ar X iv : g r - qc / 0 10 80 44 v 1 1 7 A ug 2 00 1 Planetary g ( t ) for which resistive atmospheric falling is rising

A Darboux-transformed surface gravitational acceleration of the constant gravitational acceleration for a body endowed with an atmospheric layer is shown to turn the atmospheric free fall with quadratic resistance in the opposite motion, i.e., a free rising. Although the atmosphere of such a body may look completely normal, it is the time dependence of its gravitational field that produces this type of motion. This result is a consequence of general, one-parameter-dependent Darboux transformations in mathematical physics. The following time-dependent gravitational acceleration g(t; λ) = g 1 − 2 d 2 dt 2 ln(I 01 (t) + λ) , (1) where I 01 (t) = t 0 cosh 2 xdx and λ is a parameter of the gravitational force, turns falling through a quadratic resistive-earth-like-atmosphere into just the opposite motion, i.e., a rising that for t → ∞ is of constant velocity. Eq. (1) is in fact the one-parameter Darboux-transformed acceleration of the constant acceleration g at the surface of a usual planet such as Earth. The result is a consequence of a mathematical scheme that has been called strictly isospectral Darboux technique (SIDT) and has been applied extensively by one of the authors to various fields of physics. 1 Briefly, SIDT is a three step procedure that for the resistive atmospheric free fall means the following. (i) One starts with the equation dv dt = g − ǫv 2 , (2) representing the normal resistive falling motion through an atmosphere of constant friction coefficient per unit of mass ǫ. (ii) One shifts to a motion of the form −dv dt = g 2 (t) − ǫv 2 , (3) where g 2 (t) = g[−1 + 2 tanh 2 (√ ǫgt)]. (4) One interpretation of Eq. (3) is that the motion is upwards through a fluid medium producing a driving force proportional to the square of the velocity of the particle (one may call it antifriction) plus another force g 2 (t) of driving character for t < T = 1 √ ǫg Arctanh 1 √ 2 and turning friction-like afterwards. The first two steps are connected to each other through the particular solution of the initial normal motion as given by Eq. (7) below. (iii) The third step is the return back to an equation similar to the initial one, namely dv dt = g(t; λ) − ǫv 2 , (5) where the time-dependent forces g(t; λ) are given …