A detailed study of the energy conditions of the universe and to find unique solutions of the Einstein’s field equations are carried out in this paper. With the reference to the power series expansion of the mass function

To study the energy conditions of the Universe in the lights of the theory of general relativity, we begin with orthogonal null tetrad vectors such as a time – like unit vector ^{1, 2}

All four vectors exist with the normalization conditions – ^{3, 4}

Thus, the metric tensor can be written with the help of these four orthogonal tetrad vectors such as,

Here we are considering four velocity vectors for non – space like observers such that

Where,

For non – space like vectors,

(i) When

(ii) When

(iii) When

(iv) When

Thus, we summarize that the above conditions can be written as

In this paper, we will present a class of exact solutions, stationary and non – stationary for the Einstein’s field equations which describe non – empty, conformally flat space – time. We will also study the very physical properties of the solutions.

Wang and Wu (1999) introduced a mass function

Where

n=0 corresponds to the mass function

n=-1 corresponds to the mass function

n=1 corresponds to the mass function

n=3 corresponds to the mass function

The Wang – Wu power series expansion turns out to be the most convenient form to generate new embedded and non – embedded solution of the Einstein’s field equation if one uses Newman – Penrose formalism ( NP formalism) ( Newman and Penrose, 1962). From the above identifications of the power n, we can observe that n=2 and n=-2 are not observed before, so far in the correspondence to the exact solutions of the Einstein’s field equations.

This is the main aim of this paper to generate exact solutions of Einstein’s field equations with n=2 and n=-2 to find the actual physical meaning and to investigate the properties of the energy momentum tensors describing the matter distributions of the space – time geometry.

It is a fact that, although there is no straightforward way of solving Einstein’s field equations, but a deliberate attempt is made with n=2 and n=-2 in power series expansion developed by Wang and Wu. It is believed that n=2 and n=-2 will provide exact solutions of Einstein’s field equations with physical interest and reasonable good interpretations of the matter distributions in the stationary and non – stationary space – time.

But first we find the exact solutions of Einstein’s field equations, then, consequently we will find the physical interpretations of the line elements by studying its properties. For this very purpose, we choose Wang – Wu function

Such that mass function takes the form

Where m is constant and u=t-r is retarded time coordinate. Using this mass function, we find a stationary line element.

Here, m is constant and regarded as the test particle to be present in the space – time. And it is non – zero in the matter distributions in the geometry. The line element in the equation (11) describes a stationary solution. It has coordinate singularity at

For deriving an embedded Schwarzschild – dark energy solution, we can consider the mass function in the power expansion series if we choose Wang – Wu function

Such that mass function takes the form

Where m is constant and u=t-r is retarded time coordinate. Using this mass function, we find a stationary line element.

Here, m is constant and regarded as the mass of the dark energy. And it is non – zero in the dark energy distribution in the Universe. The line element in the equation (14) will be reduced to that of Schwarzschild black holes when m=0 with singularities at r=2M . Also, it will be that dark energy when M=0 with singularities at

Now we find the exact solutions of Einstein’s field equations with n=-2, then, consequently we will find the physical interpretations of the line elements by studying its properties. For this very purpose, we choose Wang – Wu function

Such that mass function takes the form

Where m is constant and u=t-r is retarded time coordinate. Using this mass function, we find a stationary line element.

Here, m is constant and regarded as the mass of the dark energy. And it is non – zero in the dark energy distribution in the Universe. The line element in the equation (14) will be reduced to that of Schwarzschild black holes when m=0 with singularities at r=2M . Also, it will be that dark energy when M=0 with singularities at

For deriving an embedded Schwarzschild – dark energy solution with n=-2, we can consider the mass function in the power expansion series if we choose Wang – Wu function

Such that mass function takes the form

Here, m is constant and regarded as the mass of the dark energy. And it is non – zero in the dark energy distribution in the Universe. The line element in the equation (14) will be reduced to that of Schwarzschild black holes when m=0 with singularities at r=2M . Also, it will be that dark energy when M=0 with singularities at

With reference to the theory of general relativity, the first solution with the line element in the equation (11) describes a stationary solution. It has coordinate singularity at

We truly acknowledge Dr. Aparna Nath, Associate Professor, The Department of Physics, National Institute of Technology Agartala for providing great inspiration and tremendous motivation to complete this study. Also we are extremely thankful to the Department staff for cooperation.