Non-perturbative scalar gauge-invariant metric fluctuations from the Ponce de Leon metric in the STM theory of gravity
aa r X i v : . [ g r- q c ] F e b Non-perturbative s alar gauge-invariant metri (cid:29)u tuations from the Pon e de Leónmetri in the STM theory of gravity , Mariano Anabitarte ∗ , , Mauri io Bellini † Departamento de Físi a, Fa ultad de Cien ias Exa tas y Naturales,Universidad Na ional de Mar del Plata, Funes 3350, (7600) Mar del Plata, Argentina. Consejo Na ional de Investiga iones Cientí(cid:28) as y Té ni as (CONICET).We study our non-perturbative formalism to des ribe s alar gauge-invariant metri (cid:29)u tuationsby extending the Pon e de León metri .I. INTRODUCTIONThe possibility that our world may be embedded in a (4 + d ) -dimensional universe with more than four largedimensions has attra ted the attention of a great number of resear hes. One of these higher-dimensional theories,where the ylinder ondition of the Kaluza-Klein theory[1℄ is repla ed by the onje ture that the ordinary matter and(cid:28)elds are on(cid:28)ned to a 4D subspa e usually referred to as a brane is the Randall and Sundrum model[2℄.Another non- ompa t theory is the so alled Spa e - Time - Matter (STM) or Indu ed Matter (IM) theory. Inthis theory the onje ture is that the ordinary matter and (cid:28)elds that we observe in 4D result from the geometryof the extra dimension[3℄. In this framework, in(cid:29)ationary models indu ed from a 5D va uum state, where theexpansion of the universe is driven by a single s alar (in(cid:29)aton) (cid:28)eld, has been subje t of great a tivity in the lastyears[4℄. The s alar metri (cid:29)u tuations related to the in(cid:29)aton (cid:28)eld (cid:29)u tuations an be studied as invariant undergauge transformations in a standard 4D osmologi al model[5℄, or from a 5D va uum theory of gravity[6℄. Theseperturbations are related with energy density perturbations. They are spin-zero proje tions of the graviton, whi honly exist in non-va uum osmologies. The issue of gauge invarian e be omes riti al when we attempt to analyzehow the s alar metri (cid:29)u tuations ψ produ ed in the very early universe in(cid:29)uen e the expansion with respe t to the4D ba kground isotropi , homogeneous and 3D spatially (cid:29)at osmologi al metri . From the osmologi al point ofview, these metri (cid:29)u tuations are produ ed by the in(cid:29)aton (cid:28)eld (cid:29)u tuations ϕ − h ϕ i , whi h des ribe the quantum(cid:29)u tuations of the in(cid:29)aton (cid:28)eld with respe t to the expe tation value of this (cid:28)eld on the 3D sphere: h ϕ i , in theabsen e of metri (cid:29)u tuations. In other words, from the relativisti point of view, the quantum (cid:28)eld (cid:29)u tuations ofthe in(cid:29)aton (cid:28)eld (whi h is a s alar (cid:28)eld) indu e the quantum (s alar) metri (cid:29)u tuations on the ba kground metri .From the mathemati al point of view the metri (cid:29)u tuations are the geometri al deformations produ ed by quantum(cid:28)eld (cid:29)u tuations of the in(cid:29)aton (cid:28)eld.We onsider the 5D ba kground line element[8℄ dS = l dt − (cid:18) tt (cid:19) p l pp − dr − t ( p − dl , (1)were dr = dx + dy + dz , and l is the non- ompa t extra dimension and p is a dimensionless onstant. Thismetri is 3D spatially isotropi , homogeneous and Riemann (cid:29)at: ¯ R ABCD = 0 ( A and B run from to ), but it is urved in four dimensions[7℄. From the physi al point of view, this metri represents an apparent va uum ¯ G AB = 0 ,whi h deserves interest in spa e-time-matter theory. The parti ular ase in whi h we take a foliation l = l it is veryimportant for osmology[8, 9℄ dS = l dt − (cid:18) tt (cid:19) p l pp − dr , (2)be ause it des ribes an e(cid:27)e tive 4D universe that expands with a s ale fa tor a ( t ) ∼ t p , with a pressure P = (2 − p ) p πGl t and an energy density ρ = p πGl t . In parti ular, in the limit ase in whi h p → ∞ , the metri (2) des ribes anin(cid:29)ationary expansion of the universe[10℄ with a va uum dominated equation of state P ≃ − ρ . Other important ases are p = 1 / , / , whi h des ribe, respe tively, radiation and matter dominated universes in absen e of va uum.In this letter we shall study a non-perturbative formalism for gauge-invariant metri (cid:29)u tuations ψ ( x ) in the STM ∗ E-mail address: anabitarmdp.edu.ar † E-mail address: mbellinimdp.edu.artheory of gravity, starting with the Pon e de León metri (1).II. FORMALISMWith the aim to study strong gauge-invariant (s alar) metri (cid:29)u tuations, we propose the following metri : dS = l e ψ dt − (cid:18) tt (cid:19) p l pp − e − ψ dr − t ( p − e − ψ dl , (3)where ψ ( t, x, y, z, l ) is a quantum s alar (cid:28)eld. This metri is a generalization for strong gauge-invariant (s alar) metri (cid:29)u tuations of one previously studied in [11℄, whi h is only valid for small metri (cid:29)u tuations: e ± ψ ≃ ± ψ . Todes ribe the system in an apparent va uum, we shall onsider the a tion (5) I = Z d x dl s(cid:12)(cid:12)(cid:12)(cid:12) (5) g (5) g (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) (5) R πG + 12 g AB ϕ ,A ϕ ,B (cid:19) , (4)where (5) g is the determinant of the ovariant metri tensor g AB : (5) g = " t (cid:18) tt (cid:19) p l p − p − e − ψ ( p − . (5)A. Lagrange equations in a 5D apparent va uumThe Ri i S alar, whi h in our ase is null, being given by the expression (5) R = 4 l − pp − (cid:18) tt (cid:19) − p e ψ ( ∇ ψ − ( ∇ ψ ) + e − ψ l p − (cid:18) tt (cid:19) p (cid:20) ψ ,t ) − ψ ,tt − t (3 p + 1) ψ ,t (cid:21) + l pp − t (cid:18) tt (cid:19) p ( p − (cid:20) ( p − ψ ,ll − ( p − ψ ,l ) + 3 pl ψ ,l (cid:21) − p t l p − (cid:18) tt (cid:19) p (cid:2) − e − ψ (cid:3)) , (6)where ψ ,A = ∂∂A . The Lagrange equations give us the relevant equations of motion for the (cid:28)elds ϕ and ψ , respe tively: ϕ ,tt + (cid:20) (3 p + 1) t − ψ ,t (cid:21) ϕ ,t − e ψ (cid:18) tt (cid:19) − p l − p − (cid:16) ∇ ϕ − ~ ∇ ψ.~ ∇ ϕ (cid:17) − l t e ψ ( p − (cid:20) ϕ ,ll + (cid:18) (4 p − p − l − − ψ ,l (cid:19) ϕ ,l (cid:21) = 0 , (7) (cid:18) ∂ (5) R∂ψ − (5) R (cid:19) − q(cid:12)(cid:12) (5) g (cid:12)(cid:12) ∂ q(cid:12)(cid:12) (5) g (cid:12)(cid:12) ∂x A ∂ (5) R∂ψ ,A + ∂∂x A (cid:18) ∂ (5) R∂ψ ,A (cid:19) = 8 πG " l − e − ψ ( ϕ ,t ) − (cid:18) tt (cid:19) − p l − pp − e ψ ( ∇ ϕ ) − e ψ ( p − t ( ϕ ,l ) . (8)The 5D energy momentum tensor for a s alar (cid:28)eld ϕ in the absen e of intera tions, is T AB = ϕ ,A ϕ ,B − g AB ϕ ,C ϕ ,C , (9)where the omponents g AB are given by the perturbed metri (3). Using the fa t that T tt is given by the expression T tt = 12 " l − e − ψ ( ϕ ,t ) + (cid:18) tt (cid:19) − p l − pp − e ψ ( ∇ ϕ ) + e ψ ( p − t ( ϕ ,l ) , (10)(8) an be written in a more ompa t manner as (cid:18) ∂ (5) R∂ψ − (5) R (cid:19) − q(cid:12)(cid:12) (5) g (cid:12)(cid:12) ∂ q(cid:12)(cid:12) (5) g (cid:12)(cid:12) ∂x A ∂ (5) R∂ψ ,A + ∂∂x A (cid:18) ∂ (5) R∂ψ ,A (cid:19) = 8 πG (cid:2) l − e − ψ ( ϕ ,t ) − T tt (cid:3) . (11)Equations (7) and (11) relate the quantum (cid:28)eld ϕ with (quantum gauge-invariant) s alar metri (cid:29)u tuations.B. 5D Einstein equations on an apparent va uumTo obtain the Einstein equations, we shall al ulate the omponents of the Einstein tensor. Their diagonal ompo-nents (we onsider G rr = G xx + G yy + G zz ) are given by G tt = − t ( t ( ψ ,t ) − (3 p + 1) t ψ ,t − e ψ l − p − (cid:18) tt (cid:19) − p t (cid:2) ( ∇ ψ ) − ∇ ψ (cid:3) + e ψ l ( p − (cid:20) ψ ,ll − ( ψ ,l ) − pl ( p − ψ ,l (cid:21) − p ( p − (cid:2) e ψ − (cid:3)(cid:27) , (12) G rr = − l t ( e − ψ l pp − (cid:18) tt (cid:19) p (cid:2) t ψ ,tt − t ( ψ ,t ) + 5(2 p + 1) tψ ,t (cid:3) + 23 l t (cid:2) ( ∇ ψ ) − ∇ ψ (cid:3) + (cid:18) tt (cid:19) p l p − p − ( p − (cid:20) ( ψ ,l ) − ψ ,ll − ( p − l ( p − ψ ,l (cid:21) + 3 (cid:18) tt (cid:19) p l p − p − p (cid:2) − e − ψ (cid:3)) , (13) G ll = − l ( p − ( e − ψ (cid:2) t ψ ,tt − t ( ψ ,t ) + 15 ptψ ,t (cid:3) + l − p − (cid:18) tt (cid:19) − p t (cid:2) ( ∇ ψ ) − ∇ ψ (cid:3) + l ( − p + 9 p − ψ ,l + 3 p (2 p − (cid:2) − e − ψ (cid:3)(cid:9) . (14)Sin e we require the Ri i s alar (5) to be null: (5) R = 0 , we obtain (cid:2) − e − ψ (cid:3) = t p l − p − (cid:18) tt (cid:19) − p ( ∇ ψ − ( ∇ ψ ) + e − ψ l p − (cid:18) tt (cid:19) p (cid:20) ψ ,t ) − ψ ,tt − t (3 p + 1) ψ ,t (cid:21) + l pp − t (cid:18) tt (cid:19) p ( p − (cid:20) ( p − ψ ,ll − ( ψ ,l ) ) + 3 pl ψ ,l (cid:21)) , (15)so that, using (15) in (12)-(14), we obtain the 5D diagonal omponents of the Einstein tensor: G tt = − t ( − t ( p + 7)3 p ( ψ ,t ) + t p + 1)3 p ψ ,tt + (3 p + 1) p t ψ ,t + e ψ l − p − (cid:18) tt (cid:19) − p t (cid:2) ( ∇ ψ ) − ∇ ψ (cid:3) + e ψ l ( p − p − p (cid:20) ψ ,ll − ( ψ ,l ) − pl (2 p −
1) (4 p + 1)( p − ψ ,l (cid:21)(cid:27) , (16) G rr = − l t ( e − ψ l pp − (cid:18) tt (cid:19) p (cid:2) t ψ ,tt − t ( ψ ,t ) + ( p + 2) tψ ,t (cid:3) − l t (cid:2) ( ∇ ψ ) − ∇ ψ (cid:3) + (cid:18) tt (cid:19) p l p − p − (2 p − p + 1) ψ ,l ) (17) G ll = − l ( p − ( e − ψ " (2 − p ) p t ψ ,tt + (5 p − p t ( ψ ,t ) − (cid:0) p − p − (cid:1) p tψ ,t − ( p − p l − p − (cid:18) tt (cid:19) − p t (cid:2) ( ∇ ψ ) − ∇ ψ (cid:3) + l (2 p − p ( p − (cid:2) ψ ,ll − ( ψ ,l ) (cid:3)) . (18)The diagonal omponents T AB of the ( ovariant) energy momentum tensor are [see the expression (9)℄ T tt = 12 ( ( ϕ ,t ) + e ψ l − p − (cid:18) tt (cid:19) − p ( ∇ ϕ ) + l t e ψ ( p − ( ϕ ,l ) ) , (19) T ll = 12 ( ( ϕ ,l ) − t l e − ψ ( p − ( ϕ ,t ) + t ( p − l − pp − (cid:18) tt (cid:19) − p ( ∇ ϕ ) ) , (20) T xx = ( ϕ ,x ) + 12 l p − (cid:18) tt (cid:19) p e − ψ ( ϕ ,t ) −
12 ( ∇ ϕ ) − l pp − (cid:18) tt (cid:19) p ( p − t ( ϕ ,l ) , (21) T yy = ( ϕ ,y ) + 12 l p − (cid:18) tt (cid:19) p e − ψ ( ϕ ,t ) −
12 ( ∇ ϕ ) − l pp − (cid:18) tt (cid:19) p ( p − t ( ϕ ,l ) , (22) T zz = ( ϕ ,z ) + 12 l p − (cid:18) tt (cid:19) p e − ψ ( ϕ ,t ) −
12 ( ∇ ϕ ) − l pp − (cid:18) tt (cid:19) p ( p − t ( ϕ ,l ) . (23)Sin e the metri (3) is 3D spatially isotropi , we an make the identi(cid:28) ation T rr = T xx + T yy + T zz , and we obtain T rr = 32 l p − (cid:18) tt (cid:19) p e − ψ ( ϕ ,t ) −
12 ( ∇ ϕ ) − l pp − (cid:18) tt (cid:19) p ( p − t ( ϕ ,l ) . (24)Finally, using the expression (12)-(14) with (19), (24) and (20), we obtain ( p + 1)3 p ψ ,tt − ( p + 7)3 p ( ψ ,t ) + (3 p + 1) p t ψ ,t + e ψ l − p − (cid:18) tt (cid:19) − p − p p (cid:2) ( ∇ ψ ) − ∇ ψ (cid:3) + e ψ l t ( p − p − p (cid:20) ψ ,ll − ( ψ ,l ) − p (4 p + 1) l (2 p − p − ψ ,l (cid:21)(cid:27) = 4 πG ( ( ϕ ,t ) + e ψ l − p − (cid:18) tt (cid:19) − p ( ∇ ϕ ) + l t e ψ ( p − ( ϕ ,l ) ) , (25) e − ψ l p − (cid:18) tt (cid:19) p (cid:20) ψ ,tt − ψ ,t ) + ( p + 2) t ψ ,t (cid:21) − (cid:2) ( ∇ ψ ) − ∇ ψ (cid:3) + (cid:18) tt (cid:19) p l p +1 p − t (2 p − p + 1) ψ ,l = 4 πG ( l p − (cid:18) tt (cid:19) p e − ψ ( ϕ ,t ) − ( ∇ ϕ ) − l pp − (cid:18) tt (cid:19) p ( p − t ( ϕ ,l ) ) , (26) t l e − ψ (cid:20) (2 − p ) p ψ ,tt + (5 p − p ( ψ ,t ) − t p − p − p ψ ,t (cid:21) + ( p − p l − p − p − (cid:18) tt (cid:19) − p t (cid:2) ( ∇ ψ ) − ∇ ψ (cid:3) − (2 p − p − p (cid:2) ψ ,ll − ( ψ ,l ) (cid:3) = − πG ( ( p − ( ϕ ,l ) − t l e − ψ ( ϕ ,t ) + t l − pp − (cid:18) tt (cid:19) − p ( ∇ ϕ ) ) . (27)Equations (25), (26) and (34) give us the diagonal Einstein equations on a 5D apparent va uum. In the followingse tion we shall use these equations to des ribe the e(cid:27)e tive 4D physi s on a urved 4D hypersurfa e whi h is embeddedon the perturbed metri (3).III. EFFECTIVE 4D DYNAMICS OF ϕ AND GAUGE-INVARIANT METRIC FLUCTUATIONS ψ In order to study the e(cid:27)e tive 4D dynami s of the (cid:28)elds ϕ and ψ , we an make a foliation l = l , so that dl = 0 and the perturbed 4D hypersurfa e of (2), results to be dS = l e ψ dt − (cid:18) tt (cid:19) p l pp − e − ψ dr . (28)An interesting limital ase of the metri (28) is that in whi h p → ∞ . In su h a ase the metri (28) des ribesan asymptoti ally va uum expansion (i.e., a de Sitter expansion), whi h should be relevant to des ribe the early(in(cid:29)ationary) universe. The parti ular ase where the gauge-invariant metri (cid:29)u tuations are weak, was studied in[11℄. The e(cid:27)e tive 4D a tion is (4) I = R d x (4) L , where (4) L is the e(cid:27)e tive 4D Lagrangian (4) L = s(cid:12)(cid:12)(cid:12)(cid:12) (4) g (4) g (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) (4) R πG + 12 g µν ϕ ,µ ϕ ,ν − V (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l = l , (29)su h that V is the e(cid:27)e tive 4D potential indu ed by the foliation l = l : V = − g ll (cid:18) ∂ϕ∂l (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l = l , (30)and (4) R is the e(cid:27)e tive 4D Ri i s alar whose origin is also geometri ally indu ed by the foliation on the (cid:29)at metri (3) (4) R = 2 e − ψ l t ( p (2 p − − t (cid:18) tt (cid:19) − p l − pp − e ψ (cid:2) ( ∇ ψ ) − ∇ ψ (cid:3) − t (cid:2) pψ ,t − t ( ψ ,t ) + tψ ,tt (cid:3)) . (31)The e(cid:27)e tive 4D (diagonal) Einstein equations for the omponents tt , rr and ll are given respe tively by ( − ( p + 7)3 p ( ψ ,t ) + 2( p + 1)3 p ψ ,tt + (3 p + 1) p t ψ ,t + e ψ l − p − (cid:18) tt (cid:19) − p − p p (cid:2) ( ∇ ψ ) − ∇ ψ (cid:3) + e ψ l t ( p − p − p (cid:20) ψ ,ll − ( ψ ,l ) − p (4 p + 1) l (2 p − p − ψ ,l (cid:21)(cid:27)(cid:12)(cid:12)(cid:12)(cid:12) l = l = 4 πG ( ( ϕ ,t ) + e ψ l − p − (cid:18) tt (cid:19) − p ( ∇ ϕ ) + l t e ψ ( p − ( ϕ ,l ) )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l = l , (32) e − ψ l p − (cid:18) tt (cid:19) p (cid:20) ψ ,tt − ψ ,t ) + ( p + 2) t ψ ,t (cid:21) − (cid:2) ( ∇ ψ ) − ∇ ψ (cid:3) + (cid:18) tt (cid:19) p l p +1 p − t (2 p − p + 1) ψ ,l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l = l = 4 πG ( l p − (cid:18) tt (cid:19) p e − ψ ( ϕ ,t ) − ( ∇ ϕ ) − l pp − (cid:18) tt (cid:19) p ( p − t ( ϕ ,l ) )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l = l , (33) t l e − ψ (cid:20) (2 − p ) p ψ ,tt + (5 p − p ( ψ ,t ) − t p − p − p ψ ,t (cid:21) + ( p − p l − p − p − (cid:18) tt (cid:19) − p t (cid:2) ( ∇ ψ ) − ∇ ψ (cid:3) − (2 p − p − p (cid:2) ψ ,ll − ( ψ ,l ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) l = l = − πG ( ( p − ( ϕ ,l ) − t l e − ψ ( ϕ ,t ) + t l − pp − (cid:18) tt (cid:19) − p ( ∇ ϕ ) )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l = l . (34)The e(cid:27)e tive Lagrangian equations are ϕ ,tt + (cid:20) (3 p + 1) t − ψ ,t (cid:21) ϕ ,t − e ψ (cid:18) tt (cid:19) − p l − p − (cid:16) ∇ ϕ − ~ ∇ ψ.~ ∇ ϕ (cid:17) − l t e ψ ( p − (cid:20) ϕ ,ll + (cid:18) (4 p − p − l − − ψ ,l (cid:19) ϕ ,l (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) l = l = 0 . (35) (cid:18) ∂ (4) R∂ψ − (4) R (cid:19) − q(cid:12)(cid:12) (4) g (cid:12)(cid:12) ∂ q(cid:12)(cid:12) (4) g (cid:12)(cid:12) ∂x A ∂ (4) R∂ψ ,A + ∂∂x A (cid:18) ∂ (4) R∂ψ ,A (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l = l = 8 πG (cid:2) l − e − ψ ( ϕ ,t ) − T tt (cid:3)(cid:12)(cid:12) l = l , (36)whi h, like the Einstein equations (25), (26) and (34), are non-linear. They give us the dynami s of the systemdes ribed by the (cid:28)elds ϕ and ψ .Sin e the (cid:28)eld ϕ ( t, ~r ) is of quantum origin, it should be des ribed by the following non- ommutative algebra [ ϕ ( t, ~r ) , Π ϕ ( t, ~r ′ )] = i g tt s(cid:12)(cid:12)(cid:12)(cid:12) (4) g (4) g (cid:12)(cid:12)(cid:12)(cid:12) e − R [ p +1 t − ψ ,t ] dt δ (3) ( ~r − ~r ′ ) , (37)where (cid:12)(cid:12) (4) g (cid:12)(cid:12) = (cid:18) e − ψ (cid:16) tt (cid:17) p l p − p − (cid:19) is the determinant of the e(cid:27)e tive 4D perturbed metri tensor g µν . The anoni almomentum Π ϕ = ∂ (4) L∂ ˙ ϕ is given by Π ϕ = g tt s(cid:12)(cid:12)(cid:12)(cid:12) (4) g (4) g (cid:12)(cid:12)(cid:12)(cid:12) ˙ ϕ. (38)Of ourse, due to the non-linear nature of the Einstein equations, it is almost impossible to resolve the (cid:28)eld equationswithout making some approximation. IV. FINAL COMMENTSWe have extended to the Pon e de León metri in the non-perturbative formalism proposed in [11℄. To do this, wehave introdu ed the metri (3), whi h, on e we take a foliation l = l , takes into a ount the gauge-invariant metri (cid:29)u tuations during the expansion of the universe at its origin [see eq. (28)℄, as a ba k rea tion e(cid:27)e t of the in(cid:29)aton(cid:28)eld (cid:29)u tuations ϕ − h ϕ i . The ba kground 4D version of the Pon e de León metri is very important for osmology,be ause it des ribes a power-law expansion for the universe. The interesting thing of this formalism is that thesystem is onsidered from the point of view of a 5D perturbed (cid:29)at metri , on whi h we assume an apparent va uumstate. Hen e, all 4D sour es ome from the geometri al foliation l = l on the 5D metri (3) (whi h is Riemann (cid:29)at).The advantage of this formalism with respe t to another one previously introdu ed[10℄ should be in the des riptionof the strong metri (cid:29)u tuations, whi h should be more important in the early universe on very small s ales. Of ourse, the results obtained in[10℄ should here be re overed in the weak (cid:28)eld approximation. In this approximationba k rea tion e(cid:27)e ts be ome negligible and the 4D version of the Pon e de León metri [see the eq. (2)℄, des ribesin the limit ase p → ∞ a va uum dominated expansion of the universe with an equation of state P /ρ = ω ≃ −1