Acausality in Nonlocal Gravity Theory

0.22 MB·

by  Ying-li Zhang

Acausality in Nonlocal Gravity Theory
Ying-li Zhanga,b, Kazuya Koyamab, Misao Sasakic and Gong-Bo Zhaoa,b
∗ † ‡ §
aNational Astronomy Observatories, Chinese Academy of Science,
6 Beijing 100012, People’s Republic of China
1
0 bInstitute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK
2
r cYukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
a
M
March 15, 2016

3
1

]
h Abstract
t
-
p We investigate the nonlocal gravity theory by deriving nonlocal equations of motion using
e
h thetraditionalvariationprincipleinahomogeneousbackground. Wefocusonaclassofmodels
[
withalinearnonlocalmodificationtermintheaction. Itisfoundthattheresultingequationsof

2
v motioncontaintheadvancedGreen’sfunction,implyingthatthereisanacausalityproblem. As
8
aconsequence,adivergencearisesinthesolutionsduetocontributionsfromthefutureinfinity
0
8
unless the Universe will go back to the radiation dominated era or become the Minkowski
3
0 spacetime in the future. We also discuss the relation between the original nonlocal equations
.
1
and its biscalar-tensor representation and identify the auxiliary fields with the corresponding
0
6 original nonlocalterms. Finally,weshowthattheacusalityproblemcannotbeavoidedbyany
1
: function of nonlocal termsin theaction.
v
i
X
YITP-16-3
r
a
∗[email protected]
†[email protected]
‡[email protected]
§[email protected]
1
Contents
1 Introduction 3
2 A simple example: scalar field with nonlocal operator 5
3 Linear Nonlocal gravity in a homogeneous geometry 6
3.1 Original equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 A simple case: the trace equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 The biscalar-tensor representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4 General case 12
5 Conclusion 13
2
1 Introduction
The nonlocalgravitytheory wasinitially proposedas a “filter”to eliminate the contributionofthe
cosmologicalconstant to the spacetime curvature so that it might provide a possible way to relieve
thecosmologicalconstantproblem[1,2,3,4]. Asfarascosmologicalstudiesareconcerned,amodel
withnonlocalmodificationswasproposedbyDeserandWoodardin2007[5]. Atthelevelofaction,
the nonlocal correctionterm takes the form of Rf(✷−1R), in which the dimensionless combination
✷−1R is tiny during the radiation-dominated era but gradually increases in the matter-dominated
epoch. Hence, this theory could help relieve the “fine-tuning problem” of the dark energy without
introducing any small mass scale. Based on this model, the cosmological correspondences were
studiedextensively(e.g. seeRefs.[6]–[26]). However,itwasfoundthatalthoughatthebackground
level, the evolution of the nonlocal gravity could be designed to be indistinguishable from that of
ΛCDM model [27], studies of the structure formation would disfavor this model [28, 29].
Nevertheless, this negative result does not totally rule out the possibility of including nonlocal
corrections in the action. Recently, there appear a series of studies of nonlocal modifications: in
Refs. [30, 32, 33], a term proportional to g ✷−1R was introduced into the field equations. It
µν
was found that in this model, a mass term could be introduced without any reference metric [30,
31]. Moreover, its equation of state (EoS) is less than 1, hence this model could mimic the
−
phantom dark energy [32, 33], while studies of its linear perturbations showed that this model was
statistically comparable with ΛCDM model [34]. Another model was proposed by introducing a
termproportionaltoR✷−2Rintotheaction[35],withitscosmologicalperturbationsstudiedin[36]
which also gave a positive result. Besides these two interesting models, there are also discussions
on other related topics, e.g. interpretations of dark matter as nonlocal effects from the General
Relativity (GR) [37, 38, 39].
On the other hand, several theoretical aspects of a theory with nonlocal terms remain to be
clarified. Forinstance,inordertotransformtheoriginalintegro-differentialequationintodifferential
equations,thenonlocalgravitytheoryisoftenwrittenintoabiscalar-tensortheorybyintroducinga
scalarfieldψ ✷−1RwithaLangrangianmultiplier. Inthiscase,thenumberofdegreesoffreedom
≡
in this theory becomes ambiguous. It was found that in the corresponding biscalar-tensor theory,
there would appear a “ghost-like” mode so that the theory could become unhealthy [2, 3, 4, 40].
However, it was argued in [32, 41] that in the biscalar-tensor theory, the Green’s function for ψ
should be defined in the way where the initial conditions remove the homogeneous solution which
satisfies ✷ψ = 0. In this sense, when ψ is quantized, the creation and annihilation coefficients
hom
vanish so that ψ is not a “free field”, hence the “ghost-like” mode is physically irrelevant.
3
Another problem is the appearance of acausality in this theory. As discussed in [5, 23, 37, 41],
′ ′
under the replacement x x, the retarded Green’s function G (x,x) becomes an advanced one
R
↔
′
G (x,x). Hence, in the Minkowskian background, for a class of theories which contain nonlocal
A
operators acting on scalar fields, it is expected that the advanced Green’s function cannot be
eliminated in the equations of motion (EOM) obtained by the traditional variation principle. One
of the consequences is that the future information is needed in order to find the solutions, which
may imply acusality problems of the theories.
In this paper, we consider the acausality problem arising from the nonlocal gravity theory. A
similarproblem mayappear in a class ofmodified gravitytheoriesthat containnonlocaloperators.
WestartfromalinearnonlocalgravityactionandderivetheEOMinitsoriginalformulationbythe
variation principle. We find that the variation principle will symmetrize the property of Green’s
function in the EOM, i.e., no matter whether the nonlocal operator is defined by the retarded
Green’s function or the advanced one in the action, both of them symmetrically appear in the
EOM. This means that the advanced Green’s function cannot be eliminated by any construction
of functions of the nonlocal operator in the action. Hence, future information is needed to find
the solutions, i.e. the acusality problem appears in the nonlocal gravity theory. This could imply
that the nonlocal gravity theory is not well-defined, or it is not a fundamental theory to derive the
causal nonlocal equations.
Inmostliterature,especiallyforthenumericalanalysis,theanalysisisdoneinthebiscalar-tensor
representation. We make a comparison of the original EOM to its biscalar-tensor representation
andidentifytheextrascalarswiththenon-localtermsintheoriginalformulation. Wefindthatone
ofthe additionalscalarswouldbeidentifiedtothetermassociatedwithadvancedGreen’sfunction.
In the biscalar-tensor representation, when we solve the EOM for the extra scalars, we reverse the
relationship ψ = ✷−1R to the second-order differential equation ✷ψ = R. Hence, in the solution
for this second-order differential equation, there appears a homogeneous solution that makes the
solution different from the original one, i.e. ψ =✷−1R+U , where U is the homogeneous
sol hom hom
solution satisfying ✷U =0, which causes the “ghost” instability problem [2, 3, 4, 40].
hom
Thispaperisorganisedasfollows. InSection2,westartourstudywithasimpleexamplewhere
thenonlocaloperatoractsonascalarfieldintheaction. Fromthissimplestudy,itisstraightforward
to show the appearanceof the advancedGreen’s function in the correspondingequationof motion.
In Section 3, we focus on a class of nonlocal gravity models where the nonlocal modification term
is linear in the action. In Section 3.1, by using the traditional variation principle, we derive the
nonlocal equations of motion in the homogeneous backgroundand find that both the retarded and
advanced Green’s function will appear symmetrically. In Section 3.2, we use the trace equation to
4
demonstrate that the advanced Green’s function cannot be eliminated from the EOM. In Section
3.3, we consider the relationship between the original nonlocal equation and its biscalar-tensor
representation. We identify the additional scalars with their original nonlocal terms. Especially,
in the homogeneous background, we demonstrate that the difference between two representations
is caused by a homogeneous solution and discuss its consequence. In Section 4, we generalise the
analysis to a model involving a general function of both advanced and retarded Green’s functions
in its action. We find that the acusality problem cannot be avoided by constructing any function
of the non-local operator in the action. In the Appendix, as a supplement to Section 2, we show
the appearance of the advanced Green’s function in the EOM assuming the FLRW metric without
′ ′ ′
using the property that G (x,x) G (x,x) under the replacement of variables x x.
R A
↔ ↔
2 A simple example: scalar field with nonlocal operator
Before studying the nonlocal gravity theory, it would be useful to consider the case where the
nonlocal operator acts on a scalar field φ(x) in the action. Let us consider the simplest example
where the nonlocal operator appears linearly in the action:
S = d4x g(x)φ(x) ✷−1φ [x]. (1)
φ
−
Z
p (cid:0) (cid:1)
where the nonlocaloperator✷−1 operatingon anarbitraryfunction f(x) is defined by the integra-
′
tion of Green’s function G(x,x) as follows
✷−1f [x] d4x′ g(x′)f(x′)G(x,x′). (2)
≡ −
Z
(cid:0) (cid:1) p
An important property of the Green’s function is that, under the replacement of variables
′ ′ ′
x x, the retarded Green’s function changes to the advanced one G (x,x) G (x,x), and
R A
↔ ↔
vice versa. Then the problem arises that when one varies the nonlocal term in (1), the advanced
Green’s function appears when the action is defined by a retarded one [5, 23, 37, 41]:
d4x g(x)φ(x) ✷−1 δφ [x]
− R δφ(y)
Z (cid:18) (cid:19)
p δφ(x′)
= d4x d4x′ g(x) g(x′)φ(x)G (x,x′)
R
− − δφ(y)
Z Z
p p δφ(x)
= d4x′ d4x g(x′) g(x)φ(x′)G (x′,x)
R
− − δφ(y)
Z Z
p p
= d4x g(x)δ(x y) d4x′ g(x′)φ(x′)G (x′,x)
R
− − −
Z Z
p p
= d4x g(x)δ(x y) d4x′ g(x′)φ(x′)G (x,x′)
A
− − −
Z Z
p p
= d4x g(x)δ(x y) ✷−1φ [x]. (3)
− − A
Z
p (cid:0) (cid:1)
5
Hence, variation of the nonlocal action (1) with respect to the scalar field φ(x) symmetries the
property of Green’s functions, regardless whether the nonlocal operator in the action is defined by
the retarded or advanced Green’s function:
δSφ = g(y) ✷−1φ+✷−1φ [y]. (4)
δφ(y) − R A
p (cid:0) (cid:1)
This implies that the advanced Green’s function cannot be eliminated in the EOM for the scalar
field. In the Appendix, the same resultis obtainedwhen the backgroundspacetime is described by
the FLRW metric without using the property of the Green’s function.
Correspondingly, one may expect the similar result in a nonlocal gravity theory described by
the action containing a non-local term
S S = d4x g(x)R(x) ✷−1R [x]. (5)
NL
⊃ −
Z
p (cid:0) (cid:1)
by casting φ(x) = R(x). The situation would seem to be different since the nonlocal operator
✷−1 contains the determinant √ g. Also when the variation principle is applied with respect to
−
g , one should vary the nonlocal operator itself from which extra terms will appear in the EOM.
µν
However,we show in the next sectionthat a direct variationof a nonlocal gravityaction will again
symmetrise the retarded and advanced Green’s function in the EOM.
3 Linear Nonlocal gravity in a homogeneous geometry
Asasimple exampleofthe nonlocalgravitytheoryproposedinRef.[5], inthis section,weconsider
a specific nonlocal gravity theory where the nonlocal modification is linear in the action, i.e.
S = d4x g(x)R(x) ✷−1R [x]. (6)
NL − R
Z
p (cid:0) (cid:1)
wherefordefiniteness,wedefinethenonlocaloperatorintheactionbytheretardedGreen’sfunction,
denoted as ✷−1.
R
3.1 Original equations of motion
Inthis subsection,in the originalframe (6), we treatthe metric g as the only variableand derive
µν
the EOM directly by the traditional variation principle. The variation can be expressed in the
following way
δS
NL =∆G(1)+∆G(2)+∆G(3), (7)
δgµν(x˜) µν µν µν
6
where
δ g(x)
∆G(1) d4x − R(x) ✷−1R [x], (8)
µν ≡ δgµν(x˜) R
Z p
δR(x)(cid:0) (cid:1)
∆G(2) d4x g(x) ✷−1R [x], (9)
µν ≡ − δgµν(x˜) R
Z
p δ ✷(cid:0)−1R [(cid:1)x]
∆G(3) d4x g(x)R(x) R . (10)
µν ≡ − δgµν(x˜)
Z (cid:0) (cid:1)
p
Wenotethatthevariationsin∆G(1) and∆G(2) areanaloguetermsinGR,sotheycanbecalculated
µν µν
straightforwardlyas
1
∆G(1) = d4x g(x) δ4(x x˜) g (x)R(x) ✷−1R [x]
µν −2 − − µν R
Z
= 1√ g gpR ✷−1R , (cid:0) (cid:1) (11)
−2 − µν R
∆G(2) = d4x g(x)(cid:0)δ4(x (cid:1)x˜) R (x) ✷−1R [x] ✷−1R [x] +g (x)R(x)
µν − − µν R −∇µ∇ν R µν
Z (cid:26) (cid:18) (cid:19) (cid:27)
p (cid:0) (cid:1) (cid:0) (cid:1)
=√ g R ✷−1R ✷−1R +g R . (12)
− µν R −∇µ∇ν R µν
(cid:20) (cid:21)
(cid:0) (cid:1) (cid:0) (cid:1)
However, ∆G(3) contains two pieces in which the property of Green’s function will be changed:
µν
∆G(3) =∆G(3−I)+∆G(3−II), (13)
µν µν µν
where
δ ✷−1R [x] δR(x )
∆G(3−I) d4x g(x) d4x δ4(x x˜)R(x) R 1
µν ≡ − 1 1− δR(x ) δgµν(x˜)
Z Z (cid:0) 1(cid:1)
p δ ✷−1R [x]
= d4x g(x) R (x˜) +g (x˜)✷ R(x) R , (14)
µν µ ν µν
Z − (cid:18) −∇ ∇ (cid:19) (cid:0) δR(x˜)(cid:1) !
p
∆G(3−II) d4x g(x)R(x) δ✷e−R1e R [x]. e (15)
µν ≡ − δgµν(x˜)
Z (cid:20)(cid:18) (cid:19) (cid:21)
p
In fact, ∆G(3−I) is analogousto the scalarfield case considered in Sec. 2 where the property of the
µν
Green’s function changes. Hence, this term can be expressed as
∆G(3−I) = R (x˜) +g (x˜)✷ d4x g(x)δ4(x x˜) ✷−1R [x]
µν µν −∇µ∇ν µν − − A
(cid:18) (cid:19)Z
p (cid:0) (cid:1)
=√ g R ✷e−1eR e✷−1R +g R , (16)
− µν A −∇µ∇ν A µν
(cid:20) (cid:21)
(cid:0) (cid:1) (cid:0) (cid:1)
which is a symmetric counterpart of ∆G(2) shown in Eq. (12). Moreover, ∆G(3−II) is an extra
µν µν
term arising from the variation of the nonlocal operator itself. Generally speaking, it is difficult to
analyse this term. In the following, we discuss it by assuming a homogeneous geometry, i.e. the
Ricci scalar is only dependent on time: R = R(t). Under this assumption, the nonlocal term can
be expressed explicitly in the following way [42]
t dt′ t′
✷−1R [t]= dt′′R(t′′) g(t′′), (17)
(cid:0) (cid:1) −Zt1 −g(t′)Zt0 p−
p
7
where t ,t corresponds to the retarded Green’s function and t ,t + gives the
0 1 0 1
→ −∞ → ∞
advancedGreen’sfunction. Takingintoaccountthatδ√ g = 1√ gg δgµν,thevariationofthe
− −2 − µν
retarded Green’s function can be expressed as
δ✷−1 1 t g (t′) t′
R [x]= dt′ µν δ(t′ t˜) dt′′ g(t′′)R(t′′)
(cid:20)(cid:18)δgµν(x˜)(cid:19) (cid:21) +− 21Zt1t pdt′−g(t′)t′dt′−′ Zgt(0t′′)gp(−t′′)R(t′′)δ(t′′ t˜). (18)
2Zt1 −g(t′)Zt0 p− µν −
Thus, ∆G(3−II) can be calculated as p
µν
δ✷−1
∆G(3−II) = d4x g(x)R(x) R R [x]
µν − δgµν(x˜)
Z (cid:20)(cid:18) (cid:19) (cid:21)
= gµνp(t˜) +∞dt g(t)R(t) t˜ dt′′ g(t′′)R(t′′) 1√ g g R ✷−1R , (19)
−2 −g(t˜)Zt˜ p− Z−∞ p− − 2 − µν (cid:0) A (cid:1)
where we used apsimilar method with that to derive Eq. (50) in Appendix to find the second term
proportional to ✷−1R , which comes from the second line of Eq. (18). We note that this term is
A
a symmetric cou(cid:0)nter te(cid:1)rm of ∆G(1) shown in Eq. (11). Hence, combining Eqs. (11), (12), (16) and
µν
(19) together, we obtain all the terms arising from the variation of the action (6) as
δSNL =√ g R 1g R ✷−1+✷−1 R +2√ g g R
δgµν − µν − 2 µν −∇µ∇ν R A − µν
(cid:18) (cid:19)
g +∞ t(cid:2)(cid:0) (cid:1) (cid:3)
µν dt′ g(t′)R(t′) dt′′ g(t′′)R(t′′). (20)
− 2√−g Zt − Z−∞ −
p p
It is obvious from this equation that for a simple class of models (6) which are linear in the
nonlocal operator in their actions, the corresponding equation of motion symmetrises the retarded
andadvancedGreen’sfunctions. Intheoriginalnonlocaltheory,itseemsdifficulttounderstandthe
implication of the second line of Eq. (20) which originates from the variation of nonlocal operator
itself(showninEq.(19)). Thistermis understoodasthe crossproductofthe firstorderderivative
of ✷−1R and ✷−1R in a biscalar-tensor presentation of the nonlocal theory as we will show in
R A
Section. 3.3.
3.2 A simple case: the trace equation
Inorderto understandwhether Eq.(20)requires informationto the infinite future ornot, it would
be useful to consider the trace equation. Taking a trace of the equation of motion, we can reduce
Eq. (20) to the form:
δS
g NL = √ gR ✷−1+✷−1 R +6√ gR
µν δgµν − − R A −
2 (cid:2)(cid:0)+∞ (cid:1) (cid:3) t
dt′ g(t′)R(t′) dt′′ g(t′′)R(t′′). (21)
− √−g Zt − Z−∞ −
p p
8
We hope that the second line could be merged to the first line and eliminate the advanced
Green’s function. To see if this is possible, we should first notice that
t
√ gR✷−1R= ∂ dt′ g(t′)R(t′) ✷−1R
− R t − R
(cid:20) ZT (cid:21)
t p t
= dt′ g(t′)R(t′) ∂ (✷−1R)+∂ ✷−1R dt′ g(t′)R(t′)
− − t R t R −
(cid:20)ZT (cid:21) (cid:20) ZT (cid:21)
1 t p t p t
= dt′ g(t′)R(t′) dt′′ g(t′′)R(t′′)+∂ ✷−1R dt′ g(t′)R(t′) ,
√−g ZT − Z−∞ − t(cid:20) R ZT − (cid:21)
p p p
(22)
where ∂ d/dt and we used Eq. (17) in the last step. So it follows that
t
≡
1 t t t t
√ gR ✷−1+✷−1 R = dt′√ gR dt′′√ gR+ dt′√ gR dt′′√ gR
− − R A −√−g (cid:20)ZT − Z−∞ − ZT˜ − Z+∞ − (cid:21)
(cid:2)(cid:0) (cid:1) (cid:3) t t
∂ ✷−1R dt′ g(t′)R(t′)+✷−1R dt′ g(t′)R(t′) , (23)
− t(cid:20) R ZT − A ZT˜ − (cid:21)
p p
where T and T˜ are two integration boundaries. For simplicity, here we let T + and T˜
→ ∞ →−∞
so that the first line of Eq. (23) cancels with the second line of Eq. (21). 1 Hence, we obtain the
following simpler expression:
δS t t
g NL =6√ gR ∂ ✷−1R dt′ g(t′)R(t′)+✷−1R dt′ g(t′)R(t′) . (24)
µν δgµν − − t(cid:20) R Z+∞ − A Z−∞ − (cid:21)
p p
As can be observed above, the information for the future evolution is always required for the
integrations in the EOM.
3.3 The biscalar-tensor representation
In this simple class of non-localgravity,we may rewrite the actioninto a local formby introducing
a scalar field ψ =✷−1R and a Langrangianmultiplier ξ as follows [2, 3, 4, 43]
R
S = 1 d4x√ gR 1+✷−1R
2κ2 − R
Z
1 (cid:0) (cid:1)
= d4x√ g[R(1+ψ) ξ(✷ψ R)]
2κ2 − − −
Z
1
= d4x√ g[R(1+ψ+ξ)+gµν∂ ξ∂ ψ] . (25)
2κ2 − µ ν
Z
1HerewenotethatifonetakesT andT˜ + ,thereappearsanextracontantterm +∞dt′√ gR 2,
→−∞ → ∞ (cid:16)R−∞ − (cid:17)
whichgivesastrongconstraintthattheRicciscalarRshouldvanishquicklywhentgoestoinfinite.
9
By varying the action with respect to g , ξ and ψ, respectively, we obtain the field equations as
µν
1
0= g R(1+ψ+ξ)+gαβ∂ ξ∂ ψ R (1+ψ+ξ)
µν α β µν
2 −
1(∂(cid:2) ξ∂ ψ+∂ ψ∂ ξ) (g ✷ (cid:3) )(ψ+ξ) , (26)
µ ν µ ν µν µ ν
− 2 − −∇ ∇
0= R ✷ψ, (27)
−
0= R ✷ξ. (28)
−
Let us clarify the (in-)equivalence between the original nonlocal action (6) and its biscalar-
tensor presentation (25). A question is how many degrees of freedom are there in the biscalar-
tensor representation. Actually, in the biscalar-tensor presentation, one has already reversed the
definition ψ =✷−1R to obtain Eq. (27). The difference between two representations thus appears
R
in the degrees of freedom since the solution given by Eq. (27) generally contains a homogeneous
solution U defined by ✷U =0, so that
hom hom
ψ =✷−1R+U , (29)
sol hom
where ψ is the solution obtained from the localized equation (27). As observed in Ref. [35], the
sol
homogeneoussolutionis uniquely fixed once we havespecified the integrationboundariesof ✷−1R.
This necessarily means that U will never be a free field which could be expanded into plane
hom
waves. ThiscanbeseenclearlyinthebackgrounddescribedbytheFLRWmetricwithapower-law
solution H(t)=s/t where s=constant. In this case, it immediately follows that
t dt′ t′
✷−1R [t]= dt′′ g(t′′)R(t′′)
(cid:0) (cid:1) −Zt∗ −g(t′)Zt∗ p− 1−3s
6s(2sp1) 1 t t
= − 1 ln , (30)
3s−1 (1−3s"(cid:18)t∗(cid:19) − #− (cid:18)t∗(cid:19))
where the integration boundary t∗ corresponds to the boundary for the integration of the Green’s
function. Ontheotherhand,inthe localizedform,the solutionforψ canbe directlyobtainedfrom
the second-order differential equation (27):
1−3s
t 6s(2s 1) t
ψ (t)=ψ − ln , (31)
sol 1
T − 3s 1 T
(cid:18) 0(cid:19) − (cid:18) 0(cid:19)
where ψ and T are two integration constants. Inserting Eqs. (30) and (31) into (29), the homo-
1 0
geneous solution U can be explicitly expressed as
hom
U =ψ ✷−1R
hom sol− R
t 1−3s 6s(2s 1) t 1−3s
=ψ + − +β, (32)
1(cid:18)T0(cid:19) (3s−1)2 (cid:18)t∗(cid:19)
10