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 , 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  that the f(R) model with the deformed nonuniform extra space is able to reproduce the 4-dim Minkowski metric