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13 pages, 343 KiB

Open AccessArticle

by Stanislav Kuptsov, Mikhail Ioffe, Sergey Manida and Sergey Paston

Universe 2022, 8(12), 635; https://doi.org/10.3390/universe8120635 - 29 Nov 2022

Cited by 3 | Viewed by 1458

We study a perturbation theory for embedding gravity equations in a background for which corrections to the embedding function are linear with respect to corrections to the flat metric. The remaining arbitrariness after solving the linearized field equations is fixed by an assumption that the solution is static in the second order. A nonlinear differential equation is obtained, which allows for finding the gravitational potential for a spherically symmetric case if a background embedding is given. An explicit form of a spherically symmetric background parameterized by one function of radius is proposed. It is shown that this function can be chosen in such a way that the gravitational potential is in a good agreement with the observed distribution of dark matter in a galactic halo. Full article

(This article belongs to the Collection Modified Theories of Gravity and Cosmological Applications)

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14 pages, 315 KiB

Open AccessArticle

Nontrivial Isometric Embeddings for Flat Spaces

by Sergey Paston and Taisiia Zaitseva

Universe 2021, 7(12), 477; https://doi.org/10.3390/universe7120477 - 4 Dec 2021

Cited by 4 | Viewed by 2259

Abstract

Nontrivial isometric embeddings for flat metrics (i.e., those which are not just planes in the ambient space) can serve as useful tools in the description of gravity in the embedding gravity approach. Such embeddings can additionally be required to have the same symmetry as the metric. On the other hand, it is possible to require the embedding to be unfolded so that the surface in the ambient space would occupy the subspace of the maximum possible dimension. In the weak gravitational field limit, such a requirement together with a large enough dimension of the ambient space makes embedding gravity equivalent to general relativity, while at lower dimensions it guarantees the linearizability of the equations of motion. We discuss symmetric embeddings for the metrics of flat Euclidean three-dimensional space and Minkowski space. We propose the method of sequential surface deformations for the construction of unfolded embeddings. We use it to construct such embeddings of flat Euclidean three-dimensional space and Minkowski space, which can be used to analyze the equations of motion of embedding gravity. Full article

(This article belongs to the Special Issue Modified Theories of Gravity and Cosmological Applications)

18 pages, 358 KiB

Open AccessArticle

Energy–Momentum Pseudotensor and Superpotential for Generally Covariant Theories of Gravity of General Form

by Roman Ilin and Sergey Paston

Universe 2020, 6(10), 173; https://doi.org/10.3390/universe6100173 - 11 Oct 2020

Cited by 3 | Viewed by 2354

Abstract

The current paper is devoted to the investigation of the general form of the energy–momentum pseudotensor (pEMT) and the corresponding superpotential for the wide class of theories. The only requirement for such a theory is the general covariance of the action without any restrictions on the order of derivatives of the independent variables in it or their transformation laws. As a result of the generalized Noether procedure, we obtain a recurrent chain of the equations, which allows one to express canonical pEMT as a divergence of the superpotential. The explicit expression for this superpotential is also given. We discuss the structure of the obtained expressions and the conditions for the derived pEMT conservation laws to be satisfied independently (fully or partially) by the equations of motion. Deformations of the superpotential form for theories with a change in the independent variables in action are also considered. We apply these results to some interesting particular cases: general relativity and its modifications, particularly mimetic gravity and Regge–Teitelboim embedding gravity. Full article

(This article belongs to the Special Issue Selected Papers from the 17th Russian Gravitational Conference —International Conference on Gravitation, Cosmology and Astrophysics (RUSGRAV-17))

18 pages, 357 KiB

Open AccessFeature PaperArticle

Non-Relativistic Limit of Embedding Gravity as General Relativity with Dark Matter

by Sergey Paston

Universe 2020, 6(10), 163; https://doi.org/10.3390/universe6100163 - 29 Sep 2020

Cited by 17 | Viewed by 2455

Abstract

Regge-Teitelboim embedding gravity is the modified gravity based on a simple string-inspired geometrical principle—our spacetime is considered here as a 4-dimensional surface in a flat bulk. This theory is similar to the recently popular theory of mimetic gravity—the modification of gravity appears in both theories as a result of the change of variables in the action of General Relativity. Embedding gravity, as well as mimetic gravity, can be used in explaining the dark matter mystery since, in both cases, the modified theory can be presented as General Relativity with additional fictitious matter (embedding matter or mimetic matter). For the general case, we obtain the equations of motion of embedding matter in terms of embedding function as a set of first-order dynamical equations and constraints consistent with them. Then, we construct a non-relativistic limit of these equations, in which the motion of embedding matter turns out to be slow enough so that it can play the role of cold dark matter. The non-relativistic embedding matter turns out to have a certain self-interaction, which could be useful in the context of solving the core-cusp problem that appears in the

Λ

-Cold Dark Matter (

Λ

CDM) model. Full article

(This article belongs to the Special Issue Selected Papers from the 17th Russian Gravitational Conference —International Conference on Gravitation, Cosmology and Astrophysics (RUSGRAV-17))

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14 pages, 313 KiB

Open AccessFeature PaperArticle

Canonical Description for Formulation of Embedding Gravity as a Field Theory in a Flat Spacetime

by Sergey Paston, Elizaveta Semenova and Anton Sheykin

Symmetry 2020, 12(5), 722; https://doi.org/10.3390/sym12050722 - 3 May 2020

Cited by 2 | Viewed by 2335

Abstract

We consider the approach to gravity in which four-dimensional curved spacetime is represented by a surface in a flat Minkowski space of higher dimension. After a short overview of the ideas and results of such an approach we concentrate on the study of the so-called splitting gravity, a form of this description in which constant value surface of a set of scalar fields in the ambient flat space-time defines the embedded surface. We construct a form of action which is invariant w.r.t. all symmetries of this theory. We construct the canonical formalism for splitting gravity. The resulting theory turns out to be free of constraints. However, the Hamiltonian of this theory is an implicit function of canonical variables. Finally, we discuss the path integral quantization of such a theory. Full article

15 pages, 283 KiB

Open AccessArticle

Modifications of Gravity Via Differential Transformations of Field Variables

by Anton Sheykin, Dmitry Solovyev, Vladimir Sukhanov and Sergey Paston

Symmetry 2020, 12(2), 240; https://doi.org/10.3390/sym12020240 - 5 Feb 2020

Cited by 12 | Viewed by 2207

Abstract

We discuss field theories appearing as a result of applying field transformations with derivatives (differential field transformations, DFTs) to a known theory. We begin with some simple examples of DFTs to see the basic properties of the procedure. In this process, the dynamics of the theory might either change or be conserved. After that, we concentrate on the theories of gravity which appear as a result of various DFTs applied to general relativity, namely the mimetic gravity and Regge–Teitelboim embedding theory. We review the main results related to the extension of dynamics in these theories, as well as the possibility to write down the action of a theory after DFTs as the action of the original theory before DFTs plus an additional term. Such a term usually contains some constraints with Lagrange multipliers and can be interpreted as an action of additional matter, which might be of use in cosmological applications, e.g., for the explanation of the effects of dark matter. Full article

(This article belongs to the Special Issue Cosmology)

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