Modifications of Gravity Via Differential Transformations of Field Variables
Abstract
:1. Introduction
2. Examples of Theories with Differential Field Transformations
2.1. Mechanics
2.2. Massive Scalar Field
2.3. The Hilbert–Palatini Approach
3. Isometric Embeddings and Regge–Teitelboim Gravity
4. Mimetic Gravity and Its Extensions
4.1. Mimetic Gravity
4.2. Disformal Transformations
5. Concluding Remarks
- Is it possible to attain an arbitrary value of the original variables by choosing new variables?
- Is it possible to attain an arbitrary value of the variations of original variables by choosing the variations of new variables?
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
- Stephani, H.; Kramer, D.; Maccallum, M.; Hoenselaers, C.; Herlt, E. Exact Solutions of Einstein’s Field Equations, 2nd ed.; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Ferraris, M.; Francaviglia, M.; Reina, C. Variational formulation of general relativity from 1915 to 1925 “Palatini’s method” discovered by Einstein in 1925. Gen. Relat. Gravit. 1982, 14, 243–254. [Google Scholar] [CrossRef]
- Brandt, F.T.; McKeon, D.G.C. Perturbative calculations with the first order form of gauge theories. Phys. Rev. D 2015, 91, 105006. [Google Scholar] [CrossRef] [Green Version]
- Kharuk, N.V.; Paston, S.A.; Sheykin, A.A. Classical Electromagnetic Potential as a Part of Gravitational Connection: Ideas and History. Gravit. Cosmol. 2018, 24, 209–219. [Google Scholar] [CrossRef] [Green Version]
- Krasnov, K.; Percacci, R. Gravity and unification: A review. Class. Quantum Gravity 2018, 35, 143001. [Google Scholar] [CrossRef] [Green Version]
- Scholz, E.E. Cartans attempt at bridge-building between Einstein and the Cosserats- or how translational curvature became to be known as torsion. Eur. Phys. J. H 2019, 44, 47–75. [Google Scholar] [CrossRef]
- Goldstein, C.; Ritter, J. The Varieties of Unity: Sounding Unified Theories 1920–1930. In Revisiting the Foundations of Relativistic Physics: Festschrift in Honor of John Stachel; Renn, J., Divarci, L., Schröter, P., Ashtekar, A., Cohen, R.S., Howard, D., Sarkar, S., Shimony, A., Eds.; Springer: Dordrecht, The Netherlands, 2003; pp. 93–149. [Google Scholar] [CrossRef]
- Fock, V.A. Geometrisierung der Diracschen Theorie des Elektrons. Z. Phys. 1929, 57, 261–277. [Google Scholar] [CrossRef]
- Hehl, F.W.; Obukhov, Y. Elie Cartan’s torsion in geometry and in field theory, an essay. arXiv 2007, arXiv:0711.1535. [Google Scholar]
- López, L.A.G.; Quiroga, G.D. Asymptotic structure of spacetime and the Newman-Penrose formalism: A brief review. Rev. Mexicana Física 2017, 63, 275–286. [Google Scholar]
- Arnowitt, R.; Deser, S.; Misner, C. The Dynamics of General Relativity. In Gravitation: An Introduction to Current Research; Witten, L., Ed.; Wiley: New York, NY, USA, 1962; Chapter 7; pp. 227–265. [Google Scholar]
- Franke, V.A. Different canonical formulations of Einstein’s theory of gravity. Theor. Math. Phys. 2006, 148, 995–1010. [Google Scholar] [CrossRef] [Green Version]
- Corichi, A.; Rubalcava-García, I.; Vukašinac, T. Actions, topological terms and boundaries in first-order gravity: A review. Int. J. Mod. Phys. D 2016, 25, 1630011. [Google Scholar] [CrossRef] [Green Version]
- Schäfer, G.; Jaranowski, P. Hamiltonian formulation of general relativity and post-Newtonian dynamics of compact binaries. Liv. Rev. Relat. 2018, 21, 7. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Sheykin, A.A.; Paston, S.A. The approach to gravity as a theory of embedded surface. AIP Conf. Proc. 2014, 1606, 400. [Google Scholar] [CrossRef] [Green Version]
- Golovnev, A. On the recently proposed mimetic Dark Matter. Phys. Lett. B 2014, 728, 39–40. [Google Scholar] [CrossRef] [Green Version]
- Takahashi, K.; Motohashi, H.; Suyama, T.; Kobayashi, T. General invertible transformation and physical degrees of freedom. Phys. Rev. D 2017, 95. [Google Scholar] [CrossRef] [Green Version]
- Pons, J.M. Substituting fields within the action: Consistency issues and some applications. J. Math. Phys. 2010, 51, 122903. [Google Scholar] [CrossRef] [Green Version]
- Dahia, F.; Romero, C. The embedding of space–times in five dimensions with nondegenerate Ricci tensor. J. Math. Phys. 2002, 43, 3097–3106. [Google Scholar] [CrossRef] [Green Version]
- Dunajski, M.; Tod, P. Conformally isometric embeddings and Hawking temperature. Class. Quantum Gravity 2019, 36, 125005. [Google Scholar] [CrossRef] [Green Version]
- Janet, M. Sur la possibilite de plonger un espace riemannien donne dans un espace euclidien. Ann. Soc. Polon. Math. 1926, 5, 38–43. [Google Scholar]
- Kartan, E. Sur la possibilité de plonger un espace riemannien donné dans un espace euclidien. Ann. Soc. Polon. Math. 1927, 6, 1–7. [Google Scholar]
- Friedman, A. Local isometric embedding of Riemannian manifolds with indefinite metric. J. Math. Mech. 1961, 10, 625. [Google Scholar] [CrossRef]
- Clarke, C.J. On the global isometric embedding of pseudo-Riemannian manifolds. Proc. R. Soc. Lond. A Math. Phys. Sci. 1970, 314, 417–428. [Google Scholar]
- Paston, S.A.; Sheykin, A.A. Embeddings for solutions of Einstein equations. Theor. Math. Phys. 2013, 175, 806–815. [Google Scholar] [CrossRef] [Green Version]
- Paston, S.A.; Sheykin, A.A. Embeddings for Schwarzschild metric: Classification and new results. Class. Quant. Grav. 2012, 29, 095022. [Google Scholar] [CrossRef]
- Goenner, H. Local Isometric Embedding of Riemannian Manifolds and Einstein’s Theory of Gravitation. In General Relativity and Gravitation: One Hundred Years after the Birth of Albert Einstein; Held, A., Ed.; Plenum Press: New York, NY, USA, 1980; Volume 1, Chapter 14; pp. 441–468. [Google Scholar]
- Deser, S.; Levin, O. Mapping Hawking into Unruh thermal properties. Phys. Rev. D 1999, 59, 064004. [Google Scholar] [CrossRef] [Green Version]
- Paston, S.A. Hawking into Unruh mapping for embeddings of hyperbolic type. Class. Quant. Grav. 2015, 32, 145009. [Google Scholar] [CrossRef] [Green Version]
- Grad, D.A.; Ilin, R.V.; Paston, S.A.; Sheykin, A. Gravitational energy in the framework of embedding and splitting theories. Int. J. Mod. Phys. D 2018, 27, 1750188. [Google Scholar] [CrossRef] [Green Version]
- Wang, M.T.; Yau, S.T. Isometric Embeddings into the Minkowski Space and New Quasi-Local Mass. Commun. Math. Phys. 2009, 288, 919–942. [Google Scholar] [CrossRef] [Green Version]
- Regge, T.; Teitelboim, C. General relativity à la string: A progress report. In Proceedings of the First Marcel Grossmann Meeting, Trieste, Italy, 7–12 July 1975; Ruffini, R., Ed.; North-Holland: Amsterdam, The Netherlands, 1977; pp. 77–88. [Google Scholar]
- Pavsic, M.; Tapia, V. Resource Letter on geometrical results for Embeddings and Branes. arXiv 2000, arXiv:gr-qc/0010045. [Google Scholar]
- Estabrook, F.B. The Hilbert Lagrangian and Isometric Embedding: Tetrad Formulation of Regge-Teitelboim Gravity. J. Math. Phys. 2010, 51, 042502. [Google Scholar] [CrossRef] [Green Version]
- Paston, S.A.; Franke, V.A. Canonical formulation of the embedded theory of gravity equivalent to Einstein’s general relativity. Theor. Math. Phys. 2007, 153, 1582–1596. [Google Scholar] [CrossRef] [Green Version]
- Deser, S.; Pirani, F.A.E.; Robinson, D.C. New embedding model of general relativity. Phys. Rev. D 1976, 14, 3301–3303. [Google Scholar] [CrossRef] [Green Version]
- Kokarev, S.S. Space-time as multidimensional elastic plate. Nuovo Cim. B 1998, 113, 1339–1350. [Google Scholar]
- Bustamante, M.D.; Debbasch, F.; Brachet, M.E. Classical Gravitation as free Membrane Dynamics. arXiv 2005, arXiv:gr-qc/0509090. [Google Scholar]
- Paston, S.A.; Semenova, A.N. Constraint algebra for Regge-Teitelboim formulation of gravity. Int. J. Theor. Phys. 2010, 49, 2648–2658. [Google Scholar] [CrossRef] [Green Version]
- Paston, S.A.; Semenova, E.N.; Franke, V.A.; Sheykin, A.A. Algebra of Implicitly Defined Constraints for Gravity as the General Form of Embedding Theory. Gravit. Cosmol. 2017, 23, 1–7. [Google Scholar] [CrossRef]
- Davidson, A. Λ = 0 Cosmology of a Brane-like universe. Class. Quant. Grav. 1999, 16, 653. [Google Scholar] [CrossRef] [Green Version]
- Davidson, A.; Karasik, D.; Lederer, Y. Cold Dark Matter from Dark Energy. arXiv 2001, arXiv:gr-qc/0111107. [Google Scholar]
- Paston, S.A.; Sheykin, A.A. From the Embedding Theory to General Relativity in a result of inflation. Int. J. Mod. Phys. D 2012, 21, 1250043. [Google Scholar] [CrossRef]
- Paston, S.A. Forms of action for perfect fluid in general relativity and mimetic gravity. Phys. Rev. D 2017, 96, 084059. [Google Scholar] [CrossRef] [Green Version]
- Paston, S.A.; Sheykin, A.A. Embedding theory as new geometrical mimetic gravity. Eur. Phys. J. C 2018, 78, 989. [Google Scholar] [CrossRef]
- Golovnev, A.; Smirnov, F. Unusual square roots in the ghost-free theory of massive gravity. J. High Energy Phys. 2017, 2017, 130. [Google Scholar] [CrossRef]
- Chamseddine, A.H.; Mukhanov, V. Mimetic dark matter. J. High Energy Phys. 2013, 2013, 135. [Google Scholar] [CrossRef] [Green Version]
- Astashenok, A.V.; Odintsov, S.D.; Oikonomou, V.K. Modified Gauss–Bonnet gravity with the Lagrange multiplier constraint as mimetic theory. Class. Quantum Gravity 2015, 32, 185007. [Google Scholar] [CrossRef] [Green Version]
- Guendelman, E.; Singleton, D.; Yongram, N. A two measure model of dark energy and dark matter. J. Cosmol. Astrop. Phys. 2012, 1211, 044. [Google Scholar] [CrossRef] [Green Version]
- Sebastiani, L.; Vagnozzi, S.; Myrzakulov, R. Mimetic Gravity: A Review of Recent Developments and Applications to Cosmology and Astrophysics. Adv. High Energy Phys. 2017, 2017, 3156915. [Google Scholar] [CrossRef]
- Chamseddine, A.H.; Mukhanov, V.; Vikman, A. Cosmology with Mimetic Matter. J. Cosmol. Astrop. Phys. 2014, 6, 017. [Google Scholar] [CrossRef]
- Mirzagholi, L.; Vikman, A. Imperfect Dark Matter. J. Cosmol. Astrop. Phys. 2015, 2015, 028. [Google Scholar] [CrossRef] [Green Version]
- Hirano, S.; Nishi, S.; Kobayashi, T. Healthy imperfect dark matter from effective theory of mimetic cosmological perturbations. J. Cosmol. Astrop. Phys. 2017, 2017, 009. [Google Scholar] [CrossRef] [Green Version]
- Firouzjahi, H.; Gorji, M.A.; Mansoori, S.A.H.; Karami, A.; Rostami, T. Two-field disformal transformation and mimetic cosmology. J. Cosmol. Astrop. Phys. 2018, 2018, 046. [Google Scholar] [CrossRef] [Green Version]
- Vagnozzi, S. Recovering a MOND-like acceleration law in mimetic gravity. Class. Quantum Gravity 2017, 34, 185006. [Google Scholar] [CrossRef]
- Nojiri, S.; Odintsov, S.D.; Oikonomou, V.K. Unimodular-Mimetic Cosmology. Class. Quantum Gravity 2016, 33, 125017. [Google Scholar] [CrossRef] [Green Version]
- Nojiri, S.; Odintsov, S.D.; Oikonomou, V.K. Modified gravity theories on a nutshell: Inflation, bounce and late-time evolution. Phys. Rep. 2017, 692, 1–104. [Google Scholar] [CrossRef] [Green Version]
- Deruelle, N.; Rua, J. Disformal transformations, veiled general relativity and mimetic gravity. J. Cosmol. Astrop. Phys. 2014, 2014, 1409. [Google Scholar] [CrossRef]
- Bekenstein, J.D. Relation between physical and gravitational geometry. Phys. Rev. D 1993, 48, 3641–3647. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Babichev, E.; Izumi, K.; Tanahashi, N.; Yamaguchi, M. Invertible field transformations with derivatives: Necessary and sufficient conditions. arXiv 2019, arXiv:1907.12333. [Google Scholar]
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Sheykin, A.; Solovyev, D.; Sukhanov, V.; Paston, S. Modifications of Gravity Via Differential Transformations of Field Variables. Symmetry 2020, 12, 240. https://doi.org/10.3390/sym12020240
Sheykin A, Solovyev D, Sukhanov V, Paston S. Modifications of Gravity Via Differential Transformations of Field Variables. Symmetry. 2020; 12(2):240. https://doi.org/10.3390/sym12020240
Chicago/Turabian StyleSheykin, Anton, Dmitry Solovyev, Vladimir Sukhanov, and Sergey Paston. 2020. "Modifications of Gravity Via Differential Transformations of Field Variables" Symmetry 12, no. 2: 240. https://doi.org/10.3390/sym12020240
APA StyleSheykin, A., Solovyev, D., Sukhanov, V., & Paston, S. (2020). Modifications of Gravity Via Differential Transformations of Field Variables. Symmetry, 12(2), 240. https://doi.org/10.3390/sym12020240