On geodesic deviation in Schwarzschild spacetime Dennis Philipp, Volker Perlick, Claus Lammerzahl¨ and Kaustubh Deshpande ZARM - Center of Applied Space Technology and Microgravity University of Bremen 28359 Bremen, Germany Email: [email protected] Abstract—For metrology, geodesy and gravimetry in space, are slightly different and the result is the so-called satellite based instruments and measurement techniques are used cartwheel orbit. and the orbits of the satellites as well as possible deviations between nearby ones are of central interest. The measurement of iii) A more general possibility that allows to change the this deviation itself gives insight into the underlying structure of orbital plane as well as the constants of motion is the the spacetime geometry, which is curved and therefore described helical orbit configuration. by the theory of general relativity (GR). In the context of GR, the deviation of nearby geodesics can be described by the Jacobi For the precise measurements of inter-satellite distances equation that is a result of linearizing the geodesic equation in these orbit configurations the general relativistic effects around a known reference geodesic with respect to the deviation must be investigated and their impact on observables taken vector and the relative velocity. We review the derivation of into account for a given accuracy level. In this work we will this Jacobi equation and restrict ourselves to the simple case of focus on such orbital configurations and refer to one satellite the spacetime outside a spherically symmetric mass distribution as the reference object moving on the reference geodesic. The and circular reference geodesics to find solutions by projecting the Jacobi equation on a parallel propagated tetrad as done orbit of the second satellite will then be modeled by means by Fuchs [12]. Using his results, we construct solutions of the of the geodesic deviation equation that describes how nearby Jacobi equation for different physical initial scenarios inspired by geodesics deviate from each other due to the geometry of satellite gravimetry missions and give a set of parameter together spacetime and given initial conditions. We will first describe with their precise impact on satellite orbit deviation. We further the situation in standard Newtonian gravity and then turn to consider the Newtonian analog and construct the full solution, the full theory of GR to uncover relativistic modifications. that exhibits a similar structure, within this theory. Keywords—Geodesy, Orbits, Satellites II. GEODESIC DEVIATION IN NEWTONIAN GRAVITY In the Newtonian theory of Gravity we have as central I. INTRODUCTION equations When satellites follow freely falling orbits around a central ∆U(~r) = −4πGρ(~r) ; (1a) massive object like the earth, their worldlines, i.e., their paths d2~r = −∇U(~r) ; (1b) through space and time must be described by the geodesic dt2 equation together with given initial conditions. While for some (past) space missions the Newtonian theory of gravity might where the first of them is the field equation that relates the be sufficient, modern and future mission scenarios certainly Newtonian gravitational potential U to the mass density ρ need relativistic effects to be taken into account. Thus, the and introduces Newton’s gravitational constant as a factor precise description in terms of Einstein’s theory of gravity, GR, of proportionality. Outside a spherically symmetric object we becomes necessary even beyond usual Post-Newtonian (PN) obtain the gravitational potential arXiv:1508.06457v1 [gr-qc] 26 Aug 2015 approximations. In this work we will describe the geodesic GM deviation, with satellite missions in mind, in the full context U(r) = (2) r of GR. as a solutions of (1a). The second equation (1b) is the As an example, the GFZ–NASA mission GRACE-Follow- equation of motion and describes how test particles move in On [1], [2] consists of two satellites which are able to measure the gravitational potential given by U. The second derivative of the change in their relative distance (about 100 km) with an the position vector is taken w.r. to universal time t that exists in accuracy of the order of 10 nm. For the orbital motion of the Newtonian gravity. We can rewrite the second equation using two satellites we can imagine different configurations: index notation and obtain i) Tilt the orbital plane of one satellite with respect x¨a = −@aU(x): (3) to the other, but keep the constants of motion (in a magnitude) the same. This orbital configuration is The coordinates x are just the usual Cartesian coordinates called a pendulum orbit. and the overdot denotes derivatives w.r. to the Newtonian time coordinate. The argument x in the potential denotes ii) For a second configuration, the orbital plane is the position of the test particle. Here and in the following, the same, but the energy and angular momentum Latin indices a; b; ::: take values 1; 2; 3. Now, we assume a situation as shown in Fig. 1, i.e., we have a reference curve Xµ(s) Xa(t) that fulfills (3). Thereupon, we consider a second curve xa(t) = Xa(t) + ηa(t) and introduce the deviation vector ηa. Thus, we have xµ(s) x¨a = X¨ a +η ¨a = −@aU(x) = −@aU(X + η): (4) dXµ(s) Now, we linearize the right hand side w.r. to the deviation ⌘µ(s) a 2 ds U(x) = U(X + η) = U(X) + η @aU(X) + O(η ) ; (5) and obtain finally the deviation equation in Newtonian gravity a b a η¨ = −η @ @bU(X) : (6) From (6) we clearly see that second derivatives of the Newto- nian potential cause non-linear deviations. If either @aU = 0 (homogeneous gravitational field) or U ≡ 0 (no gravitational field) the deviation equation has the solution Fig. 1. The deviation of two nearby geodesics Xµ(s) and xµ(s) = Xµ(s)+ ηµ(s). Note: the deviation vector is always defined orthogonal (as measured a µ η (t) = C1t + C2 (7) with the metric g) on the four-velocity dX (s)=ds of the reference geodesic. and the deviation vector grows only linearly in time. In the following, we use the Newtonian gravitational poten- III. GEODESIC DEVIATION EQUATION IN GR tial (2) outside a spherically symmetric mass distribution and we change to usual spherical coordinates (x; y; z) = In the spirit of general relativity we model the spacetime (r sin # cos '; r sin # sin '; r cos #). We further specialize the geometry, i.e., the universe we live in (or at least the relevant reference geodesic (R(t); Θ(t); Φ(t)) to be a circular orbit in part for our model) using a pseudo-Riemannian metric tensor the equatorial plane. Thus, we have µ ν g = gµν dx dx (12) R(t) = R0 = const. ; (8a) on a four dimensional manifold M. The set fM; gg describes ≡ (8b) Θ(t) = Θ0 π=2 ; the four-dimensional spacetime and local coordinates on M r GM are xµ ; µ = 0; 1; 2; 3. We have an affine connection r, which Φ(t) = t =: !0t ; (8c) r3 is fully defined by the Christoffel symbols where the motion is oscillating with the Keplerian period σ r@ @µ = Γ @σ : (13) p 3 ν µν T0 = 2π=!0 = 2π r =(GM). For this situation the deviation equation (6) reduces to three differential equations for the On a pseudo-Riemannian manifold we can specialize r to be components of the deviation vector (ηr; η#; η'), see [7], the Levi-Civita connection and we get 2 # 1 d η 2 # σ σλ + ! η = 0 ; (9a) Γµν = g (@ν gµλ + @µgνλ − @λgµν ) : (14) dt2 0 2 2 r ' d η dη r Then, the geodesic equation that describes the motion of freely − 2!0 − 3!0η = 0 ; (9b) dt2 dt falling particles reads 2 ' r d η dη 2 µ ν σ (9c) d x µ dx dx 2 + 2!0 = 0 : + Γ (x) = 0 (15) dt dt ds2 νσ ds ds The first equation decouples from the other two and the µ solution is given by and gives as solution curves the geodesics x (s) of the spacetime. The affine parameter s along such a geodesic # η (t) = C5 cos !0t + C6 sin !0t ; (10) can be interpreted as proper time and is, thus, related which is an oscillation with the Keplerian period T = T = to the reading of a clock that is transported along the # 0 µ . In [7] the author derived only the oscillating solutions geodesic. Now, we fix a certain (known) geodesic (X (s)) = 2π=w0 0 1 2 3 for the remaining two equations. However, we extend this and (X (s);X (s);X (s);X (s)) and this curve will be called give the general solution that yields the reference geodesic in the following. To consider a neigh- µ r boring geodesic x (s) in a given coordinate system we make η (t) = C1 + C2 sin !0t + C3 cos !0t ; (11a) the ansatz 3 ' µ µ µ η (t) = − !0C1t + 2(C2 cos !0t − C3 sin !0t) + C4 : x (s) = X (s) + η (s) (16) 2 (11b) and define, thereupon, the deviation vector ηµ(s) that connects There are several possibilities to perturb the circular reference both geodesics. We assume, as sketched in Fig. 1, the four geodesic. One can incline the orbital plane, add a constant velocity of the reference geodesic and the deviation vector to radius or cause an eccentricity in the motion. All these effects be always orthogonal to each other, are due to the choice of the six parameter Ci. We must have ν µ dX precisely six free parameter to set the initial position and gµν η = 0 : (17) velocity as starting conditions.