Geometrothermodynamics of a gravitating system with axially symmetric metric

Theory of the first-order phase transition in contact statistical manifold has been proposed to describe interface systems. The theory has not limitations of known Van der Waals-like phase transitions theories. Based on this approach, geometrothermodynamics of gravitating systems with axially symmetric metric is investigated. It has been shown that such geometrothermodynamics occurs in space-ti,e with Newman-Unti-Tamburino metric.


Introduction
To date, there are known important experimental facts in cosmology: a nonlinear dependence of galaxies recession on magnitude of the red shift, a nonlinear dependence of supernova luminosity on the red shift, gravitational lensing of galaxies which does not associated with fluctuations, the existence of voids in galaxies distribution [1].These facts can be explained through the existence of space-time universe regions which is expanding at an accelerating rate.Gravitating matter in such regions does not interact (or interacts weakly) with ordinary substance and is known as dark matter.Beside the fact that our universe is in the phase of accelerated expansion, the universe is a flat one, and therefore vacuum energy cosmological constant  has to be nonzero: 0  [2][3][4][5].-term within Einstein equations describes the dark energy (see, for example, [6] and references therein] which can be interpreted as a fluid with negative pressure p , 0 p  [7].The pressure takes on a negative value at the first-order phase transition while the matter is in a metastable state.Because of this, it has been proposed in [8][9] to use the thermodynamics at gravity modeling.CDM (Lambda cold dark matter) cosmological models [10] and electrically charged AdS (anti de Sitter) black holes [11][12] exhibit a liquid-gas like first order (1 st -order) phase transition culminating in critical points that resemble the phase diagram of a Van der Waals fluid [13].
However, the Van der Waals-Maxwell gas model is not invariant with respect to the physically meaningful Legendre transformations [14][15].Moreover, Einstein equations are not conformally invariant because of Weyl tensor does not vanish.The thermodynamics of cosmological models, such as Friedmann-Lemaître-Robertson-Walker ones, includes effects of the non-null Weyl tensor (anisotropic pressure) by introducing a viscosity parameter and relaxation time (inverse expansion coefficient) [16][17].But, relaxation processes are absent in the Van der Waals-Maxwell gas model.Thus, a more realistic theory of the 1 st -order phase transition has to be used to develop geometrothermodynamic approaches to cosmology.
In [18] we have proposed a theory of the 1 st -order phase transition in a contact statistical manifold that describes interface systems with an electrocapillary mechanism of the energy dissipation and with relaxation time distributions; and this theory has not limitations which are in the Van der Waals-like phase transition theories.The triad   vector field of the contact manifold (see [22][23][24] and references therein).Therefore, T is a fifthdimension (5D) contact manifold.Because the 1 st -order phase transition proceeds on the interface, the metric is an axially symmetric one.
In the paper, utilizing approach considered above we study the geometrothermodynamics of gravitating systems with an axially symmetric metric which is similar to Newman-Unti-Tamburino (NUT) one.

A geometro-thermodynamical background of problem
An entropy production S  in the first-order phase transition proceeding on the interface [18] reads where  a S is the entropy production for the a -th process,  3) are determined the Helmholtz free energy F and the internal energy E for the first-order phase transition as 1 1 ln Expanding the expression (1) in series over small parameter s  as In a case of electrocapillary loss, the Lagrangian ( ) and, respectively, the action dl on the statistical manifold is written in the following form: where dr ds r  and V is a modulus of compression (or expansion) rate, p and m are model parameters.

Generalized NUT-metric
The NUT metric reads where n is a NUT parameter, rg is a gravitational radius, x0 is a time coordinate; r, θ,  are spatial spherical coordinates.In the case of 22   8( / 3) ( / 3) 5 n q g rn rn to the metric (10).Then the NUT-metric transforms to the form: where ,  is a cosmological constant.
While 22  , g r r r n , r ~ rg, the substitution 2 , dr = dR leads to the metric (11) obtains the features of anti de Sitter metric.

An axially symmetric metric of thermodynamic phase space entering a 5-dimensional contact manifold
Based on the metric function ( 8) one can construct the following metric: A space-time metric can be obtained from the metric (12) by the Wick rotation i   and where . Then, the metric (13) can be considered as a metric of 4D-surface θ = 2  in the contact 5D-manifold:  .
Let us choose some another 4D-surface in the contact 5D-manifold in a such way to the angle θ takes arbitrary values, while ranges of variables r,  are dynamically limited by the variable τ.This can be realized in the following way.Let us impose a gauge condition on the scalar field We consider the case of small values of rate 0 V  when the function . Then, while the condition ( 15) is imposed, the scalar field  oscillates strongly,   At   the expression ( 18) is an expansion over a small parameter Comparison between ( 11) and ( 20) gives the expression for the NUT-parameter, which becomes an imaginary constant at θ , 0   : Geodesic in the contact manifold with the metric (12) and an estimation of dependence of  at 0 V  are shown in fig. 1.The simulation results and the analytic calculations performed above prove that such Schwarzschild-like geodesics behave similarly to that of the NUT metric.

Discussion and conclusion
So, using of the contact statistical 5-dimensional manifold, which describes phase transitions proceeding in the interface systems with an electrocapillary mechanism of energy dissipation, allows us to construct the theory of cosmological models with axially symmetric metrics.These metrics are that of 4-dimensional surfaces in the 5D manifold.We has shown that the NUT-theory parameter n is the gauge parameter of the scalar field which plays a role of fifth dimension.The metrics in 5D world are similar to the (anti) de Sitter metrics at large values of n.

Fig. 1 .
Fig. 1.(a) Geodesic; (b) the dependence of () s  (blue solid curve) and its approximation by 1 ( ) 0.8 ( ) 0.45 s rs     V is the vertical subspace generated by  , and H is the horizontal distribution.The bundle T of our problem can be redefined by the dynamical-system phase-