The Physical Significance of Time Conformal Minkowski Spacetime

The Minkowsiki spacetime is flat and there is no source of gravitation. The time conformal factor is adding some cuvature to this spacetime which introduces some source of gravitation to the spacetime. For the Minkowski spacetime the Einstein Field equation tells nothing, because all the components of the Ricci curvature tensor are zero, but for the time conformal Minkowski spacetime some of them are non zero. Calculating the components of the Ricci tensor and using the Einstein field equations, expressions for the cosmological constant are cacultaed. These expressions give some information for the cosmological constant. Generally, the Noether symmetry generator corresponding to the energy content in the spacetime disapeares by introducing the time conformal factor, but our investigations in this paper reveals that it appears somewhere with some re-scale factor. The appearance of the time like isometry along with some re-scaling factor will rescale the energy content in the corresponding particular time conformal Minkowski spacetime. A time conformal factor of the form ( ) is introduced in the Minkowski spacetime for the invistigation of the cosmological constant. The Noether symmetry equation is used for the Lagrangian of general time conformal Minkowski spacetime to find all those particular Minkowski spacetimes that admit the time conformal factor. Besides the Noether symmetries the cosmology constant is calculated in the corresponding spacetimes.


INTRODUCTION
manifold resembles the Euclidean space locally and can be covered by patches of Euclidean space. However, the global behavior of the manifold is entirely different from the Euclidean geometry. The manifold of Minkowski spacetime plays an important role in both special as well as general relativity. The knowledge of differential geometry and differential manifold is necessary to explain Minkowski spacetime geometry.
In the following, a brief discussion of manifold and causality has been given before discussing the Minkowski metric. Let = (x , x , … , x ) ∈ . A collection is defined to be a manifold if every member of has a continuous open neighborhood which forms a bijective function to an open set of for some n. Mathematically, [1][2]  Symmetries play an important role in the solution of the Einstein field equations [3][4][5]: We used Noether symmetries to classify plane symmetric, cylindrically symmetric and spherically symmetric spacetimes according to their Noether symmetries [6][7][8][9][10]. Plane symmetric and cylindrically symmetric spacetime admitting the time conformal factor is studied in [11][12], while the dynamics of neutral and charge particles in the time conformal Schwarzschild spacetime is studied in [13]. Different types of spacetimes symmetries are given in references [14][15][16].
Here we consider the time conformal Minkowski spacetime: u(t) is a general function of time and the Noether symmetry equation to find particular classes for the spacetime (2). Different values of u(t) give different symmetry structure of spacetime (2). There are seventeen Noether symmetries admitting by the exact Minkowski spacetime: Ten of the seventeen Noether symmetries form the Poincare group, four are the Galilean transformations, one is scaling, one is the conformal transformation, and one is the symmetry corresponding to the Lagrangian of spacetime given in Equation (2). e 8(9) , the conformal factor, will reduce the Noether symmetries' number for the spacetime (2). Decreasing the number of Noether symmetries mean decreasing conservation laws. The study of particular classes for particular values of function u(t) will tell which conservation law holds and which one violates. Specifically, the symmetry generator ∂ @ will disappear due to the time conformal factor A B(@) , which is linked to the energy content in the specified space time. But this symmetry will occur along with some re-scale term, as our investigation in this work confirms it. The symmetries correspond to the boosts and Galilean transformations will also disappear, because of the introduction of the same conformal factor e 8 (9) , which can not be recovered. In this work, the cosmological constant ᴧ is also calculated in each spacetime. Furthermore, the effect of time on this constant is investigated, too.

NOETHER SYMMETRY AND THE CONSERVATION LAWS
The Lagrangian corresponding to Equation (2) is: Using this Lagrangian in the Noether symmetry equation: and obtaining system of Noether symmetry determining partial differential equations. The solution of which will give different particular values for the function J(K) along with Noether symmetries and conservation laws. This solution will consist of all time conformal Minkowski spacetime, the symmetry structure of each class will be discussed correspondingly. Where represents extension of the first order of the Noether symmetry operator: S is the total differential operator of the form:  (4) in Equation (5), a system of nineteen determining PDEs (Partial Differential Equation) are obtained as follows: The solution of the system Equation (9) will give the spacetimes of our interest. This solution will be consists of all the time conformal Minkowski spacetimes. The conservation law corresponding to each Noether symmetry is: This calculation will classify the spacetime given in Equation (2). The detail of which is given below.

SOLUTION-I
The general form of the time conformal Minkowski spacetime is: The system (9) has the following solution for any general function J(K), A = c , u = u(t), χ = c , η = 0, η = c T y + c W z + c ] η T = −c T x + c^z + c _ , η W = −c W x − c^y + c / The action corresponding to the spacetime (11) admits the following Noether symmetries: L corresponds to the Lagrangian (4), L , L T and L W correspond to the linear momentum along b, c and d axes, respectively. Hence, the linear momentum along these axes conserved. L ] , L^ and L _ correspond to the conservation laws of angular momentum in bc, cd and db planes respectively. The energy in the spacetime (11) is not conserved as we do not see the generator ∂ @ in the set (12). The Einstein fields equations given in equation (1) for the spacetime given in Equation (11) takes the form:

SOLUTION-II
The second solution of system (9) is: L , L , L T , L W , L ] , L^ and L _ do the same job as in Solution-I. m / is the Noether symmetry generator corresponds to the scaling or the similarity transformation for the spacetime (14), it is the homothety of spacetime (14). The Einstein fields equations for the spacetime (14) take the form: Equation (17) shows that for positive | the cosmological constant Λ is a decreasing function of time t.

SOLUTION-III
The third solution of system (9) where ‡ ƒ is the re-scale energy of the test particle in spacetime (18), while E is the exact energy of the test particle and ˆ is the Lagrangian density given in Equation (4). We see from Equation (21)  will move forever with same energy given at the beginig. The curvature introduced by the time conformal factor will either accelerate the test particle or de-accelerate it. In our case it de-accelerate the test particle when it enters in the corresponding spacetime.
Equation (23) shows that the cosmological constant decreases as time elapse, which agrees with the accelerated expansion of the universe.

CONCLUSION
The classification of the time conformal Minkowski spacetime is presented in this article. There are four classes of the time conformal Minkowski spacetime according to the Noether symmetries. These types of spacetimes admit, seven, eight, nine or eleven Noether symmetries. Therefore, there exist seven, eight, nine or eleven conservation laws for the Lagrangian of such spacetimes. In Equation (12), the Noether symmetry ∂s corresponds to the Lagrangian L and the remaining symmetries are all isometries of the spacetime given in Equation (11). The Noether symmetry N9 given in Equation (15) corresponds to the scaling transformation in the corresponding spacetime (14) and is called homothety or homothetic vector field of the corresponding spacetime. The Noether symmetry N8 in Equation (20) is the Noether symmetry generator corresponds to the energy in spacetime (18) with the re-scaling term s∂s, while N9 is the symmetry generator corresponding to the conformal transformation. The time conformal spacetime given in Equation (24) is the one which admits maximum Noether symmetries and the corresponding Lagrangian admits maximum conservation laws. The symmetry generators N8, N9 and N10 correspond to the conformal transformations in the manifold of the spacetime (24). The Noether symmetry generator N11 given in Equation (26) is the symmetry genrator corresponding to the similarity transformation (scaling).

ACKNOWLEDGEMENT
Authors are thankful to the unknown refrees who's comments and suggestions highly improved the text.