Aspects of (d+D) -dimensional Anisotropic Conformal Gravity
aa r X i v : . [ h e p - t h ] F e b Aspects of ( d + D ) -dimensional AnisotropicConformal Gravity Jae-Hyuk Oh and Phillial Oh Department of Physics, Hanyang University, Seoul 04763, Korea Department of Physics and Institute of Basic Science, SungkyunkwanUniversity, Suwon 16419, Korea [email protected], [email protected]
Abstract
We discuss various aspects of anisotropic gravity in ( d + D )-dimensionalspacetime where D -dimensions are treated as extra dimensions. It isbased on a foliation preserving diffeomorphism(FPD) invariance andan anisotropic conformal invariance. The anisotropy is embodied byintroducing a factor z which characterizes the scaling degree of theextra D -dimensions against the d -dimensional base spacetime. Thereis no intrinsic scale in our model but a physical scale M ∗ emerges asa consequence of spontaneous conformal symmetry breaking of Weylscalar field which mediates the anisotropic scaling symmetry. Somevacuum solutions are obtained and we discuss an issue of ‘size separa-tion’ between the base spacetime and the extra dimensions. The sizeseparation means large hierarchy between the scales appearing in thebase spacetime and the extra dimensions respectively. We also dis-cuss interesting theories obtained from our model. In a ( d, D ) = (2 , Introduction
One of the main motivations to consider spacetime dimensions other thanfour in theoretical physics is to tackle with the quantum gravity. In string the-ory which is regarded as the most promising theory of quantum gravity[1, 2],the spacetime is enlarged to accommodate extra dimensions at short dis-tances. The higher dimensional gravity theories like Kaluza-Klein theory[3]or the brane-world models[4] are also based on the assumption of extra di-mensions. The reverse-path of reducing the spacetime dimensionality to twoat high energy also has been studied [5].In the higher dimensional theories, the dimensions are scale-dependent.At small enough distances, one can probe extra dimensions. However, ata distance larger than the scale, the spacetime becomes effectively four-dimensional, the (compact) extra dimensions are no longer being observed.Under the circumstance that the exact nature of spacetime is still far frombeing accessible, it is worthwhile to consider diverse possibilities and oneof the natural things is to question the isotropy of spacetime. Is spacetimeisotropic in all its directions? The current Universe favors this and it is wellaccepted that the four-dimensional spacetime we are living in is uniform inall directions [6, 7] . However as far as extra dimensions are concerned, itleaves us with the possibility that they might not share the same property asthe four-dimensional spacetime. Therefore, it is worthwhile to undertake thetask of investigating whether the higher dimensional spacetime has uniformphysical properties in all directions.Recently, there is an attempt to construct a higher dimensional gravitytheory where the four dimensional spacetime and the extra dimensions arenot treated as an equal footing[9]. The essential feature is given by an intro-duction of a real parameter z which measures the degree of anisotropy of thefour-dimensional spacetime and the extra dimensions and make our modelenjoy the anisotropic conformal invariance. There are no restrictions on thevalue of z at the classical level. However, the Planck and WMAP’s cosmicmicrowave background data[10, 11] can restrict its value. A cosmologicaltest in a five-dimensional theory[12] was presented in order to check whichspecific value of z is preferred by applying the data to the theory.In this work, we generalize it to ( d + D )-dimensional spacetime where D -spatial dimensions are being treated as the extra dimension. The extensionprovides opportunity to discuss not only for d = 4 theory but also for otherinteresting cases of d like d = 2 and d = 5 among others. The generaliza- Abandoning the Lorentz invariance and equal-footing treatment of time and space inUV is advocated in Ref.[8] for quantum gravity. d, D )-ADM decomposition by keep-ing the general covariance only for the d - dimensional spacetime. This can beachieved by adopting FPD in which the foliation is adapted along the extradimensions. The theory is extended to conformal gravity by introductionof a Weyl scalar field. The scalar field is introduced as a Stueckelberg fieldof a scale transformation of metric, where the base spacetime and the extradimension transform in different ways.The resulting action does not contain any dimensionful parameters thanksto conformal invariance and so exhibits some interesting properties due tothe scaling parameter of z . For example, in (2 , D ) case, when z → , theeffective theory reduces to two-dimensional conformal gravity theory coupledwith non-linear sigma model whose scalar fields are furnished by the metriccomponents of the extra dimensions. This can be achieved with a suitablegauge fixing of the Weyl scalar field to a constant. All the other interactionscoming from the metric components involving the shift vector vanishes in thislimit regardless of the value of D. The derivative along the extra dimensionsall disappears and the extra dimensions become virtually ‘deactivated’ in thislimit. This effective two-dimensional gravity theory can be renormalizablewith a suitable choice of parameter in the Lagrangian.Spontaneous conformal symmetry breaking brings in the physical massscale M ∗ , even if we starts with no scales. Then, depending on the value of z , there could be a large separation between the physical mass parametersand M ∗ . This property has been used to generate the vanishingly smallcosmological constant [9] and photon mass [13, 14]. It is demonstrated thatit can result in a huge size separation between the d -spacetime dimensionand D -extra dimension. Since anisotropic factor z enters in the action as afree parameter, the isotropic case can be readily recovered by setting z = 1.It seems that the anisotropic approach provides a road to a single frameworkto describe aspects of both low dimensional and higher dimension gravitytheories which cannot be unraveled in the isotropic approach.We also suggest a possible scenario for a resolution of hierarchy problemand discuss comparison with the Randall-Sundrum(RS) model. It turns outthat our model can provide a large scale distinction between the original massscale given in certain ( d + D ) dimensional field theories and the physical massscale in their effective d -dimensional theories. The ratio of the two scales aresensitively depends on the parameter z and so fine tuning of the z is needed tofit the phenomenological results. In the RS-model, they suggest a space with M × S with two different 3-branes put at certain locations on S (visible andhidden branes), where M is the Minkowskian 3+1 dimensional spacetime3nd the S is a circle being an extra dimension. The ratio of the theory’smass scale on the hidden brane and the effective mass scale on the visiblebrane in the RS-model depends on the circle size of the extra dimensionwhereas ours relies on the parameter z .The paper is organized as follows; In Section II, a brief review of FPDgeometry is given and gravity theory in ( d, D ) spacetime is constructed. InSection III, anisotropic conformal extension is performed with the use ofthe Weyl scalar field. In Section IV, vacuum solutions are obtained withspontaneous conformal symmetry breaking scale M ∗ and various cases arediscussed. Section V contains summary and discussions. ( d, D ) Gravity
We start from a brief review of ( d + D )-dimensional gravity which possessesFPD invariance [15, 16]. Then, we will extend to the theory with anisotropicconformal invariance [9]. The ( d + D ) dimensional metric can be written inADM decomposition by ds = g µν ( dx µ + N µm dy m )( dx ν + N νn dy n ) + γ mn dy m dy n , (1)( µ, ν, · · · = 0 , , · · · , d ; m, n, · · · = d + 1 , · · · , d + D ) . Here, N µm is the shift vector along the base spacetime corresponding to the D -direction [17]. The inverse metric is obtained as g MN = (cid:18) g µν + g mn N µm N νn − N µn − N mν g mn (cid:19) , (2)where g µν and g mn are inverse metrics of g µν and g mn respectively. Also, N µm = g mn N µn . Let us consider FPD given by x µ → x ′ µ ≡ x ′ µ ( x, y ) , y n → y ′ n ≡ y ′ n ( y ) , (3)whose infinitesimal transformations can be written as x ′ µ = x µ + ξ µ ( x, y ) , y ′ m = y m + η m ( y ); ∂ µ η m = 0 . (4)4ere η m ( y ) is a function of y m only. The finite coordinate transformationsof each component of the metric under FPD of (3) are given as follows [9]; g ′ µν ( x ′ , y ′ ) = ∂x ρ ∂x ′ µ ∂x σ ∂x ′ ν g ρσ ( x, y ) , (5) N ′ µm ( x ′ , y ′ ) = (cid:16) ∂y n ∂y ′ m (cid:17)h ∂x ′ µ ∂x ν N νn ( x, y ) − ∂x ′ µ ∂y n i , (6) γ ′ mn ( x ′ , y ′ ) = ∂y p ∂y ′ m ∂y q ∂y ′ n γ pq ( x, y ) . (7)One can check that the above transformations (5)-(7) leave the line element(1) invariant.The computation of the curvatures scalar R is greatly simplified by goingto orthogonal basis [18], and let us rewrite line element (1) as ds = G AB dx A dx B = ˆ g µν d ˆ x µ d ˆ x ν + ˆ γ mn d ˆ y m d ˆ y n , ( A, B, ... = 1 , ..., d + D ) (8)where the coordinate transformations are given by d ˆ x µ = dx µ + N µm dy m , d ˆ y m = dy n . (9)These transformations are non-integrable and the basis is non-coordinatebasis. First we have ˆ g µν = g µν , ˆ γ mn = γ mn (10)Going from orthogonal basis to coordinate basis can be formally done using ∂ ˆ x ν ∂x µ = δ νµ , ∂ ˆ y m ∂x µ = 0 , ∂ ˆ x ν ∂y m = N νm , ∂ ˆ y m ∂y n = δ mn , (11)or ∂x ν ∂ ˆ x µ = δ νµ , ∂y m ∂ ˆ x µ = 0 , ∂x ν ∂ ˆ y m = − N νm , ∂y m ∂ ˆ y n = δ mn . (12)Therefore, we have ∂∂ ˆ x µ = ∂∂x µ , ∂∂ ˆ y m = ∂∂y m − N µm ∂∂x µ . (13)We also have ∂∂ ˆ x µ ∂∂ ˆ x ν = ∂∂ ˆ x ν ∂∂ ˆ x µ , ∂∂ ˆ x µ ∂∂ ˆ y m − ∂∂ ˆ y m ∂∂ ˆ x µ = − ∂N νm ∂ ˆ x µ ∂∂ ˆ x ν (14)5hich defines the structure functions[ ˆ ∂ A , ˆ ∂ B ] = f CAB ˆ ∂ C (15)with f mµν = f σµν = 0 , f νµm = − f νmµ = − ∂N νm ∂ ˆ x µ , f µmn = − F µmn , (16)where F µmn = ∂ m N µn − ∂ n N µm − N νm ∂ ν N µn + N νn ∂ ν N µm . (17)Geometrically, the field strength F µmn is a Diff(d) vector-valued curvatureassociated with the holonomy of the shift vector N µn along the extra dimen-sions.To define the connection, introduce vector notation ~r = ( x , x , · · · , x d + D )for convenience and consider a vector ~V in the basis ~ ˆ e A ≡ ∂~r∂ ˆ x A , ~V = ˆ V A ~ ˆ e A .Parallel transport is defined as d~Vdu = d ˆ x A du " ∂ ˆ V B ∂ ˆ x A ~ ˆ e B + ˆ V B ∂~ ˆ e B ∂ ˆ x A = 0 (18)along a curve parametrized by u. Introducing the connection via ∂~ ˆ e B ∂ ˆ x A = ˆΓ CAB ~ ˆ e C (19)and using ˆ e A · ˆ e B = ˆ g AB , one findsˆΓ CAB = ˆΓ C ( C ) AB + 12 ˆ g CD (cid:16) f DAB − f ABD − f BAD (cid:17) , ( f ABC = ˆ g AD f DBC ) , (20)where ˆΓ C ( C ) AB is the Chrisoffel connection. The covariant derivative is definedas ˆ ∇ A ˆ V B = ∂ ˆ V B ∂ ˆ x A + ˆΓ BAC ˆ V C . (21)Then, one can check explicitly the metricity:ˆ ∇ A ˆ g BC = 0 (22)The Riemann tensor is defined as[ ˆ ∇ A , ˆ ∇ B ] ˆ V C = ˆ R CDAB ˆ V D , (23)6hich givesˆ R ABCD = ˆ ∂ C ˆΓ ADB − ˆ ∂ D ˆΓ ACB + ˆΓ
ACE ˆΓ EDB − ˆΓ ADE ˆΓ ECB − f ECD ˆΓ AEB . (24)A straightforward computation yields the scalar curvature [19] (up to totalderivative terms) R = G BD ˆ R ABAD = R ( d ) + ˆ R ( D ) + R F + R K + R R , (25)where R ( d ) is the the scalar curvature constructed with Christoffel connectionΓ µνρ ( g αβ ), and ˆ R ( D ) is constructed with ˆ ∂ m = ∂ m − N µm ∂ µ ;ˆ R mn = ∂ p ˆΓ pnm − ∂ n ˆΓ ppm + ˆΓ qpq ˆΓ pnm − ˆΓ qpn ˆΓ pqm , (26)with the connection ˆΓ pmn given byˆΓ pmn = 12 γ pq ( ˆ ∂ m γ nq + ˆ ∂ n γ qm − ˆ ∂ q γ mn ) . (27)We also have FPD invariants R F = − γ mn γ pq g µν F µmp F νnq R K = − γ mn g µν g αβ ( D m g µα D n g νβ − D m g µν D n g αβ ) R R = − g µν γ mn γ pq ( ∂ µ γ mp ∂ ν γ nq − ∂ µ γ mn ∂ ν γ pq ) , (28)where the Diff(d)-covariant derivative is given by [15, 16] D m g µν = ∂ m g µν − N ρm ∂ ρ g µν − ( ∂ µ N ρm ) g ρν − ( ∂ ν N ρm ) g µρ . Geometrically, D m g µν is a generalized extrinsic curvature along the m -thextra dimension. One can check that under FPD of (3), D m g µν and F µmn transform covariantly as( D m g µν ) ′ = ∂y n ∂y ′ m ∂x ρ ∂x ′ µ ∂x σ ∂x ′ ν D n g ρσ , (29)( F µmn ) ′ = ∂y p ∂y ′ m ∂y q ∂y ′ n ∂x ′ µ ∂x ν F νpq . (30)One also has the spacetime derivative of the metric γ mn transforming as( ∂ µ γ mn ) ′ = ∂x ν ∂x ′ µ ∂y p ∂y ′ m ∂y q ∂y ′ n ( ∂ ν γ pq ) (31)7nder FPD of (3), because ∂ ′ µ = (cid:16) ∂x ν ∂x ′ µ (cid:17) ∂ ν . Therefore, we find that each ofthe five terms in (28) are separately FDP invariant. Also, usingˆ ∂ ′ m = (cid:18) ∂y n ∂y ′ m (cid:19) ˆ ∂ n (32)under FPD, we find that both the curvature scalar R (4) = g µν R µν and ˆ R ( D ) = γ mn ˆ R mn are FPD invariants. Thus, we can construct a general FPD invariant( d + D ) dimensional action as follows: S ( d + D ) = M d − D ∗ Z d xd D y √− g √ γ h (cid:0) R ( d ) − (cid:1) + α ˆ R ( D ) − α γ mn γ pq g µν F µmp F νnq − α γ mn g µν g αβ ( D m g µα D n g νβ − α D m g µν D n g αβ ) − α g µν γ mn γ pq ( ∂ µ γ mp ∂ ν γ nq − α ∂ µ γ mn ∂ ν γ pq ) i + S m , (33)where α ∼ are arbitrary constants and M d − D ∗ is the higher dimensionalgravitational constant. S m is the matter action which will not be consideredin this work. One can show that when α = · · · = α = 1, terms in theaction (33) combine into the ( d + D ) dimensional Einstein-Hilbert actionwith cosmological constant with the FPD being elevated to the full d + D general covariance. When d = 4 and D = 1, we have α = α = 0 and m = n = 5 . With γ = N , the action can be written as S (5) = M ∗ Z d xdy √− gN " (cid:0) R (4) − (cid:1) − β (cid:16) B µν B µν − λB (cid:17) + β ∇ µ N ∇ µ NN , where K µν is the extrinsic curvature tensor, K µν = ( ∂ y g µν −∇ µ N ν −∇ ν N µ ) / (2 N ). In this section, we consider conformal extension of the action (33) which canbe achieved with additional Weyl scalar field φ [20]. To enforce anisotropybetween the d -dimensional spacetime and extra D -dimensions, one introducea parameter z which characterizes the anisotropy of the extra dimensionsand consider the following conformal transformation g µν → e ω ( x,y ) g µν , γ mn → e zω ( x,y ) γ mn , φ → e − v ( z ) ω ( x,y ) φ, N µm → N µm . (34)Here, v ( z ) can be an arbitrary function of z which can be adjusted with asuitable redefinition of the field φ . We take v = d − zD in order to set the8onformal factor in front of R ( d ) to be φ . z = 1 corresponds to the isotropicconformal transformations [20].Taking into account of the additional scalar field φ , one can construct theanisotropic invariant action in ( d + D ) dimensions by considering conformallyinvariant combinations˜ g µν = φ d − zD g µν , ˜ γ mn = φ zd − zD γ mn , (35)and one obtains S = Z d d xd D y √− g √ γ " L d + L D + L F + L K + L R , (36) L d = φ (cid:16) R ( d ) − d − d − zD ∇ µ ∇ µ φφ + 4 zD ( d − d − zD ) ∇ µ φ ∇ µ φφ (cid:17) − V φ d + zDd − zD , L D = α D φ d + z ( D − d − zD (cid:16) ˆ R ( D ) − z D − d − zD ˆ ∇ m ˆ ∇ m φφ + 4 z ( D − d − z )( d − zD ) ˆ ∇ m φ ˆ ∇ m φφ (cid:17) , L F = − α F φ d +2+ z ( D − d − zD γ mn γ pq g µν F µmp F νnq , L K = − α K φ d + z ( D − d − zD γ mn g µν g αβ (cid:16) ˜ D m g µα ˜ D n g νβ − α ˜ D m g µν ˜ D n g αβ (cid:17) , L R = − α R φ g µν γ mn γ pq (cid:16) ˜ ∇ µ γ mp ˜ ∇ ν γ nq − β ˜ ∇ µ γ mn ˜ ∇ ν γ pq (cid:17) . In the above, V is a constant and the hat covariant derivative is given byˆ ∇ m j m = ˆ ∂ m j m + ˆΓ mmp j p and˜ D m g µν = D m + 4 d − zD ˆ ∇ m φφ ! g µν , ˜ ∇ µ γ mn = (cid:18) ∇ µ + 4 zd − zD ∇ µ φφ (cid:19) γ mn .α D , α F , α K , α R , α and β are also constants but they could in principle have z -dependence with normalization such that they all become unity at z = 1 . With this, the action (36) becomes ( d + D )-dimensional Weyl gravity [20].Note that the invariance under (34) is built in the above action (36). Theaction does not display any dimensionful parameters and is invariant underthe following scaling x → b − x, y → b − z x, φ → b d − zD x, N µm → b z − N µm . (37)This gives z -dependent scaling dimensions as[ x ] s = − , [ y ] s = − z, [ φ ] s = d − zD , [ N µn ] s = z − , (38)others being zero. On the other hand, the anisotropic conformal invarianceis given by (34) and conformal weights are[ g µν ] = 2 , [ γ mn ] = 2 z, [ φ ] = − d − zD , [ N µm ] = 0 . (39)9 Gravity in Diverse Dimensions
In this section, we discuss several aspects of the anisotropic gravity theoryand compare it with the isotropic case in diverse dimensions. We restrictto the case where the coefficients α D = α F = α K = α R = 1 so that gauge-fixed φ = constant action of (36) in the isotropic case z = 1 with α = β = 1 reduces to the ( d + D )-dimensional Einstein-Hilbert action. We firstconsider the lower dimensional gravity with d ≤
3. Then, we discuss higherdimensional spacetime with d ≥ . In either case, we introduce a scale M ∗ which sets the scale for the conformal symmetry breaking of the vacuumsolution. We assume that the field φ and the coordinate y recover theircanonical dimensions with the vacuum solution, and therefore re-scale themvia y → M − z +1 ∗ y, φ → M d − zD ∗ φ, (40)after which the coordinate y has inverse-mass dimension and φ becomes adimensionless field. Let us first consider the case of d = 2 . We consider a rather simple casethat the L D , L F , L K and the term of the scalar potential in L d =2 (beingproportional to V ) vanish. This can be achieved in the limit that z → φ <
1, where the φ is the vev of the scalar field φ as φ = φ M d − zD ∗ . The remaining L and L R terms do not contain any y -derivatives and do not yield towers of massive modes. The spacetime di-mensions is reduced effectively to two. The effective two-dimensional gravitytheory after integrating over y can be written as S (2 ,D ) = φ (cid:16) M ∗ M e (cid:17) D Z d x p − g (2) h f ( X ) R (2) − g µν G IJ ∂ µ X I ∂ ν X J i , (41)where M e is the length associated with scale of the extra dimensions’ co-ordinate volume, [ R dy ] D ≡ M − De . The matter-looking like fields X I ( I =1 , · · · , ( d + D − d + D − / γ mn by γ = X , γ (= γ ) = X , . . . , γ d + Dd + D = X ( d + D − d + D − / . The factor in frontof the curvature scalar R (2) comes from √ γ . S (2 ,D ) can be regarded as two-dimensional gravity coupled with nonlinear sigma model of scalar matterfields.Einstein gravity in 2 + ǫ spacetime dimensions and this type of extensionhas been studied extensively and the most interesting aspect is its renormal-izabilty [21]. This aspect may have some application in the quantum theory10f (2,D) gravity theory. For D = 2, one can show that action (41) can beput into a more familiar expression S (2 , = φ (cid:16) M ∗ M e (cid:17) Z d x p − g (2) h ϕ (cid:16) R (2) − g µν ˜ γ mp ˜ γ nq ∇ µ ˜ γ mn ∇ ν ˜ γ pq (cid:17) −
12 (1 − β ) g µν ϕ − ∂ µ ϕ∂ ν ϕ i , (42)where ϕ ≡ √ γ and ˜ γ mn = γ mn / √ γ with √ ˜ γ = 1 . For 2 β ≤
1, this theory isghost free.One may consider such a theory as a UV-completion of certain low energytheories. The coupling β and the parameter z are able to be scale dependentand may show z = 1 and β = 1 fixed point in IR, where the theory becomesEinstein theory. This scenario is deserved to be studied further.When D = 1 , L D and L F terms vanish identically. For β = 1, L R contains a single scalar given by γ d +1 d +1 . L K term which involve dynamicsalong y -direction can again be made to vanish with z = 2 − d ( φ < . The resulting effective d -theory is gravity coupled with a single scalar field γ d +1 d +1 ≡ N ( x ). More explicitly, for d = 3 and z = − S (3 , = φ (cid:16) M ∗ M e (cid:17) Z d x p − g (3) h R (3) −
12 ( 32 − β ) g µν ∂ µ ϕ∂ ν ϕ i . (43)In the above action, we rescaled g µν → N − g µν and redefined ϕ ≡ √ N .For β < / d = 3 anisotropicgravity furnishes alternative perspective to the three-dimensional gravity.Another interesting limit appears in ( d, D ) = (2 , z → V → ∞ by keeping that V φ d + zDd − zD ≡ Λ is held fixed.In this case, the effective 2-dimensional action is given by S Λ(2 , = φ M ∗ M e Z d xe − ψ (cid:2) R (2) − − β ) g µν ∂ µ ψ∂ ν ψ − Λ (cid:3) , (44)where √ γ = √ γ d +1 ,d +1 ≡ e − ψ . When we choose that the β = , then thisaction becomes “Callan–Giddings–Harvey–Strominger model” or in shortCGHS model action without matters[22]. In this subsection, we search for the solutions of equation of motion derivedfrom (36). We start from a simple ansatz which solves them instead of11riting down the tedious looking equations . For our purpose, the detailedequations of motion are not necessary. Consider the following ansatz g AB = (cid:18) g µν ( x ) 00 γ mn ( y ) (cid:19) , φ = φ , N µm = 0 , (45)which yields F µmp = ˜ D m g µα = ˜ ∇ µ γ mp = 0 and the constant φ is dimension-less.Since L F , L K , and L R are quadratic in F µmp , ˜ D m g µα , and ˜ ∇ µ γ mp respec-tively, the ansatz (45) will automatically solve the equations of motion de-rived from these terms. Then, effectively one can consider S ′ = M d − D ∗ Z d d xd D y √− g √ γ " φ R ( d ) + φ d + z ( D − d − zD ˆ R ( D ) − V M ∗ φ d + zDd − zD . (46)Note that the over-all gravitational scale M ∗ appears as a consequence ofsymmetry breaking in (46). The equations of motion are derived as follows: φ (cid:18) R ( d ) µν − g µν R ( d ) (cid:19) = − V M ∗ φ d + zDd − zD g µν + 12 φ d + z ( D − d − zD ˆ R ( D ) g µν ,φ d + z ( D − d − zD (cid:18) ˆ R ( D ) mn − γ mn ˆ R ( D ) (cid:19) = − V M ∗ φ d + zDd − zD γ mn + 12 φ ˆ R ( d ) γ mn ,φ R ( d ) + d + z ( D − d − zD φ d + z ( D − d − zD ˆ R ( D ) = V d + zDd − zD M ∗ φ d + zDd − zD , (47)where the terms containing derivatives of φ is all ignored since we assumethat φ = φ . To solve the above equations, we make further ansatz R ( d ) µν = Λ ( d ) g µν ( x ) , R ( D ) mn = Λ ( D ) γ mn ( y ) . (48)Then, one obtains the following values of Λ ( d ) and Λ ( D ) :Λ ( d ) = V d − D φ d − zD M ∗ , Λ ( D ) = V d − D φ zd − zD M ∗ . ( D ≥
2) (49)When z = 1, there is no size separation between d dimensional spacetime and D dimensional extra space which means that Λ ( d ) = Λ ( D ) . Large separationcan occur when z is near z = (2 − d ) /D. We obtain the ratio of the size asΛ ( d ) = φ − zd − zD Λ ( D ) (50) See Ref. [9] in some special case with d = 4 and D = 1 he relation between Planck length and the size of the extra di-mension We define the d -dimensional Planck length l P from (46) (with l F = M − ∗ ) l P = (cid:2) φ l De l − d +2 − DF (cid:3) − d +2 ( D ≥
1) (51)with Newton constant G F = l P . Then, We have size of the extra dimensiongiven by l e = l F φ − D (cid:18) l F l P (cid:19) d − D . (52)Note expression (52) is independent of anisotropic factor z and almostidentical to the one given in the brane-world models except the factor φ necessary for conformal invariance. Another crucial difference is that M ∗ which we introduced for anisotropic conformal symmetry breaking cannot beconsidered as ( d + D )-dimensional Planck scale; we do not have the notion of( d + D )-dimensional Planck length (at least classically), because the theory isintrinsically formulated in anisotropic spacetime. But we have d -dimensionalNewton constant. We assume l F ∼ − cm which is somewhat lower thanthe present-day accelerators can explore. (52) displays many possibilitiesincluding large extra dimensions. When d = 4 and D = 1, l p = 10 − cm, l e ∼ cm for φ ∼ . However, for φ ∼ . , we can have l e ∼ − cm.When d − D , we have a strange coincidence that the size of the extradimensions is always given by l e = 10 − cm for φ ∼ l e = 10 − cm is stillenough to detect by experiment. However, in string theory, by assumingthat we are living on a D3-brane, only closed string can probe the extradimensions, whose massless excitaions correspond to gravitational fields. Thegravity is not certain if it obeys inverse-equare law below or near about10 − cm yet. If the gravity force becomes weaker below this length scale asgravity force ∼ r − k , where k >
2, then that could be an indication of extradimensions. Of course, l e = 10 − cm in dimension other than d = 4 is notwell motivated by experiment.From (49), d -dimensional spacetime can be characterized by a positivecosmological constant if V > V < ( D ) determines whether the extra dimensionsare compact (Λ ( D ) >
0) or non-compact (Λ ( D ) < : For a general value of α D different from 1, we have Λ ( D ) = α − D V d − D φ zd − zD M ∗ .Then, we have two more possibilities; (iii) de Sitter( d ) × N C D for V > , α D < d ) × C D for V < , α D < . d ) × C D for V > , α D > . (ii) anti-de Sitter( d ) × N C D for V < , α D > . Here, C D ( N C D ) denotes compact (non-compact) D -dimensional extra space. We investigated various aspects of anisotropic gravity in which the scalingproperties of the d -dimensional spacetime and the extra D -dimensions aredifferent and characterized by the parameter z . The parameter z is a crucialfactor in our model. The most salient feature is that z can act as an agentto (de)activate the extra dimensions for some specific value and induce thedimensional reduction. When z = − d − D , the kinenamics along the extradimensions are completely suppressed and the extra dimensions are virtu-ally obsolete. This suggests a quantum gravity where the four-dimensionalEinstein gravity reduces to a certain two-dimension theory: Einstein gravitystarts effectively in two-dimensional spacetime at z = 0 in UV where it isrenormalizable, and it flows to isotropic z = 1 in IR recovering the four-dimensional spacetime. It also provide a possibility to address the originof large scale separations between d -dimensional spacetime and the extra D -dimension in physics.Our model also suggests a resolution to the hierarchy problem. Let us con-sider a scalar matter field Φ(for simplicity) under the (foliation preserving)diffeomorphism and the scale transformation and it lives in d-dimensionalmanifold, i.e. Φ → Φ( x ). Assume that the matter field action is given by S matter = Z d d xd D y √− g √ γ (cid:20) φ g µν ∂ µ Φ ∂ ν Φ − M ∗ φ d + Dz ) d − Dz Φ (cid:21) , (53)where the natural mass scale of the matter field Φ is M ∗ which is the onlyscale appearing in our model. This action is manifestly invariant under thediffeomorphism and the scale transformation. As we discuss above, let usassume that the field φ has a vev, φ . To keep the kinetic part to be usual,take a field redefinition of the Φ as Φ = ( √ V D φ ) − ˜Φ, where V D = R √ γd D y .By integrating the D -dimensional volume, one can construct d -dimensionaleffective action of the field ˜Φ. After all, one realizes that the physical mass ofthe field ˜Φ becomes M ∗ exp (cid:0) d − Dz log φ (cid:1) . The factor, exp (cid:0) d − Dz log φ (cid:1) can give large scale distinction between M ∗ and the physical mass of the field˜Φ for z = − dD ± ǫ where ǫ is a small(but not very much small) and positivenumber.Randall-Sundrum model[24, 25] gives rise to a large hierarchy betweentwo scales appearing in their theory. The geometric setting is pretty much14ifferent from ours. They consider a 5-dimensional space of which topology isgiven by M × S , where the space M is a 4-dimensional Minkowskian man-ifold. The circle S is parameterized by an angle ψ and they put extendedobjects, which are called branes at ψ = 0 and ψ = π . The brane at ψ = 0is given a name as ‘visible brane’, where we live on and the one at φ = π iscalled ‘hidden brane’, which we cannot observe. The solution of the Einsteinequation of the system is ”so called” warp geometry which is characterizedby a factor e − kr c | ψ | , and we call it ‘ warp factor ’. The k = q − Λ24 M , the Λ is5-dimensional cosmological constant and the M is the 5-dimensional energyscale which is related to 5-dimensional Newton constant as G ∼ M − . Itturns out that any physical mass scale m phy in 4-dimensional effective the-ory on the visible brane is suppressed by the warp factor from the massparameter, m appearing in the visible sector theory as m phy = e − kr c π m The resolution to the hierarchy problem in Randall-Sundrum model ismediated by the large kr c . If kr c = 15 /π ∼ .
8, the we get electro-weak scalefrom m by assuming that the m is in Planck scale. On the other hand,the large hierarchy occurs in our model due to the scaling factor z , which issomewhat tuned. Appendices
We consider the following matter field action which is invariant under the(foliation preserving) diffeomorphism and the scale transformation. S matter = Z d d xd D y √− g √ γ (cid:20) φ g µν ∂ µ Φ ∂ ν Φ + 12 φ d + zD − zd − zD γ mn ∂ m Φ ∂ n Φ − φ d + Dz ) d − Dz f (Φ) (cid:21) , (54)where the f (Φ) is an arbitrary function of the matter field Φ. As we discussed,the field φ has a vev, φ . With a rescaling as Φ = ( √ V D φ ) − ˜Φ, the actionbecomes S matter = Z d d x √− g Z d D y √ γV D (cid:20) g µν ∂ µ ˜Φ ∂ ν ˜Φ + 12 φ − z ) d − zD γ mn ∂ m ˜Φ ∂ n ˜Φ − V eff (cid:21) . (55)What we want to get is an effective d − dimensional action from this by sup-posing that the matter field ˜Φ does not depend on the coordinate of extradimension, y m , i.e. ˜Φ → ˜Φ( x µ ). This causes that the second term in (55)vanishes and it lets the factor, R d D y √ γV D →
1. The effective potential V eff is15iven by V eff = V D M D + d ∗ φ d + Dz ) d − Dz f M − D − d ∗ ˜Φ √ V D φ ! (56) → ∞ X n =1 a n n ! (cid:0) V D M D + d ∗ φ (cid:1) − n M n ∗ φ d + zD − ˜Φ n . For the last line in (56), we assume that the f ( X ) = P ∞ n =2 a n n ! X n with con-stants, a n . Recall (51) and (52) then we realize that V D M D + d ∗ φ = M ∗ M d − p .Therefore, the effective potential becomes V eff = ∞ X n =1 a n n ! (cid:0) M ∗ M d − p (cid:1) − n M n ∗ φ d + zD − ˜Φ n . (57)The effective potential depends on the d -dimensional Planck energy scale, M ∗ and the z . In fact, it relies on the D -dimensional volume implicitly since M p is determined once M ∗ , φ and l e are to be chosen(See equations, (51)and (52)). One may truncate the effective potential upto an order of ˜Φ and demandthat a = −| a | , a = 0 and a is positive to form a Mexican-hat potential.The spontaneous symmetry breaking occurs at ∂ V eff ∂ ˜Φ = 0, where the field ˜Φhas its vev in the real vacuum in the presence of such a potential. The valueof the vev is given by Φ = − (cid:18) a a (cid:19) M d − p . (58)The radial directional perturbation around the vacuum in the field space of˜Φ produces a massive particle and its physical mass also shows a large scaledifference from the scale M ∗ as M phys = exp (cid:0) d − Dz log φ (cid:1) when z = − dD ± ǫ where ǫ is a small(but not very much small) and positive number. Acknowledgement
J.H.O would like to thank his W .J. and Y .J. This work was supported bythe National Research Foundation of Korea(NRF) grant funded by the Ko-rea government(MSIP) (No.2016R1C1B1010107). This work is also partiallysupported by Research Institute for Natural Sciences, Hanyang University.16 eferences [1] C. V. Johnson, doi:10.1142/9789812799630 0002 [arXiv:hep-th/0007170[hep-th]].[2] “String Theory and M-Theory: A Modern Introduction”, by KatrinBecker, Melanie Becker, John H. Schwarz, Cambridge University Press(December 7, 2006); String Theory, Vol. 1 and Vol. 2, by Joseph Polchin-ski, Cambridge University Press (June 2, 2005)[3] T. Appelquist, A. Chodos and P. G. O. Freund, “MODERN KALUZA-KLEIN THEORIES,” ADDISON-WESLEY (1987)[4] S. Raychaudhuri and K. Sridhar, “Particle Physics of Brane Worlds andExtra Dimensions,” doi:10.1017/CBO9781139045650[5] S. Carlip, Universe (2019), 83 doi:10.3390/universe5030083[arXiv:1904.04379 [gr-qc]].[6] J. C. Long, H. W. Chan, A. B. Churnside, E. A. Gulbis, M. C. M. Var-ney and J. C. Price, Nature , 922 (2003) doi:10.1038/nature01432[hep-ph/0210004].[7] A. Joyce, B. Jain, J. Khoury and M. Trodden, Phys. Rept. , 1 (2015)doi:10.1016/j.physrep.2014.12.002 [arXiv:1407.0059 [astro-ph.CO]].[8] P. Horava, Phys. Rev. D (2009), 084008doi:10.1103/PhysRevD.79.084008 [arXiv:0901.3775 [hep-th]].[9] T. Moon and P. Oh, JCAP (2017), 024 doi:10.1088/1475-7516/2017/09/024 [arXiv:1705.00866 [hep-th]].[10] G. Hinshaw et al. [WMAP], Astrophys. J. Suppl. , 19 (2013)doi:10.1088/0067-0049/208/2/19 [arXiv:1212.5226 [astro-ph.CO]].[11] P. A. R. Ade et al. [Planck], Astron. Astrophys. , A16 (2014)doi:10.1051/0004-6361/201321591 [arXiv:1303.5076 [astro-ph.CO]].[12] S. Kouwn, P. Oh and C. G. Park, Phys. Dark Univ. (2018), 27-37doi:10.1016/j.dark.2018.08.003 [arXiv:1709.08499 [astro-ph.CO]].[13] T. Kim and P. Oh, [arXiv:2007.01551 [hep-th]].[14] S. Kouwn, P. Oh and C. G. Park, Phys. Rev. D , no. 8, 083012 (2016)doi:10.1103/PhysRevD.93.083012 [arXiv:1512.00541 [astro-ph.CO]].1715] Y. M. Cho, K. S. Soh, J. H. Yoon and Q. H. Park, Phys. Lett. B (1992) 251. doi:10.1016/0370-2693(92)91771-Z[16] J. H. Yoon, Phys. Lett. B , 296 (1999) doi:10.1016/S0370-2693(99)00202-6 [gr-qc/0003059].[17] C. W. Misner, K. S. Thorne and J. A. Wheeler, San Francisco 1973,1279p[18] Y. M. Cho and P. G. O. Freund, Phys. Rev. D , 1711 (1975).doi:10.1103/PhysRevD.12.1711[19] J. H. Yoon, [arXiv:gr-qc/9611050 [gr-qc]].[20] See T. Y. Moon, J. Lee and P. Oh, Mod. Phys. Lett. A , 3129(2010) doi:10.1142/S0217732310034201 [arXiv:0912.0432 [gr-qc]], andreferences therein.[21] H. Kawai, Y. Kitazawa and M. Ninomiya, Nucl. Phys. B , 313-331(1996) doi:10.1016/0550-3213(96)00119-8 [arXiv:hep-th/9511217 [hep-th]].[22] C. G. Callan, Jr., S. B. Giddings, J. A. Harvey and A. Strominger,Phys. Rev. D , no.4, 1005 (1992) doi:10.1103/PhysRevD.45.R1005[arXiv:hep-th/9111056 [hep-th]].[23] B. Zwiebach, “A first course in string theory,”Cambridge Univ. Pr.(2004).[24] L. Randall and R. Sundrum, Phys. Rev. Lett. , 4690-4693 (1999)doi:10.1103/PhysRevLett.83.4690 [arXiv:hep-th/9906064 [hep-th]].[25] L. Randall and R. Sundrum, Phys. Rev. Lett.83