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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.

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 [

Consider an ordered set of N real variables_{i} the i-component of the radius vector joining the points before the deformation, and_{i} [

so that,

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

where

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

where

are the components of the small deformation strain tensor [

where

are the components of the fundamental metric tensor, and the summation convention was used. Since further,

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

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

where

On the other hand, if instead of the expression for

form a covariant four-vector. In that case,

Since this, with the arbitrary choice of the vector dh_{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

On the other hand

in such a way that

Besides, from Equation (12) it is obtained that

Now, and given that

Finally, if

it is fulfilled that

In that case,

In the general theory of relativity, it is used to write _{ik}, quantity which is always real; because of the hyperbolic character of the space-time continuum [

A curve in space is defined as the locus of a point whose coordinates depend on a single parameter [^{r}. Let Y^{r} be the components of the vector-field in this coordinate system [^{r} are all constants along the curve and consequently the derivatives of Y^{r} with respect to t are zero [

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 [

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

Hence if we write by brevity

and substitute this in (19), it is obtained the following result

If on the other hand, we write

it is obtained Equation (16) in the final form

and the parallel vector-field along the given curve must satisfy this differential equation [

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 [

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 [

This absolute track is of fundamental importance in dynamics. Keeping the beginning and the end o the path fixed, we give every intermediate point an arbitrary infinitesimal displacement δx_{σ} so as to deform the path. According to definition (11), it has that

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 [

now, it is clear that [

and

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 [

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

is invariant along a geodesic.

Now, if Equation (27) is used, we have that along a geodesic

With this result in (29) 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

we have by (30)

due to

So that

because [

Hence multiplaying through by u^{μυ} and given that u^{μσ}u_{σε} is a substitution-operator, we have

This is called the covariant derivative of A^{μ}. The tensors ^{μυ} are called the contravariant derivative of A_{μ} and A^{μ}.

The covariant derivatives of tensors of the second rank are formed as follows [

Thus, the general rule for covariant differentiation with respect to x_{σ} is ilustrated by the next example [

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

as invariant along a geodesic, we obtain

Then substituting for the second derivatives from (27) the expression reduces to

showing that A_{μυσ} is a tensor.

Applying (34) to the fundamental tensor, we have

due to the fact that

Hence, the covariant derivatives of the fundamental tensors vanish identically and the fundamental tensors can be treated as constants in covariant differentiation. The utility of the covariant derivative arises from the fact that, when the u_{μυ} are constants, the Christoffel symbols vanish and the covariant derivative reduces to the ordinary derivative. Now, in general, the physical equations have been stated for the case of Galilean coordinates in which the u_{μυ} are constants; and we may in Galilean equations replace the ordinary derivative by the covariant derivative without altering anything. This is a necessary step in reducing such equations to the general tensor for which holds true for all coordinates systems [

The second covariant derivative of A_{μ} is found by inserting in (34) the value of A_{μυ} given in (30). That is to say

The first five terms are unaltered when υ and α are interchangel. The last two terms may be written, by changing the dummy suffix α to ε in the last term, in such a way that we get

Hence

The quotient theorem shows that the co-factor of A_{ε} must be a tensor; so that, it is fulfilled that [

where

This is called the Riemann-Christoffel Tensor. It is only when this tensor vanishes that the order of covariant differentitation is permutable [

The suffix ε may be lowered. Thus

where ε has been replaced by α in the last two terms. Hence,

where the relationships (36) and (20) has been used.

It will be seen from (42) that B_{μυσρ}, beside being antisymmetrical in υ and σ, is also antisymmetrical in μ and ρ. Also it is symmetrical for the double interchange μ and υ, ρ and σ.

It has the further cyclic property [

as is easily verified from (42). The Riemann-Christoffel tensor has 20 independent components [_{μ}_{ }_{< υ}, and also from the g_{μυ}, and therefore belongs to the class of fundamental tensors [

In order to get the physical significance of the fundamental tensor fields u_{μυ} and g_{μυ}, let us consider a region of space-time in which the gravitational field vanishes. If we introduce a non-inertial coordinate system, free bodies will be accelerated with respect to the chosen coordinate system, although they move along straight world lines [_{a}/ds = 0, or dw_{a} = 0; where w_{a} = dx_{a}/ds is the four-velocity. Clearly, in curvilinear coordinates we have that

Dividing this equation by ds we get

This is the required equation of motion. But, this is the same relationship (27) of a geodesic. We see that the motion of a particle in a gravitational field is determined by the quantities {μυ, α}. The derivative

the four-force, acting on the particle in the gravitational field. Here, the tensor u_{μυ} plays the role of the potential of the gravitational field; in such a way that, its derivatives determine the field intensity {μυ, α} [

That law can also be obtained from the fundamental tensors u_{μν} and B_{μνσρ}. The later has been expressed in terms of the former, and its first and second derivatives. Thus, the contracted Riemann-Christoffel tensor, denoted by D_{μν}, is formed setting ε = σ in the relationship (41). That is to say

The 3-index symbols containing a duplicated suffix can be simplified by means of the expression (A.7). Hence, with some changes of dummy suffixes, it has that

The expression

in empty space, is chosen as the law of gravitation in a Dynamic Theory of Gravitation.

Through the present paper, it was possible to demonstrate that the small deformation strain tensor could be used as a fundamental metric tensor, instead of the usual fundamental metric tensor. Also, it was possible to prove that from that tensor, not only other mathematical structures could be constructed, but also another fundamental tensor was obtained; that was to say, we had constructed two of them, u_{μυ}, and B_{μυσρ}. It is through these tensors that the gap between pure geometry and physics is bridged. In particular, u_{μυ} relates the observed interval ds to the mathematical coordinate specification dx_{μ}. Also, the u_{μυ} appear as the potentials of the inertial field [_{μυ} is again the potential which determines the accelerations of free bodies; in other words, the u_{μυ} is the potential of the gravitational field. Thus, a stage has been reached at which the results obtained can be applied to the theory of gravitation [

Angel FierrosPalacios, (2015) The Small Deformation Strain Tensor as a Fundamental Metric Tensor. Journal of High Energy Physics, Gravitation and Cosmology,01,35-47. doi:

Since

where

it has that

Hence

In a similar way,

Now, and multiplying by a contravariant tensor

For any tensor C_{a}_{b} other than the fundamental small deformation strain tensor, the corresponding formula would be the following

by the expressions

The exception for u_{α}_{b} arises because a change du_{α}_{b} has an additional indirect effect through alterning the operation of raising and lowering suffixes.

On the other hand, du is formed taking the differential of each u_{µv} and multiplying by its co-factor u・u^{µv} in the determinant. So that

in such a way that

In that case, the contracted 3-index Christoffel symbol becomes

The other two terms cancel each other by interchange of the dummy suffixes σ and l. Hence, by (A. 6) it has that

this is so, because for real coordinates u is always negative.