We rewrite these coefficients in the 4-tuple form as ([[gamma].sup.1], [[gamma].sup.2], [[gamma].sup.3], [[gamma].sup.0]) which may be summarized using the

Minkowski metric on space-time as follows:

In terms of geodesic normal coordinates (adapted to the motion of free moving particles in space-time) the Riemann tensor at a point encodes the information of the difference up to second order (as a function of the distance to that point) between the standard

Minkowski metric and the space-time metric.

The results confirmed the relevance of using the Engle-Granger methodology in all previous surveys, but it also suggested some interesting properties related to the estimate of regression coefficients based on different variants of the

Minkowski metric or to estimate regression equation without intercept.

In the earlier work [1], we introduced deviation to the flat

Minkowski metric due to the gravitational field in the form,

The main difficulty associated with such 4-d spacetime formulation is that the

Minkowski metric is indefinite (Lorentzian manifold).

where p is the

Minkowski metric order, and it can take values from 0 to infinity (and can even be a real value between 0 and 1).

For higher dimensional data, a popular measure is the

Minkowski metric, where is the dimensionality of the data, and are two comparable parameters, and are -values of these parameters.

To deal with this question note that any vacuum solution must be found in the absence of matter, strictly speaking, only the

Minkowski metric can be considered as a vacuum solution.

The paper first considers a spacetime without gravity, as described by the

Minkowski metric. The

Minkowski metric can be rewritten as a summation of velocities and as an apportionment of energy equivalence.

In addition to the gravitational refraction index (14), in the Standard Model Lagrangian one can take into account the constant gravitational factor (13) by introduction of the new time parameter, t [right arrow] St, and conducting the conformal transformation of the

Minkowski metric,

Let us consider the pseudo-Riemannian space with the signature (+ - --) and select the

Minkowski metric tensor [[??].sub.ij] in the metric tensor [g.sub.ij](x), of this space explicitly

It has been shown recently [9] that the f(R) model with the deformed nonuniform extra space is able to reproduce the 4-dim

Minkowski metric.