Saueressig, Quantum gravity and the functional renormalization group: The road towards asymptotic safety (Cambridge Monographs on Mathematical Physics), Cambridge: Cambridge University Press (2019).
[4] A. Bonanno and M. Reuter, Phys. Rev. D 62, 043008 (2000), A. Bonanno and M. Reuter, Phys. Rev. D 65, 043508 (2002).
[5] A. Bonanno and M. Reuter, Phys. Rev. D 73, 083005 (2006).
[6] M. Reuter and H. Weyer, Phys. Rev. D 69, 104022 (2004).
[7] R. Moti and A. Shojai, Phys. Lett. B 793, 313 (2019).
[8] M. Reuter and H. Weyer, Phys. Rev. D 70, 124028 (2004), S. Weinberg, Phys. Rev. D 81, 083535 (2010),
A. Bonanno, A. Contillo and R. Percacci, Class. Quant. Grav. 28, 145026 (2011), R. Moti and A. Shojai, Euro. Phys. J. C 78, 32 (2018).
[9] M. Reuter and E. Tuiran, Phys. Rev. D, 83, 044041 (2011),
D. Becker and M. Reuter, JHEP 1207, 172 (2012) , K. Falls, F. Litim and A. Raghuraman, Int. J. Mod. Phys. A 27, 1250019 (2012), T. Ramon, Phys. Rev. D 95, 124004 (2017).
13 [10] R. Moti and A. Shojai, arXiv:1802.06553 (2018).
[11] F. Shojai and A. Shojai, Phys. Rev. D 78, 104011 (2008).
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