M-PROJECTIVELY FLAT SPACETIMES

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F ¨USUN ¨OZEN ZENGIN

The object of the present paper is to study the M-projectively flat Riemannian manifold. It is shown that the M-projectively flat Riemannian manifold is an Einstein manifold. In addition, some theorems about the energy-momentum tensor satisfying the Einstein field equations with a cosmological constant of the M-projectively flat spacetime are proved.

AMS 2010 Subject Classification: 53C15, 53C25, 53B15, 53B20.

Key words: M-projective curvature tensor, energy-momentum tensor, Killing vector field, conformal Killing vector field, symmetry inheritance property, quadratic Killing tensor, quadratic conformal Killing tensor.

1. INTRODUCTION

Let (M_n, g) be ann-dimensional differentiable manifold of classC^∞with the metric tensor gand the Riemannian connection∇. The M-projective curvature tensor of Mn defined by G.P. Pokhariyal and R.S. Mishra in 1971 (see [15]) is the following form

W(X, Y)Z =R(X, Y)Z− 1 2(n−1)

h

S(Y, Z)X−S(X, Z)Y+ (1.1)

+g(Y, Z)QX−g(X, Z)QY i

,

whereR(X, Y)Z andS(X, Y) are the curvature tensor and the Ricci tensor of M_n, respectively. Such a tensor fieldW(X, Y)Z is known as the M-projective curvature tensor. Some authors studied the properties of this tensor [12–13].

In 2010, S.K. Chaubey and R.H. Ojha investigated the M-projective curvature tensor of a Kenmotsu manifold [2].

In this paper, the M-projectively flat Riemannian manifold will be investigated and after that, defining the M-projectively flat spacetime, the con- ditions at which the energy-momentum tensor has the symmetry inheritance property and the Lie derivative of this tensor vanishes will be found.

MATH. REPORTS14(64),4 (2012), 363–370

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2. ON M-PROJECTIVELY FLAT RIEMANNIAN MANIFOLD

Now, we consider that (Mn, g) is M-projectively flat Riemannian manifold. Since W(X, Y)Z = 0 then we have from (1.1)

(2.1) S(X, Y) =g(X, Y).

Thus, we also get

(2.2) r=n.

Theorem 2.1. M-projectively flat Riemannian manifold is Ricci symmetric.

Proof. Taking the covariant derivative of (2.2) side by side, we get

(2.3) (∇_ZS)(X, Y) = 0.

Thus, from (2.3), the proof is completed.

3. M-PROJECTIVELY FLAT SPACETIMES

In cosmology, the spaces with the constant curvature play a significant role. The assumption that the universe is isotropic and homogeneous is given the simplest cosmological model. It is known as cosmological principle. This principle, when translated into the language of Riemannian geometry, asserts that the three dimensional position space is a space of maximal symmetry [17], that is, a space of constant curvature whose curvature depends upon time. The cosmological solutions of Einstein equations which contain a three dimensional space-like surface of a constant curvature are the Robertson-Walker metrics, while four dimensional space of constant curvature is the de Sitter model of the universe [11], [17].

The general theory of relativity, which is a field theory of gravitation, is described by the Einstein field equations. These equations whose fundamental constituent is the space-time metric g, are highly non-linear partial differen- tial equations and therefore it is very difficult to obtain exact solutions. They become still more difficult to solve if the spacetime metric depends on all coor- dinates [1], [7], [17]. This problem however, can be simplified to some extent if some geometric symmetry properties are assumed to be possessed by the metric tensor. These geometric symmetry properties are described by Killing vector fields and lead to conservation laws in the form of first integrals of a dynamical system [3]. There exists, by now, a reasonably large number of solutions of the Einstein field equations possessing different symmetry structure [14]. These solutions have been further classified according to their properties and groups of motions admitted by them [10].

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Katzin et al. [9] were the pioneers in carrying out a detailed study of curvature collineation, (CC), in the context of the related particle and field conservation laws that may be admitted in the standard form of general relativity [8].

The geometrical symmetries of the spacetime are expressed through the equation

(3.1) £_ξA−2ΩA= 0,

where A represents a geometrical/physical quantity £_ξ denotes the Lie derivative with respect to the vector field ξ and Ω is a scalar.

One of the most simple and widely used example is the metric inheritance symmetry for which A= g in (3.1); and for this case, ξ is the Killing vector field if Ω is zero.

A Riemannian manifoldM_n is said to admit a symmetry called a curvature collineation (CC) provided there exists a vector field ξ such that [4–5]

(3.2) (£_ξR)(X, Y)Z = 0,

where R(X, Y)Z is the Riemannian curvature tensor [6].

In this paper, we denote the M-projectively flat spacetime by (M P F S).

Now, we shall investigate the role of such symmetry inheritance for (M P F S).

Theorem3.1. A Riemannian manifold with M-projective curvature tensor admits (CC) if and only if the Lie derivative of the M-projective curvature tensor along the Killing vector field ξ vanishes.

Proof. By taking the Lie derivative of (1.1) and taking the vector fieldξ as a Killing vector field, we get

(3.3)

(£ξW)(X, Y)Z = (£ξR)(X, Y)Z− 1

2(n−1)[(£ξS(Y, Z))X−(£ξS(X, Z))Y].

If we consider that our manifold admits CC then from (3.3), we find

(3.4) (£_ξW)(X, Y)Z = 0.

This leads the proof.

Theorem 3.2. For (M P F S)4, the energy-momentum tensor satisfying the Einstein field equations with a cosmological constant is in the form

T(X, Y) = 1

k(λ−1)g(X, Y).

Proof. Let S(X, Y) 6= 0 and the Einstein field equations with a cosmological term are

(3.5) S(X, Y)−r

2g(X, Y) +λg(X, Y) =kT(X, Y)

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for all vector fieldsXandY. Hence,S(X, Y) denotes the Ricci tensor,T(X, Y) is the energy momentum tensor, λ is the cosmological constant and k is the non-zero gravitational constant.

From (3.5), we get

(3.6) S(X, Y) +

λ−r

2

g(X, Y) =kT(X, Y).

Assume that our manifold is (M P F S)₄, by using (2.1), the equation (3.6) reduces to

(3.7) T(X, Y) = 1

k(λ−1)g(X, Y).

This completes the proof.

Theorem3.3. For(M P F S)₄ satisfying the Einstein field equations with a cosmological term, there exists a Killing vector field ξ if and only if the Lie derivative of the energy-momentum tensor along that vector field is zero.

Proof. Taking the Lie derivative of both sides of (3.7), we find (3.8) (λ−1)(£_ξg)(X, Y) =k(£_ξT)(X, Y).

If ξ is a Killing vector field then we have

(3.9) (£_ξg)(X, Y) = 0.

By using (3.8) and (3.9), we get

(3.10) (£_ξT)(X, Y) = 0.

Conversely, if (3.10) holds, from (3.8), we obtain the condition (3.9). Thus, we can say that ξ is a Killing vector field. The proof is completed.

Theorem 3.4. (M P F S)4 obeying the Einstein field equations with a cosmological term, there exists a conformal Killing vector field ξ if and only if the energy-momentum tensor has the symmetry inheritance property.

Proof. Ifξ satisfies the condition

(3.11) (£_ξg)(X, Y) = 2Ωg(X, Y)

then it is called a conformal Killing vector field. Now, we assume that ξ is a conformal Killing vector of (M P F S)₄. Thus from (3.8) and (3.11), we find (3.12) (£_ξT)(X, Y) = 2ΩT(X, Y).

In this case, it can be said that the energy-momentum tensor has the symmetry inheritance property. Conversely, if the condition (3.12) holds, from (3.8), we can get that the equation (3.11) is satisfied. This completes the proof.

Now, we will give a definition to use in the other theorems:

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Definition. Let (Mn, g) be a spacetime manifold with Levi-Civita connection ∇. A quadratic Killing tensor is a generalization of a Killing vector and is defined as a second order symmetric tensor T satisfying the condition (3.13) (∇_XT)(Y, Z) + (∇_YT)(Z, X) + (∇_ZT)(X, Y) = 0.

A quadratic conformal Killing tensor is analogous generalization of a conformal Killing vector and is defined as a second order symmetric tensor T satisfying the condition

(∇_XT)(Y, Z) + (∇_YT)(Z, X) + (∇_ZT)(X, Y) = (3.14)

=a(Y)g(Y, Z) +a(Y)g(Z, X) +a(Z)g(X, Y)

for a smooth 1-formaonM_n. The above equation is equivalent to the require- ment that T(l, l) be constant along null geodesic with parallely propagated tangent vector l^[16],^[18],[19].

Theorem 3.5. The energy-momentum tensor of (M P F S)4 satisfying the Einstein field equations with a cosmological term is locally symmetric.

Proof. Let us consider that our space is (M P F S)4. If we take the covariant derivative of (3.7) then we find

(3.15) (∇_XT)(Y, Z) = 0.

Thus, we can see that the energy-momentum tensor satisfies the equation (3.13) is locally symmetric. In this case, the proof is completed.

Theorem 3.6. (M P F S)₄ cannot admit a quadratic conformal Killing energy-momentum tensor satisfying the Einstein field equations with a cosmological term.

Proof. If we put (3.15) in (3.14), we get

(3.16) a(Y)g(Y, Z) +a(Y)g(Z, X) +a(Z)g(X, Y) = 0.

A contraction of (3.16) over X and Y leads to a(X) = 0.

Thus, we can say that the energy-momentum tensor of this manifold cannot be a quadratic conformal Killing tensor. This result completes the proof.

In a perfect fluid spacetime, the energy-momentum tensor is in the form (3.17) T(X, Y) = (σ+p)u(X)u(Y) +pg(X, Y),

where σ is the energy density, p is the isotropic pressure andu(X) is a non- zero 1-form such that g(X, V) = u(X) for allX, V being the velocity vector field of the flow, that is, g(V, V) =−1. Also, σ+p6= 0.

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Theorem3.7. In(M P F S)4 satisfying the Einstein field equations with a cosmological term, the matter contents of the spacetime satisfy the vacuum- like equation of state.

Proof. For (M P F S)4, with the help of (3.7) and (3.17), the Einstein field equations are found as in the following

(3.18) (λ−1−kp)g(X, Y) =k(σ+p)u(X)u(Y).

By contraction of (3.18) over X and Y leads to

(3.19) λ= 1−k

4(σ−3p).

If we put X=Y =V in (3.18) then we find

(3.20) λ= 1−kσ.

Combining the equations (3.19) and (3.20), we get

(3.21) σ+p= 0.

The proof is completed.

Theorem3.8. The(M P F S)₄ admitting a dust for a perfect fluid is filled with radiation.

Proof. If we assume a dust in a perfect fluid, we have

(3.22) σ = 3p.

By putting (3.22) in (3.21), we get

p= 0.

Thus, this leads the proof.

Theorem3.9. A relativistic(M P F S)4satisfying the Einstein field equations with a cosmological term is vacuum.

Proof. In a relativistic spacetime, the energy-momentum tensor is in the form

(3.23) T(X, Y) =µu(X)u(Y).

From (3.7) and (3.23), we obtain

(3.24) (λ−1)g(X, Y) =kµu(X)u(Y).

A contraction of (3.24) over X and Y leads to

(3.25) λ= 1−1

4kµ.

And, if we put X=Y =V in (3.24), we get

(3.26) λ= 1−kµ.

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Thus, combining the equations (3.25) and (3.26), we finally get that

(3.27) µ= 0.

From (3.23) and (3.27), we find

T(X, Y) = 0.

This means that the spacetime is devoid of the matter. This result completes the proof.

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Received 22 February 2011 Istanbul Technical University Faculty of Sciences and Letters

Department of Mathematics Istanbul, Turkey fozen@itu.edu.tr