_{1}

^{*}

We present a class of axially symmetric and stationary spaces foliated by a congruence of surfaces of revolution. The class of solutions considered is that of Carter’s family [A] of spaces and we try to find a solution to Einstein’s equations in the presence of a perfect fluid with heat flux. This approach is an attempt to find an interior solution that could be matched to a corresponding exterior solution across a surface of zero hydrostatic pressure. The presence of a congruence of surfaces of revolution, described as the quotient space of the commoving observers, can be useful to the determination of the surface of zero pressure. Finally we present two formal solutions representing ellipsoids of revolutions.

The contents of this paper summarize and improve a series of papers in which I have tried to solve the problem of constructing an interior Kerr solution, [

We suppose that the space-time admits an Abelian isometry group G_{2}, invertible with non-null surface of transitivity [

The components of the metric tensor (2.1) depend on x and y, the G_{2} group is generated by the time-like Kill-

ing field

The invertibility of the group is obvious from the fact that the transformation

The functions L, M, N, P are real and depend only on x and y. This metric has been introduced by Carter and used extensively by Debever [

This family of spaces is characterized by the fact that the Hamilton-Jacobi equation for the geodesics is solvable by separation of variables (x, y), which takes place in a particular way and gives rise to a fourth constant of motion, quadratic in the velocities integral or equivalently to the existence of a Killing tensor with two double eigenvalues, see [

The metric is:

where:

The null vectors n, l are real and

The Carter’s family [A] of solutions includes many interesting metrics, we present some of them in order to make easier the comparison with our solution and justify some technical details of calculations.

The Einstein’s vacuum equations imply that:

where b, c, d, p are constants of integration. The Kerr metric is obtained if we set:

m is the mass, a is the parameter of angular momentum per unit mass and the Kerr metric can be written in the Boyer-Lindquist coordinates:

The Einstein’s equations in the presence of a perfect fluid reduce to two relations:

These equations ensure that the energy-momentum tensor admits one simple and one triple eigenvalues. The solution of (2.11) permits us to determine the following functions:

where

We get Wahlquist solution [

We consider a 3-dimensional Euclidean space foliated by a congruence of surfaces of revolution, obtained by revolving a plane curve C about a line L in its plane. This line will be identified with the axis of rotation of the fluid configuration. In a Cartesian coordinate system

where t is the corresponding parameter. The elimination of this parameter between x_{1}, x_{3} gives the Cartesian equation of the curve:

The Cartesian equation of the surface is:

Then a parametric representation of a surface of revolution is:

We can now define a system of coordinates, adapted to the surface of revolution. The azimuthal angle

And the equation for each member of the congruence of the surfaces of revolution is given by:

The corresponding metric in the Euclidean space assumes the expression:

where

The coordinates are orthogonal if:

We will consider only orthogonal coordinate systems and we generalize (3.7) to a three-dimensional axisymmetric Riemannian space by considering a supplementary function

Obviously when _{1} containing the parametric representation of the curve C.

Finally we introduce a coordinate change of great utility for the integration of Einstein’s equations. Instead of r and θ we will use y and x that appear in (2.3). This coordinate transformation is given by (2.9), the coordinate used to obtain Kerr metric from the Carter’s family [A] of spaces.

Definition 3.1.

The metric of a three-dimensional Riemannian space foliated by a congruence of surfaces of revolution, in an orthogonal coordinate system is given by:

And the orthogonality condition holds:

The observers who will describe the shape of the two-dimensional surfaces embedded in the four-dimensional Lorentzian space as rotating surfaces of revolution (defining the equipotential or equipressure surfaces for the vacuum and the interior case respectively), follow the trajectories of the symmetry group. The velocity field of

this family of observers is then a linear combination of the Killing vector fields

Definition 3.2.

The vector field of the family of the observers describing the surfaces of revolution is given by:

And as usual the norm is given by:

where

The observers defined above are at rest on their quotient space that has to be identified with the three-dimen- sional Riemannian space of the Definition (3.1):

This equation and the relations (3.12), (3.13) imply the following conditions:

The general stationary and axially symmetric metric (2.3) can be written as:

where

And condition (3.11) holds. Compare with Krasinski {(3.14) in [

The functions U, V, K, f and

Now we impose that metric (3.16) is identified with metric (2.3), considering only the special case of Carter’s family [A] of spaces and not the general stationary, axisymmetric spaces. Then we can state the following theorem:

Theorem 3.1.

In Carter’s family [A] of spaces, we can define observers with quotient space identified with a congruence of surfaces of revolution if the following conditions are satisfied:

The steps that we have followed in Section 3, give the outline of the proof of Theorem 3.1. Equations (3.18) permit to define completely U and V as well as the functions h_{1}, F(x) and the ratio G(y)/f(x, y). We can define the functions U and V by direct calculation, using (3.18a)-(3.18f):

In the null tetrad (2.5) we have that:

The remaining Equations (3.18d), (3.18e), (3.18g) permit us to determine partially the shape of the surface of revolution and the functions F^{2} and the ratio

q and g are constants and

The expression of h_{1} given by (3.24) is very restrictive for the shape of the surfaces of revolution identified with the quotient space of the observers. There are two possible cases, according if g is zero or different of zero:

(I) g = 0

The surfaces of revolution are oblate ellipsoids of revolution. The constant q has to be positive and can be put equal to one:

In Cartesian coordinates the equation of the ellipsoids is:

Obviously the ellipsoids are oblate due to the rotation (a is the angular-momentum per unit mass in the Kerr metric).

The corresponding surfaces are tori of revolution. Again q has to be positive and can be put equal to one. The revolved curve is an oblate ellipse with center of symmetry located at a distance g from the origin:

If we consider the Kerr metric applying (2.7)-(2.10) we can state that the class of observers defined by (3.19)- (3.21) and (3.24) can characterize Kerr metric as ellipsoidal or toroidal space. The function f(y) is given by:

Clearly

In this section we are going to solve the Einstein’s equations in the presence of a perfect fluid with heat flux, for Carter’s family [A] of spaces imposing that the commoving observers are those with quotient spaces the family of surfaces of revolution.

The energy-momentum tensor of a perfect fluid with heat flux has the following form:

where e is the mass-energy density of the fluid, p is the hydrostatic pressure, u_{i} is the vector field of the commoving observer and q_{i} is the vector of the heat flux. These two vector fields can be written in the complex vectorial formalism:

The norm of

The Carter’s class of family [A] of spaces written in the form (2.3) belongs to the class of symmetric null tetrads [

Expressions (4.6)-(4.10) depend on the functions G(y), E(y), F(x), H(x) of Carter’s metric (2.3). The Einstein’s equations reduce to a single equation:

And after replacing the expressions of the Ricci traceless tensor:

where:

We can prove that after successive differentiations with respect to x and y, the functions W(y) and Z(x) can be defined completely:

If we substitute (4.14) in (4.12) and we differentiate again successively with respect x and y, we obtain two equations each depending on one variable only. After successive integrations we get the final decoupled equations for E^{2} and H^{2}:

The constants of integration are k, k_{4},_{ }k_{0}, k_{2}, l_{4}, l_{2}, l_{0} and m (compare with (2.13)), P_{6}(y) and Q_{6}(x) are arbitrary polynomials on x and y respectively of sixth degree. If we put k equal to zero (without the heat flux) we obtain a generalization of the Wahlquist solution [

cond degree with respect to the ratio

The two solutions of (4.17) are the following:

The case (4.18a) gives a Wahlquist like solution, so we are going to consider only the case (4.18b). Find the general solution or even partial solutions of (4.15), (416) is a project of its own, but the purpose of this paper is to solve these equations under the assumption of the existence of a congruence of surfaces of revolution presented in Sections 1-3.

First we try to define the function H(x):

The corresponding equations are (3.22), (4.14) and (4.16). The function H(x) satisfies an equation of fourth degree [(3.22) and (4.14)]:

Also H(x) has to satisfy the differential Equation (4.16). The system is over determined but we can find two possible cases:

The remaining equations to consider are the differential Equations (4.15), (3.23), (4.14) and (4.18) where the

ratio

and the following equations:

where:

Two partial formal solutions can be given, characterized by

S_{1}:

S_{2}:

For both solutions f(y) is given by (5.3). The x dependence of the two solutions is Kerr-like; a detailed study and the calculation of the physical quantities will be presented in a subsequent paper. The quotient spaces are ellipsoids of revolution (g = 0).

Taxiarchis Papakostas, (2015) Surfaces of Revolution in the General Theory of Relativity. Journal of Modern Physics,06,2000-2010. doi: 10.4236/jmp.2015.614206