A scalar geodesic deviation equation and a phase theorem (Q791881)

The well known scalar form of the equation of geodesic deviation is derived (*) \({\ddot \eta}+(K-\Omega^ 2)\eta =0\) where \(\eta\) denotes the distance between two test particles falling freely in a gravitational field, and where \(K=-R_{ijk\ell}u^ i\mu^ ju^ k\mu^{\ell},\quad \Omega^ 2=-{\dot \mu}_ i{\dot \mu}^ i,\) in which \(u^ i\) is the unit tangent vector to the geodesic under consideration and \(\mu^ i\) represents the direction cosines of the deviation vector in the local rest space. By considering a nonvanishing solution \(\rho\) of a nonhomogeneous counterpart of equation (*), in which the right-hand side consists of the term \(\rho^{-3}\), a conserved quantity \(a^ 2\) of (*) is obtained. An explicitly defined phase variable \(\Phi\), that depends on \(\rho\), gives rise to the equation \(\eta =a\rho \cos(\Phi)\), and hence to an amplitude \(a\rho\). For the case that \(K-\Omega^ 2>0\), a corresponding frequency and energy for the relative motion is introduced. The problem of phase stability is discussed in some detail. Possible applications to the theory of gravitational waves are briefly outlined.