Covariant Canonical Gauge theory of Gravitation resolves the Cosmological Constant Problem

Covariant Canonical Gauge theory of Gravitation resolves 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-Universität, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany
GSI Helmholtzzentrum für Schwerionenforschung GmbH, Planckstrasse 1, 64291 Darmstadt, Germany
July 3, 2019

The covariant canonical transformation theory applied to the relativistic theory of classical matter fields in dynamic space-time yields a new (first order) gauge field theory of gravitation. The emerging field equations embrace a quadratic Riemann curvature term added to Einstein’s linear equation. The quadratic term facilitates a momentum field which generates a dynamic response of space-time to its deformations relative to de Sitter geometry, and adds a term proportional to the Planck mass squared to the cosmological constant. The proportionality factor is given by a dimensionless parameter governing the strength of the quadratic term.

In consequence, Dark Energy emerges as a balanced mix of three contributions, (A)dS curvature plus the residual vacuum energy of space-time and matter. The Cosmological Constant Problem of the Einstein-Hilbert theory is resolved as the curvature contribution relieves the rigid relation between the cosmological constant and the vacuum energy density of matter.

1 Introduction

Covariant canonical transformation theory is a rigorous framework to enforce symmetries of relativistic physical systems of fields with respect to arbitrary transformation (Lie) group [1]. The imposed requirement of invariance of the original action integral with respect to local transformations is “cured” via the introduction of additional degrees of freedom, the gauge fields. This cure closes the system with the gauge fields emerging as the mediators of forces acting upon the original fields (or particles). Those forces can be regarded as pseudo forces, like the centrifugal force “felt” by particles moving on a circle. However, here these force fields acquire the status of dynamic entities and, in the language of field theory, of independent fundamental fields. The canonical transformation theory applied to matter fields embedded in dynamic space-time leads to a novel gauge theory of gravitation (Covariant Canonical Gauge theory of Gravitation, CCGG) that contains also the linear Einstein-Hilbert (EH) term [2]. It emerges naturally in the Palatini formalism with the metric tensor and affine connection being independent fundamental (“co-ordinate”) fields. The theory is shown to be inherently inconsistent unless a quadratic invariant in the momentum field – conjugate to the affine connection – is added to the covariant Hamiltonian111Such a term translates into a squared Riemann tensor invariant in the Lagrangian that was indeed anticipated by Einstein already hundred years ago, and suggested in a letter to Weyl [3]. However, the rigorous math in CCGG shows that the Einstein model must be corrected when spin-carrying fields are considered [4] which generates a dynamic response of space-time to deformations relative to the de Sitter geometry. The strength of that invariant relative to the EH term is adjustable by a dimensionless coupling (deformation [5]) parameter .

This letter addresses the impact of the CCGG theory on the cosmological constant. We briefly review why the canonical gauge theory of gravitation invokes a “quadratic” Riemann invariant in the space-time Hamiltonian, and discuss its interpretation as deformation of the space-time geometry relative to de Sitter. In Einstein’s general theory of relativity only a linear term, the Ricci scalar, appears in the Lagrangian. Therefore space-time lacks a momentum field which enables its dynamic response to deformations of the metric in the presence of matter. This is different in the Covariant Canonical Gauge theory of Gravitation where a “restraining force to maximum symmetry” is reminiscent of the restoration force due to a strained string, and space-time itself appears to be a non-gravitating element of Dark Energy. We show how the de Sitter deformation adds a term to the cosmological constant that relieves the rigid relation between the cosmological constant and the vacuum energy density of matter encountered in the Einstein-Hilbert theory. This resolves the so called Cosmological Constant Problem [6, 7, 8], namely the huge discrepancy (of the order of ) between the theoretically predicted and the observed current value of the cosmological constant.

2 The Covariant Canonical Gauge Gravitation

The CCGG equation generalizing the Einstein-Hilbert equation has been derived in [2] from the “gauged” generic regular Hamiltonian density


where the tensor density -tilde, , is the canonical “momentum” field conjugate to the affine connection field. includes interaction terms coupling matter to space-time. (Our conventions are metric signature (+, - , - , - ) and natural units .) The quadratic term ensures regularity, i.e. the existence and reversibility of Legendre transformations between the Hamiltonian and Lagrangian pictures222Other contractions of the quadratic tensor product in Eq. (1) would inhibit a unique relation between the “momentum” field and the dual (“velocity”) field .. The coupling constants and come with the dimensions and .

The canonical equations of motion [2] are obtained by variation with respect to metric, connection, and matter fields, and the related conjugate (momentum) fields. Solving then for the momentum field yields




is the Riemann tensor of the maximally symmetric 4-dimensional space-time (denoted in Ref. [2]) with a constant Ricci curvature scalar . 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 , respectively. The CCGG theory thus enforces (via the required existence of the Legendre transform) both, the quadratic Ansatz for the Lagrangian and the (A)dS symmetry of the ground state. The momentum field corresponds to the deformation tensor relative to that ground state.

Legendre-transforming the above Hamiltonian yields the Lagrangian density


The canonical (CCGG) equation is derived in [2] by variation. It includes the Einstein tensor as a linear term:


Here is the energy-momentum tensor of matter 333 is according to the CCGG theory the canonical energy-momentum tensor rather than Hilbert’s metric energy-momentum tensor that in the Einstein-Hilbert theory of general relativity [9] appears in the Einstein equation. It is in general different from the canonical energy-momentum tensor. In the context of this letter that subtlety is irrelevant, though.. The l.h.s. of this equation can be interpreted as the canonical energy-momentum tensor of space-time [2], such that the energy and momentum of matter and space-time are locally balanced, analogously to the stress-strain relation in elastic media, according to


is the gravitational coupling constant and is the cosmological constant as known from Einstein’s general relativity. The CCGG equation boils down to the Einstein equation only in the special case 444Notice though that the canonical transformation framework forbids that . This is seen directly in Eq. (1) as the contribution of the quadratic term diverges with . In that case the Hesse matrix of the Lagrangian would become singular, and the Legendre transformations and the expressions of the conjugate fields in terms of derivatives of the gauge fields are undefined. . The relations between these two physical constants and the coupling constants and of the CCGG theory are unambiguously fixed:


Combining these relations yields


The dimensionless coupling constant controls the magnitude of the quadratic Riemann term. is the (A)dS curvature and scale parameter in this theory. Because of Eq. (7) the sign of and must be equal.

3 Dark Energy revisited

The vacuum solution of Eq. (3) is a vanishing “momentum” field, , the ground state of space-time. With Eq. (2) this corresponds to the maximally symmetric Riemann curvature tensor


In absence of any matter fields and vacuum fluctuations the energy-momentum tensor vanishes identically. With the trace of Eq. (5) yields the Ricci scalar


Recall now that the energy-momentum tensor on the r.h.s. of the Einstein equation vanishes in vacuum. The homogeneous and isotropic vacuum energy density is — in the Einstein-Hilbert Ansatz — identified with the cosmological constant, . The CCGG theory, on the other hand, exhibits complex relations between the involved fundamental constants modifying 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


where is the vacuum energy density derived from a renormalizable field theory of matter, and the “normal ordered” stress tensor that vanishes in vacuum. Similarly, under the reasonable assumption that the vacuum state of space-time is globally isotropic, the quantity representing the vacuum fluctuations of space-time (“graviton condensate”) will be proportional to the metric and can be separated from the space-time tensor in Eq. (6):


represents the homogeneous graviton condensate, and is the normalized strain tensor. The vacuum term can now in Eq. (6) be absorbed in an effective cosmological “constant”. Referring to Eq. (9) that effective cosmological constant is then


denotes the residual value from the contributions of the vacuum energy densities of matter and space-time. The space-time continuum responds to any residual gravitating vacuum fluctuations of matter fields by adjusting its curvature, relative to the maximally symmetric ground state, by


For a maximally symmetric 4D space-time, , and must hold. Dark Energy can then be identified with the curvature of (A)dS space-time. A finite value of , on the other hand, signals a violation of the (A)dS symmetry. To decide whether this is the case or not requires a comparison with data based on an appropriate analysis of the Friedman equation [7, 10].

Based on the above assumptions and definitions, the normal ordered version of the CCGG equation (5) can now be recast into


In Einstein’s General Theory of Relativity, the vacuum energy term cannot be modified. The mismatch between , as derived from observations, and from field theoretical calculations of the vacuum energy density of matter, does not give rise to a “Cosmological Constant Problem” in the CCGG theory since the cosmological term is composed of three terms of different origins: The vacuum fluctuations of the space-time fabric - we call “graviton condensate” - balance via Eq. (6) the vacuum energy density of matter, and the kinetic term in the CCGG Lagrangian gives rise to a non-gravitating term related to the (A)dS curvature (or deformation) parameter . The isotropic and homogeneous residual stress may possibly be even compatible with zero, if the graviton condensate completely shields away the gravitating effect of the matter vacuum, see below.

4 The momentum of space-time

The values of the parameters of the CCGG theory can be estimated by treating the equations (7) and (14) as functions of the yet undetermined fundamental coupling constant . Equation (14) shows that and determine the value of the cosmological constant . The gravitational coupling constant, , on account of Eq. (7), relates to . If the value of is close to used in standard cosmology [11, 12], two extreme cases help to illustrate the range of possible values of :

Case 1:

If the vacuum energy densities of space-time and matter cancel each other, i.e. for , the desired value, , can be matched when

( is the reduced Planck mass.) The famous factor of has in this case been absorbed in the dimensionless coupling constant of the CCGG theory. Dark Energy is then exclusively generated by the (A)dS curvature.

If the global deformation of space-time relative to the ground state is non-zero but small, will be close to zero. is then large, in the range . It naturally explains why Dark Energy does not gravitate.

Case 2:

The opposite extreme is when the graviton condensate vanishes, and the vacuum portion of the cosmological constant equals the vacuum energy density of the matter fields: . Field theoretical calculations estimate . The cosmological constant needed to counteract that large vacuum density would require a large value of the (A)dS curvature,

For the coupling constant this would mean

Obviously, the relative strength of the quadratic term ranges in .

5 Summary

The application of the canonical transformation framework to the classical covariant field theory of matter and space-time yields the Covariant Canonical Gauge theory of Gravitation (CCGG) with a quadratic “kinetic” term in the Hamiltonian and Lagrangian densities, representing the inertia of the dynamic space-time. That term, absent in the Einstein theory, leads to a dynamic response of space-time to deformations away from maximal symmetry, and adds the (A)dS curvature parameter to the cosmological constant. The cosmological constant is then composed of three independent portions: A balanced contribution from the vacuum energies of space-time and matter, and the “(A)dS curvature term”. The latter term provides an additional freedom to align the theoretical and observational values of the cosmological constant. This resolves the long-standing so called “Cosmological Constant Problem”. In fact, any vacuum energy density can, by a “suitable” choice of the necessarily non-vanishing (A)dS deformation parameter, be made compatible with the standard value of the cosmological constant. However, since the deformation parameter enters also other observables, a more comprehensive study is required to identify the optimal choice.

6 Conclusion and Outlook

We conclude that the CCGG theory extends the theory of gravitation and facilitates, in a mathematically rigorous way, a new understanding of the dynamics of space-time and of the evolution of the universe. The theory unambiguously fixes the coupling of space-time to matter fields [2]. This distinguishes the CCGG theory from other theories where the effects of Dark Energy can only been modeled with help of ad hoc assumptions about the underlying Lagrangian density and auxiliary fields (see for example [13, 14, 15, 16]). The approach naturally introduces a new fundamental constant proportional to the Planck mass. Such a construction of space-time [17] has been discussed earlier under the heading of de Sitter relativity to calculate the cosmological constant, and explain cosmic coincidence and time delays of extragalactic gamma-ray flares (see for example [18]). A detailed investigation of the CCGG modifications is underway using the Friedman model [19, 20] in the spirit of modified gravity models (cf. for example [21, 22, 23]). The objective is to analyze the geometrically induced portion of Dark Energy (“perfect geometrical fluid”), and to identify observables to detect the presence and measure the relative contribution of the quadratic term (to be published in [24]).


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