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Classification of parallel tensors and its application in general relativity.

1. INTRODUCTION

We critically examine here the basic properties of theory of gravitation Noun 1. theory of gravitation - (physics) the theory that any two particles of matter attract one another with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them  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 general relativity
n.
The geometric theory of gravitation developed by Albert Einstein, incorporating and extending the theory of special relativity to accelerated frames of reference and introducing the principle that gravitational and inertial forces
 which co-exist for both gravitational grav·i·ta·tion  
n.
1. Physics
a. The natural phenomenon of attraction between physical objects with mass or energy.

b. The act or process of moving under the influence of this attraction.

2.
 and electromagnetic waves. We show here that any parallel tensor tensor, in mathematics, quantity that depends linearly on several vector variables and that varies covariantly with respect to some variables and contravariantly with respect to others when the coordinate axes are rotated (see Cartesian coordinates).  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 ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. .] where [[nabla].sub.i] denotes covariant derivative In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with .

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 The term curvature tensor is ambiguous in its generality. It could refer to:
  • the Riemann curvature tensor of a Riemannian manifold — see also Curvature of Riemannian manifolds;
  • the curvature of an affine connection or covariant derivative (on tensors);
 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 (programming) invariant - A rule, such as the ordering of an ordered list or heap, that applies throughout the life of a data structure or procedure. Each change to the data structure must maintain the correctness of the 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
For null vectors as used in special relativity, see Minkowski space#Causal structure.


In linear algebra, the null vector or zero vector is the vector (0, 0, …, 0) in Euclidean space, all of whose components are zero.
 [[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 generalize /gen·er·al·ize/ (-iz)
1. to spread throughout the body, as when local disease becomes systemic.

2. to form a general principle; to reason inductively.
 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 tensor analysis

Branch of mathematics concerned with relations or laws that remain valid regardless of the coordinate system used to specify the quantities. Tensors, invented as an extension of vectors, are essential to the study of manifolds.
 and Nonlinear function. Kluer Academic Publishers, Dordrecht.

[3.] Ebin. 1970. The mainifold of Reimannian matrices. Proc. Symp. Pure Math., AMS AMS - Andrew Message System  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 calculus, branch of mathematics that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limit—the notion of tending toward, or approaching, an ultimate value.  and differential geometry differential geometry, branch of geometry in which the concepts of the calculus are applied to curves, surfaces, and other geometric entities. The approach in classical differential geometry involves the use of coordinate geometry (see analytic geometry; Cartesian .

[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
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Author:Sinha, Nalin; Sinha, Tarun Kumar
Publication:Bulletin of Pure & Applied Sciences-Physics
Date:Jan 1, 2006
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