Scalar-Tensor Theory of Gravitation in Minkowski Space–Time

Nonlinear Phenomena in Complex Systems, vol. 17, no. 4 (2014), pp. 423 - 425

A. Leonovich, A. Tarasenko, and Yu. Vyblyi∗ University of Informatics and Radioelectronics, Minsk, BELARUS and B.I.Stepanov Institute of Physics of NAS of Belarus, Minsk, BELARUS (Received 05 September, 2014) In Minkowski space–time we consider the scalar and tensor fields, which form together the effective Riemann metric for a matter lagrangian. The demand of minimal interaction, including tensor selfinteraction, leads to Einstein equations for the tensor field and the nonlinear equation for the scalar field. The cosmological scenario to a given theory leads to the existence of the slow-roll regime of cosmological expansion in agreement with modern observations.

PACS numbers: 04.30.Db; 02.30.Gp Keywords: cosmic acceleration, cosmological term, dark energy, Minkowski space–time, scalar ﬁeld

As known now, we live in the universe dynamical variables. The gravitational fields where there is an abundance of dark matter and equations of RTG massless variant are the dark energy and the acceleration of cosmological Einstein ones for this effective metric. The gauge expansion is observed [1, 2]. One of possible invariance demand allows the generalization of approach to theoretical description of this space–time metric in the form observations is the modification of General Relativity. In this connection the interest in Brans–Dicke scalar theories is resumed [3]. In such f˜ik = f(φ)˜gik, (1) an approach the gravitational field is depended on the metric and scalar potential, moreover the f(φ) is certain function of the field φ, latter may be interpreted as dark energy or dark g˜ik = √ γ(γik + qψik) is the effective RTG matter [4]. − metric, forming from Minkowski metric γik and Scalar fields are commonly used as ik gravitational tensor potential ψ ; g = det gik,γ = candidates for the dark energy. Nevertheless, det γik, q is gravitational interaction constant. there is no unambiguous criterion for the choice In the framework of Brans–Dicke theory such of the field lagrangian in scalar field theories. approach was proposed in [6]. Moreover, any scalar field in the slow-roll regime Let us consider in Minkowski space the most can model the cosmological constant and hence general form of scalar field Lagrangian, leading to leads to an appropriate cosmological scenario. linear field equations In present paper we consider the minimal possible scalar-tensor generalization of Relativistic Theory of Gravitation (RTG) φ 1 ik 1 2 2 [5]. In this theory the tensor gravitational L = φ,i φ,k γ m φ + aφ b √ γ 0 2 − 2 − − field is regarded in Minkowski space–time and (2) effective metric arises due to the demand of where m, a, b are constants. Gauge invariant gauge invariance under Lie transformations of Lagrangian for interacting scalar and tensor fields may be obtained with the help of replacement in ∗E-mail: [email protected] the expression (2) Minkowski metric by metric f ik and adding field ψik Lagrangian of RTG

423 424 A. Leonovich, A. Tarasenko, and Yu. Vyblyi

1 G φ √ c d L = L + L = g 2 R V (φ)= (8) − h−16πq φ − φ2 1 ik 1 2 2 2 2 2 + fφ,i φ,k g f m φ +af φ bf (3) 2 2 2 − 2 − i where c = a/4k ,d = b/4k . This Lagrangian possesses the property of spontaneously broken where R is the scalar curvature for metric gik. At presence of matter we must add to (3) symmetry, if constant d is positive, and has the 2 the matter Lagrangian LM (Q ,f ik), Q are energy minimum equals to 3c /4d at φ0 = 2d/c. A A − ik dynamical variables of matter. We assume that the metric f coincides ik For minimal interaction tensor and scalar with the metric g in the vacuum state, then φ0 = fields with a matter, the scalar field equations 1/2k. In linear approximation for the vacuum have the form state, the field equations corresponding to used Lagrangian, describe a massive gravitational field with graviton mass equal to qc 12π/d δLφ 1 f, = √ g φ T M (4) Consider the scenariop of the Universe δφ −2 − f evolution in the given scalar-tensor theory. The M M ik metric of the homogenous and isotropic Universe where T = Tik g is the trace of the energy- momentum tensor in space with metric gik. is the Robertson-Walker metric with a flat 3- We choose the function f in the form dimensional space. The reason for choosing the flat model is mostly the observational evidence. In addition, if − f(φ)=(2kφ) 1, (5) the problem is solved in the framework of RTG, the flat model is the only acceptable one [5]. k is the scalar interaction constant. We are interested in evolution of the The field equations are Universe at the epoch following the annihilation of electron-positron pairs. At the given epoch we can 2 M mn φ mn treat the matter as consisting of the cold matter Gik = 8πq [Tik (QA,φ,g )+ Tik(φ,g )], (6) that includes barionic and dark matter and the radiation that includes photons and three types of i ik DiφD φ M mn neutrino and antineutrino. The equation of state g DiDkφ = kT (QA,φ,g ). (7) CM r r − 2φ is p = 0 for the cold matter and p = ǫ /3 for radiation. The equations of matter evolution The equations, restricting spin states of the ik ik follow from the gravitational field equations and field ψ Dig˜ = 0 [5] and matter equations must have the form be added (Di is covariant derivative in Minkowski space). (CM)k (r)k ϕk (T i + T i + T i );k = 0 (9) Thus, the usage in this approach the effective metric f ik allows to introduce in a universal where the indices CM, r, ϕ denote the quantities manner the interaction between a matter with related to the cold matter, the radiation and the nonzero trace of the energy-momentum tensor scalar field respectively. and scalar field. As a result the scalar field does The performed numeric simulations of not interact with electromagnetic one, that allows the full system of field equations, describing to regard it as a candidate for dark matter. the cosmological solution for the homogenous If the tensor field ψ equals to zero, the and isotropic Universe, shows that for certain Lagrangian of the nonlinear scalar field in restrictions on the parameters the slow-roll regime Minkowski space has the potential V given as is possible. In the slow-roll regime the scalar field

Нелинейные явления в сложных системах Т. 17, № 4, 2014 Scalar-Tensor Theory of Gravitation in Minkowski Space–Time 425 models the dark energy in agreement with modern observational data.

References

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Nonlinear Phenomena in Complex Systems Vol. 17, no. 4, 2014