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Massive Kaluza-Klein Gravity

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Massive Kaluza-Klein Gravity Massive Kaluza-Klein Gravity Maia/Disrael UnB Brasïlia Massive Kaluza-Klein Gravity [email protected] 1-Massive Gravity 2-Kaluza-Klein Maia/Disrael 3-Cosmology UnB 4-Quantum Gravity Brasïlia [email protected] Quantum Gravity in the Southern Cone ICTP-SAIFR, Maresias, São Paulo, September, 2013 Massive Kaluza-Klein Gravity Maia/Disrael UnB Brasïlia 1 Massive and massless spin-2 fields. [email protected] Early in the 70’s, a debate emerged on the possibility that 1-Massive Gravity the massless gravitation described by the Einstein-Hilbert 2-Kaluza-Klein 3-Cosmology action could be considered as the zero mass limit m → 0 of 4-Quantum Gravity the Pauli-Fierz action. However, it was soon noted by van Dam, Veltmann and Zakharov (vDVZ), that such limit would lead to a gravitational theory that contradicts by 1/4 of the result of the light bending measurements of 1919. Massive Kaluza-Klein Gravity Maia/Disrael In 1972 Vainshtein suggested that the vDVZ limit problem UnB Brasïlia could be solved by considering a Fierz-Pauli theory with [email protected] non-linear terms in the equation of motion. However, in that 1-Massive Gravity same year Boulware and Deser showed that such non-linear 2-Kaluza-Klein 3-Cosmology theory would be haunted by non-physical states with 4-Quantum negative energy(ghosts). Gravity More recently the subject emerged again as a possible explanation for the acceleration of the universe in a higher-dimensional program, mostly in connection with the DGP model of the brane-world program, with a possible intermediate massive spin-2 field. Massive Kaluza-Klein Gravity Maia/Disrael The zero mass limit problem suggests a conflict between two UnB Brasïlia variational principles for massless spin-2 fields: the [email protected] Einstein-Hilbert and the Fierz-Pauli actions. 1-Massive Gravity 2-Kaluza-Klein However, a theorem due to Soraj Gupta in 1960 and later 3-Cosmology reviewed by Stanley Deser an others in 1970, proved that 4-Quantum Gravity any massless spin-2 field defined by a symmetric rank-2 tensor is necessarily a solution of an Einstein-like equation. The Gupta theorem essentially reverses the linear approximation of Einstein equations, applied to the Fierz-Pauli theory, in practice reconstructing the non-linear terms. Massive Kaluza-Klein Therefore, the non-linearity of a massless spin-2 field would Gravity Maia/Disrael appear when the the Fierz-Pauli Lagrangian UnB 1 Brasïlia L= [H H,µ− H Hνρ,µ− 2H ,µH,ν+ 2H Hµ,ρ] [email protected] 4 ,µ νρ,µ µν νρ,µ is written in the geometric language as 1-Massive Gravity 2-Kaluza-Klein √ 3-Cosmology L = R g 4-Quantum Gravity This conclusion suggests an interesting corolary to Gupta’s theorem: The Einstein-Hilbert action is not a guess or an accident: The non-linear geometric expression for massless gravitation is derived from the Einstein-Hilbert principle, built with the Ricci scalar R itself, and not with an unknown function F (R). Massive Kaluza-Klein On the other hand, a massive spin-2 field does not lead to Gravity Maia/Disrael an Einstein-like equation and its physical properties are UnB Brasïlia different when compared with massless gravitation. Indeed, [email protected] such massive field may have a short range interaction with 1-Massive Gravity 2-Kaluza-Klein gauge fields at the Tev scale. 3-Cosmology Such possibility emerged in 1971 when Isham, Salam and 4-Quantum Gravity Stradthee, suggested the existence of a massive spin-2 field with mass, acting as an intermediate field between Einstein’s gravitation and hadrons. They never found it because they were insisting on Gupta’s theorem in General Relativity. In the next sections we will show that ghost-free massive spin-2 field, appears in a modified massive Kaluza-Klein theory. Massive Kaluza-Klein 2 Kaluza-Klein Gravity Maia/Disrael The non-Abelian Kaluza-Klein theory proposed in 1963 was UnB Brasïlia defined in a higher-dimensional space with product topology [email protected] V4 × BN , N > 1, where BN is a compact space with Planck 1-Massive Gravity size, and the 4 + N geometry is defined by the 2-Kaluza-Klein 3-Cosmology Einstein-Hilbert action. 4-Quantum By writing the metric in the form (the Kaluza-Klein ansatz) Gravity ab ! gµν + g AµaAνb Aµa GAB = Aνb gab The action decomposes as √ √ 1 √ R G = R −g + trF F µν −g 4 µν a where Aµ = Aµaη^ , Fµν = [Dµ, Dν] and Dµ = ∂µ + Aµ Massive Kaluza-Klein Gravity Maia/Disrael UnB Brasïlia In 1984 it was noted that the predictions of the Kaluza-Klein [email protected] theory at the electroweak limit did not agree with the 1-Massive Gravity observed chiral motion of fermions. 2-Kaluza-Klein 3-Cosmology The fermion chirality problem was soon understood to be a 4-Quantum Gravity consequence of the small size of Bn, which contributed to a large fermionic mass proportional to the Planck mass. In spite of the many efforts presented at that time to save the theory, it was literally abandoned. Massive Kaluza-Klein Gravity In the 60’s Joseph and Maia/Disrael UnB Ne’eman conjectured Brasïlia that the internal (gauge) [email protected] symmetry could be 1-Massive Gravity geometrically explained 2-Kaluza-Klein by the group of rotations 3-Cosmology 4-Quantum of the space orthogo- Gravity nal to the space-time embedded in a flat space. regarded as the ground state of the high-dimensional gravitational field, defined by the Einstein’s equations 1 R − RG = κ∗ T ∗ AB 2 AB g AB which are hyperbolic equations, compatible with the product 4 N topology IR × IR in place of the V4 × Bn. In 1985 we have checked the Joseph-Ne’eman conjecture, in Massive Kaluza-Klein a modified Kaluza-Klein theory, where the higher-dimensional Gravity Maia/Disrael space was the embedding space of the space-time, and found UnB Brasïlia that the geometry of the embedded space-time is given by [email protected] ab ! 1-Massive Gravity g~µν + g AµaAνb Aµa GAB = 2-Kaluza-Klein Aνb gab 3-Cosmology 4-Quantum where Gravity a a b αβ g~µν =g ¯µν− 2y k¯µνa + y y g k¯µαak¯νβb b αβ k~µνa(x, y) = k¯µνa − y g¯ k¯µαak¯νβb b Aµa = gµa(x, y) = y A¯µab 1 ∂g~ k~ = − µν µνa 2 ∂y a Massive Kaluza-Klein Notice that GAB looks like the Kaluza-Klein metric ansatz, Gravity with the difference that the space-time metric is replaced by Maia/Disrael UnB Brasïlia a a b αβ [email protected] g~µν =g ¯µν− 2y k¯µνa + y y g¯ k¯µαak¯νβb | {z } 1-Massive Gravity 2-Kaluza-Klein so that the Kaluza-Klein Lagrangian decomposes as 3-Cosmology 4-Quantum √ 1 Gravity R G = R~p−g~ + trF F µνp−g~ 4 µν where Fµν is defined by the second fundamental form Aµab, written in the Lie algebra of the rotations group: b b b b a Dµa =g ~a + Aµa , Dµ = Dµa Lb, Fµν = [Dµ, Dν] Since the internal space is non-compact, the Chiral fermions problem does not appear. Massive Kaluza-Klein Gravity Rewriting the above Kaluza-Klein Lagrangian, but now Maia/Disrael UnB Brasïlia replacing g~µν in terms of the extrinsic curvature [email protected] √ √ 1 √ 1-Massive Gravity L = R G = [R + (K 2 − h2)] g − trF F µν g 4 µν 2-Kaluza-Klein 3-Cosmology 2 µνa 2 a µν where K = kµνak and h = hah , ha = g kµνa 4-Quantum Gravity Showing that kµνa corresponds to a spin-2 field which intermediates between the gravitational field gµν and the gauge fields Aµ. As we see the term K 2 − h2 is proportional to the mass term of the the Fierz-Pauli action with mass (written for kµνa): 1 L= h ha,µ− k kρνa;µ− 2k ;µh ,ν+ 2k ha;ρ− m2(K 2− h2) 4 a,µ ρνa;µ µνa a νρa;µ Massive Kaluza-Klein Gravity Maia/Disrael UnB Brasïlia Therefore, the redefined Kaluza-Klein in accordance with the [email protected] Joseph-Ne’emann conjecture leads to a chiral fermion 1-Massive Gravity compatible theory at the electroweak level, including a 2-Kaluza-Klein 3-Cosmology massive, short range spin-2 term which corresponds to the 4-Quantum second fundamental form. Gravity The gauge symmetries are identified with the group of rotations of the normal vectors of the embedded space-time and the gauge fields are given by the third fundamental form. Massive Kaluza-Klein 3 Cosmology Gravity Maia/Disrael In particular for N=1, the third fundamental form vanishes so UnB Brasïlia that there is no unification. Einstein’s equations simplifies to [email protected] 1 1-Massive Gravity R − Rg − Q = κ∗ GT ∗ µν 2 µν µν g µν 2-Kaluza-Klein ρ ∗ ∗ 3-Cosmology kµ;ρ − h,µ = α Tµ5 4-Quantum 1 Gravity R − Rg = α∗ T ∗ 55 2 55 55 55 where 1 Q = g 55(kρk − hk ) − (K 2 − h2)g , µν µ ρν µν 2 µν Qµν is conserved in the sense of Noether and it reproduces the cosmological constant term in the particular case where kµν = αgµν. Massive Applying to the FLRW metric Kaluza-Klein Gravity 2 2 2 2 ( 2 2 2 2 2 Maia/Disrael ds = −dt +a(t) [dr +f r)(dθ +sin (θ)dφ )]+φ (r, t)dy UnB | {z } Brasïlia [email protected] where f (r) = sin r, r, sinh r for k = 1, 0, -1 respectively and 1-Massive Gravity 2-Kaluza-Klein φ = g55.
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