# Broken Scale Invariance, Gravity Mass, and Dark Energy inModified Einstein Gravity with Two Measure Finsler like Variables

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## Abstract

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## 1. Introduction

- (a)
- The (modified) Einstein equations with some effective and/or matter field sources consist of very sophisticated systems of nonlinear partial derivative equations (PDEs). The bulk of most known and important physical applications (of black hole, cosmological and other type solutions) were elaborated for the ansatz of metrics which can be diagonalized by certain frame/coordinate transforms, and when physically important systems of nonlinear PDEs can be reduced to systems of decoupled nonlinear ordinary differential equations (ODEs). In such cases, the generated exact or parametric solutions (i.e., integrals, with possible non-trivial topology, singularities, of different smooth classes, etc.) depend on one space, or time, like the coordinate, being determined by certain imposed symmetries (for instance, spherical/axial ones, which are invariant on some rotations, with Lie algebras symmetries, etc). The integration constants can be found in an explicit form by considering certain symmetry/Cauchy/boundary/asymptotic conditions. In this way, tvarious classes of black/worm hole and isotropic and anisotropic cosmological solutions were constructed;
- (b)
- The AFDM allows us to decouple and integrate physically important systems of nonlinear PDEs in more general forms than (a) when the integral varieties are parameterized not only by integration constants but also by generating and integration functions subjected to nonholonomic constraints and functional/nonlinear dependence on sources and data for certain classes of “prime metrics and connections”. The resulting “target” off-diagonal metrics and generalized connections depend, in general, on all space–time coordinates. It is important to note that, at the end, we can impose additional nonholonomic constraints and consider “smooth” limits or various type non-trivial topology and/or parametric transitions to Levi–Civita configurations (with zero torsion) and/or diagonal metrics. In this way, we can reproduce well-known black hole/cosmological solutions, which can have deformed horizons (for instance, ellipsoid/toroid symmetries), anisotropic polarized physical constants, for instance, imbedding into nontrivial gravitational vacuum configurations. These new classes of solutions cannot be constructed if we impose particular types of ansatz for diagonalizable metrics, frames of references and/or sources at the beginning, depending on only one spacetime coordinate. This is an important property of nonlinear parametric physical systems subjected to certain nonholonomic constraints. More general solutions with geometric rich structure and various applications for a nonlinear gravitational and matter field dynamics can be found if we succeed in directly solving certain generic nonlinear systems of PDEs which are not transformed into systems of ODEs. Having constructed such general classes of solutions, one might analyze the limits to diagonal configurations and possible perturbative effects. We “loose” the bulk of generic nonlinear solutions with multi-variables if we consider certain “simplified” ansatz for “higher-symmetries”, resulting in ODEs, from the beginning.

## 2. Nonholonomic Variables and (Modified) Einstein and Lagrange–Finsler Equations

#### 2.1. Geometric Objects and GR and MGTs in Nonholonomic Variables

**V**.

**V**, we can always introduce a nonholonomic $2+2$ splitting, which is determined by a non-integrable distribution

**V**, the Withney sum ⊕ defines a conventional splitting into horizontal (h), $h\mathbf{V},$ and vertical (v), $v\mathbf{V},$ subspaces. In local cooridinates

#### 2.2. Finsler–Lagrange Variables in GR and MGTs

## 3. TMTs and Other MGTs in Canonical Nonholonomic Variables

#### 3.1. Nonholonomic Ghost–Free Massive Configurations

#### 3.2. Tmt Massive Configurations with (Broken) Global Scaling Invariance

## 4. Cosmological Solutions in Effective Einstein Gravity and Fmgts

- The first equation for $\psi $ is the 2D Laplace/d’ Alambert equation which can be solved for any given $\phantom{\rule{3.33333pt}{0ex}}\tilde{\mathrm{Y}},$ which allows us to find ${g}_{1}={g}_{2}={e}^{\psi ({x}^{k})}.$
- Using the second equation in (44) and (41), the coefficients ${h}_{a}$ can be expressed as functionals on $(\Psi ,\mathrm{Y}).$ We redefine the generating function as in (43) and consider an effective source$$\Xi :=\int dt\mathrm{Y}{({\tilde{\Psi}}^{2})}^{\ast}={\phantom{\rule{4pt}{0ex}}}^{ef}\Xi +{\phantom{\rule{4pt}{0ex}}}^{m}\Xi +{\phantom{\rule{4pt}{0ex}}}^{f}\Xi +{\phantom{\rule{4pt}{0ex}}}^{\mu}\Xi ,$$$${h}_{3}=\frac{{\tilde{\Psi}}^{2}}{4({\phantom{\rule{4pt}{0ex}}}^{ef}\Lambda +{\phantom{\rule{3.33333pt}{0ex}}}^{m}\Lambda +{\phantom{\rule{3.33333pt}{0ex}}}^{f}\Lambda +{\phantom{\rule{3.33333pt}{0ex}}}^{\mu}\Lambda )}\mathrm{and}{h}_{4}=\frac{{({\tilde{\Psi}}^{\ast})}^{2}}{({\phantom{\rule{4pt}{0ex}}}^{ef}\Xi +{\phantom{\rule{4pt}{0ex}}}^{m}\Xi +{\phantom{\rule{4pt}{0ex}}}^{f}\Xi +{\phantom{\rule{4pt}{0ex}}}^{\mu}\Xi )}.$$
- We have to integrate t twice in order to find it in the 3D subset of equations in (44)$${n}_{i}={\phantom{\rule{4pt}{0ex}}}_{1}{n}_{i}+{\phantom{\rule{4pt}{0ex}}}_{2}{n}_{i}\int dt\frac{{({\tilde{\Psi}}^{\ast})}^{2}}{{\tilde{\Psi}}^{3}({\phantom{\rule{4pt}{0ex}}}^{ef}\Xi +{\phantom{\rule{4pt}{0ex}}}^{m}\Xi +{\phantom{\rule{4pt}{0ex}}}^{f}\Xi +{\phantom{\rule{4pt}{0ex}}}^{\mu}\Xi )}$$
- The 4th set of equations in (44) are algebraic ones, which allows us to compute$${w}_{i}={\left[{\varpi}^{\ast}\right]}^{-1}{\partial}_{i}\varpi ={\left[{\Psi}^{\ast}\right]}^{-1}{\partial}_{i}\Psi ={\left[{({\Psi}^{2})}^{\ast}\right]}^{-1}{\partial}_{i}{(\Psi )}^{2}={\left[{\Xi}^{\ast}\right]}^{-1}{\partial}_{i}\Xi .$$
- We can satisfy the conditions for $\omega $ in the second line in (44) if we keep the Killing symmetry on ${\partial}_{i}$ and take, for instance, ${\omega}^{2}={\left|{h}_{4}\right|}^{-1}.$Different types of inhomogeneous cosmological solutions of the system (33) are determined by corresponding classes of effective sources$$\begin{array}{ccc}\hfill \mathrm{generating}\mathrm{functions}:& & \psi ({x}^{k}),\tilde{\Psi}({x}^{k},t),\omega ({x}^{k},{y}^{3},t)\hfill \\ \hfill \mathrm{effective}\mathrm{sources}:& & \tilde{\mathrm{Y}}({x}^{k});{\phantom{\rule{4pt}{0ex}}}^{ef}\Xi ({x}^{k},t),{\phantom{\rule{4pt}{0ex}}}^{m}\Xi ({x}^{k},t),{\phantom{\rule{4pt}{0ex}}}^{f}\Xi ({x}^{k},t),{\phantom{\rule{4pt}{0ex}}}^{\mu}\Xi ({x}^{k},t),\hfill \\ & & or{\phantom{\rule{4pt}{0ex}}}^{ef}\mathrm{Y}({x}^{k},t),{\phantom{\rule{4pt}{0ex}}}^{m}\mathrm{Y}({x}^{k},t),{\phantom{\rule{4pt}{0ex}}}^{f}\mathrm{Y}({x}^{k},t),{\phantom{\rule{4pt}{0ex}}}^{\mu}\mathrm{Y}({x}^{k},t)\hfill \\ \hfill \mathrm{integration}\mathrm{cosm}.\mathrm{constants}:& & {\phantom{\rule{4pt}{0ex}}}^{ef}\Lambda ,{\phantom{\rule{3.33333pt}{0ex}}}^{m}\Lambda ,{\phantom{\rule{3.33333pt}{0ex}}}^{f}\Lambda ,{\phantom{\rule{3.33333pt}{0ex}}}^{\mu}\Lambda \hfill \\ \hfill \mathrm{integration}\mathrm{functions}:& & {\phantom{\rule{4pt}{0ex}}}_{1}{n}_{i}({x}^{k})\mathrm{and}{\phantom{\rule{4pt}{0ex}}}_{2}{n}_{i}({x}^{k})\hfill \end{array}$$We can generate solutions with any nontrivial ${\phantom{\rule{4pt}{0ex}}}^{ef}\Lambda ,{\phantom{\rule{3.33333pt}{0ex}}}^{m}\Lambda ,{\phantom{\rule{3.33333pt}{0ex}}}^{f}\Lambda ,{\phantom{\rule{3.33333pt}{0ex}}}^{\mu}\Lambda $ even any, or all, effective source ${\phantom{\rule{4pt}{0ex}}}^{ef}\mathrm{Y},{\phantom{\rule{4pt}{0ex}}}^{m}\mathrm{Y},{\phantom{\rule{4pt}{0ex}}}^{f}\mathrm{Y},{\phantom{\rule{4pt}{0ex}}}^{\mu}\mathrm{Y}$ can be zero.

#### 4.1. Inhomogeneous FTMT and MGT Configurations with Induced Nonholonomic Torsion

#### 4.2. Extracting Levi–Civita Cosmological Configurations

## 5. Locally Anisotropic Effective Scalar Potentials and Flat Regions

#### 5.1. Off-Diagonal Interactions and Associated Tmt Models with Two Flat Regions

#### 5.2. Limits to Diagonal Two Flat Regions

## 6. Reproducing Modified Massive Gravity as TMTS and Effective GR

#### 6.1. Massive Gravity Modifications of Flat Regions

#### 6.2. Reconstructing Off-Diagonal Tmt and Massive Gravity Cosmological Models

## 7. Results and Conclusions

#### 7.1. Modified Gravity and Cosmology Theories with Metric Finsler Connections on (Co) Tangent Lorentz Bundles or for Nonholonmic Einstein Manifolds

- We defined nonholonomic geometric variables for which various classes of modified gravity theories (MGTs), (generally with nontrivial gravitational mass) can be modelled equivalently as respective two-measure (TMT) [55,56,57,60,61,62], bi-connection and/or bi-metric theories. For well-defined nonholonomic constraint conditions, the corresponding gravitational and matter field equations are equivalent to certain classes of generalized Einstein equations with nonminimal connection to effective matter sources and nontrivial nonholonomic vacuum configurations;
- We stated the conditions when nonholonomic TMT models encode ghost-free massive configurations with (broken) scale invariance and such interactions can modelled by generic off-diagonal metrics in effective general relativity (GR) and generalizations with induced torsion. Such a nonholonomic geometric technique was elaborated in Finsler geometry in gravity theories and, for a corresponding 2 + 2 splitting, we can consider Finsler-like variables and work with so-called FTMT models;
- We developed the anholonomic frame deformation method [30,31,32,33], AFDM, in order to generate off-diagonal, generally inhomogeneous and locally anisotropic cosmological solutions in TMT snd MGTs. It was proven that the effective Einstein equations for such gravity and cosmological models can be decoupled in general form, which allows for the construction of various classes of exact solution depending on generating functions and integration functions and constants;
- We analysed a very important re-scaling property of generating functions with association of effective cosmological constants for different types of modified gravity and matter field interactions, which allow for the definition of nonholonomic variables into which the associated systems of nonlinear partial differential equations (PDEs) can be integrated in explicit form when the coefficients of generic off-diagonal metrics and (generalized) nonlinear and linear connections depend on all space–time coordinates;
- There were stated conditions for generating functions and effective sources when zero-torsion (Levi–Civita, LC) configurations can be extracted in general form, with possible nontrivial limits to diagonal configurations in ΛCDM cosmological scenarios, encoding dark energy and dark matter effects, possible nontrivial zero mass contributions, effective cosmological constants induced by off-diagonal interactions and constrained nonholonomically, to result in nonlinear diagonal effects;
- We studied possible massive gravity modifications of flat regions and the possible reconstruction of off-diagonal TMT and massive gravity cosmological models. Through corresponding frame transforms and the re-definition of generating functions and nonholonomic variables, we proved that the same geometric techniques are applicable in all such MGTs.

#### 7.2. Alternative Finsler Gravity Theories with Metric Non-Compatible Connections

- In the abstract and introduction, as well as Section 2.2 of this article, it is emphasized that we do not elaborate a typical work on Finsler gravity and cosmology, but rather provide a cosmological work on Einstein gravity and MGTs, TMTs ones, with two measures/two connections and/or bi-metrics, mass terms, etc., when the constructions are modelled on a Lorentz manifold V of signature (+ + + −) with conventional nonholonomic 2 + 2 splitting. For such theories, the spacetime metrics ${g}_{\alpha \beta}({x}^{i},{y}^{a})$ (with $i,j,\cdots =1,2$ and $a,b,\cdots =3,4$) are generic off-diagonal and, together with the coefficients of other fundamental geometric objects, depend on all space–time conventional fibred coordinates. Lagrange–Finsler-like variables are introduced to V for “toy” models, when ${y}^{a}$ are treated similarly to (co) fiber coordinates on a (co) tangent manifold (${T}^{\ast}V$) $TV,$ for a prescribed a fundamental Lagrange, $L(x,y)$ (or Finsler, for certain homogeneity conditions $F(x,\beta y)=\beta F(x,y),x=\left\{{x}^{i}\right\}$ etc., for a real constant $\beta >0,$ when $L={F}^{2}$). This states, for V, a canonical Finsler-like N-connection and nonholonomic (co-)frames structure, which can also be described in coordinate bases, using additional constraints to extract the LC-connection or distorting it to other linear connections determined by the same metric structures. In dual form, we can consider momentum, like ${p}_{a}$-dependencies in ${g}_{\alpha \beta}({x}^{i},{p}_{a}),$ for a conventional Hamiltonian $H(x,p)$, which can be related to an L via corresponding Legendre transforms. The reason for introducing Finsler-like and other types of nonholonomic variable to a manifold $V,$ or on a tangent bundle $TV$ is that, in so-called nonholonomic canonical variables (with hats on geometric objects), the modified Einstein Equation (9) can be decoupled and integrated in vary general forms. We have to consider some additional nonholonomic constraints (10) in order to extract LC-configurations. This is the main idea of the AFDM [30,31,32,33], which was applied in a series of works for constructing a locally anisotropic black hole and cosmological solutions defied by generic off-diagonal metrics and (generalized) connections in Lagrange–Finsler–Hamilton gravity in various limits of (non-)commutative/supersymmetric string/brain theories, massive gravity, TMT models, etc., as we consider in partner works [26,27,28,29,30,31,32,33].
- One of the formal difficulties in modern Finsler geometry and gravity is that some authors (usually mathematicians) use a different terminology compared to that elaborated by physicists in GR, MGTs, TMTs etc. For instance, a theory of “standard static Finsler spaces”, with a time like Killing field and/or for static solutions of a type of filed equation in Finsler gravity is elaborated in [69,70,71]. Of course, it is possible to prescribe a class of static and a corresponding smooth class of Finsler-generating functions, $F(x,y)$, when semi-spray, N-connections and d-connections, and certain Finsler–Ricci generalized tensors, etc., can be computed for static configurations embedded in locally anisotropic backgrounds. Such constructions can be chosen to have spherical symmetry. However, by introducing and computing corresponding “standard static” Sasaki type metrics of type (16), and their off-diagonal coordinate base equivalents, involving N-coefficients (see the total (phase) space–time metric (17)), we can check that such geometric d-objects (and their corresponding canonical d-connection, or LC-connection) do not solve the (modified) Einstein Equation (9) if the data are the general ones considered in [69,70,71]. If the d-metric coefficients ${g}_{\alpha \beta}({x}^{i},{y}^{a})$ are generic off-diagonal with nontrivial N-connection coefficients, such metrics can be only quasi-stationary following the standard terminology in mathematical relativity and MGTs (when coefficients do not depend on time-like variable, i.e., ${\partial}_{t}$ is a Killing symmetry d-vector), but there are nontrivial off-diagonal metric terms because of rotation, N-connections, etc. Stationary metrics of type (16) and/or (17) can be prescribed to describe, for instance, black ellipsoids, which are different from the solutions for Kerr black holes, BHs, because of their more general Finsler local anisotropy. Static configurations with diagonal metrics of Schwarzschild type BHs can be introduced for some trivial N-connection structures (but, in Finsler geometry, this is a cornerstone geometric object). For Finsler-like gravity theories, there are no proofs of BH uniqueness theorems, and it is not clear if such static configurations (for instance, with spherical symmetry) can be stable. Such proofs are sketched for black ellipsoids; see details in [26,27,28,29,30,31,32,33]. Therefore, the existing concepts, definitions, and proofs of “standard” static/stationary/cosmological /stable/nonlinear evolution models, etc., depend on the type of postulated principles for respective concepts and theories of Finsler spacetime.
- In [72,73,75], certain attempts to elaborate models of Finsler spacetime, geometry and gravity are considered for some types of N-connection and chosen classes of Finsler metric compatible and non-compatible d-connections. In many cases, the Berwald–Finsler d-connection is considered, which is generally noncompatible but can be subjected to certain metrization procedures. Different geometric constructions, with a non-fixed signature for Hessians and sophisticate causality conditions via semi-sprays and generalized nonlinear geodesic configurations, have been proposed and analyzed. In such approaches, there are a series of fundamental unsolved physical and geometric problems in the development of such Finsler theories in self-consistent and viable physical forms. Here, we focus only on the most important issues (for details, critiques, discussions, and motivation regarding Finsler gravity principles, we cite [17,24,25,76,79]):
- For theories with metric noncompatible connections, for instance, of Chern or Berwald type, there are no unique and simple possibilities to define spinors, conservation laws of type ${D}_{i}{T}^{jk}$, elaborate on supersymmetric and/or noncommutative/nonassociative generalizations, or to consider generalized type classical and quantum symmetries, considering only Finsler type d-connections proposed by some prominent geometers like E. Cartan, S. Chern, B. Berwald etc., and physically un-motivated (effective) energy-momentum tensors with local anisotropy;
- Physical principles and nonlinear causality schemes elaborated on a base manifold with undetermined lifts, without geometric and physical motivations, on total bundles, depend on the type of Finsler-generating function. Hessians and nonlinear and linear connections are chosen for elaborating geometric and physical models. A Finsler geometry is not a (pseudo/semi-) Riemannian geometry, where all constructions are determined by the metric and LC-connection structures. For instance, certain constructions with cosmological kinetic/statistical Finsler spacetime in [73,75] are subjected to very complex type conservation laws and nonlinear kinetic/diffusion equations. Those authors have not cited and or applied earlier, locally anisotropic, generalized Finsler kinetic/diffusion/statistical constructions performed for the metric compatible connections studied in [77,78,79] (N. Voicu was at S. Vacaru’s seminars in Brashov in 2012, on Finsler kinetics, diffusion and applications in modern physics and information theory; see also [33], but, together with her co-authors, do not cite, discuss, or apply such locally anisotropic, metric, compatible and solvable geometric flow, kinetic and geometric thermodynamic theories);
- Various variational principles and certain versions of Finsler modified Einstein equations were proposed and developed in [72,73,75], but such theories have been not elaborated on total bundle spaces, for certain metric compatible Finsler connections. Usually, metric non-compatible Finsler connections were used, when it is not possible to elaborate on certain general methods for the construction of exact and parametric solutions to nonlinear systems of PDEs; for instance, describing locally anisortopic interactions of modified Finsler–Einstein–Dirac–Yang–Mills–Higgs systems. In S. Vacaru and co-authors’ axiomatic approach to relativistic Finsler–Lagrange–Hamilton theories [17,24,25,76], such generalized systems can be studied—for instance, on (co) tangent Lorentz bundles (and on Lorentz manifolds with conventional nonhlonomic fibred splitting)—when the AFDM was applied to generate exact and parametric solutions, and certain deformation quantization, gauge-like, etc., schemes were developed;

- As a result, the authors of [74] concluded their work in a pessimistic fashion: “Finsler geometry is a very natural generalisation of pseudo-Riemannian geometry and there are good physical motivations for considering Finsler spacetime theories. We have mentioned the Ehlers-Pirani-Schild axiomatic and also the fact that a Finsler modification of GR might serve as an effective theory of gravity that captures some aspects of a (yet unknown) theory of Quantum Gravity. We have addressed the somewhat embarrassing fact that there is not yet a general consensus on fundamental Finsler equations, in particular on Finslerian generalisations of the Dirac equation and of the Einstein equation, and not even on the question of which precise mathematical definition of a Finsler spacetime is most appropriate in view of physics. We have seen that the observational bounds on Finsler deviations at the laboratory scale are quite tight. By contrast, at the moment we do not have so strong limits on Finsler deviations at astronomical or cosmological scales.” In that work, there was no discussion or analysis of the approach developed for Lorentz–Finsler–Lagrange–Hamilton, and the nonholonomic manifolds developed by authors of this paper, beginning in 1994 and published in more than 150 papers in prestigious mathematical and physical journals, as well as summarized in three monographs (for reviews, see [24,25,79]).

## Author Contributions

## Funding

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## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Stavrinos, P.; Vacaru, S.I.
Broken Scale Invariance, Gravity Mass, and Dark Energy inModified Einstein Gravity with Two Measure Finsler like Variables. *Universe* **2021**, *7*, 89.
https://doi.org/10.3390/universe7040089

**AMA Style**

Stavrinos P, Vacaru SI.
Broken Scale Invariance, Gravity Mass, and Dark Energy inModified Einstein Gravity with Two Measure Finsler like Variables. *Universe*. 2021; 7(4):89.
https://doi.org/10.3390/universe7040089

**Chicago/Turabian Style**

Stavrinos, Panayiotis, and Sergiu I. Vacaru.
2021. "Broken Scale Invariance, Gravity Mass, and Dark Energy inModified Einstein Gravity with Two Measure Finsler like Variables" *Universe* 7, no. 4: 89.
https://doi.org/10.3390/universe7040089