On geodesic deviation in Schwarzschild spacetime

Abstract

For metrology, geodesy and gravimetry in space, satellite based instruments and measurement techniques are used and the orbits of the satellites as well as possible deviations between nearby ones are of central interest. The measurement of this deviation itself gives insight into the underlying structure of the spacetime geometry, which is curved and therefore described by the theory of general relativity (GR). In the context of GR, the deviation of nearby geodesics can be described by the Jacobi equation that is a result of linearizing the geodesic equation around a known reference geodesic with respect to the deviation vector and the relative velocity. We review the derivation of this Jacobi equation and restrict ourselves to the simple case of the spacetime outside a spherically symmetric mass distribution and circular reference geodesics to find solutions by projecting the Jacobi equation on a parallel propagated tetrad as done by Fuchs. Using his results, we construct solutions of the Jacobi equation for different physical initial scenarios inspired by satellite gravimetry missions and give a set of parameter together with their precise impact on satellite orbit deviation. We further consider the Newtonian analog and construct the full solution, that exhibits a similar structure, within this theory.Comment: 6 pages, 3 figures, contribution to conference on Metrology for Aerospace 2015 - relativistic metrology sessio

On the Applicability of the Geodesic Deviation Equation in General Relativity

“…We now recall the derivation of the Newtonian deviation equation, see, e.g., Ref. [4] and references therein. For a given reference curve Y µ (t) that fulfills the equation of motion we construct a second curve X µ (t) = Y µ (t) + η µ (t) and introduce the deviation η.…”

“…The meaning of these parameters and their impact on the perturbed orbit was studied briefly in Ref. [4], and the analysis will be extended in section 4 in the context of the general relativistic results. Note that the only frequency appearing in the solution so far is the Keplerian frequency Ω K .…”

On geodesic deviation in static spherically symmetric situations

“…Where E and L are constants of motion given by R. 3,4,10 The most general solution of the deviation equation (9) in a spherically symmetric and static spacetime was found by Fuchs.…”

“…3 If we set the constants C (1,2,3,4) equal to zero, we get a perturbed orbit that has a radius r = R and azimuthal motion ϕ(s) = Φ(s) = Ωs. However, the polar motion is given by Since the polar motion has the same frequency as the azimuthal motion T = 2π/Ω we have ϕ(2π/Ω) = 2π and ϑ(2π/Ω) = ϑ(0).…”

“…One possibility is to allow only the constant C (3) to be unequal to zero. The result is a perturbed orbit with elliptical shape…”