Gravity from Pre-geometry

Gravity is derived from an entropic action coupling matter fields with geometry. The fundamental idea is to relate the metric of Lorentzian spacetime to a quantum operator, playing the role of an and to describe the matter fields topologically, according to a Dirac-Kähler formalism, as the direct sum of a 0-form, a 1-form and a 2-form. While the geometry of spacetime is defined by its metric, the matter fields can be used to define an alternative metric, the metric induced by the matter fields, which geometrically describes the interplay between spacetime and matter. The proposed entropic action is the quantum relative entropy between the metric of spacetime and the metric induced by the matter fields. The modified Einstein equations obtained from this action reduce to the Einstein equations with zero cosmological constant in the regime of low coupling. By introducing the , which acts as a set of Lagrangian multipliers, the proposed entropic action reduces to a dressed Einstein-Hilbert action with an emergent small and positive cosmological constant only dependent on the G-field. The obtained equations of modified gravity remain second order in the metric and in the G-field. A canonical quantization of this field theory could bring new insights into quantum gravity while further research might clarify the role that the G-field could have for dark matter.

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A quantization of Einstein-Cartan's theory is discussed in Dirac's Hamiltonian formalism. The spinor field is regarded as an anticommutingc-number. To find the simplest Hamiltonian formalism for this case, we choose one leg of the vierbein field to be normal to the hypersurfacex0=const. Since this theory has two kinds of gauge symmetries related to general co-ordinate transformations and to local Lorentz transformations, these gauge symmetries must be fixed. After the gauge fixing, the anticommutator for the spinor field is derived.

We study the equivalence principle and its violations by quantum effects in scalar-tensor theories that admit a conformal frame in which matter only couples to the spacetime metric. These theories possess Ward identities that guarantee the validity of the weak equivalence principle to all orders in the matter coupling constants. These Ward identities originate from a broken Weyl symmetry under which the scalar field transforms by a shift, and from the symmetry required to couple a massless spin two particle to matter (diffeomorphism invariance). But the same identities also predict violations of the weak equivalence principle relatively suppressed by at least two powers of the gravitational couplings, and imply that quantum corrections do not preserve the structure of the action of these theories. We illustrate our analysis with a set of specific examples for spin zero and spin half matter fields that show why matter couplings do respect the equivalence principle, and how the couplings to the gravitational scalar lead to the weak equivalence principle violations predicted by the Ward identities.

This article gives a pedagogical review of recent work in which the Einstein-Hilbert gravitational action is obtained as a symmetry-breaking effect in quantum field theory. Particular emphasis is placed on the case of renormalizable field theories with dynamical scale-invariance breaking, in which the induced gravitational effective action is finite and calculable. A functional integral formulation is used throughout, and a detailed analysis is given of the role of dimensional regularization in extracting finite answers from formally quadratically divergent integrals. Expressions are derived for the induced gravitational constant, for the induced cosmological constant, and for quantized matter theories on a background manifold, and a strategy is outlined for computing the induced constants in the case of an SU(n) gauge theory. By use of the background field method, the formalism is extended to the case in which the metric is also quantized, yielding a derivation of the semiclassical Einstein equations as an approximation to quantum gravity, as well as general formulas for the induced (or renormalized) gravitational and cosmological constants.

Einstein-Cartan theory is an extension of the standard formulation of general relativity characterized by a nonvanishing torsion. The latter is sourced by the matter fields via the spin tensor, and its effects are expected to be important at very high spin densities. In this work, we analyze in detail the physics of Einstein-Cartan theory with Dirac and Maxwell fields minimally coupled to the spacetime torsion. This breaks the $U(1)$ gauge symmetry, which is suggested by the possibility of a torsion-induced phase transition in the early Universe. The resulting Dirac-like and Maxwell-like equations are nonlinear with self-interactions as well as having fermion-boson nonminimal couplings. We discuss several cosmological aspects of this theory under the assumption of randomly oriented spin densities (unpolarized matter), including bounces, acceleration phases, and matter-antimatter asymmetry in the torsion era, as well as late-time effects such as the generation of an effective cosmological constant, dark energy, and future bounces within cyclic solutions.

H. Weyl's proposal of 1918 for generalizing Riemannian geometry by local scale gauge (later called {\em Weyl geometry}) was motivated by mathematical, philosophical and physical considerations. It was the starting point of his unified field theory of electromagnetism and gravity. After getting disillusioned with this research program and after the rise of a convincing alternative for the gauge idea by translating it to the phase of wave functions and spinor fields in quantum mechanics, Weyl no longer considered the original scale gauge as physically relevant. About the middle of the last century the question of conformal and/or local scale gauge transformation were reconsidered by different authors in high energy physics (Bopp, Wess, et al.) and, independently, in gravitation theory (Jordan, Fierz, Brans, Dicke). In this context Weyl geometry attracted new interest among different groups of physicists (Omote/Utiyama/Kugo, Dirac/Canuto/Maeder, Ehlers/Pirani/Schild and others), often by hypothesizing a new scalar field linked to gravity and/or high energy physics. Although not crowned by immediate success, this ``retake'' of Weyl geometrical methods lives on and has been extended a century after Weyl's first proposal of his basic geometrical structure. It finds new interest in present day studies of elementary particle physics, cosmology, and philosophy of physics.

In this paper noncommutative gravity is constructed as a gauge theory of the noncommutative SO(2,3) group, while the noncommutativity is canonical (constant). The Seiberg-Witten map is used to express noncommutative fields in terms of the corresponding commutative fields. The commutative limit of the model is the Einstein-Hilbert action with the cosmological constant term and the topological Gauss-Bonnet term. We calculate the second order correction to this model and obtain terms that are of zeroth to fourth power in the curvature tensor and torsion. Trying to relate our results with $f(R)$ and $f(T)$ models, we analyze different limits of our model. In the limit of big cosmological constant and vanishing torsion we obtain a $x$-dependent correction to the cosmological constant, i.e. noncommutativity leads to a $x$-dependent cosmological constant. We also discuss the limit of small cosmological constant and vanishing torsion and the teleparallel limit.

The following topics are considered: 1. Time as coordinate, introduction of proper time through a metric of space time. 2. Geometrical description of matter, the cosmical constant as a second constant demanded by the logical structure of the theory. 3. Detailed computation of the gravitational equations for a general non-static field of spherical symmetry. 4. Schwarzschild's and de Sitter's solutions free from artificial singularities. 5. Map representation of elliptical space, expanding space, Friedmann's equation, reduction to elliptic functions, numerical results. 6. Geodesics of space time, application to proper velocities and red-shift. Discussion of the time scale, numerical value of the cosmical constant, data on density. 7. Use of statical coordinates, the instability of Einstein's universe when no restriction is imposed on the symmetry of fluctuations, formation of the nebulae from fast-moving clouds of gases. 8. Discussion of the problem of the clusters of nebulae. 9. The cosmic rays as remnants of primitive matter. Explanation of the large amount of hydrogen and helium in the stars.

In this review article, the electromagnetic coupling constant (fine structure), a microscopic dimensionless constant is connected to the macroscopic world using a corrected Valev’s proposed general solution for the mass dimension expression suggesting a link between the evolution of fundamental constants and cosmological parameters, like the cosmological constant. From the corrected and modified mass dimension expression M=mpγ1/2 (where mp is the Planck mass and γ = 4.4702596×10121 is a large dimensionless quantity defined as a quantum cosmological dimensionless constant), we calculate the total mass of ordinary matter in the universe in agreement with that found by the Planck(2018) collaboration and the Lambda CDM. In section six, we connect the fine structure constant to the cosmological parameters (the present root mean square matter fluctuation averaged over a sphere of radius 8h-1Mpc, the baryon density, the dark energy density and the reduced Hubble dimensionless parameter).In the results, the S8 tension is calculated and found to be in agreement with the recently observed CMB satellite value. A modification of the Einstein constant in the field equations also connects the fine structure constant and the cosmological parameters from which a new calculation of the S8 tension without the amplitude of the mass fluctuations is derived. Also In the results, exact equation of the Dirac large number 6.686×1060 has been found, connecting cosmological parameters and Planck units. Thus a precise formulation and evidence of Dirac large number hypothesis and fine structure constant has been found, connecting the microscopic worlds and the macroscopic worlds.

An exact solution is obtained for a massive scalar field conformally coupled with the curvature in a cosmological model with the scale factor $a(t) = a_0 / |t|$, corresponding to background matter with the equation of state $ p = - 5 \varepsilon /3 $. An expression for the number density of created particles is obtained, and its behavior is studies as the model approached the instant of a Big Rip. Renormalization of the energy-momentum tensor is considered, and it is shown that back reaction of the quantum effects of a conformally coupled scalar field on the space-time metric can be neglected if the field mass is much smaller than the Planck mass and if the time left to the Big Rip is greater than the Planck time.

We review the current status of Finsler–Lagrange geometry and generalizations. The goal is to aid non-experts on Finsler spaces, but physicists and geometers skilled in general relativity and particle theories, to understand the crucial importance of such geometric methods for applications in modern physics. We also would like to orient mathematicians working in generalized Finsler and Kähler geometry and geometric mechanics how they could perform their results in order to be accepted by the community of "orthodox" physicists. Although the bulk of former models of Finsler–Lagrange spaces where elaborated on tangent bundles, the surprising result advocated in our works is that such locally anisotropic structures can be modeled equivalently on Riemann–Cartan spaces, even as exact solutions in Einstein and/or string gravity, if nonholonomic distributions and moving frames of references are introduced into consideration. We also propose a canonical scheme when geometrical objects on a (pseudo) Riemannian space are nonholonomically deformed into generalized Lagrange, or Finsler, configurations on the same manifold. Such canonical transforms are defined by the coefficients of a prime metric and generate target spaces as Lagrange structures, their models of almost Hermitian/Kähler, or nonholonomic Riemann spaces. Finally, we consider some classes of exact solutions in string and Einstein gravity modeling Lagrange–Finsler structures with solitonic pp-waves and speculate on their physical meaning.

We study the small speed of light expansion of general relativity, utilizing the modern perspective on non-Lorentzian geometry. This is an expansion around the ultra-local Carroll limit, in which light cones close up. To this end, we first rewrite the Einstein--Hilbert action in pre-ultra-local variables, which is closely related to the 3+1 decomposition of general relativity. At leading order in the expansion, these pre-ultra-local variables yield Carroll geometry and the resulting action describes the electric Carroll limit of general relativity. We also obtain the next-to-leading order action in terms of Carroll geometry and next-to-leading order geometric fields. The leading order theory yields constraint and evolution equations, and we can solve the evolution analytically. We furthermore construct a Carroll version of Bowen--York initial data, which has associated conserved boundary linear and angular momentum charges. The notion of mass is not present at leading order and only enters at next-to-leading order. This is illustrated by considering a particular truncation of the next-to-leading order action, corresponding to the magnetic Carroll limit, where we find a solution that describes the Carroll limit of a Schwarzschild black hole. Finally, we comment on how a cosmological constant can be incorporated in our analysis.

We investigate topology change in (1 + 1) dimensions by analysing the scalar-curvature action at the points of metric-degeneration that (with minor exceptions) any non-trivial Lorentzian cobordism necessarily possesses. In two dimensions any cobordism can be built up as a combination of only two elementary types, the `yarmulke' and the `trousers.' For each of these elementary cobordisms, we consider a family of Morse-theory inspired Lorentzian metrics that vanish smoothly at a single point, resulting in a conical-type singularity there. In the yarmulke case, the distinguished point is analogous to a cosmological initial (or final) singularity, with the spacetime as a whole being obtained from one causal region of Misner space by adjoining a single point. In the trousers case, the distinguished point is a `crotch singularity' that signals a change in the spacetime topology (this being also the fundamental vertex of string theory, if one makes that interpretation). We regularize the metrics by adding a small imaginary part, whose sign is fixed to be positive by the condition that it lead to a convergent scalar field path integral on the regularized spacetime. As the regulator is removed, the scalar density approaches a delta-function, whose strength is complex: for the yarmulke family the strength is , where is the rapidity parameter of the associated Misner space; for the trousers family it is simply . This implies that in the path integral over spacetime metrics for Einstein gravity in three or more spacetime dimensions, topology change via a crotch singularity is exponentially suppressed, whereas appearance or disappearance of a universe via a yarmulke singularity is exponentially enhanced. We also contrast these results with the situation in a vielbein-cum-connection formulation of Einstein gravity.

We propose an adynamical, background independent approach to quantum gravity and unification whereby the fundamental elements of Nature are graphical units of space, time and sources (in parlance of quantum field theory). The transition amplitude for these elements of “spacetimesource” is computed using a path integral with discrete Gaussian graphical action. The unit of action for a spacetimesource element is constructed from a difference matrix K and source vector J on the graph, as in lattice gauge theory. K is constructed from graphical relations so that it contains a non-trivial null space (whence gauge invariance), and J is then restricted to the column space of K which ensures it is distributed in a divergence-free fashion over the spacetime defined by the element. This rule for the relational construct of K and J is our proposed fundamental axiom of physics and results in a self-consistency relationship between sources, the spacetime metric, and the stress-energy-momentum content of the element, rather than a dynamical law for time-evolved entities. In its most general form, the set of fundamental elements employed by lattice gauge theory contains scalar fields on nodes and links, and vector fields on nodes. To complete the fundamental set (unification in this view), we propose the addition of scalar fields on plaquettes (basis for graviton) and vector fields on links. We use this approach via modified Regge calculus to correct proper distance in the Einstein-deSitter cosmology model yielding a fit of the Union2 Compilation supernova data that matches ɅCDM without having to invoke accelerating expansion or dark energy.