On the Curvature and Torsion of the Gravity Field

Abstract

Considering two points P₁ and P₂ of the Earth’s external gravity field, sufficiently close to each other, and the two (vertical) versors n₁ and n₂, directed upwards which are tangent to the lines of force at P₁ and P₂, then the axes of two such versors will not, in general, be coplanar. When the two points lie on the same equipotential surface ∑, such a defect of coplanarity appears, as is well known, as the presence of the geodesic torsion of ∑ in the direction P₁P₂.If we put P₂—P₁ = st, where t is a unit vector, and s the length of the segment P₁P₂ which we suppose to be small to a first order, we then have from the generalized formulae of Frenet

$$\frac{{n_2 - n_1 }}{s}\; = \;x_\alpha t + \;\tau _\alpha b, $$

(1)

where b is a versor such that (t,b,n) form a positive right-handed triad, x _α is the first curvature or flexion of the surface in the direction t considered, and τ _α the geodesic torsion of ∑ in the same direction. The index α characterizes the direction t, for the reason which will later be apparent.

Originally published as: Marussi A (1952) Sulla curvatura e torsione del campo di gravità. Ann Geofis 5: no 2