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The importance of Einstein’s geometrization philosophy, as an alternative to the least action principle, in constructing general relativity (GR), is illuminated. The role of differential identities in this philosophy is clarified. The use of Bianchi identity to write the field equations of GR is shown. Another similar identity in the absolute parallelism geometry is given. A more general differential identity in the parameterized absolute parallelism geometry is derived. Comparison and interrelationships between the above mentioned identities and their role in constructing field theories are discussed.

In the second decade of the twentieth century, Einstein constructed a successful theory for gravity, the general theory of relativity (GR) [

In the third decade of the twentieth century, Einstein [

In the sixties of the past century, Dolan and McCrea (1963, private communication to the first author in 1973) have developed a variational method to get fundamental identities in Riemannian geometry, without using an action principle. Unfortunately, this method is not published although it is shown to be of importance when applied in geometries other than the Riemannian one.

In the seventies of the twentieth century, two important results have been obtained by developing AP-geometry in a certain direction [

It is to be noted that identity (

In the last decade of the twentieth century, another important result has been obtained as a consequence of another development in AP-geometry. It has been discovered that AP-geometry admits a hidden parameter [

In the past ten years or so, AP-geometry has gained a lot of attention in constructing field equations for what is called

It is the aim of the present work to find out differential identities in PAP-geometry and to study their relations to similar identities in AP-geometry (

In the present section, we are going to give basic mathematical machinery and formulae of PAP-geometry necessary for the present work. For more details, the reader is referred to [

As mentioned in Section

An AP-space is a pair

It is to be noted that all these linear connections have nonvanishing curvature except for (

The parameter

In the case of

In the case of

Between the limits

In what follows, we will decorate the tensors containing the parameter

Now, since the parameterized canonical connection

It is of importance to note that the BB of PAP-geometry are the same as those of AP-geometry. In other words, the parameter

We are going to apply the Dolan-McCrea variational method to obtain differential identities in PAP-geometry. This method is an alternative to the least action method. It was originally suggested in 1963 to derive Bianchi and other identities in the context of Riemannian geometry. It has been generalized [

Since, as stated above, the BB of PAP-geometry are the same as those of AP-geometry, let us define the Lagrangian function

Now, we express

Using the vector transformation law:

Using Taylor expansion:

Now, we treat the Lagrangian density in the two different ways mentioned above.

(a) Using Taylor expansion,

The Lagrangian density

(b) Using scalar transformation,

By (

So, (

Now, the integral of the first term can be treated as follows:

The three differential identities, (

If

Using the definition of the parameterized dual connection

Finally, it is clear that if

If

The above two theorems imply the following result.

(a) For a symmetric tensor

In the present work, we have used the Dolan-McCrea variational method to derive possible differential identities in PAP-geometry. The importance of this work for physical applications can be discussed in the following points:

It is well known that the field equations of GR can be obtained using either one of the following approaches:

The Einstein approach, in which the geometrization philosophy plays the main role in constructing the field equations of the theory. One of the principles of this philosophy is that “

Hilbert approach, in which the standard method of theoretical physics, the action principle, has been used to derive the field equations of the theory.

It is to be noted that the first approach is capable of constructing a complete theory, not only the field equations of the theory. This will be discussed in the following point.

Einstein geometrization philosophy can be summarized as follows [

“To understand

In applying this philosophy, one has to consider its main principles:

There is a one-to-one correspondence between geometric objects and physical quantities.

Curves (paths) in the chosen geometry are trajectories of test particles.

Differential identities represent laws of nature.

In view of the above principles, Einstein has used Riemannian geometry to construct a full theory for gravity, GR, in which the metric and the curvature tensors represent the gravitational potential and strength, respectively. He has also used the geodesic equations to represent motion in gravitational fields. Finally, he has used Bianchi identity to write the field equations of GR.

The above mentioned philosophy can be applied to any geometric structure other than the Riemannian one. It has been applied successfully in the context of conventional AP-geometry (cf. [

PAP-geometry is more general than both Riemannian and conventional AP-geometry. It is shown in Section

To use PAP-geometry as a medium for constructing field theories, the above three principles are to be considered. The curves (paths) of the geometry have been derived [

The present work is done as a step to complete the use of the geometrization philosophy in PAP-geometry. The general form of the differential identity characterizing this geometry is obtained in Section

The results obtained from the two theorems given in Section

Now, for any symmetric tensor defined by (

Due to the structure of the parameterized connection (

where

As the first term vanishes, due to Bianchi identity, then we get the identity

This implies that the physical entity represented by

Finally, the following properties are guaranteed in any field theory constructed in the context of PAP-geometry:

The theory of GR, with all its consequences, can be obtained upon taking

As the parameterized connection (

One can always define a geometric material-energy tensor in terms of the BB of the geometry.

Conservation is not violated in any of such theories.

The authors declare that they have no competing interests.

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