On deviation in Schwarzschild

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 (GR). In the context of GR, the deviation of nearby 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 ϑ cos ϕ, r sin ϑ sin ϕ, r cos ϑ). 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 {M, g} 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 ∇, which Φ(t) = t =: ω0t , (8c) r3 is fully defined by the Christoffel symbols where the motion is oscillating with the Keplerian period σ ∇∂ ∂µ = Γ ∂σ . (13) p 3 ν µν T0 = 2π/ω0 = 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 ∇ to be components of the deviation vector (ηr, ηϑ, ηϕ), see [7], the Levi-Civita connection and we get 2 ϑ 1 d η 2 ϑ σ σλ + ω η = 0 , (9a) Γµν = g (∂ν gµλ + ∂µgνλ − ∂λgµν ) . (14) dt2 0 2 2 r ϕ d η dη r Then, the geodesic equation that describes the motion of freely − 2ω0 − 3ω0η = 0 , (9b) dt2 dt falling particles reads 2 ϕ r d η dη 2 µ ν σ (9c) d x µ dx dx 2 + 2ω0 = 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 ϑ η (t) = C5 cos ω0t + C6 sin ω0t , (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 ϑ 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 ω0t + C3 cos ω0t , (11a) the ansatz 3 ϕ µ µ µ η (t) = − ω0C1t + 2(C2 cos ω0t − C3 sin ω0t) + 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. The meaning of these parameter ds and their impact on the perturbed orbit will be given in section Inserting (16) into the geodesic equation (15) gives a second IV-A in one go with those for the GR results. order differential equation that is quadratic in the deviation vector. When this deviation vector is assumed to be very These constants can be derived using the Euler Lagrange small, we can linearize with respect to the deviation vector equations µ µ itself η (s) and its derivative dη (s)/ds if we further assume d ∂L ∂L small relative velocities. Thus, we obtain the so called standard − (23) µ µ = 0 Jacobi equation ds ∂x˙ ∂x for the Lagrangian D2ηµ(s) dXτ (s) dXσ(s) = −Rµ (X) ην , (18) 1 ds2 τνσ ds ds L = g x˙ µx˙ ν . (24) 2 µν where the D/ds and the tensor µ Since the metric (20) does neither depend on the time coor- components R τνσ are given by dinate t nor on the angle φ we get for the reference geodesic D ηµ(s) dηµ(s) dXσ = + Γµ (X) ην , (19a) ds ds νσ ds E := A(R(s)) T˙ (s) = const. , (25a) 2 µ µ µ µ λ µ λ L := R(s) Φ(˙ s) = const. , (25b) R τνσ(X) = ∂ν Γτσ − ∂τ Γνσ + ΓνλΓτσ − ΓτλΓνσ . (19b) where the overdot denotes the derivative with respect to the In (18) we can clearly see that the curvature of spacetime in- proper time s. The general solution of the geodesic deviation duces a possible non-linear deviation between two neighboring equation (18) in the spacetime (20) was given by Fuchs [9] in geodesics. We have seen in the Newtonian case before that terms of first integrals of the Jacobi equation. This solution the same effect of non-linear deviation was caused by non- is, however, not applicable for circular reference geodesics vanishing second derivatives of the Newtonian gravitational since in this case the condition dR(s)/ds = 0 holds and potential and we have, thus, an intuitive understanding of the causes singularities in the equations in this reference. Shirokov role of the curvature tensor; The derived periodic solutions for the Schwarzschild spacetime and includes second derivatives of the metric as well. A somewhat circular reference geodesics [4]. One different possibility to detailed discussion of the equation of geodesic deviation can solve the geodesic deviation equation for the case of circular be found in standard textbooks on general relativity like [5] reference geodesics in the spacetime (20) is to refer it to a and [6]. It should be mentioned that if we do not linearize w.r. parallel propagated tetrad and solve the resulting differential to the relative velocity dη(s)/ds a generalized version of the equation in this reference system [12]. We will use the results Jacobi equation is obtained, see, e.g., [3], [10], [14], [15]. of this method here. In the following we specialize the spacetime to be spheri- The simplest model that we could use to describe the cally symmetric and static, described by the metric motion of satellites in orbit around the earth is to specialize 2 2 2 2 2 2 to the case of Schwarzschild spacetime g = −A(r)dt + B(r)dr + r (dϑ + sin ϑdϑ ) (20)  2M  and introduce the coordinates x0 = t, x1 = r, x2 = ϑ, x3 = ϕ. A(r) = 1 − ,B(r) = A(r)−1 (26) r The angles ϑ and ϕ are the usual polar and azimuthal angles as in spherical polar coordinates and the radial coordinate r and the circular reference geodesic is essentially determined is defined such that circles at a distance r have circumference through its radius R0 and the mass M of the earth that enters 2πr. In these coordinates the reference geodesic is given by the metric coefficients 0 1 X (s) = T (s),X (s) = R(s), R(s) ≡ R0 = const. (27a) π X2(s) = Θ(s),X3(s) = Φ(s) . (21) Θ(s) ≡ = const. (27b) 2 For a metric of the form (20) we can always, without loss ˙ L Φ(s) = Φ s = 2 s =: ωs (27c) of generality, assume that the reference geodesic is fixed in R0 the equatorial plane by choosing the coordinate system to E E T (s) = T˙ s = s = s . (27d) match this condition. This is due to the spherical symmetry A(R ) 1 − 2M/R of the spacetime exhibited in (20) and implies i) Θ(s) = 0 0 π/2 = const. and ii) dΘ(s)/ds = 0. Since we wish to actually We should mention that the mass of the earth in the units that describe the motion of satellites, we must restrict to timelike we use is about M ≈ 0.5 cm. After some lengthy calculations geodesics, which describe the motion of massive test particles one arrives at the solution of the Jacobi equation using the at subluminal speed. For such geodesics we can identify the result of Fuchs in [12] parameter s along the reference geodesic with the proper time LE η0(s) = f(s) (28a) according to the normalization of the four velocity p 2 2 (1 − 2M/R0) L + R0 µ ν dX (s) dX (s) 1 ER0 −1 = gµν . (22) η (s) = g(s) (28b) ds ds p 2 2 L + R0 Here, we use natural units in which Newtons gravitational 2 C5 C6 constant and the speed of light take the value . η (s) = cos ωs + sin ωs (28c) G c c = G = 1 R0 R0 For such timelike geodesics in the metric (20) we obtain p 2 2 constants of motion that correspond to the conservation of 3 L + R0 η (s) = 2 f(s) , (28d) energy E and angular momentum L, see for example [8]. R0 where the two proper time dependent function f(s) and g(s) C4: The constant C4 simply describes an offset in the are given by azimuthal and temporal components, e.g., when only   r C4 =6 0 we have a motion on the same circle as 4R3 4R3 the reference geodesic but with a constant azimuthal f(s) = C1 1 − s + (C2 cos ks − C3 sin ks) k2 k2 separation

+ C4 , (29a) p 2 2 L + R0 φ(s) = Φ(s) + 2 C4 = ωs + δφ r R0 C1 4R3 g(s) = + C2 sin ks + C3 cos ks . (29b) k k2 C5,6: These two constants incline the orbit w.r. to the reference geodesic, e.g., if only C5 =6 0 one obtains a The constants of motion E and L as well as the remaining circular orbit with radius R0 but with a polar angle quantities k and R3 are uniquely defined by the radius R0 of C cos ωs the circular reference geodesic θ(s) = Θ(s) + 5 R0 M π R3 = 3 , (30a) = + δθ cos ωs R0 2 2 M(R0 − 6M) By choosing more than one constant unequal to zero at the k = 3 , (30b) R0(R0 − 3M) same time, we get combinations of the described effects. We 2 2 (R0 − 2M) have named the constants Ci in the Newtonian calculation in E = , (30c) the same way, such that the before mentioned effects of these R0(R0 − 3M) constants are qualitatively the same. MR2 L2 = 0 . (30d) R0 − 3M In the following we examine two examples for possible orbit deviations in some detail and give a visual representation All the other constants Ci, i = 1..6 can be used to model in Fig. (2) and (3). different initial conditions. B. Pendulum orbits IV. RESULTS One possible application of the above results is to model a A. The initial conditions pendulum orbit, where the orbital planes of two satellites are To describe different orbital configurations we have to inclined w.r. to each other but we keep the constants of motion the same. For a circular reference geodesic this means that the examine the meaning of the constants Ci. A proper way to do this is to investigate the impact of each constant separately, radius of the second satellite’s orbit is the same and the orbit i.e., having only one of them unequal to zero. The parameter is therefore circular again. To achieve this, we have chosen C1 = C2 = C3 = 0. The precise choice of C5,C6 determines Ci then yield the following meaning: the line of nodes and we can include an azimuthal offset using C1: In the case that only C1 =6 0 we obtain again a circular C4 to prevent both satellites from colliding. A result of this orbit in the equatorial plane with radius kind is shown in Fig. (2) for different values of the elapsed proper time. The perturbed motion is again circular with x1(s) = r(s) = R(s) + η1(s) √ r(s) = R0 , (31a) ER0 2 R3 = R0 + C1 p 2 2 2 θ(s) = π/2 + C5/R0 cos ωs + C6/R0 sin ωs , (31b) L + R0 k φ(s) = ωs + C4 . (31c) = R0 + δr This result holds equally well in the Newtonian case, there we We have to ensure that the perturbation δr is small in can also get a perturbed orbit like this and the result shown in comparison to the reference radius R0 by means of Fig. (2) is the same. choosing C1 in a proper way.

C2,3: These two constants will cause an elliptical motion in C. Cartwheel orbits the r − φ plane, e.g., if only C3 =6 0 we get a radial and azimuthal motion of the form In this section we show the results for deviation from a circular reference geodesic through an eccentricity. To con- ER0 struct such solutions of the Jacobi equation (18) we choose r(s) = R0 + p C3 cos ks = R0 + δr cos ks L2 + R2 the parameter according to 0 √ p 2 2 √ L + R0 2 R3 R3 φ(s) = Φ(s) − C3 sin ks 2 C1 = arbitrary constant ,C3 = −2C1 2 . (32) R0 k k = ωs − δφ sin ks This choice ensures that both objects start from the same point in space and the radial components of the deviation vector For a perturbed motion of this kind, the eccentricity e η1(s) as well as its derivative dη1/ds are initially (at s = 0) and the semi major axis a are then given by equal to zero. C1 will then determine the maximal possible δr radial deviation from the circular reference orbit. Note that e = , a = R0 . R0 this example was considered by Fuchs in [9] as well, but there Fig. 2. A pendulum orbit: the deviating geodesic (black) is now inclined but circular again with the same radius as the reference geodesic (red) to keep the energy and angular momentum the same since these are purely determined by the radius of the circular motion. We included an azimuth angle offset via C4 to prevent both objects from colliding.

the choice of the constant C3 has the wrong sign and there are V. CONCLUSIONAND OUTLOOK several misprints at the indices of the constants. The perturbed orbit is given by Modeling satellite configurations by means of the Jacobi equation produces qualitatively good results in the sense that r known effects are reproduced. However, it has to be carefully ER0 C1 4R3 r(s) = R0 + (1 − cos ks) determined to which accuracy the results can be used compared p 2 2 k k2 L + R0 with two direct solutions of the geodesic equation since the = R0 + δr(1 − cos ks) , (33a) Jacobi equation was linearized w.r. to relative distance and θ(s) = π/2 , (33b) velocity between the two objects. In an upcoming paper we  4R  4R  will give results for the comparison between solutions of the 3 3 Jacobi equation and the direct solution of the geodesics equa- φ(s) = ωs + C1 1 − 2 s + 3 sin ks . (33c) k k tion for simple . For higher accuracy the generalized For a motion of that kind we derive the semi major axis a that Jacobi equation might be used to achieve better accuracy. If is simply given by the distance between both satellites is small but the relative velocity is not, one might do the linearization only w.r. to the a = R0 + δr (34) deviation itself and keep higher order terms in the derivative. If the relative velocity is small instead but the spatial deviation and an eccentricity of is not, one should do it vice versa. δr For the future import steps will be to obtain solutions e = . (35) R of the Jacobi equation in more realistic, but therefore more complicated spacetimes as models of the real earth. The The result is shown in figure (3) for different values of considered Schwarzschild spacetime is the simplest model that the elapsed proper time. We can clearly see the effect of does not include the rotation or even higher multipole moments perigee precession that was examined already by Fuchs for this of the earth. For these more realistic spacetimes the solution of specific example [12]. The r and φ-motions involve different the Jacobi equation will certainly need numerical integration frequencies, i.e., ω and k. For a full orbit that starts at radius or approximation methods. r(s = 0) = R0 and ends at r(s = 2π/k) = R0 the elapsed proper time is s = 2π/k and yields an azimuthal angle φ(s = 2π/k) = ω 2π/k. The difference to 2π is now called ACKNOWLEDGMENT the perigee precession ∆φ. Thus, we get, see e.g. [9], The authors would like to thank Dirk Putzfeld¨ for insightful ω 2π ω  discussions. The first author acknowledges financial support ∆φ = − 2π = 2π − 1 . (36) from the German Collaborative Research Center (SFB) 1128. k k Support from the Research Training Group Models of Gravity Inserting ω and k that were given before in terms of the circular is gratefully acknowledged. VP was financially supported radius R0 gives the known value during this work by Deutsche Forschungsgemeinschaft. ! r R ∆φ = 2π 0 − 1 . (37) REFERENCES R − 6M 0 [1] (May 2015) The Grace Follow-On Mission website [Online] Available: http://gracefo.jpl.nasa.gov However, in the Newtonian case there is no such quantity as [2] Dehne, M. et al., Laser interferometer for spaceborne mapping of the k and only the Keplerian frequency ω0 appears instead of k. Earths gravity field, Journal of Physics: Conference Series 154 (2009) Thus, for the Newtonian case we get no perigee precession [3] Hodgkinson, D.E., A modified equation of geodesic deviation, General and the behavior is qualitatively different in comparison to Relativity and Gravitation 3 (1972) 351-375 that shown in Fig. (3) - the perturbed orbit, i.e., the ellipse [4] Shirokov, M.F., On one new effect of the Einstein theory of gravitation, just remains unchanged. General Relativity and Gravitation 4 (1973) 131-136 Fig. 3. Solution of the Jacobi equation (18) for satellite geodesics around the earth. The circular reference geodesic is shown in red and the deviating geodesic that corresponds to an elliptical orbit in black. The trajectories of both are shown for different values of the elapsed proper time on the reference orbit. The 3D orbits are shown in the top, while in the bottom row we display a top view to illustrate the eccentricity.

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