# Background Independence and Gauge Invariance in General Relativity Part 1—The Classical Theory

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- It should fulfill the principle of general covariance (PGC) in arbitrary GR-frames that are related by means of local point transformations (see discussion below in Section 2).
- It should have the goal of determining the structure of space-time, based on the identification of a Riemannian differential manifold associated with it. In particular, this refers to the prescription of its local metric tensor, namely the so-called background metric field tensor, to be represented with respect to an arbitrary GR-frame.
- It should determine the Hamiltonian structures, both classical and quantum ones, respectively, realized in the frameworks of SF-GR and QG, both associated with the Einstein field equations (EFE).

- To achieve an admissible candidate for a QG theory worth of this name, the constructions of a, possibly non-unique, Hamiltonian representation for GR, namely of EFE, is required. The crucial characteristics of the same representation that are consistent with the basic principles of GR will be investigated.
- Besides PGC, the consequences of the principle of manifest covariance (PMC) [4] and its relationship with the principle of objectivity must be addressed. The latter principle concerns the determination of the classical background space-time and the prescription of the appropriate variational treatments to be adopted for the gravitational field.
- Furthermore, comparisons with literature are necessary both in GR and QG and must be discussed separately, given the different and peculiar properties of currently available theories.

## 2. The General Covariance Principle

- (a)
- The Riemann distance in the two space-times $\left\{{\mathbf{Q}}^{4},\widehat{g}\left(r\right)\right\}$ and $\left\{{\mathbf{Q}}^{\prime 4},{\widehat{g}}^{\prime}\left({r}^{\prime}\right)\right\}$ is the same; namely, it is realized by means of a $4-$scalar, so that $d{s}^{2}={\widehat{g}}_{\mu \nu}\left(r\right)d{r}^{\mu}d{r}^{\nu}={\widehat{g}}_{\mu \nu}^{\prime}\left({r}^{\prime}\right)d{r}^{\prime \mu}d{r}^{\prime \nu}$.
- (b)
- The fields $\widehat{g}\left(r\right)$ and ${\widehat{g}}^{\prime}\left({r}^{\prime}\right)$ are $4-$tensors. Hence, their covariant components ${\widehat{g}}_{\mu \nu}\left(r\right)$ and ${\widehat{g}}_{\mu \nu}^{\prime}\left({r}^{\prime}\right)$ are related via the corresponding covariant $4-$tensor transformation laws. In tensor and symbolic form, the direct and inverse transformations $\widehat{g}\left(r\right)\equiv \left\{{\widehat{g}}_{\mu \nu}\left(r\right)\right\}\to {\widehat{g}}^{\prime}\left({r}^{\prime}\right)\equiv \left\{{\widehat{g}}_{\mu \nu}^{\prime}\left({r}^{\prime}\right)\right\}$ and ${\widehat{g}}^{\prime}\left({r}^{\prime}\right)\equiv \left\{{\widehat{g}}_{\mu \nu}^{\prime}\left({r}^{\prime}\right)\right\}\to \widehat{g}\left(r\right)\equiv \left\{{\widehat{g}}_{\mu \nu}\left(r\right)\right\}$ read, respectively,$$\left\{\begin{array}{c}{\widehat{g}}_{\alpha \beta}^{\prime}\left({r}^{\prime}\right)={\widehat{g}}_{\mu \nu}\left(r\left({r}^{\prime}\right)\right)\frac{\partial {r}^{\mu}}{\partial {r}^{\prime \alpha}}\frac{\partial {r}^{\nu}}{\partial {r}^{\prime \beta}}\\ {\widehat{g}}_{\mu \nu}\left(r\right)={\widehat{g}}_{\alpha \beta}^{\prime}\left({r}^{\prime}\left(r\right)\right)\frac{\partial {r}^{\prime \alpha}}{\partial {r}^{\mu}}\frac{\partial {r}^{\prime \beta}}{\partial {r}^{\nu}}\end{array}\right.,$$$$\left\{\begin{array}{c}{\widehat{g}}^{\prime}\left({r}^{\prime}\right)=M\left(r\left({r}^{\prime}\right)\right)\u2022\widehat{g}\left(r\left({r}^{\prime}\right)\right)\u2022M\left(r\left({r}^{\prime}\right)\right)\\ \widehat{g}\left(r\right)={M}^{-1}\left({r}^{\prime}\left(r\right)\right)\u2022\widehat{g}\left({r}^{\prime}\left(r\right)\right)\u2022{M}^{-1}\left({r}^{\prime}\left(r\right)\right)\end{array}\right.,$$
- (c)
- The tensor fields $\widehat{g}\left(r\right)$ and ${\widehat{g}}^{\prime}\left({r}^{\prime}\right)$ are metric tensors so that they are required to satisfy the orthogonality conditions$$\begin{array}{ccc}\hfill {\widehat{g}}_{\mu \nu}\left(r\right){\widehat{g}}^{\mu \eta}\left(r\right)& =& {\delta}_{\nu}^{\eta},\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\widehat{g}}_{\mu \nu}^{\prime}\left({r}^{\prime}\right){\widehat{g}}^{\prime \mu \eta}\left({r}^{\prime}\right)& =& {\delta}_{\nu}^{\eta}.\hfill \end{array}$$
- (d)
- Finally, the Ricci and Riemann tensors ${R}_{\mu \nu}\left(\widehat{g}\left(r\right)\right),{R}_{\mu p\nu q}\left(\widehat{g}\left(r\right)\right)$ and ${R}_{\mu \nu}^{\prime}\left({\widehat{g}}^{\prime}\left({r}^{\prime}\right)\right),{R}_{\mu p\nu q}^{\prime}\left({\widehat{g}}^{\prime}\left({r}^{\prime}\right)\right)$, which are associated, respectively, with the two structures $\left\{{\mathbf{Q}}^{4},g\left(r\right)\right\}$ and $\left\{{\mathbf{Q}}^{\prime 4},{g}^{\prime}\left({r}^{\prime}\right)\right\}$, are transformed in each other in accordance with the covariance $4-$tensor transformation laws indicated above by Equations (4) and (8).

#### 2.1. The Principle of Manifest Covariance and Its Extension

#### 2.2. The Principle of Objectivity

- Consequence #1—the validity of the Objectivity Principle implies, therefore, the existence of a classical background space-time, characterized by a prescribed background metric tensor $\widehat{g}\left(r\right)\equiv \left\{{\widehat{g}}_{\mu \nu}\left(r\right)\right\}\equiv \left\{{\widehat{g}}^{\mu \nu}\left(r\right)\right\}$.
- Consequence #2— all observables depending on $\widehat{g}\left(r\right)$ must be regarded as objective observables as well. Equivalently, the geometry of the background space-time $\left\{{\mathbf{Q}}^{4},\widehat{g}\left(r\right)\right\}$ depends uniquely on $\widehat{g}\left(r\right)$. This implies, in particular, that the Riemann distance and the invariant $4-$volume element of the background space-time should be unique functions of $\widehat{g}\left(r\right)$, namely of the form$$\begin{array}{ccc}\hfill d{s}^{2}& =& {\widehat{g}}_{\mu \nu}\left(r\right)d{r}^{\mu}d{r}^{\nu},\hfill \end{array}$$$$\begin{array}{ccc}\hfill d\widehat{\Omega}& =& {d}^{4}r\sqrt{-\left|\widehat{g}\left(r\right)\right|}.\hfill \end{array}$$Similarly, the space-time Riemann and Ricci $4-$tensors associated with the same background space-time, i.e.,$$\begin{array}{ccc}& & {\widehat{R}}_{\mu p\nu q}={R}_{\mu p\nu q}\left(\widehat{g}\left(r\right)\right),\hfill \end{array}$$$$\begin{array}{ccc}& & {\widehat{R}}_{\mu \nu}={R}_{\mu \nu}\left(\widehat{g}\left(r\right)\right)={\widehat{g}}^{pq}\left(r\right){R}_{\mu p\nu q}\left(\widehat{g}\left(r\right)\right),\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\widehat{\nabla}}_{\eta}{\widehat{g}}_{\mu \nu}\left(r\right)& =& {\partial}_{\eta}{\widehat{g}}_{\mu \nu}-{\widehat{\Gamma}}_{\eta \mu}^{p}{\widehat{g}}_{p\nu}-{\widehat{\Gamma}}_{\eta \nu}^{p}{\widehat{g}}_{\mu p}=0,\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\widehat{\nabla}}_{\eta}{\widehat{g}}^{\mu \nu}\left(r\right)& =& {\partial}_{\eta}{\widehat{g}}^{\mu \nu}+{\widehat{\Gamma}}_{\eta p}^{\mu}{\widehat{g}}^{p\nu}+{\widehat{\Gamma}}_{\eta p}^{\nu}{\widehat{g}}^{\mu p}=0,\hfill \end{array}$$
- Consequence #3— In the following, we intend to prove that, once the objective viewpoint is taken, then necessarily all relevant dynamical variables and physical observables can always be identified with $4-$tensors. This will be referred to here as the universal $4-$tensor property of classical field variables.

## 3. Asynchronous and Synchronous Lagrangian Variational Approaches

#### 3.1. The Asynchronous and Synchronous Functional Settings

#### 3.2. Asynchronous Lagrangian Action Principle (Non-Objective Viewpoint)

- The first one is that, consistent with the non-objective viewpoint discussed above, the varied functions, i.e., the Lagrangian variables $g\left(r\right)\equiv \left\{{g}_{\mu \nu}\left(r\right)\right\}\equiv \left\{{g}^{\mu \nu}\left(r\right)\right\}$ appearing in Equation (25) belong by assumption to ${\left\{g\right\}}_{C}$.
- The second aspect is that, since the space-time $4-$volume element $d\Omega $ by definition depends on the determinant of the variational metric tensor, necessarily its variation $\delta d\Omega $ is non-vanishing, since$$\delta d\Omega ={d}^{4}r\delta \sqrt{-\left|g\right|}\ne 0,$$
- The third feature is that the variational Lagrangian $\overline{L}\left(g\right)$, defined by Equation (26), is not a $4-$scalar, but rather a $4-$scalar-density. This confirms, therefore, that the same $4-$tensor-density appears as a consequence of the chosen non-objective viewpoint, implicit in the choice of the functional setting ${\left\{g\right\}}_{C}$. Indeed, in Equations (25) and (26)$$L(g,r)={V}_{EH}\left(g\right)+{V}_{Fg}(g,r)$$$${V}_{EH}={\alpha}_{L}\left[{g}^{\mu \nu}{R}_{\mu \nu}\left(g\right)-2\Lambda \right]$$
- The fourth notable feature lies in the EH variational principle (or EH action principle). This is obtained by requiring that for arbitrary variations $\delta g\left(r\right)$ belonging to ${\left\{g\right\}}_{C}$, it must be$${\left.\delta {S}_{EH}\left(g\left(r\right)\right)\right|}_{g=\widehat{g}\left(r\right)}={\left.\frac{d}{d\theta}{S}_{EH}({g}_{extr}\left(r\right)+\theta \delta g\left(r\right))\right|}_{\theta =0}=0,$$$${\left.\delta {S}_{EH}\left(g\left(r\right)\right)\right|}_{g=\widehat{g}\left(r\right)}={\left(\delta {S}_{EH}\left(g\right)\right)}_{\mathrm{expl}}+{\left(\delta {S}_{EH}\left(g\right)\right)}_{\mathrm{impl}},$$$${\left(\delta {S}_{EH}\left(g\right)\right)}_{\mathrm{impl}}={\alpha}_{L}{\int}_{{\mathbf{Q}}^{4}}{d}^{4}r\sqrt{-\left|g\right|}{\widehat{g}}^{\mu \nu}\delta {R}_{\mu \nu},$$$${\left(\delta {S}_{EH}\left(g\right)\right)}_{\mathrm{expl}}={\alpha}_{L}{\int}_{{\mathbf{Q}}^{4}}{d}^{4}r\sqrt{-\left|g\right|}\left[{\widehat{R}}_{\mu \nu}-\left(\frac{1}{2}\widehat{R}-\Lambda \right){\widehat{g}}_{\mu \nu}\left(r\right)-\kappa {\widehat{T}}_{\mu \nu}\right]\delta {g}^{\mu \nu}.$$Here, as usual, ${\widehat{R}}_{\mu \nu}={R}_{\mu \nu}\left(\widehat{g}\left(r\right)\right)$ and $\widehat{R}={\widehat{g}}^{\mu \nu}\left(r\right){R}_{\mu \nu}\left(\widehat{g}\left(r\right)\right)\equiv R(\left(\widehat{g}\left(r\right)\right)$ denote, respectively, the background Ricci $4-$tensor and Ricci $4-$scalar, while ${\widehat{T}}_{\mu \nu}={T}_{\mu \nu}\left(\widehat{g}\left(r\right)\right)$ is the background stress-energy tensor associated with the external source fields described by the external-field Lagrangian density ${L}_{F}\left(g\right)$. Here, as anticipated, the universal constant ${\alpha}_{L}$ which multiplies the rhs (right hand side) of Equation (33), does not affect the EH-action principles, while $\kappa $ denotes the universal constant$$\kappa =\frac{8\pi G}{{c}^{4}},$$
- Finally, as a fifth notable feature, in order to exactly recover the Einstein field equations, namely$${\widehat{R}}_{\mu \nu}-\left(\frac{1}{2}\widehat{R}-\Lambda \right){\widehat{g}}_{\mu \nu}=\kappa {\widehat{T}}_{\mu \nu},$$$${\left(\delta {S}_{EH}\left(g\right)\right)}_{\mathrm{impl}}=0$$

#### 3.3. Synchronous Metric-Lagrangian Action Principle (Objective Viewpoint)

#### 3.4. Search of a Metric-Hamiltonian Action Principle

## 4. Extended Lagrangian Variables: The Metric-Ricci Action Principle

## 5. Manifestly Covariant Metric-Ricci Hamiltonian Approach

#### 5.1. Gauge Properties of the metric-Ricci Lagrangian and Hamiltonian Action Principles

#### 5.2. Discussion and Comparisons

## 6. Conclusions

- The first one is that the asynchronous EH-action principle does not permit the construction of a manifestly covariant Hamiltonian approach for EFE. In fact, such a possibility is simply ruled out by the prescription of the functional setting for the same action principle, the reason being that the “generalized velocities”, i.e., the covariant derivatives of the variational metric tensors $g\left(r\right)$ vanish.
- On the contrary, a manifestly–covariant $4-$tensor Hamiltonian theory exists, which is based on the adoption of a synchronous action variational principle. As shown here, such a result can be achieved by means of suitably prescribed Lagrangian and Hamiltonian (synchronous) action principles.
- As a third issue, we have shown that based on the same type of synchronous action principles, an extended-variable manifestly-covariant Hamiltonian variational formulation can also be reached, in which Lagrangian coordinates are represented by the set of the metric-Ricci variational fields $\left\{g\left(r\right),R\left(s\right)\right\}$. The resulting Euler–Lagrange equations have been obtained in manifestly covariant $4-$tensor form. In terms of the corresponding metric-Ricci action principle, the Einstein field equations have been recovered as particular solutions.
- Fourth, all the synchronous variational principles considered here have been shown to fulfill the fundamental gauge property characteristic of all classical field theories in flat space-time. Namely, the $4-$scalar variational Lagrangian, which characterizes these theories, is gauge-invariant with respect to an arbitrary additive constant of the same variational Lagrangian.
- Fifth, all synchronous variational principles obtained here are background independent in the sense that they hold for an arbitrary background metric field tensor $\widehat{g}\left(r\right)$, namely an arbitrary particular solution of the Einstein field equations.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Functional Settings

**:**

## Appendix B. The Metric-Lagrangian and Hamiltonian Principles

## References

- Frolov, A.M. General Principles of Hamiltonian Formulations of the Metric Gravity. Phys. At. Nucl.
**2021**, 84, 750. [Google Scholar] [CrossRef] - Kiriushcheva, N.; Kuzmin, S.V. The Hamiltonian of Einstein affine-metric formulation of General Relativity. Eur. Phys. J. C
**2010**, 70, 389. [Google Scholar] [CrossRef] [Green Version] - Ali, S.A.; Cafaro, C.; Capozziello, S.; Corda, C. On the Poincaré Gauge Theory of Gravitation. Int. J. Theor. Phys.
**2009**, 48, 3426–3448. [Google Scholar] [CrossRef] [Green Version] - Cremaschini, C.; Tessarotto, M. Classical variational theory of the cosmological constant and its consistency with quantum prescription. Symmetry
**2020**, 12, 633. [Google Scholar] [CrossRef] - Cremaschini, C.; Tessarotto, M. Synchronous Lagrangian variational principles in General Relativity. Eur. Phys. J. Plus
**2015**, 130, 123. [Google Scholar] [CrossRef] [Green Version] - Tessarotto, M.; Cremaschini, C. The Principle of Covariance and the Hamiltonian Formulation of General Relativity. Entropy
**2021**, 23, 215. [Google Scholar] [CrossRef] [PubMed] - Arnowitt, R.; Deser, S.; Misner, C. Dynamical Structure and Definition of Energy in General Relativity. Phys. Rev.
**1959**, 116, 1322. [Google Scholar] [CrossRef] - Arnowitt, R.; Deser, S.; Misner, C.W. Gravitation: An Introduction to Current Research; Witten, L., Ed.; Wiley: New York, NY, USA, 1962. [Google Scholar]
- Cremaschini, C.; Tessarotto, M. Hamiltonian approach to GR – Part 1: Covariant theory of classical gravity. Eur. Phys. J. C
**2017**, 77, 329. [Google Scholar] [CrossRef] [Green Version] - Cremaschini, C.; Tessarotto, M. Coupling of quantum gravitational field with Riemann and Ricci curvature tensors. Eur. Phys. J. C
**2021**, 81, 548. [Google Scholar] [CrossRef] - De Donder, T. Théorie Invariantive Du Calcul des Variations; Gaultier-Villars & Cia.: Paris, France, 1930. [Google Scholar]
- Weyl, H. Geodesic Fields in the Calculus of Variation for Multiple Integrals. Ann. Math.
**1935**, 36, 607. [Google Scholar] [CrossRef] - Saunders, D.J. The Geometry of Jet Bundles; Cambridge University Press: Cambridge, UK, 1989. [Google Scholar]
- Sardanashvily, G. Generalized Hamiltonian Formalism for Field Theory; World Scientific Publishing: Singapore, 1995. [Google Scholar]
- Echeverría-Enríquez, A.; Mu noz-Lecanda, M.C.; Román-Roy, N. Geometry of Lagrangian first-order classical field theories. Fortschr. Phys.
**1996**, 44, 235. [Google Scholar] [CrossRef] [Green Version] - Echeverría-Enríquez, A.; Mu noz-Lecanda, M.C.; Román-Roy, N. Geometry of multisymplectic Hamiltonian first-order field theories. J. Math. Phys.
**2000**, 41, 7402. [Google Scholar] [CrossRef] [Green Version] - Struckmeier, J.; Redelbach, A. Covariant Hamiltonian Field Theory. Int. J. Mod. Phys. E
**2008**, 17, 435. [Google Scholar] [CrossRef] [Green Version] - Struckmeier, J.; Muench, J.; Vasak, D.; Kirsch, J.; Hanauske, M.; Stoecker, H. Canonical transformation path to gauge theories of gravity. Phys. Rev. D
**2017**, 95, 124048. [Google Scholar] [CrossRef] [Green Version] - Cremaschini, C.; Tessarotto, M. Hamiltonian approach to GR—Part 2: Covariant theory of quantum gravity. Eur. Phys. J. C
**2017**, 77, 330. [Google Scholar] [CrossRef] - Tessarotto, M.; Cremaschini, C. Generalized Lagrangian path approach to manifestly-covariant quantum gravity theory. Entropy
**2018**, 20, 205. [Google Scholar] [CrossRef] [Green Version] - Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity; Wiley: New York, NY, USA, 1972. [Google Scholar]
- Wald, R. General Relativity; University of Chicago Press: Chicago, MI, USA, 1984. [Google Scholar]
- Landau, L.D.; Lifschitz, E.M. Field Theory, Theoretical Physics Volume 2; Addison-Wesley: New York, NY, USA, 1957. [Google Scholar]
- Lanczos, C. The Variational Principles of Mechanics, 4th paperback ed.; Courier Corporation: Dover, UK, 1975; p. 351. ISBN 0-8020-1743-6. [Google Scholar]
- Cremaschini, C.; Kovář, J.; Stuchlík, Z.; Tessarotto, M. Variational theory of the Ricci curvature tensor dynamics. Eur. Phys. J. C
**2021**, 81, 1030. [Google Scholar] [CrossRef] - Stuchlík, Z.; Kološ, M.; Kovář, J.; Tursunov, P.S.A. Influence of Cosmic Repulsion and Magnetic Fields on Accretion Disks Rotating around Kerr Black Holes. Universe
**2020**, 6, 26. [Google Scholar] [CrossRef] [Green Version] - Ovalle, J.; Contreras, E.; Stuchlík, Z. Kerr-de Sitter black hole revisited. Phys. Rev. D
**2021**, 103, 084016. [Google Scholar] [CrossRef] - Palatini, A. Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton. Rend. Circ. Mat. Palermo
**1919**, 43, 203. [Google Scholar] [CrossRef] - Olmo, G.J. Palatini Approach to Modified Gravity: f(R) Theories and Beyond. Int. J. Mod. Phys. D
**2011**, 20, 413. [Google Scholar] [CrossRef]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tessarotto, M.; Cremaschini, C.
Background Independence and Gauge Invariance in General Relativity Part 1—The Classical Theory. *Symmetry* **2022**, *14*, 2083.
https://doi.org/10.3390/sym14102083

**AMA Style**

Tessarotto M, Cremaschini C.
Background Independence and Gauge Invariance in General Relativity Part 1—The Classical Theory. *Symmetry*. 2022; 14(10):2083.
https://doi.org/10.3390/sym14102083

**Chicago/Turabian Style**

Tessarotto, Massimo, and Claudio Cremaschini.
2022. "Background Independence and Gauge Invariance in General Relativity Part 1—The Classical Theory" *Symmetry* 14, no. 10: 2083.
https://doi.org/10.3390/sym14102083