# Classification of parallel tensors and its application in general relativity.

1. INTRODUCTION

We critically examine here the basic properties of theory of gravitation and role of parallel tensors in obtaining exact solution of Einstein field equation. The study of the parallel tensors (Gerretsen 1962) has wide applications in solving the field equations in general relativity which co-exist for both gravitational and electromagnetic waves. We show here that any parallel tensor of order ([less than or equal to]3) are expressible in terms of ([g.sub.ij], [[eta].sub.ijlm]) with suitable application of outer products and law of contraction of tensors (Gerretsen, 1962).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] where [[nabla].sub.i] denotes covariant derivative.

The non-vanishing independent components of the christoffel symbol [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Where (i,j, k =1,......4) and [f.sub.x] = [delta]f/[delta]x, [f.sub.y] = [delta]f/[delta]y, [delta]f/[delta]y = f

And curvature tensor in O-system are as follows :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

2. CLASSIFICATION OF PARALLEL TENSORS

We show here that classification of H, Ha and Hb are invariant in the co-ordinate system in which [ds.dup.2] = -[dl.sup.2] - [dz.sup.2] + [dt.sup.2] ([dl.sup.2] = [Adx.sup.2] + 2Ddxdy + [Bdy.sup.2])

where A, B and D are functions of the single variable Z = z - t being known as O'-System.

Proof :

We derive the solution of the problem of the parallel tensors in the O'-System.

We introduce here an orthogonal ennuple [H.sub.2] [h.sup.a.sub.j] (a = 1,4) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Where m = AB - [D.sup.2]. Then the non-vanishing independent components of the coefficients of rotation are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

The equations to be solved are as follows :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The most remarkable properties of the O-System is that the actual form of the parallel null vector [[lambda].sub.i] is (0,0,-1,1), which is of the same form as that O-System.

In particular, when D = 0 hold the expression may be reduced to simpler form. In this case, we have

[[gamma].sub.131] = i[[gamma].sub.141] = i [bar.A]/2A, [[gamma].sub.232] = i[[gamma].sub.242] = i[bar.B]/2B (3)

in place of (2) and the non-vanishing independent components of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We cannot express the condition D = 0 in an invariant form.The condition that the H (whose D= 0) be an [H.sub.1] is [bar.[alpha]][bar.[beta]] [not equal to] 0 and the H be an [H.sub.2] is ([bar.[alpha]] [not equal to] 0, [bar.[beta]] [not equal to] 0) or ([bar.[alpha]] [not equal to] 0, [bar.[beta]] [not equal to] 0. We deal with an [H.sub.2] if and only if only if [bar.[alpha]] = [bar.[beta]] = 0, H is flat. We denote by [H.sup.*.sub.1] and [H.sup.*.sub.2] and the [H.sub.1] and [H.sub.2] satisfying D [not equal to] 0 respectfully. We write the results for n = 1. The parallel vector in [H.sup.*.sub.1] is given by C[[lambda].sub.1], while that is [H.sup.*.sub.2] by, [[gamma].sub.ijkl] [[lambda].sup.-l] = [[mu].sub.i][V.sub.jk] + [[mu].sub.j][V.sub.ki] + [[mu].sub.k][V.sub.ij] [less than or equal to]2 [C.sub.1][[[gamma].sub.i] + [C.sub.2][[mu].sub.i], where C's are arbitrary constants and [[mu].sub.i] = (0,-[square root of (B)], -[beta] - [beta][gamma], [beta][gamma] and we have [[mu.sub.i][[mu].sup.i] = -1. We find that [H.sup.*.sub.2] belongs to [H.sub.b].

PROPOSITION : Properties of Semi-parallel Tensors.

We consider the case of n = 2. Let H be an. Thus, in the O'-System under restricted consideration. We take in (1) as follows:

[V.sub.ij] : [V.sub.13] = -[V.sub.14] = -[V.sub.31] = [V.sub.41] = [square root of (A)], Other [V.sub.ij] = O. and hence [V.sup.*.sub.ij] : [V.sup.*.sub.23] = -[V.sup.*.sub.24] = -[V.sup.*.sub.32] = [V.sup.*.sub.42] = [square root of (B)], Other [V.sup.*.sub.ij] = O

Here we add, Concerning these [V.sub.ij] and [V.sup.*.sub.ij],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then we obtain the following expressions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We obtain [[nabla].sub.k][v.sub.ij] = [[nabla].sub.k][v.sup.*.sub.ij] = 0. If we consider the case of [H.sup.*.sub.2], it follows that [v.sub.ij] and [v.sup.*.sub.ij] are parallel again.

In this case, we find that the [v.sub.ij] given by (1) is equivalent but the tensor [[eta].sub.ijml][[lambda].sup.i][[mu].sup.m] where [[mu].sup.m] is the contravariant components of the parallel vector [[mu].sub.i] = (0, [square root of (B)], -[beta][gamma], [beta][gamma]).

We now consider the case when n = 3. In the same way we may determine the general form of the parallel tensor [V.sub.ijk] for [H.sup.*.sub.1] and [H.sup.*.sub.2]. The equation to be solved in this case is composed of [4.sup.4] = 256 equations.

1.3 PROPOSITION :

In [H.sup.*.sub.1], the general form of [V.sub.ij] is given by [([g.sub.ij], [[eta].sub.ijml], [[lambda].sub.i][V.sub.ij]).sub.3] where [v.sub.ij] and [v.sup.*.sub.ij] are those in (2) and (1).

PROOF:

In actual form of the [V.sub.ij] is a linear combination with constant co-defficients of [g.sub.ijkl(U)][[lambda].sub.i] [[lambda].sub.j][[lambda].sub.k], [V.sub.ij] [[lambda].sub.k](U) and [[eta].sub.ijkl][[lambda].sup.l]. The symbol [g.sub.ij][[lambda].sub.k] (U) means that three terms obtained by the cyclic changes of the indices

i.e. [g.sub.ij][[lambda].sub.k] [g.sub.ij][[lambda].sub.i], [g.sub.ki] [[lambda].sub.j] are equivalent. However we show that three are some significant identities connecting the above quantities given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

In [H.sup.*.sub.2] the general form of [V.sub.ijk] is a linear combination with constant co-efficient [[lambda].sub.i] [[lambda].sub.j][[lambda].sub.k], [[mu].sub.i] [[mu].sub.j][[mu].sub.k], [g.sub.ij] [[lambda].sub.k] (U), [[lambda].sub.i] [[mu].sub.j][[mu].sub.k] (U) [[lambda].sub.i] [[lambda].sub.j][[mu].sub.k] (U), [[lambda].sub.i] [[lambda].sub.j][V.sub.jk] (U), of [[mu].sub.i] [V.sub.jk](U),[[lambda].sub.i] [[V.sup.*.sub.j].sub.k] (U), [[mu].sub.i] [[V.sup.*.sub.j].sub.k](U) and [[eta].sub.ijkl][mu] where. Again we have may identities. For example,

We have

[[gamma].sub.ijkl][[lambda].sup.-l] = [[mu].sub.i][V.sub.jk] + [[mu].sub.j][V.sub.ki] + [[mu].sub.k][V.sub.ij] (4)

We consider here the identifies (3) and (4). From (2) we find that [V.sub.ijk] may be expressed in terms of the parallel vector, and parallel tensors of the 2nd Order, while (4) shows that can be expressed in terms of are parallel vectors. In both case, we do not have new tensor of the 3rd order which cannot be expressed in terms of and the parallel tensors of order [less than or equal to]2.

CONCLUSION

The general form of the parallel tensor in H is determined completely for [less than or equal to]2. But the form for n = 3 is given only for some restricted types of H.

We have generalize these results in [H.sub.2] by assuming that all of [g.sub.ij][V.sub.i] and [V.sub.ij] are real. This assumption is most natural and proper. But [H.sub.2lb] satisfying l + [s.sup.2] = 0 has very interesting properties. We denote such as [H.sub.2lb] by [H.sub.i]. Hence the line elements of [H.sub.i] is not real.

REFERENCES

[1.] Christopher ,Kohler. 2000. Semi parallel theories of gravitation. Vol. 32 (7) :1301-1317

[2.] Dimitrienko, Yu I. 2002.Tensor Analysis and Nonlinear function. Kluer Academic Publishers, Dordrecht.

[3.] Ebin. 1970. The mainifold of Reimannian matrices. Proc. Symp. Pure Math., AMS Vol. XVP 11-40.

[4.] Fischer, A.E. Marsten. 1972. The Einstein equations of evolution--A Geometric approach "Journ. Math. Phys. 13 :568.

[5.] Gerretsen, J.H.C. 1962.Tensor Calculus and differential geometry.

[6.] Noordhoff. N.V. 1962. Hakim, Remi. 1999. An introduction to Relativities gravitation, Combridge University Press.

[7.] James B. Hartle. 2003.An introduction to Einsteins General Relativity, Addison Westey. San Fransis.

[8.] Zahar,E.1980. Einstein, Meyerson and the Role of Mathematical Physical Discovery, British Jouunal of Philosophy of Science, 31:1-43.

Nalin Sinha and Tarun Kumar Sinha *

Department of Physics, K.S.R.College, Sarairanjan, Samastipur-(Bihar) 848101-India

We critically examine here the basic properties of theory of gravitation and role of parallel tensors in obtaining exact solution of Einstein field equation. The study of the parallel tensors (Gerretsen 1962) has wide applications in solving the field equations in general relativity which co-exist for both gravitational and electromagnetic waves. We show here that any parallel tensor of order ([less than or equal to]3) are expressible in terms of ([g.sub.ij], [[eta].sub.ijlm]) with suitable application of outer products and law of contraction of tensors (Gerretsen, 1962).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] where [[nabla].sub.i] denotes covariant derivative.

The non-vanishing independent components of the christoffel symbol [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Where (i,j, k =1,......4) and [f.sub.x] = [delta]f/[delta]x, [f.sub.y] = [delta]f/[delta]y, [delta]f/[delta]y = f

And curvature tensor in O-system are as follows :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

2. CLASSIFICATION OF PARALLEL TENSORS

We show here that classification of H, Ha and Hb are invariant in the co-ordinate system in which [ds.dup.2] = -[dl.sup.2] - [dz.sup.2] + [dt.sup.2] ([dl.sup.2] = [Adx.sup.2] + 2Ddxdy + [Bdy.sup.2])

where A, B and D are functions of the single variable Z = z - t being known as O'-System.

Proof :

We derive the solution of the problem of the parallel tensors in the O'-System.

We introduce here an orthogonal ennuple [H.sub.2] [h.sup.a.sub.j] (a = 1,4) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Where m = AB - [D.sup.2]. Then the non-vanishing independent components of the coefficients of rotation are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

The equations to be solved are as follows :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The most remarkable properties of the O-System is that the actual form of the parallel null vector [[lambda].sub.i] is (0,0,-1,1), which is of the same form as that O-System.

In particular, when D = 0 hold the expression may be reduced to simpler form. In this case, we have

[[gamma].sub.131] = i[[gamma].sub.141] = i [bar.A]/2A, [[gamma].sub.232] = i[[gamma].sub.242] = i[bar.B]/2B (3)

in place of (2) and the non-vanishing independent components of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We cannot express the condition D = 0 in an invariant form.The condition that the H (whose D= 0) be an [H.sub.1] is [bar.[alpha]][bar.[beta]] [not equal to] 0 and the H be an [H.sub.2] is ([bar.[alpha]] [not equal to] 0, [bar.[beta]] [not equal to] 0) or ([bar.[alpha]] [not equal to] 0, [bar.[beta]] [not equal to] 0. We deal with an [H.sub.2] if and only if only if [bar.[alpha]] = [bar.[beta]] = 0, H is flat. We denote by [H.sup.*.sub.1] and [H.sup.*.sub.2] and the [H.sub.1] and [H.sub.2] satisfying D [not equal to] 0 respectfully. We write the results for n = 1. The parallel vector in [H.sup.*.sub.1] is given by C[[lambda].sub.1], while that is [H.sup.*.sub.2] by, [[gamma].sub.ijkl] [[lambda].sup.-l] = [[mu].sub.i][V.sub.jk] + [[mu].sub.j][V.sub.ki] + [[mu].sub.k][V.sub.ij] [less than or equal to]2 [C.sub.1][[[gamma].sub.i] + [C.sub.2][[mu].sub.i], where C's are arbitrary constants and [[mu].sub.i] = (0,-[square root of (B)], -[beta] - [beta][gamma], [beta][gamma] and we have [[mu.sub.i][[mu].sup.i] = -1. We find that [H.sup.*.sub.2] belongs to [H.sub.b].

PROPOSITION : Properties of Semi-parallel Tensors.

We consider the case of n = 2. Let H be an. Thus, in the O'-System under restricted consideration. We take in (1) as follows:

[V.sub.ij] : [V.sub.13] = -[V.sub.14] = -[V.sub.31] = [V.sub.41] = [square root of (A)], Other [V.sub.ij] = O. and hence [V.sup.*.sub.ij] : [V.sup.*.sub.23] = -[V.sup.*.sub.24] = -[V.sup.*.sub.32] = [V.sup.*.sub.42] = [square root of (B)], Other [V.sup.*.sub.ij] = O

Here we add, Concerning these [V.sub.ij] and [V.sup.*.sub.ij],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then we obtain the following expressions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We obtain [[nabla].sub.k][v.sub.ij] = [[nabla].sub.k][v.sup.*.sub.ij] = 0. If we consider the case of [H.sup.*.sub.2], it follows that [v.sub.ij] and [v.sup.*.sub.ij] are parallel again.

In this case, we find that the [v.sub.ij] given by (1) is equivalent but the tensor [[eta].sub.ijml][[lambda].sup.i][[mu].sup.m] where [[mu].sup.m] is the contravariant components of the parallel vector [[mu].sub.i] = (0, [square root of (B)], -[beta][gamma], [beta][gamma]).

We now consider the case when n = 3. In the same way we may determine the general form of the parallel tensor [V.sub.ijk] for [H.sup.*.sub.1] and [H.sup.*.sub.2]. The equation to be solved in this case is composed of [4.sup.4] = 256 equations.

1.3 PROPOSITION :

In [H.sup.*.sub.1], the general form of [V.sub.ij] is given by [([g.sub.ij], [[eta].sub.ijml], [[lambda].sub.i][V.sub.ij]).sub.3] where [v.sub.ij] and [v.sup.*.sub.ij] are those in (2) and (1).

PROOF:

In actual form of the [V.sub.ij] is a linear combination with constant co-defficients of [g.sub.ijkl(U)][[lambda].sub.i] [[lambda].sub.j][[lambda].sub.k], [V.sub.ij] [[lambda].sub.k](U) and [[eta].sub.ijkl][[lambda].sup.l]. The symbol [g.sub.ij][[lambda].sub.k] (U) means that three terms obtained by the cyclic changes of the indices

i.e. [g.sub.ij][[lambda].sub.k] [g.sub.ij][[lambda].sub.i], [g.sub.ki] [[lambda].sub.j] are equivalent. However we show that three are some significant identities connecting the above quantities given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

In [H.sup.*.sub.2] the general form of [V.sub.ijk] is a linear combination with constant co-efficient [[lambda].sub.i] [[lambda].sub.j][[lambda].sub.k], [[mu].sub.i] [[mu].sub.j][[mu].sub.k], [g.sub.ij] [[lambda].sub.k] (U), [[lambda].sub.i] [[mu].sub.j][[mu].sub.k] (U) [[lambda].sub.i] [[lambda].sub.j][[mu].sub.k] (U), [[lambda].sub.i] [[lambda].sub.j][V.sub.jk] (U), of [[mu].sub.i] [V.sub.jk](U),[[lambda].sub.i] [[V.sup.*.sub.j].sub.k] (U), [[mu].sub.i] [[V.sup.*.sub.j].sub.k](U) and [[eta].sub.ijkl][mu] where. Again we have may identities. For example,

We have

[[gamma].sub.ijkl][[lambda].sup.-l] = [[mu].sub.i][V.sub.jk] + [[mu].sub.j][V.sub.ki] + [[mu].sub.k][V.sub.ij] (4)

We consider here the identifies (3) and (4). From (2) we find that [V.sub.ijk] may be expressed in terms of the parallel vector, and parallel tensors of the 2nd Order, while (4) shows that can be expressed in terms of are parallel vectors. In both case, we do not have new tensor of the 3rd order which cannot be expressed in terms of and the parallel tensors of order [less than or equal to]2.

CONCLUSION

The general form of the parallel tensor in H is determined completely for [less than or equal to]2. But the form for n = 3 is given only for some restricted types of H.

We have generalize these results in [H.sub.2] by assuming that all of [g.sub.ij][V.sub.i] and [V.sub.ij] are real. This assumption is most natural and proper. But [H.sub.2lb] satisfying l + [s.sup.2] = 0 has very interesting properties. We denote such as [H.sub.2lb] by [H.sub.i]. Hence the line elements of [H.sub.i] is not real.

REFERENCES

[1.] Christopher ,Kohler. 2000. Semi parallel theories of gravitation. Vol. 32 (7) :1301-1317

[2.] Dimitrienko, Yu I. 2002.Tensor Analysis and Nonlinear function. Kluer Academic Publishers, Dordrecht.

[3.] Ebin. 1970. The mainifold of Reimannian matrices. Proc. Symp. Pure Math., AMS Vol. XVP 11-40.

[4.] Fischer, A.E. Marsten. 1972. The Einstein equations of evolution--A Geometric approach "Journ. Math. Phys. 13 :568.

[5.] Gerretsen, J.H.C. 1962.Tensor Calculus and differential geometry.

[6.] Noordhoff. N.V. 1962. Hakim, Remi. 1999. An introduction to Relativities gravitation, Combridge University Press.

[7.] James B. Hartle. 2003.An introduction to Einsteins General Relativity, Addison Westey. San Fransis.

[8.] Zahar,E.1980. Einstein, Meyerson and the Role of Mathematical Physical Discovery, British Jouunal of Philosophy of Science, 31:1-43.

Nalin Sinha and Tarun Kumar Sinha *

Department of Physics, K.S.R.College, Sarairanjan, Samastipur-(Bihar) 848101-India