Covariant Canonical Gauge theory of Gravitation resolves the Cosmological Constant Problem
aa r X i v : . [ g r- q c ] A ug Covariant Canonical Gauge theory of Gravitationresolves the Cosmological Constant Problem
D Vasak , J Struckmeier , J Kirsch , and H Stoecker Frankfurt Institute for Advanced Studies (FIAS),Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main, Germany Goethe-Universit¨at, Max-von-Laue-Strasse 1, 60438 Frankfurtam Main, Germany GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH,Planckstrasse 1, 64291 Darmstadt, GermanyAugust 8, 2018
Abstract
The covariant canonical transformation theory applied to the relativis-tic theory of classical matter fields in dynamic space-time yields a new(first order) gauge field theory of gravitation. The emerging field equa-tions embrace a quadratic Riemann curvature term added to Einstein’slinear equation. The quadratic term facilitates a momentum field whichgenerates a dynamic response of space-time to its deformations relativeto de Sitter geometry, and adds a term proportional to the Planck masssquared to the cosmological constant. The proportionality factor is givenby a dimensionless parameter governing the strength of the quadraticterm.In consequence, Dark Energy emerges as a balanced mix of three con-tributions, (A)dS curvature plus the residual vacuum energy of space-timeand matter. The Cosmological Constant Problem of the Einstein-Hilberttheory is resolved as the curvature contribution relieves the rigid rela-tion between the cosmological constant and the vacuum energy density ofmatter.
Covariant canonical transformation theory is a rigorous framework to enforcesymmetries of relativistic physical systems of fields with respect to arbitrarytransformation (Lie) group [1]. The imposed requirement of invariance of theoriginal action integral with respect to local transformations is “cured” via theintroduction of additional degrees of freedom, the gauge fields. This cure closesthe system with the gauge fields emerging as the mediators of forces acting uponthe original fields (or particles). Those forces can be regarded as pseudo forces,like the centrifugal force “felt” by particles moving on a circle. However, herethese force fields acquire the status of dynamic entities and, in the language of1eld theory, of independent fundamental fields. The canonical transformationtheory applied to matter fields embedded in dynamic space-time leads to a novelgauge theory of gravitation (Covariant Canonical Gauge theory of Gravitation,CCGG) that contains also the linear Einstein-Hilbert (EH) term [2]. It emergesnaturally in the Palatini formalism with the metric tensor and affine connectionbeing independent fundamental (“co-ordinate”) fields. The theory is shown tobe inherently inconsistent unless a quadratic invariant in the momentum field– conjugate to the affine connection – is added to the covariant Hamiltonian which generates a dynamic response of space-time to deformations relative tothe de Sitter geometry. The strength of that invariant relative to the EH termis adjustable by a dimensionless coupling (deformation [5]) parameter 1 /g .This letter addresses the impact of the CCGG theory on the cosmologi-cal constant. We briefly review why the canonical gauge theory of gravitationinvokes a “quadratic” Riemann invariant in the space-time Hamiltonian, anddiscuss its interpretation as deformation of the space-time geometry relative tode Sitter. In Einstein’s general theory of relativity only a linear term, the Ricciscalar, appears in the Lagrangian. Therefore space-time lacks a momentum fieldwhich enables its dynamic response to deformations of the metric in the pres-ence of matter. This is different in the Covariant Canonical Gauge theory ofGravitation where a “restraining force to maximum symmetry” is reminiscentof the restoration force due to a strained string, and space-time itself appearsto be a non-gravitating element of Dark Energy. We show how the de Sitterdeformation adds a term ∼ M p /g to the cosmological constant that relieves therigid relation between the cosmological constant and the vacuum energy densityof matter encountered in the Einstein-Hilbert theory. This resolves the so calledCosmological Constant Problem [6, 7, 8], namely the huge discrepancy (of theorder of 10 ) between the theoretically predicted and the observed currentvalue of the cosmological constant. The CCGG equation generalizing the Einstein-Hilbert equation has been derivedin [2] from the “gauged” generic regular Hamiltonian density˜ H = 14 g √− g ˜ q αξβη ˜ q ητλα g ξτ g βλ − g ˜ q αηβη g αβ + ˜ H matter , (1)where the tensor density q -tilde, ˜ q αξβη = q αξβη √− g , is the canonical “momen-tum” field conjugate to the affine connection field. ˜ H matter includes interactionterms coupling matter to space-time. (Our conventions are metric signature(+, - , - , - ) and natural units ~ = c = 1.) The quadratic term ensures regu-larity, i.e. the existence and reversibility of Legendre transformations betweenthe Hamiltonian and Lagrangian pictures . The coupling constants g and g come with the dimensions [ g ] = 1 and [ g ] = L − . Such a term translates into a squared Riemann tensor invariant in the Lagrangian thatwas indeed anticipated by Einstein already hundred years ago, and suggested in a letter toWeyl [3]. However, the rigorous math in CCGG shows that the Einstein model must becorrected when spin-carrying fields are considered [4] Other contractions of the quadratic tensor product in Eq. (1) would inhibit a uniquerelation between the “momentum” field ˜ q ητλα and the dual (“velocity”) field R αητλ . q ηαξβ = g (cid:16) R ηαξβ − ˆ R ηαξβ (cid:17) . (2)Here ˆ R ηαξβ = g ( g ηξ g αβ − g ηβ g αξ ) (3)is the Riemann tensor of the maximally symmetric 4-dimensional space-time(denoted R ηαξβ | max in Ref. [2]) with a constant Ricci curvature scalar ˆ R = 12 g .It is the “ground state geometry of space-time” [7] which is the de Sitter (dS)or the anti-de Sitter (AdS) space-time for the positive or the negative sign of g , respectively. The CCGG theory thus enforces (via the required existence ofthe Legendre transform) both, the quadratic Ansatz for the Lagrangian and the(A)dS symmetry of the ground state. The momentum field q ηαξβ correspondsto the deformation tensor relative to that ground state.Legendre-transforming the above Hamiltonian yields the Lagrangian density˜ L = (cid:2) g R αξβη R ητλα g ξτ g βλ + g g R αηβη g αβ − g g (cid:3) √− g + ˜ L matter . (4)The canonical (CCGG) equation is derived in [2] by variation. It includes theEinstein tensor as a linear term: g (cid:0) R αβγµ R αβγν − δ µν R αβγδ R αβγδ (cid:1) − πG (cid:0) R µν − δ µν R + δ µν Λ (cid:1) = θ µν . (5)Here θ µν is the energy-momentum tensor of matter . The l.h.s. of this equa-tion can be interpreted as the canonical energy-momentum tensor − Θ µν ofspace-time [2], such that the energy and momentum of matter and space-timeare locally balanced, analogously to the stress-strain relation in elastic media,according to Θ µν + θ µν = 0 . (6) G is the gravitational coupling constant and Λ is the cosmological constant asknown from Einstein’s general relativity. The CCGG equation boils down to theEinstein equation only in the special case g = 0. The relations between thesetwo physical constants and the coupling constants g , g and g of the CCGGtheory are unambiguously fixed: g g ≡ πG (7)6 g g ≡ πG Λ . (8) θ µν is according to the CCGG theory the canonical energy-momentum tensor ratherthan Hilbert’s metric energy-momentum tensor that in the Einstein-Hilbert theory of generalrelativity [9] appears in the Einstein equation. It is in general different from the canonicalenergy-momentum tensor. In the context of this letter that subtlety is irrelevant, though. Notice though that the canonical transformation framework forbids that g = 0. Thisis seen directly in Eq. (1) as the contribution of the quadratic term diverges with g → g (9)The dimensionless coupling constant g controls the magnitude of the quadraticRiemann term. g is the (A)dS curvature and scale parameter in this theory.Because of Eq. (7) the sign of g and g must be equal. The vacuum solution of Eq. (3) is a vanishing “momentum” field, q ηαξβ = 0,the ground state of space-time. With Eq. (2) this corresponds to the maximallysymmetric Riemann curvature tensor R vac ηαξβ = ˆ R ηαξβ . (10)In absence of any matter fields and vacuum fluctuations the energy-momentumtensor vanishes identically. With Θ µν ≡ R = 4Λ = 12 g . (11)Recall now that the energy-momentum tensor on the r.h.s. of the Einsteinequation vanishes in vacuum. The homogeneous and isotropic vacuum energydensity h vac | θ µν | vac i = δ µν θ mat vac is — in the Einstein-Hilbert Ansatz — iden-tified with the cosmological constant, Λ. The CCGG theory, on the other hand,exhibits complex relations between the involved fundamental constants modify-ing the relation between the cosmological constant and vacuum energy density.The matter term on the r.h.s. of Eq. (5) can be decomposed as θ µν = δ µν θ mat vac + ˆ θ µν , (12)where θ mat vac is the vacuum energy density derived from a renormalizable fieldtheory of matter, and ˆ θ µν the “normal ordered” stress tensor that vanishes invacuum. Similarly, under the reasonable assumption that the vacuum state ofspace-time is globally isotropic, the quantity representing the vacuum fluctua-tions of space-time (“graviton condensate”) will be proportional to the metricand can be separated from the space-time tensor in Eq. (6):Θ µν = δ µν (cid:18) Θ st vac + Λ8 πG (cid:19) + ˆΘ µν . (13)Θ st vac represents the homogeneous graviton condensate, and ˆΘ µν is the normal-ized strain tensor. The vacuum term θ mat vac can now in Eq. (6) be absorbed inan effective cosmological “constant”. Referring to Eq. (9) that effective cosmo-logical constant is thenΛ eff := Λ + 8 πG (cid:0) θ mat vac + Θ st vac (cid:1) = 3 g + 8 πG Θ res vac . (14)Θ res vac denotes the residual value from the contributions of the vacuum energydensities of matter and space-time. The space-time continuum responds to4ny residual gravitating vacuum fluctuations of matter fields by adjusting itscurvature, relative to the maximally symmetric ground state, by △ R = R − ˆ R = 4Λ eff − g = 32 πG Θ res vac . (15)For a maximally symmetric 4D space-time, △ R = 0, and Θ res vac = 0 must hold.Dark Energy can then be identified with the curvature of (A)dS space-time.A finite value of Θ res vac , on the other hand, signals a violation of the (A)dSsymmetry. To decide whether this is the case or not requires a comparison withdata based on an appropriate analysis of the Friedman equation [7, 10].Based on the above assumptions and definitions, the normal ordered versionof the CCGG equation (5) can now be recast into g (cid:0) R αβγµ R αβγν − δ µν R αβγδ R αβγδ (cid:1) − πG (cid:0) R µν − δ µν R + δ µν Λ eff (cid:1) = ˆ θ µν . (16)In Einstein’s General Theory of Relativity, the vacuum energy term Λ can-not be modified. The mismatch between Λ, as derived from observations, andfrom field theoretical calculations of the vacuum energy density of matter, doesnot give rise to a “Cosmological Constant Problem” in the CCGG theory sincethe cosmological term Λ eff is composed of three terms of different origins: Thevacuum fluctuations of the space-time fabric - we call “graviton condensate” -balance via Eq. (6) the vacuum energy density of matter, and the kinetic term inthe CCGG Lagrangian gives rise to a non-gravitating term related to the (A)dScurvature (or deformation) parameter g . The isotropic and homogeneous resid-ual stress may possibly be even compatible with zero, if the graviton condensatecompletely shields away the gravitating effect of the matter vacuum, see below. The values of the parameters of the CCGG theory can be estimated by treatingthe equations (7) and (14) as functions of the yet undetermined fundamentalcoupling constant g . Equation (14) shows that g and Θ res vac determine thevalue of the cosmological constant Λ eff . The gravitational coupling constant,8 π G = 1 . × − GeV − , on account of Eq. (7), relates g to g . If the valueof Λ eff is close to Λ exp ∼ . × − GeV used in standard cosmology [11, 12],two extreme cases help to illustrate the range of possible values of g : Case 1:
If the vacuum energy densities of space-time and matter cancel eachother, i.e. for Θ res vac = 0, the desired value, Λ eff = Λ exp ≪ M p , can be matchedwhen g = 316 πG Λ exp = 3 M p exp = 1 . × . ( M p = 1 / √ πG ∼ . × GeV is the reduced Planck mass.) The famousfactor of 10 has in this case been absorbed in the dimensionless couplingconstant g of the CCGG theory. Dark Energy is then exclusively generated bythe (A)dS curvature.If the global deformation of space-time relative to the ground state is non-zero but small, Θ res vac will be close to zero. g is then large, in the range 10 –10 . It naturally explains why Dark Energy does not gravitate.5 ase 2: The opposite extreme is when the graviton condensate vanishes, andthe vacuum portion of the cosmological constant equals the vacuum energydensity of the matter fields: Θ res vac = θ mat vac . Field theoretical calculations estimate θ mat vac ≈ M . The cosmological constant needed to counteract that large vacuumdensity would require a large value of the (A)dS curvature, g = Λ exp + 8 πG M ≈ M For the coupling constant g this would mean g = 116 πG g = 3 M p (cid:0) Λ exp + M (cid:1) ≈ . Obviously, the relative strength of the quadratic term ranges in g ∈ ( , ). The application of the canonical transformation framework to the classical co-variant field theory of matter and space-time yields the Covariant CanonicalGauge theory of Gravitation (CCGG) with a quadratic “kinetic” term in theHamiltonian and Lagrangian densities, representing the inertia of the dynamicspace-time. That term, absent in the Einstein theory, leads to a dynamic re-sponse of space-time to deformations away from maximal symmetry, and addsthe (A)dS curvature parameter g = M p / g to the cosmological constant. Thecosmological constant is then composed of three independent portions: A bal-anced contribution from the vacuum energies of space-time and matter, andthe “(A)dS curvature term”. The latter term provides an additional freedomto align the theoretical and observational values of the cosmological constant.This resolves the long-standing so called “Cosmological Constant Problem”. Infact, any vacuum energy density Θ res vac can, by a “suitable” choice of the neces-sarily non-vanishing (A)dS deformation parameter, be made compatible withthe standard value of the cosmological constant. However, since the deforma-tion parameter enters also other observables, a more comprehensive study isrequired to identify the optimal choice. We conclude that the CCGG theory extends the theory of gravitation and facili-tates, in a mathematically rigorous way, a new understanding of the dynamics ofspace-time and of the evolution of the universe. The theory unambiguously fixesthe coupling of space-time to matter fields [2]. This distinguishes the CCGGtheory from other theories where the effects of Dark Energy can only been mod-eled with help of ad hoc assumptions about the underlying Lagrangian densityand auxiliary fields (see for example [13, 14, 15, 16]). The approach naturallyintroduces a new fundamental constant proportional to the Planck mass. Sucha construction of space-time [17] has been discussed earlier under the headingof de Sitter relativity to calculate the cosmological constant, and explain cosmiccoincidence and time delays of extragalactic gamma-ray flares (see for example618]). A detailed investigation of the CCGG modifications is underway using theFriedman model [19, 20] in the spirit of modified gravity models (cf. for exam-ple [21, 22, 23]). The objective is to analyze the geometrically induced portionof Dark Energy (“perfect geometrical fluid”), and to identify observables to de-tect the presence and measure the relative contribution of the quadratic term(to be published in [24]).
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