The Small Deformation Strain Tensor as a Fundamental Metric Tensor

In the general theory of relativity, the fundamental metric tensor plays a special role, which has its physical basis in the peculiar aspects of gravitation. The fundamental property of gravitational fields provides the possibility of establishing an analogy between the motion in a gravitational field and the motion in any external field considered as a noninertial system of reference. Thus, the properties of the motion in a noninertial frame are the same as those in an inertial system in the presence of a gravitational field. In other words, a noninertial frame of reference is equivalent to a certain gravitational field. This is known as the principle of equivalence . From the mathematical viewpoint, the same special role can be played by the small deformation strain tensor, which describes the geometrical properties of any region deformed because of the effect of some external agent. It can be proved that, from that tensor, all the mathematical structures needed in the general theory of relativity can be constructed.


Introduction
Within the theoretical frame of classical fluid dynamics, the effect of applied forces to any continuous medium is studied. Under the action of applied forces, the region occupied by the continuous medium exhibits deformations to some extent, that is to say, the region changes in shape and volume. Those deformations can be described mathematically by the small deformation strain tensor. When the deformation is the result of a process of hydrostatic or volumetric compression or expansion, the small deformation strain tensor is reduced to the sum of the elements of its principal diagonal, that is, its trace. This trace is likewise in this case, the fractional change of the volume element of the region occupied by the continuous medium. Thus, when the deformations are small, the trace of the small deformation strain tensor is nearly equal to the reciprocal of the mass density. Then, that characteristic property of the matter is contained in that tensor [1].

The Small Deformation Strain Tensor and the Fundamental Metric Tensor
Consider an ordered set of N real variables 1 2 , , , N x x x . These variables are called the coordinates of a point. Thus, all the points corresponding to all values of the coordinates are said to form an N-dimensional space. Let R be any region in that space, and let us consider two points very close together. If over the boundary surface of R an external force is applied, the geometry of the region changes inform and size; that is to say, it is deformed. In order to mathematically describe the deformation, the procedure is as follows. Be dx i the i-component of the radius vector joining the points before the deformation, and d d d  [1]. The distances between the points before and after deformation respectively where the following expansion was used Since the summation is taken over both suffixes i and k, the second term on the right of (2) can be written as In the third term on the right of (2), the surffixes i and m can be interchanged, in order to finally obtain that are the components of the strain tensor. From its definition, it is clear that it is a symmetrical tensor, that is to say For small deformations it is possible to neglect the last term in (2) and write that are the components of the small deformation strain tensor [1]. On the other hand, after deformation the distance between the near by points, can be written as follows are the components of the fundamental metric tensor, and the summation convention was used. Since further, is a simmetrical covariant tensor of the second rank [2]. Now, if instead considering that the points are separate we make them coincide in the undeformed initial situation, it is clear that In that case, in (7) it is obtainedw that If Equations (9) and (11) are compared we have that that is to say, Equation (12) must be considered as a relation of congruence between physics and geometry more than equality. Since further ik ki u u = is a simmetrical covariant tensor of the second rank, also; and it can be considered that it is equivalent to the fundamental metric tensor, apart from the unessential factor 1/2. That means that both tensors have the same properties [3]. Now, if in the determinant formed by the elements u ik , and taking into account the co-factor of each of the u ik and divide by the determinant u, certain quantities ik ki u u = are obtained, as we shall demonstrate soon, form a contravariant tensor. In fact, by a well known property of determinants, it can be obtained that Since this, with the arbitrary choice of the vector dη n , ds 2 is a scalar, and u nr by its definition is symmetrical, it follows that u nr is a contravariant tensor. It further follows from (14) that k i δ is also a tensor, which we will call the miked fundamental tensor [4]. Now, by the rule for the multiplication of determinants On the other hand in such a way that Besides, from Equation (12) it is obtained that Now, and given that Finally, if ( ) In that case, In the general theory of relativity, it is used to write g − inestead of g , with g the determinant of g ik , quantity which is always real; because of the hyperbolic character of the space-time continuum [3]. Due to the same past arguments, it is possible to propose the use of u − instead of u . This is so, because really, for all coordinates connected with a real space-time, the determinant g, and also, the determinant u, are negative [5].

The Christoffel Symbols
A curve in space is defined as the locus of a point whose coordinates depend on a single parameter [2]. Consider a given curve and let us suppose that the coordinates of any point on it, are functions of the parameter t. If we take any vector at a given point of the curve and at every other point on it, take the vector equal to it in magnitude and parallel to it in direction, we obtain a vector Χ defined at each point of the curve, and the components of X, will be functions of t. In other words, we have a parallel field of vectors along the given curve [2]. Our objective is to find the differential equations which such a vector-field must satisfy. In order to do so, it is necessary to consider Cartesian coordinates y r . Let Y r be the components of the vector-field in this coordinate system [2]. Since the components of parallel vectors are equal in Cartesian Systems, it is easy to see that the Y r are all constants along the curve and consequently the derivatives of Y r with respect to t are zero [2]. Now, it is clear that Therefore, differentiating with respect to t we have that dY i /dt = 0, and then, it is fulfilled that Now, let us consider the following transformation If we multiply Equations (16) by the last transformation, the relationships (8) and (10) are used, and sum i from 1 to 3, it is obtained that [2].
where the relationship (8) was used, and clearly, the factor 1/2 was sup-pressed. Next, let us consider the expression and referring again the relationship (8), we get, on differentiating partially with respect to x r that These equations are true when m, n, p take any of the values 1, 2, 3. If now we take any of the two equations obtained by permuting m, n, p cyclically in (18), and substract (18) from their sum, we obtain and the parallel vector-field along the given curve must satisfy this differential equation [2]. The quantitites [m n, p] and {m n, r}, defined by (20) and (21), in terms of the components of the small deformation tensor, we will call the Christoffel symbols of the first and second kinds, or they are sometimes referred as the three-index symbols [2] [4]. It is seen at once that they are symmetrical in m, n; an important property [2].

Equations of a Geodesic
In order to obtain the equations of a geodesic or path between two points in the Riemannian space, we will use the calculus of variations, and the following condition [ The stationary condition is in such a way that in (23) we get where we will use greek index instead of latin index, and for convenience, a factor 1/2 has been eliminated Changing dummy suffixes in the last two terms, it is obtained that Consider the terms enclosed in the round parenthesis. Applying the method of partial integration, we get The first term is an exact differential. It is zero because the δx σ varishes at both limits of the integral. Hence, it is obtained that This must hold for all values of the arbitrary displacements δx σ at all points, hence the coefficient in the integrand must vanish at all points on the path [ Also, in the last two terms we replace the dummy suffixes μ and υ by ε. The equation then becomes We can get rid of the factor u εσ by multiplying through by u σα in such a way that However, is one of Christoffel's 3-index symbols. Finally, in (26) we obtain that This is the looked for differential equation. For α = 1, 2, 3, 4 that relationship gives the four equations determining a geodesic [4].

Covariant Derivative of a Vector
Since dx μ is a contravariant vector, and ds an invariant, dx μ /ds a kind of velocity, is a contravariant vector. Hence if A μ is any covariant vector, the inner product The rate of change of this expression per unit interval ds along any assigned curve must also be independent of the coordinate system; that is to say This assumes that we keep to the same absolute curve however the coordinate system is varied. The result (28) is therefore only of practical use if it is applied to a curve which is defined independently of the coordinate system; and then, we shall apply it to a geodesic. Performing the differentiation, we get that The result is now general since the curvature, which distinguishes the geodesic, has been eliminated by using Equations (27), and only the gradient of the curve, dx μ /ds and dx υ /ds, has been left in the expression.
Since dx μ /ds and dx υ /ds are contravariant vectors, their co-factor is a covariant tensor of second rank. We therefor write and the tensor A μυ is called the covariant derivative of A μ . By raising a suffix we obtain two associated tensors A µ υ and A υ µ which must be distinguished since the two suffixes are not symmetrical. The first of these is the most important, and is to be understood when the tensor is written simply as A µ υ without distinction of its original position [4]. Since

Covariant Derivative of a Tensor
The covariant derivatives of tensors of the second rank are formed as follows [4] Thus, the general rule for covariant differentiation with respect to x σ is ilustrated by the next example [4].
The above formula is primarly definitions. We have to prove that the quantities on the right are actually tensors. This is done by a generalization of the method of the preceding section. Thus if in place of (28) we use the following expression .