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We discuss the possibility of explaining observations usually related to the existence of dark matter by passing from the general relativity (GR) theory to a modified theory of gravity, the embedding theory proposed by Regge and Teitelboim. In this approach, it is assumed that our space-time is a four-dimensional surface in a ten-dimensional flat ambient space. This clear geometric interpretation of a change of a variable in the GR action leading to a new theory distinguishes this approach from the known alternatives: mimetic gravity and other variants. After the passage to the modified theory of gravity, additional solutions that can be interpreted as GR solutions with additional fictitious matter appear besides the solutions corresponding to GR. Just in that fictitious matter, one can try to see dark matter, with no need to assume the existence of dark matter as a fundamental object; its role is played by the degrees of freedom of modified gravity. In the embedding theory, the number of degrees of freedom of fictitious matter is sufficiently large, and hence an explanation of all observations without complicating the theory any further can be attempted., Comment: LaTeX, 16 pages, based on a talk given at the VII International Conference "Models of quantum field theory" (MQFT-2022) dedicated to the 82th anniversary of Professor Alexandr Nikolaevich Vasiliev and to the 80th anniversary of Professor Vladimir Dmitrievich Lyakhovski, Saint Petersburg, Russia, 10-14 October 2022

The description of gravity in the form of an embedding theory is based on the hypothesis that our space-time is a four-dimensional surface in a flat ten-dimensional space. The choice of standard Einstein-Hilbert action leads in this case to more general field equations than Einstein's equations. By writing them in the form of Einstein's equations with the contribution of additional fictitious matter, one can try to interpret this matter as dark matter. In order to study the behavior of this fictitious matter near the centers of real galaxies, we discuss an analytical method of obtaining corresponding matter density profiles. This method is based on the consideration of the distribution function of particles over all possible trajectories and allows us to estimate the type (core or cusp) of the emerging density profile., Comment: LaTeX, 8 pages, based on a talk given at the International Conference on Quantum Field Theory, High-Energy Physics, and Cosmology, Dubna, Russia, 18-21 July 2022

We propose an approach that allows one to reformulate $n$-dimensional $p$-form mimetic gravity (including usual mimetic gravity as particular case $p = 0$) as nonlinear $(n-p-1)$-form electrodynamics via electric-magnetic duality. The resulting dual Lagrangian density is just the square root of the ordinary quadratic Lagrangian density of $(n-p-1)$-form electrodynamics. By applying field transformation in the action, we show that for the arbitrary $p$ this dual theory transforms into the $(p+1)$-brane fluid: the model of the stack of the parallel $(p+1)$-dimensional branes foliating physical spacetime. As the structure of the field transformations depends on $p$, the sets of solutions in these models are related differently. We prove, that for $p = 0$ and $p = n-2$ dual mimetic models describe usual particle fluid with the potential flow and to the $(n-1)$-brane fluid respectively. For other values of $p$ not all mimetic solutions behave like that, in general, so we restrict ourselves only to the case $n = 4$, $p = 1$. In this case, we show, that mimetic formulation is dual to the well-known Nielsen-Olesen theory of "dual strings" and discuss the criterion indicating whether its solutions behave like string fluid. The cosmological solutions for these models in are also discussed., Comment: 9 pages, published version

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 arbitrariness remaining 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 makes it possible to find 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., Comment: LaTeX, 14 pages. This version corresponds to the published one

We study embedding gravity, a modified theory of gravity, in which our space-time is assumed to be a four-dimensional surface in flat ten-dimensional space. Based on a simple geometric idea, this theory can be reformulated as General Relativity with additional degrees of freedom and contribution to action, which can be interpreted as describing dark matter. We study the canonical formalism for such a formulation of embedding gravity. After solving simple constraints, the Hamiltonian is reduced to a linear combination of four first class constraints with Lagrange multipliers. There still remain six pairs of second class constraints. Possible ways of taking these constraints into account are discussed. We show that one way of solving the constraints leads to the canonical system going into the previously known canonical formulation of the complete embedding theory with an implicitly defined constraint., Comment: LaTeX, 15 pages. This version corresponds to the published one

We propose an analytical method to describe a matter density profile near a galaxy center. The description is based on the study of the distribution function of particles over possible trajectories. We establish a relation between the central slope of density profile and the near-origin behavior of the angular momentum distribution function. We consider both a spherically symmetric (on average) matter distribution as well as deviations from it. If the density profile forms in a background of spherical gravitation potential then a core-type distribution arises. A regular matter may behave in such way if the background potential was formed by the dark matter. In the presence of deviation from spherical symmetry the formation of cusp-type distribution is possible. Moreover, a reduction of spherical symmetry to the axial one leads to a less steep cusp profile. The complete symmetry breaking (which corresponds, in particular, to the common setup of numerical simulations), leads to a steeper cusp profile., Comment: LaTeX, 13 pages. In this version, the study of possible deviations from spherical symmetry was added, the present version corresponds to the published one

Faddeev variant of embedding theory is an example of using the embedding approach for the description of gravity. In the original form of the embedding approach, the gravity is described by an embedding function of a four-dimensional surface representing our spacetime. In Faddeev variant, the independent variable is a non-square vielbein, which is a derivative of embedding function in embedding theory. We study the possibility of the existence of extra solutions in Faddeev variant, which makes this theory non-equivalent to GR. To separate the degrees of freedom corresponding to extra matter, we propose a formulation of this theory as GR with an additional contribution to the action. We analyze the equations of motion for a specific class of solutions corresponding to a weak gravitational field. We construct a simple exact solution corresponding to arbitrary matter and nontrivial torsion, which is an extra solution in Faddeev variant in the absence of real matter., Comment: LaTeX, 7 pages, based on a talk given at the International Conference PhysicA.SPb/2021, Saint Petersburg, Russia, 18-22 October 2021

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., Comment: LaTeX, 15 pages. This version corresponds to the published one

It is often easier to study pseudo-Riemannian manifolds by presenting them as surfaces in some ambient space. We propose an algorithm for construction of explicit isometric embeddings of pseudo-Riemannian manifolds with symmetries into an ambient space of higher dimension. While most of the existing methods are based on Gauss-Codazzi-Mainardi-Peterson equations, we do not use them and instead concentrate on a system of equations which connects the metric on the manifold and the embedding function of the surface. Our algorithm is based on the group theoretical method of separation of variables that we developed earlier (arXiv:1202.1204). The algorithm makes this method more convenient and simple to use. It allowed us to simplify the construction of many known embeddings as well as obtain some new ones. In particular, we obtain explicit global embeddings of spinning BTZ black hole in 7-dimensional flat space., Comment: 16 pages

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) of the equations of motion. Deformations of the superpotential form for theories with a change of the independent variables in the 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., Comment: LaTeX, 19 pages

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 LambdaCDM model., Comment: LaTeX, 19 pages; in v2 some references were added

The work is devoted to the construction of explicit embeddings for the metrics of the black holes, formed by nonsingular matter distribution. One of the possible examples of such type of solutions is regular black hole. Using the existing classification of minimal symmetric embeddings of the Schwarzschild metric as a base, we construct embeddings for regular black holes with de Sitter interior. Another simple example is black hole, formed by collapsing homogeneous spherically symmetric cloud of dustlike matter. We discuss embeddings for two variants of such black holes - the one with the eternally existing horizon, when dust ball never leaves the interior of the horizon, and another variant with the dynamically forming horizon., Comment: LaTeX, 8 pages

We study the possibility to explain the mystery of the dark matter through the transition from General Relativity to embedding gravity. This modification of gravity, which was proposed by Regge and Teitelboim, is based on a simple string-inspired geometrical principle: our spacetime is considered here as a 4-dimensional surface in a flat bulk. We show that among the solutions of embedding gravity, there is a class of solutions equivalent to solutions of GR with an additional contribution of non-relativistic embedding matter, which can serve as cold dark matter. We prove the stability of such type of solutions and obtain an explicit form of the equations of motion of embedding matter in the non-relativistic limit. According to them, 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 LambdaCDM model., Comment: LaTeX, 11 pages. This version corresponds to the published one

We study the problem of construction of explicit isometric embeddings of (pseudo)-Riemannian manifolds. We discuss the method which is based in the idea that the exterior symmetry of the embedded surface and the interior symmetry of the metric on it must be the same. In case of high enough symmetry of the metric such method allows to transform the metric inducedness condition, which is the one to be solved in order to construct an embedding, into a system of ODEs. It turns out that this method can be generalized to allow the surface to have lower symmetry as long as the above simplification occurs. This generalization can be of use in the construction of embeddings for metrics whose symmetry group is hard to analyze, as well for the construction of isometrically deformed (bended) surface. We give some examples of application of this method. In particular, we construct the embedding of spatially-flat Friedmann model and isometric bendings of sphere, 3-sphere and squashed AdS universe, which is connected to the Godel universe., Comment: 7 pages. The minor typo in eq. (10) of published version is corrected here, minor clarifications are also made

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., Comment: 16 pages

The work is devoted to the search for explicit isometric embeddings of a metric corresponding to the collapse of spherically symmetric matter with the formation of a black hole. Two approaches are considered: in the first, the embedding is constructed for the whole manifold at once; in the second, the idea of a junction of solutions, obtained separately for areas inside and outside the dust ball, is used. In the framework of the first approach, a global smooth embedding in 7D space with a signature (2 + 5) was constructed. It corresponds to the formation of the horizon as a result of matter falling from infinity. The second approach generally leads to an embedding in 7D space with the signature (1 + 6). This embedding corresponds to the case when matter flies out of a white hole with the disappearance of its horizon, after which the radius of the dust ball reaches its maximum, and then a collapse occurs with the formation of the horizon of a black hole. The embedding obtained is not smooth everywhere --- it contains a kink on the edge of the dust ball, and {also, it is} not quite global. In the particular case, when the maximum radius of the dust ball coincides with the radius of the horizon, it is possible to construct a global smooth embedding in a flat 6D space with a signature (1 + 5)., Comment: LaTeX, 18 pages

We discuss field theories appearing as a result of applying field transformations with derivatives (differential field transformations, DFT) 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 conserve. After that we concentrate on the theories of gravity which appear as a result of various DFT applied to general relativity, namely the mimetic gravity and Regge-Teitelboim embedding theory. We review 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 DFT as the action of the original theory before DFT 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., Comment: 18 pages

We study the procedure that allows to rewrite the actions of some modified gravity theories like mimetic gravity and Regge-Teitelboim embedding theory as Einstein-Hilbert actions with additional matter contributions. It turns out that this procedure can be applied to brane action, which lead to a polynomial brane action. We also examine the connection between this procedure and Polyakov trick in string theory., Comment: LaTeX, 8 pages, minor corrections. This version matches the published one

We study the problem of construction of global isometric embedding for spherically symmetric black holes with negative cosmological constant in various dimensions. Firstly, we show that there is no such embedding for 4D RN-AdS black hole in 6D flat ambient space, completing the classification which we started earlier. Then we construct an explicit embedding of non-spinning BTZ black hole in 6D flat ambient space. Using this embedding as an anzats, we then construct a global explicit embedding of (d)-dimensional Schwarzschild-AdS black hole in a flat (d+3)-dimensional ambient space., Comment: 15 pages

We investigate the connection between stress-energy tensor (SET) arising from Noether's theorem and Belinfante SET which can be obtained as a right-hand side of the Einstein's equation in the flat metric limit. This question is studied in the wide class of Poincar\`e-invariant field theories with actions which depend on the tensor fields of arbitrary rank and their derivatives of arbitrary order. For this class we derive the relation between these SET and present the exact expression for the difference between them. We also show that the difference between corresponding integrals of motion can be expressed as a surface integral over 2-dimensinal infinitely remote surface., Comment: LaTeX, 6 pages, based on a talk given at the International Conference PhysicA.SPb/2018, Saint Petersburg, Russia, 23-25 October 2018

We study some particular modifications of gravity in search for a natural way to unify the gravitational and electromagnetic interaction. The certain components of connection in the appearing variants of the theory can be identified with electromagnetic potential. The methods of adding matter in the form of scalar and spinor fields are studied. In particular, the expansion of the local symmetry group up to $GL(2,C)$ is explored, in which equations of Einstein, Maxwell and Dirac are reproduced for the theory with Weyl spinor., Comment: LaTeX, 6 pages, based on a talk given at the XXth International Seminar on High Energy Physics (QUARKS-2018), Valday, Russia, May 27 - June 2, 2018

We discuss the relation between canonical and metric energy-momentum tensors for field theories with actions that can depend on the higher derivatives of tensor fields in a flat spacetime. In order to obtain it we use a modification of the Noether's procedure for curved space-time. For considered case the difference between these two tensors turns out to have more general form than for theories with no more than first order derivatives. Despite this fact we prove that the difference between corresponding integrals of motion still has the form of integral over 2-dimensional surface that is infinitely remote in the spacelike directions., Comment: LaTeX, 11 pages

It is known that recently proposed model of mimetic gravity can be presented as general relativity with an additional mimetic matter. We discuss a possibility to analogously reformulate the embedding theory, which is the geometrical description of gravity proposed by Regge and Teitelboim, treating it also as general relativity with some additional matter. We propose a form of action which allows to describe this matter in terms of conserved currents. This action turns out to be a generalization of the perfect fluid action, which can be useful in the analysis of the properties of the additional matter. On the other side, the action contains a trace of the root of the matrix product, which is similar to the constructions appearing in bimetric theories of gravity. The action is completely equivalent to the original embedding theory, so it is not just some artificial model, but has a clear geometric sense. We discuss the possible equivalent forms of the theory and ways of study of the appearing equations of motion., Comment: LaTeX, 11 pages

We study the definitions of energy, naturally arising in the splitting theory, which is the field theoretic formulation of the Regge-Teitelboim gravity. The latter regards our spacetime as a surface embedded in a flat bulk. The splitting theory describes embedded spacetime in the language of the some field theory in a flat bulk. We consider the Noether energy-momentum tensor (EMT) and the metric EMT defined by the variation with respect to the metric of a flat bulk. We discuss a localizability of energy. Then using these EMTs we calculate the full energy of an isolated massive body. We compare the results with the standard general relativity results obtained from the Einstein energy-momentum pseudotensor (pEMT) and from the M{\o}ller pEMT. Finally, we propose the several ways of correction of the definitions of the energy in the splitting theory., Comment: This version matches the published one

A generalization of General Relativity is studied. The standard Einstein-Hilbert action is considered in the Palatini formalism, where the connection and the metric are independent variables, and the connection is not symmetric. As a result of variation with respect to the metric Einstein equations are obtained. A variation with respect to the connection leads to an arbitrariness in the determination of connection, i.e. the presence of gauge invariance. Then a matter in a form of point particle which interacts with field of connection is introduced. Also the action is complemented by a kinetic term for field of the connection to avoid incompatible equation of motion. Thus after the variation procedures we obtain the Einstein equations, the geodesic equation and the Maxwell`s equations for electromagnetism, where some components of the connection play the role of the electromagnetic potential. Thereby the electromagnetic potential is obtained from the geometry of space-time., Comment: LaTeX, 6 pages, based on a talk given at the 13th International Workshop High Energy Physics and Quantum Field Theory, Yaroslavl, Russia, June 26 - July 3, 2017

We consider a natural form of unified theory of gravity and electromagnetism which was somehow missed at the time of intense search for such an unification, and was noticed only in 1978 but remained quite unknown. The basic idea of this unification is to use the metric and non-symmetric connection as independent variables, which generalizes the so-called Palatini formalism. The certain components of connection in the appearing theory can be naturally identified with electromagnetic potential, and with the proper choice of action the Einstein-Maxwell equations are reproducing. In this paper we compare such an approach with the known ideas of unification. Also we propose the more consistent way of including matter (in the form of classical particles) in the theory and briefly discuss the perspectives of further development of this approach., Comment: LaTeX, 17 pages

We consider various forms of action allowing us to describe a perfect fluid in the framework of General Relativity. First we consider a potential motion without pressure. Starting from an action in terms of the current density vector built similarly to the action of a single particle, we build a series of equivalent actions through various changes of variables. The results are then generalized to the cases of motion with vorticity and the presence of pressure. The obtained forms of action are compared to the previously known variants. We discuss the possibilities to use the obtained results to formulate the theories containing a description of a perfect fluid. A new form of action is proposed for mimetic gravity, which allows a motion with vorticity for mimetic matter as a perfect fluid., Comment: LaTeX, 14 pages. Some discussions and references were added in this version

We study various definitions of the gravitational field energy based on the usage of isometric embeddings in the Regge-Teitelboim approach. For the embedding theory we consider the coordinate translations on the surface as well as the coordinate translations in the flat bulk. In the latter case the independent definition of gravitational energy-momentum tensor appears as a Noether current corresponding to global inner symmetry. In the field-theoretic form of this approach (splitting theory) we consider Noether procedure and the alternative method of energy-momentum tensor defining by varying the action of the theory with respect to flat bulk metric. As a result we obtain energy definition in field-theoretic form of embedding theory which, among the other features, gives a nontrivial result for the solutions of embedding theory which are also solutions of Einstein equations. The question of energy localization is also discussed., Comment: 19 pages. This version matches the published one. The text has been shortened and revised

We consider the embedding theory, the approach to gravity proposed by Regge and Teitelboim, in which 4D space-time is treated as a surface in high-dimensional flat ambient space. In its general form, which does not contain artificially imposed constraints, this theory can be viewed as an extension of GR. In the present paper we study the canonical description of the embedding theory in this general form. In this case, one of the natural constraints cannot be written explicitly, in contrast to the case where additional Einsteinian constraints are imposed. Nevertheless, it is possible to calculate all Poisson brackets with this constraint. We prove that the algebra of four emerging constraints is closed, i.e., all of them are first-class constraints. The explicit form of this algebra is also obtained., Comment: 11 pages

Theory of gravity is considered in the Regge-Teitelboim approach in which the pseudo-Riemannian space is treated as a surface isometrically embedded in an ambient Minkowski space of higher dimension. This approach is formulated in terms of a field theory in which the original pseudo-Rimannian space is defined by the field constant-value surfaces. The symmetry properties of the proposed theory are investigated, and possible structure of the field-theoretical Lagrangian is discussed., Comment: This version contains some additions in comparison with the journal version: the derivation of the Jacobian was added

We present results relevant to the relation between quantum effects in a Riemannian space and on the surface appearing as a result of its isometric embedding in a flat space of a higher dimension. We discuss the mapping between the Hawking effect fixed by an observer in the Riemannian space with a horizon and the Unruh effect related to an accelerated motion of this observer in the ambient space. We present examples for which this mapping holds and examples for which there is no mapping. We describe the general form of the hyperbolic embedding of the metric with a horizon smoothly covering the horizon and prove that there is a Hawking into Unruh mapping for this embedding. We also discuss the possibility of relating two-point functions in a Riemannian space and the ambient space in which it is embedded. We obtain restrictions on the geometric parameters of the embedding for which such a relation is known., Comment: LaTeX, 18 pages

We consider the problem of the existence of global embeddings of metrics of spherically symmetric black holes into an ambient space with the minimal possible dimension. We classify the possible types of embeddings by the type of realization of the metric symmetry by ambient space symmetries. For the Schwarzschild, Schwarzschild-de Sitter, and Reissner-Nordstrom black holes, we prove that the known global embeddings are the only ones. We obtain a new global embedding for the Reissner-Nordstrom-de Sitter metrics and prove that constructing such embeddings is impossible for the Schwarzschild-anti-de Sitter metric. We also discuss the possibility of constructing global embeddings of the Reissner-Nordstrom-anti-de Sitter metric., Comment: LaTeX, 12 pages

This paper is devoted to the approach to gravity as a theory of a surface embedded in a flat ambient space. After the brief review of the properties of original theory by Regge and Teitelboim we concentrate on its field-theoretic reformulation, which we call splitting theory. In this theory embedded surfaces are defined through the constant value surfaces of some set of scalar fields in high-dimensional Minkowski space. We obtain an exact expressions for this scalar fields in the case of Friedmann universe. We also discuss the features of quantisation procedure for this field theory., Comment: LaTeX, 7 pages, based on a talk given at 9th Alexander Friedmann International Seminar, St. Petersburg, Russia, June 21-27, 2015

The formulation of gravity theory is considered where space-time is a 4-dimensional surface in flat ten-dimensional space. The possibility of using the "external" time (the time of ambient space) in this approach is investigated. The transition to the "external" time is realized with the help of partial gauge fixing, the coordinate condition which equates the timelike coordinate of the surface and time of the ambient space. It is shown that by using such a gauge condition in the action, the loss of any equations of motion does not take place, although it can happen in the general case. A version of the canonical formalism of the theory is studied in which certain additional constraints are imposed, providing the equivalence of the approach under consideration and general relativity. The corresponding first-class constraint algebra is obtained. It is proved that using the gauge directly in the action leads to the same result as gauge fixing in the constraint algebra, despite the artificial introduction of some of the constraints into the theory. Application of the "external" time may be useful for attempts to quantize the theory as well as for studies in embedding theory where 4-dimensional surfaces fill the whole ambient space and coordinates on the surfaces are not introduced., Comment: LaTeX, 17 pages

The renormalization problem of (2+1)-dimensional Yang-Mills theory quantized on the light front is considered. Extra fields analogous to those used in Pauli-Villars regularization are introduced to restore perturbative equivalence between such quantized theory and conventional formulation in Lorentz coordinates. These fields also provide necessary ultraviolet regularization to the theory. Obtained results allow to construct renormalized Hamiltonian of the theory on the light front., Comment: LaTeX, 12 pages, 2 figures. Article will be published in Theoretical and Mathematical Physics, the final publication will be available at http://link.springer.com/journal/11232

The advantage of Pauli-Villars regularization in quantum field theory quantized on the light front is explained. Simple examples of scalar $\lambda\varphi^4$ field theory and Yukawa-type model are used. We give also an example of nonperturbative calculation in the theory with Pauli-Villars fields, using for that a model of anharmonic oscillator modified by inclusion of ghost variables playing the role similar to Pauli-Villars fields., Comment: LaTeX, 10 pages, 2 figures. Article will be published in AIP Conference Proceedings, the final publication will be available at http://scitation.aip.org/content/aip/proceeding/aipcp

We study the conditions of the existence of Hawking into Unruh mapping for hyperbolic (Fronsdal-type) embeddings of metric into the Minkowski space, for which timelines are hyperbolas. Many examples are known for global embeddings into the Minkowskian spacetime (GEMS) with such mapping for physically interesting metrics with some symmetry. However the examples of embeddings, both smooth and hyperbolic, for which there is no mapping, were also given. In the present work we prove that Hawking into Unruh mapping takes place for a hyperbolic embedding of an arbitrary metric with a time-like Killing vector and a Killing horizon if the embedding of such type exists and smoothly covers the horizon. At the same time we do not assume any symmetry (spherical for example), except the time translational invariance which corresponds to the existence of a time-like Killing vector. We show that the known examples of the absence of mapping do not satisfy the formulated conditions of its existence., Comment: LaTeX, 13 pages, in this version small corrections made

We discuss for which smooth global embeddings of a metric into a Minkowskian spacetime the Hawking into Unruh mapping takes place. There is a series of examples of global embeddings into the Minkowskian spacetime (GEMS) with such mapping for physically interesting metrics. These examples use Fronsdal-type embeddings for which timelines are hyperbolas. In the present work we show that for some new embeddings (non Fronsdal-type) of the Schwarzschild and Reissner-Nordstrom metrics there is no mapping. We give also the examples of hyperbolic and non hyperbolic type embeddings for the de Sitter metric for which there is no mapping. For the Minkowski metric where there is no Hawking radiation we consider a non trivial embedding with hyperbolic timelines, hence in the ambient space the Unruh effect takes place, and it follows that there is no mapping too. The considered examples show that the meaning of the Hawking into Unruh mapping for global embeddings remains still insufficiently clear and requires further investigations., Comment: LaTeX, 10 pages. This is extended version of the paper

We study the approach to gravity in which our curved spacetime is considered as a surface in a flat ambient space of higher dimension (the embedding theory). The dynamical variable in this theory is not a metric but an embedding function. The Euler-Lagrange equations for this theory (Regge-Teitelboim equations) are more general than the Einstein equations, and admit "extra solutions" which do not correspond to any Einsteinian metric. The Regge-Teitelboim equations can be explicitly analyzed for the solutions with high symmetry. We show that symmetric embeddings of a static spherically symmetric asymptotically flat metrics in a 6-dimensional ambient space do not admit extra solutions of the vacuum Regge-Teitelboim equations. Therefore in the embedding theory the solutions with such properties correspond to the exterior Schwarzchild metric., Comment: LaTeX, 10 pages. Proceedings of "II Russian-Spanish Congress Particle and Nuclear Physics at all Scales and Cosmology", Saint-Petersburg, October 1-4, 2013

We study the explicit embeddings of static black holes. We obtain two new minimal embeddings of the Schwarzchild-de Sitter metric which smoothly cover both horizons of this metric. The lines of time for these embeddings are more complicated than hyperbolas. Also we shortly discuss the possibility of using non-hyperbolic embeddings for calculation of the black hole Hawking temperature in the Deser and Levin approach., Comment: LaTeX, 7 pages. Proceedings of "The XXI International Workshop High Energy Physics and Quantum Field Theory" (QFTHEP 2013), Saint Petersburg Area, Russia, 23-30 June, 2013

We address the problem of nonperturbative calculations on the light front in quantum field theory regularized by Pauli-Villars method. As a preliminary step we construct light front Hamiltonians in (2+1)-dimensional $\lambda\phi^4$ model, for the cases without and with spontaneous symmetry breaking. The renormalization of these Hamiltonians in Pauli-Villars regularization is carried out via comparison of all-order perturbation theory, generated by these Hamiltonians, and the corresponding covariant perturbation theory in Lorentz coordinates., Comment: LaTeX, 18 pages, 2 figures. Appendix B has been replaced. Some changes have been made in Introduction and Conclusion. The new list of references is more complete. Article will be published in International Journal of Theoretical Physics, the final publication is available at link.springer.com

We study isometric embeddings of some solutions of the Einstein equations with suffciently high symmetries into a flat ambient space. We briefly describe a method for constructing surfaces with a given symmetry. We discuss all minimal embeddings of the Schwarzschild metric obtained using this method and show how the method can be used to construct all minimal embeddings for the Friedmann models. We classify all the embeddings in terms of realizations of symmetries of the corresponding solutions., Comment: LaTeX, 13 pages. This version contains some additions in comparison with the journal version

We study isometric embeddings of non-extremal Reissner-Nordstr\"om metric describing a charged black hole. We obtain three new embeddings in the flat ambient space with minimal possible dimension. These embeddings are global, i.e. corresponding surfaces are smooth at all values of radius, including horizons. Each of the given embeddings covers one instance of the regions outside the horizon, one instance between the horizons and one instance inside the internal horizon. The lines of time for these embeddings turn out to be more complicated than circles or hyperbolas. The obtained embeddings are also smooth at all values of radius for extremal and hyperextremal black holes., Comment: LaTeX, 12 pages, in this version we stressed that obtained new embeddings are also global and smooth for extremal and hyperextremal black holes

We suggest a method to search the embeddings of Riemannian spaces with a high enough symmetry in a flat ambient space. It is based on a procedure of construction surfaces with a given symmetry. The method is used to classify the embeddings of the Schwarzschild metric which have the symmetry of this solution, and all such embeddings in a six-dimensional ambient space (i.e. a space with a minimal possible dimension) are constructed. Four of the six possible embeddings are already known, while the two others are new. One of the new embeddings is asymptotically flat, while the other embeddings in a six-dimensional ambient space do not have this property. The asymptotically flat embedding can be of use in the analysis of the many-body problem, as well as for the development of gravity description as a theory of a surface in a flat ambient space., Comment: LaTeX, 19 pages