Cylindrical solutions in mimetic gravity
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Abstract
This paper is devoted to investigate cylindrical solutions in mimetic gravity. The explicit forms of the metric of this theory, namely mimeticKasner (say) have been obtained. In this study we have noticed that the Kasner’s family of exact solutions needs to be reconsidered under this type of modified gravity. A nogo theorem is proposed for the exact solutions in the presence of a cosmological constant.
Keywords
Scalar Field Cosmological Constant Constant Curvature Cosmic String Einstein Gravity1 Introduction
Exact solutions play a crucial role in general relativity (GR) and modified gravity. The thermodynamic and dynamics of the gravitational model are often attributed to the presence of an exact solution, which solves the equations of motion. An introduction of a new technique to find the solutions is also an important and natural way to build modify gravity models [1].
In cosmology of early Universe, we investigate the generally accepted doctrine that the universe is affecting to what we termed as “topological defects” through exhaustion of all sources of matter, and suggest that by virtue of a cosmic string mechanism which maintains its available energy is selfgravitating. Energy is being “degraded” in objects which are in the cosmos, but “elevated” or raised to a higher level in strings [2, 3]. One main motivation for us to study exact cylindrical solutions in gravitational theories is to describe such topological defects by Riemannian geometry. A simple description of the above topological defects is to find the cylindrical solution by solving highly non linear field equations. In GR, the simplest cylindrical model described by the class of exact cylindrical solutions were found by Kasner and later on studied by several authors [4, 5, 6, 7, 8].
This type of cylindrical solution has remained popular in the literature for some time with the name of cosmic string as a model to describe topological defects of early cosmology and closed time like curves.
Modified gravity has received much attention in recent years due to its interesting properties, which offer important solutions to cosmological queries about the origin of the Universe [9, 10, 11]. Modified gravity is an alternative theory for gravity obtained from action principle and is produced by the replacement of the Einstein–Hilbert action with general function of the curvature and higher derivative terms of it (see [12, 13, 14, 15] for reviews). It has many possible uses in the gravitational physics and has also been investigated as a potential candidate for the formation of cosmic strings. For example cosmic strings investigated in f(R) gravity [16, 17], teleparallel theories [18, 19, 20], brane worlds [21], Kaluza–Klein models [22], Lovelock Lagrangians[23], Gauss–Bonnet [24, 25, 26], Born–Infeld [27, 28], bimetric theories [29], nonrelativistic models of gravity [30], in scalartensor theories [31, 32, 33, 34, 35, 36], Brans–Dicke theory [37, 38, 39, 40, 41, 42], dilation gravity [43, 44], nonminimally coupled models of gravity [45] and recently the Bose–Einstein condensate strings [46].
Recent research has allowed a prescribed number of models to propose, by what is called the “Mimetic Gravity” (MG) [47], which are devoted to resolve the dark matter problem using a class of restricted disformal transformations \(g_{\mu \nu }\rightarrow \hat{g}_{\mu \nu }=\Omega (\phi )g_{\mu \nu }\) (\(\phi \) is an auxiliary field which can be a complex function) of the physical metric \(g_{\mu \nu }\). In the multiple remarks people proposed to follow this main idea from different points of view [48, 49, 50, 51, 52, 53, 54, 55]. The general structure of the cosmic strings in MG has not been investigated to render any comparison of this structure of strings with that of other classical possible. The subject naturally divides itself into two sections, which we here propose to treat separately; cylindrical solutions in empty space, and passing on to the presence of the non zero cosmological constant.
Our plan in this work is as the following: In Sect. (2) we briefly review the basis of mimetic gravity, action and equations of motion. In Sect. (3) we prove a possible equivalency between constant Ricci scalar solutions of this theory and Einsteinmassless scalar field theory. In Sect. (4) we study static spacetimes in cylindrical form. In Sect. (5) we study the possibility to have Kasner solutions as the known solutions in GR in this type of modified gravity. In Sect. (6) we study solutions with time dependent scalar field. In Sect. (7) we investigate extensions of static solutions in the presence of a cosmological constant term. We’ll conclude in Sect. (8).
2 Brief review of mimetic gravity
3 Notes on constant curvature MG
In this section we prove a general theorem about the exact solutions in MG with the case in which Ricci scalar \(R=R_{\mu }^{\mu }\) remains constant. Such type of solutions could be used to explain late time behavior of cosmos in de Sitter epoch as well as solutions with cosmological constant which lead to the Schwarzschild(Anti) de Sitter spacetime with a wide class of different applications from cosmology to string theory. Calculations show that the conformal degree of freedom can be eliminated by adjusting the constant curvature condition, providing conditions to compare with the exact solutions of MG and GR.
Theorem
It is universally found that the MG with constant curvature geometries describes the same theory of one proposed by Buchdahl, These two models are equivalent in their actions and dynamical features.
Proof
For the dynamical point of view (for example in MG scenario for inflation) it makes no difference between the action of GR with a massless scalar field and the one in MG. This equivalency between constant Ricci scalar R solutions of MG and Einsteinmassless scalar field theory is an essential feature of any purely kinetic (only function of \(\partial _{\alpha }\phi \partial ^{\alpha }\phi \) ) form of this type of disformal deformation of GR.
4 Field equations for a static cylindrical spacetime
The particular problem is to solve a system of non linear differential equations (15–18) and to find the metric functions A, B, C.
5 Realization of Kasner’s solution

Quasi–Kasner solutions in MG: \(A=(kr)^{2a},B=\beta ^{2}r^{2} (kr)^{2(b1)}, C=(kr)^{2c}\):

Non Kasner type of the exact solutions:
6 Solutions for time dependent scalar field \(\phi =\phi (r,t)\)

a = 0: This case corresponds to the static and cylindrical solutions which we investigated in the previous section.
 \(A(r)=\pm a\ne \{0,\frac{\pm 1}{2}\}\): In this case, the modified forms of the EOMs given in Eqs. (15–18) are obtained by the following system:$$\begin{aligned}&\left( a\pm \frac{ 1}{2}\right) \left[ \,{\frac{{ B''}}{B}}+{\frac{{ C''}}{C}}\frac{1}{2}\left( \,{\frac{{{ B'}}^{2}}{{B}^{2}}}+\,{\frac{{{ C'}}^{2}}{{C}^{2}}}\right) \right] =0, \end{aligned}$$(39)$$\begin{aligned}&B' C'=0, \end{aligned}$$(40)and$$\begin{aligned}&\frac{B''}{B}+\frac{C''}{C}\frac{1}{2}\left( \left( \frac{B'}{B}\right) ^2+\left( \frac{C'}{C}\right) ^2\right) =0. \end{aligned}$$(41)When \(B'=0,C'\ne 0\) (or vice versa, without disturbing the generality of discussion), we obtain:$$\begin{aligned}&\frac{{B''}}{B}\frac{1}{2}\left( \frac{B'}{B}\right) ^2=0. \end{aligned}$$(42)the metric is obtained as follows:$$\begin{aligned}&C(r)=\frac{1}{2}\left( \frac{c^2}{2}r^2+dc r+\frac{ d^2}{2}\right) . \end{aligned}$$(43)Where B is an arbitrary constant. Another choice is obtained when \(B'\ne 0,C'=0\):$$\begin{aligned}&\mathrm{d}s^2=(\mp a)\mathrm{d}t^2\mathrm{d}r^2B^2\mathrm{d}\varphi ^2\nonumber \\&\quad \quad \quad \frac{\mathrm{d}z^2}{2}\left( \frac{c^2}{2}r^2+dc r+\frac{d^2}{2}\right) . \end{aligned}$$(44)here \(\{C,d,c,d',c'\}\) denote a new set of parameters and \(\pm a>0\) (plus sign for \(a\in \mathcal {R}^{+}\) and minus for \(a\in \mathcal {R}^{}\)). It is remarkable that the class (44) represents a cosmic string when \(d=0\) , and the class (45) when we set \(d'=0\).$$\begin{aligned}&\mathrm{d}s^2=(\pm a)\mathrm{d}t^2\mathrm{d}r^2 \nonumber \\&\quad \quad \quad \frac{1}{2}\left( \frac{c'^2}{2}r^2+d'c' r+\frac{d'^2}{2}\right) \mathrm{d}\varphi ^2C^2\mathrm{d}z^2. \end{aligned}$$(45)
7 Case with cosmological constant

If \(\xi ^2>0\), the scalar field is real \(\phi \in \mathcal {R}\) , and the Kretschmann scalar \(\mathcal {K}=R_{\mu \nu \alpha \beta }R^{\mu \nu \alpha \beta }\) is free of any naked singularity.

When \(\xi \in \mathcal {C}\), we have a naked singularity located at \(r=r_{0}=\frac{1}{3}a\ln (\xi )\) where \( 0<\xi <1\).

When \(\xi >1\) the solution is free of naked singularities.
This argument is principally based on the following general theorem, which is a remarkable extension of Nogo theorems.
Nogo Theorem: If we consider the MG with cosmological constant, i.e. the system of differential equations given by Eqs. (15–18), this theorem clearly state that the solution given by Eq. (46) is not a solution to the MG theory.
Proof
To obtain the corresponding mathematical proof concerning the general form of equations of motion (15–18) , we eliminate A, C throughout by (15–18) and obtain (24). If we want to perform an elimination of A, C, and then B respectively, we can seek this through use of a lex ranking for the algebraic problem. Paying attention merely to the other equations \([B'CC'B\ne 0 , B'\ne 0]\),we see that there is no solution for the metric functions \(\{A,B,C\}\) with cosmological constant. Indeed, the only solution with constant curvature is when \(\Lambda =0\) and this solution generally does not coincide with the LT solution.
The general theorem is manifest, and yields a development in any attempt to generalize LT family of corresponding MG Lagrangian.
8 Discussions and final remarks

Constant curvature vacuum solutions in mimetic gravity are equivalent to the solutions in Einstein gravity with a massless scalar field.

Quasi Kasner solution doesn’t exist in mimetic gravity.

A family of exact solutions with variable R is found which are different from the LeviCivita or Kasner family in GR. This solution has four Kretschmann’s singularities, one is naked singularity on string’s axis \(\rho =0\), and three cylindrical “horizons” as \(\rho _{+}\le \rho _{*}\le \rho _{l}\).

Finding new solution, we relaxed \(\phi (r,t) \) to be static and began finding solutions in the two cases, namely (c, d),\((c',d')\).

When \(\Lambda \ne 0\), we proved the following theorem: Nogo Theorem: Mimetic gravity doesn’t have LinetTian family of cosmic strings. The only possible solution is when \(\Lambda =0\).
Notes
Acknowledgments
We thank the referee for good observations and kind guide lines.
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