A natural metric of the gravity force field.
by
Zafar Usmanov
Institute of Mathematics, Tajik Ac. of Sci.

In this paper the energy integral for motion equations of a free mass point in the gravity force field is interpreted as an asometric curve of a Riemann's space, whose metric is constracted by a natural way.

Let Oxyz be a Cartesian coordinate system on the Earth surface and axis Oz is vertically upwards directed. It is known that the energy integral of the above equations may be written in the following form:

v2=h-2gz,

(1)

where v is a velocity modulus of the mass point, g is the gravity accelaration and h is a constant defined by initial terms. Since

v2 dt2 = dx2+dy2+dz2,

formula (1) is transformed into an equivalent expression

dt2 = \fracdx2+dy2+dz2h-2gz.

(2)

This formula may serve for definition of an intrinsic time infinitesimal element for the motion of a free mass point under the gravity force. As is obvious, a value of dt depends on a location and an infinitesimal moving of the mass point in the gravity field.

Now the following quadratic differential form, called a natural space-time metric of the gravity field, is introduced:

ds2 = dt2 - \fracdx2 + dy2 + dz2h - 2gz

(3)

With respect to (3) formula (2) defines two isotropic curves with null lengths, ds² = 0.

This metric enables to investigate the gravity field using geometric methods.

Date received: January 18, 2000