Redshift and Frequency Comparison in Schwarzschild Spacetime

Redshift and frequency comparison in Schwarzschild spacetime

Dennis Philipp, Eva Hackmann, Claus L¨ammerzahl ZARM, University of Bremen, 28359 Bremen, Germany

We derive exact expressions for the relativistic redshift between an Earth–bound observer, that is meant to model a standard clock on the Earth’s surface, and various (geodesic) observers in the Schwarzschild spacetime. We assume that the observers exchange radial light signals to compare the frequencies of standard clocks, which they transport along their respective worldlines. We calculate the redshift between an Earth–bound clock and static observers, observers in radial free fall, on circular geodesics, and on arbitrary bound quasi–elliptical orbits. For the latter case, we consider as examples an almost circular orbit, the Schwarzschild analog of Galileo satellites 5 orbits with a moderate eccentricity, and a highly elliptical orbit as special examples. Furthermore, we also use orbits close to a Schwarzschild black hole to highlight the influence of the relativistic perigee precession on the redshift signal. Calculating a post–Newtonian expansion of our results, the total redshift is decomposed into its special relativistic Doppler parts and the gravitational part due to the theory of General Relativity. To investigate the impact of higher order relativistic multipole moments on the gravitational redshift, we consider static observers in a general Weyl spacetime. We give a general expression for the mutual redshift of their standard clocks and consider in particular the effect of the relativistic quadrupole as a modification of the Schwarzschild result.

I. INTRODUCTION Clock Ensemble in Space (ACES) will test the gravitational redshift at the 10−6 level [7, 8]. In the past, sev- The theory of General Relativity (GR) predicts that eral satellite test of SR and GR that involve space–based the frequencies of clocks are influenced by the clocks’ clocks have been proposed, see, e.g., [9, 10]. A recent motion as well as their respective positions in the gravi- approach to test the GR prediction of the gravitational redshift uses the RadioAstron satellite and the authors tational field. From Special Relativity (SR), the parallel −5 and transverse Doppler effects on the redshift between are confident to reach an accuracy in the 10 regime moving observers are well–known. However, GR predicts [11]. Further prospects for future satellite missions using yet another redshift effect related to gravity, i.e. to the spacecraft clocks are discussed in Ref. [12]. spacetime curvature. Since only relative measurements For a general overview of ”The Confrontation between are meaningful clocks shall be linked and their mutual General Relativity and Experiment“, we refer the reader redshift is to be determined, either by real optical fiber to the living review article [13]. Details on relativistic ef- links when the clocks are close to each other or by ex- fects in the Global Positioning System (GPS), in partic- changing electromagnetic signals. ular on timing and redshifts, can be found in the review The first experiment in this respect was conducted by in Ref. [14] and references therein. For time transfer in Pound and Rebka in 1960 [1]. The authors verified the the vicinity of the Earth and definitions of different time change of a photon’s frequency during propagation in the scales we refer to Ref. [15]. gravitational field of the Earth. In 1966, the GEOS- The relativistic redshift effects can also serve to test 1 satellite was used to observe the relativistic Doppler different theories of relativistic gravity and, therefore, effects [2]. To date, the best test of the gravitational to test GR [16, 17]. For instance, scalar–tensor theo- redshift is still given by the Gravity Probe A (GPA) ex- ries and their parametrized post–Newtonian form where periment, see [3–5]. The measurement was conducted in considered in Ref. [18], in which the authors derived the arXiv:1711.01237v1 [gr-qc] 3 Nov 2017 1976 and used two hydrogen masers, one of which was difference to the GR redshift signals for artificial satellite carried to a height of 103 km by a Scout D rocket to be orbits around the Earth. compared with the hydrogen maser on ground. To quote All the articles mentioned so far do only take into ac- from the results of this seminal experiment [4]: “The count the first order post–Newtonian approximation of agreement of the observed relativistic frequency shift with the full relativistic redshift, which contains the Doppler prediction is at the 70 × 10−6 level.” parts known from SR and the gravitational redshift, The gravitational redshift, however, was confirmed which is a genuine effect of GR. This consideration is with an accuracy of 1.4 × 10−4, see Ref. [5]. At present, certainly sufficient to meet present technological capabil- the ESA–funded study GREAT and DLR–funded study ities and to provide the framework for clocks in space RELAGAL aim to improve the accuracy of the gravi- with contemporary accuracies for frequency comparison. tational redshift test by analyzing clock data from the However, in this work we derive an exact expression for Galileo satellites 5 and 6, which were fortunately lunched the general relativistic redshift in static simple space- into elliptic orbits [6]. The authors claim that the data times, in particular the Schwarzschild spacetime. To con- analysis and integration over one year can improve the struct a framework for future high–precision experiments GPA limit to about 4 × 10−5. The upcoming Atomic in a top–down approach we find it is necessary to investi- 2 gate all notions in full GR first to enable a thorough un- Now, we consider two different observers, which shall derstanding and an undoubtedly correct interpretation of serve as emitter and receiver of a light signal sent from measurement results. one to the other. Let the two observers be described by their respective four–velocities u andu ˜, with integral curves γ(τ) andγ ˜(˜τ) that are the observers’ worldlines, II. SETTING respectively. We define the redshift z between the two observers by A. Geometry, notation and redshift definition ν dτ˜ ∆˜τ z + 1 := = = lim . (5) We use Einstein’s summation convention, greek indices ν˜ dτ ∆τ→0 ∆τ are spacetime indices and run from 0 to 3, and latin in- In GR, there is a universal formula for the redshift [20] dices are purely spatial indices running from 1 to 3. Our metric signature convention is (−, +, +, +). ν g k, u γ The Schwarzschild spacetime is the simplest solution of 1 + z = = . (6) ν˜ gk, u˜ Einstein’s vacuum field equation and describes the space- γ˜ time outside a spherically symmetric mass distribution. For the purpose of this work, it shall serve as an approxi- For the definition of the redshift according to Eq. (5) see mation of the spacetime outside the Earth. The geometry the sketch of the general situation in Fig. 1. of the Schwarzschild solution is described by the metric Equation (6) allows to determine the redshift z between two observers as a function of their respective g = −f(r)dt2 + f(r)−1dr2 + r2 dϑ2 + sin2 ϑ dϕ2 , worldlines and the exchanged light signal. The precise (1) way of how this signal is exchanged from emitter to receiver still needs to be prescribed, i.e. the lightlike tan- where we use spherical coordinates (t, r, ϑ, ϕ), and the gent vector k needs to be specified. We will treat this metric function f(r) is given by issue in the next section. 2m In studies such as RELAGAL and GREAT [6] however, f(r) = 1 − . (2) usually clock residual are investigated. Hence, we also r have to relate the proper times τ andτ ˜. Using Eqs. (5) The Schwarzschild radius is rs := 2m, and it is related and (6), we obtain to the SI-mass M of the source by d˜τ = dτ (z + 1) , (7) GM m = , (3) c2 and therefore where G is Newton’s gravitational constant, and c is the Z vacuum speed of light. For the Earth, the Schwarzschild τ˜ = (z + 1) dτ . (8) radius is about 2m⊕ ≈ 8.8 mm. In the following, We use natural units such that G = 1 and c = 1, if not stated oth- Thus, an analytic model of the redshift can be used to erwise. However, to calculate the post–Newtonian limit model the relation between the two proper times, and for of our results, we introduce SI units again. a constant redshift we obtain the simple relation In GR, there is a universal formula for the frequency ν of a light ray that a given observers measures. Let a τ˜ = (z + 1) τ . (9) light signal be described by a curve λ(s), where s is an affine parameter along its trajectory. Furthermore, let the tangent vector to the light ray’s worldline be k := dλ/ds. For any such lightlike signal, we have g(k, k) = 0. B. Observers and signal transfer Let an observer’s worldline be described by xµ(τ), and its four-velocity dx/dτ =:x ˙ =: u. The proper time τ is We introduced two different kinds of observers, of defined by the requirement that g(u, u) = −1. Hence, the which the worldlines γ(τ) andγ ˜(˜τ) are described as inte- observer’s worldline is parametrized by proper time and, gral curves of their respective four-velocities u andu ˜. The therefore, its clock is a standard clock. Such standard worldline γ shall now corresponds to a moving (geodesic) clocks can indeed be characterized operationally as shown observer which is the emitter of a signal, andγ ˜ corre- in Ref. [19]. sponds to an observer that we think of being the receiver The frequency associated with the lightlike signal λ, attached to the Earth’s surface. Within the approxima- which a given observer measures, is the scalar product tion given by the Schwarzschild spacetime, the Earth (w.r.t. the metric) of the observer’s four-velocity and the is described by a sphere with radius r⊕. We assume light ray’s tangent vector, evaluated at the observers po- τ andτ ˜ to be the observers’ respective proper times. sition x, i.e. We can, without loss of generality, choose the equatorial plane to be the plane of motion. Even though we ν = g k(x), u(x) , (4) model the spacetime by the Schwarzschild geometry, we 3

FIG. 1. Sketch of the signal transmission between emitter and receiver for the definition of the redshift. For any family of observers, we pick two wordlines γ andγ ˜ with tangent vectors u andu ˜, respectively. The continuous exchange of FIG. 2. Schematic description of the signal transmission: lightlike signals λ gives rise to the definition of the redshift z radial lightlike signals are sent within the equatorial plane. according to Eq. (5). We assume a continuous distribution of receiver clocks along the Earth’s equator with no mutual redshift. The clock comparison is done between the moving clocks on various orbits can of course consider observers fixed to the surface of and the respective Earth–bound clock which is reached first by the radial lightlike signal. a rotating Earth. This allows to accurately describe all Doppler contributions to the redshift but ignores all grav- itomagnetic effects such as the clock effect, see [21] and Above, we introduced v = r Ω2 as the rotational ve- references therein. Hence, we have in general ⊕ ⊕ locity of the Earth–bound clock. For v⊕ ≡ 0, we recover (uµ) = (ut, ur, uϕ, 0) , (10a) the case of static observers fixed to the surface of a non– µ t ϕ rotating Earth. (˜u ) = (˜u , 0, u˜ , 0) . (10b) We can now already give a general expression for the Observers on the Earth’s surface are fixed at spatial co- redshift between the Earth–bound observer and a second arbitrary observer that sends radial light signals for the ordinates (r = r⊕, ϑ = π/2), and their worldlines are in- frequency comparison. The result is tegral curves of the Killing vector field ∂t +Ω ∂ϕ, where Ω is the Earth’s angular velocity in natural units. Hence, p ϕ ν ur(r, ϕ) f(r ) we require thatu ˜ = Ω to have the Earth–bound ob- 1 + z = = u0(r, ϕ) + √ ⊕ , (14) servers forming a Born-rigid congruence. The value of Ω ν˜ f(r) 1 + v⊕ is roughly calculated by Ω c ≈ 2π/86400 s. Both observers, γ(τ) andγ ˜(˜τ), shall exchange light sig- and is parametrized by the momentary position (r, ϕ) at nals to compare the frequencies of their standard clocks, the emission of the signal. Besides of being parametrized which they transport along their respective worldlines. by proper time, no assumption is made so far on the emit- We assume that the frequency comparison is realized via ter’s worldline and four–velocity u. Hence, in Eq. (14), t r radial lightlike geodesics λ(s). Such a geodesic connects u and u can be chosen also to correspond to an arbi- events on the wordlines of γ andγ ˜. For radial frequency trary (non–geodesic) motion. However, in the following comparison, the tangent vector k of the light ray’s world- we will mainly consider geodesic observers and specify line must lie within the t − r–plane in the tangent space. these velocity components. Therefore, we have Since the emitter of the signal shall be a geodesic observer, the geodesic equation (kµ) = (kt, kr, 0, 0) . (11) µ µ ν σ x¨ + Γ νσx˙ x˙ = 0 (15) For both observers, the four-velocities are normalized according to must be fulfilled along the worldline γ. The Lagrangian to describe the observer’s motion is g(u, u) γ = −1 , g(˜u, u˜) γ˜ = −1 , (12) 1 L = g(u, u) , (16) and their worldlines are, therefore, parametrized by their 2 respective proper times. For the Earth–bound observers, we can immediately calculate the t−component of the and there are two constants of motion related to the two four–velocity: Killing vector field ∂t and ∂ϕ

t 2 2 2 t g(˜u, u˜) γ˜ = −f(r⊕)(˜u ) + r⊕Ω = −1 E := f(r)u = −ut , (17a) s s 2 ϕ 1 + r2 Ω2 1 + v2 L := r u = uϕ . (17b) ⇒ u˜t = ⊕ = ⊕ . (13) f(r⊕) f(r⊕) The normalization of the four-velocity then contains 4 the same information as the r-component of the Euler- A. Static observer Lagrange equations, and we obtain s This first scenario is an extension of the standard text- L2 ur = ± E2 − + 1 f(r) . (18) book example in which the redshift between two static r2 observers is calculated, see, e.g., Ref. [23]. Consider the Earth–bound observer at radius r and a second observer For any geodesic light ray λ, however, there exist con- ⊕ hovering at r . Assume that a radial light ray with tan- stants of motion as well 0 gent k is emitted at the larger radius r0 > r⊕ tangential t Eλ := f(r) k = −kt , (19a) to an r-line. The motion of the two observers and the 2 ϕ light ray is then described by Lλ := r k = kϕ . (19b) s  For radial light rays in particular, which we consider for 2 2 µ 1 + r⊕Ω signal exchange between the observers, the angular mo- u˜ =  , 0, Ω, 0 , (22a) f(r⊕) mentum vanishes identically, Lλ = 0. Since for any lightlike geodesic g(k, k) = 0 holds along the orbit, we further uµ = 1/pf(r ), 0, 0, 0 , (22b) obtain 0 µ −1 r k = Eλ f(r) , −1, 0, 0 , (22c) k = ±Eλ . (20) To summarize, we have in total for the respective four– respectively. According to the general equations (6) and velocities of the observers and the light ray’s tangent vec- (14), the redshift between emitter and receiver is tor s s ν f(r⊕) 1 2 ! 1 + z = = µ L 2 stat q u = E/f(r), ± E2 − + 1 f(r), L/r , 0 , ν˜ f(r0) 2 2 r2 1 + r⊕Ω s (21a) f(r ) 1 = ⊕ . (23) s  q 2 2 f(r0) 1 + v2 µ 1 + r⊕Ω ⊕ u˜ =  , 0, Ω, 0 , (21b) f(r⊕) Since f(r) is a monotonically increasing function and q kµ = E f(r)−1, ±1, 0, 0 . (21c) 2 λ 1/ 1 + v⊕ < 1, we have Note that we could also consider a co–rotating coordinate system to describe the rotating Earth and observers 1 + zstat < 1 , (24) on its surface by the transformation ϕ → ϕ¯ = ϕ − ωt, where ω :=u ˜ϕ/u˜t is the angular frequency w.r.t. co- and the redshift is negative. Hence, the frequency ν ordinate time. However, we choose to keep the non– is smaller thanν ˜ and the clock at smaller radius ’runs rotating coordinates and we use these coordinates exclu- slower’. In Fig. 3, the redshift is shown as a function 2 −1/2 sively throughout the rest of the present paper. of the static satellite height r0. The factor (1 + v⊕) in Eq. (23) is related to the transverse Doppler contribution, already known in SR. It modulates the gravita- III. CLOCK COMPARISON AND REDSHIFT tional redshift and is caused by the circular motion of the Earth–bound receiver clock. In the following sections, we determine the red- Calculating the first order post–Newtonian (weak– shift between an Earth–bound observerγ ˜ and different field) limit of Eq. (23), we recover the well–known limit (geodesic) observers γ in orbits around the Earth. We for the gravitational redshift and Doppler redshift in the consider static observers hovering in space, the case of post–Newtonian framework free radial infall, circular geodesic motion, and arbitrary ! v2 bound quasi–elliptical geodesic orbits. We determine the ν 1 ⊕,SI 1 + zstat = = 1 + ∆U − , respective redshift for a radial signal transfer, see Fig. 2 pN ν˜ pN c2 2 for a sketch of the situation. We assume a continuous dis- (25) tribution of receiver clocks along the Earth’s equator with no mutual redshift since they move on an isochronomet- see, e.g., Ref. [14] and references therein for a compar- ric surface [22] of the Schwarzschild spacetime and these ison. Here, we use U(r) = −GM/r, which is the New- Earth–bound clocks form an isometric congruence. The tonian gravitational potential of a spherically symmetric respective clock comparison is done between the moving source, and ∆U := U(r⊕) − U(r0). For the limit pro- clocks on various orbits and the respective Earth–bound cedure, we introduced SI–units and v⊕,SI = c v⊕ is the clock which is reached first by the emitted radial light Earth–bound clock’s velocity in [m/s]. We see that the signal. gravitational redshift, to lowest order, is given by ∆U/c2. 5

0 tion of proper time is given by the solution of

-1 2

] dr = E2 − f(r) . (28) - 10 -2 dτ

-3 The redshift between the infalling observer γ and the observerγ ˜ on the Earth, parametrized by the momentary -4 radial position r at the time of signal emission, is given redshift z [ 10 -5 by s 6 p 2 - ν f(r⊕) E − E − f(r) 0 5 10 15 20 25 30 36 1 + zrad = = ν˜ f(r) p q 2 satellite height above Earth's surface[10 3km] f(r) 1 + v⊕ E − pE2 − f(r) FIG. 3. The redshift between an Earth–bound observer and = (1 + zstat) p , (29) an observer hovering at radius r0. We show the redshift for f(r) different heights above the Earth’s surface up to the geostationary height of about 36000 km. The total redshift contains where the constant of motion E is related to the initial the transverse Doppler contribution due to the Earth’s rota- conditions. Assuming that the motion of the infalling tion as well as the gravitational effect. observer starts at a certain radius r0 with initial radial velocityr ˙0, we obtain q 2 Redshift experiments are therefore sensitive to potential E = r˙0 + f(r0) . (30) differences. We can estimate the error made by using only the first order approximation above by considering In the Fig. 4, we show an example of such a scenario, for which assume the radial infall to begin from a geosta- the next order term in the week field expansion of 1+zstat, which is tionary height with zero initial velocityr ˙0 = 0. When the satellite’s motion starts from rest, a post– ! 1 r + 3r ∆Uv2 3v4 Newtonian expansion in inverse powers of c leads to −∆U 2 0 ⊕ − ⊕,SI + ⊕,SI . (26) 4 ! c r0 − r⊕ 2 8 v 1 v2 v2 1 + z = 1 − sat + ∆U − ⊕,SI + sat , rad pN c c2 2 2 The last two terms are even several orders of magnitude smaller than the leading term which is proportional to (31) ∆U 2/c4. The entire second order contribution is evi- where v := p2GM/r − 2GM/r is the (Newtonian) dently proportional to c−4, and for a satellite height of sat 0 satellite’s velocity at radius r after being dropped from about 5000 km above the Earth’s surface, this contribu- rest at radius r . We recognize parallel Doppler effect tion is approximately −3 × 10−19 and therefore 9 orders 0 terms, which are proportional to v /c, and the gravi- of magnitude smaller than the first order contribution, sat tational contribution proportional to the potential differ- which is about −3 × 10−10 for this situation. Hence, for ence ∆U/c2 between the Earth and the momentary satel- redshift experiments sensitive to the 10−19–level, at least lite position. The transverse Doppler effect ∼ v2 /c2 second order contributions must be taken into account. ⊕,SI is due to the Earth’s rotation. However, this exceeds the present accuracy by some orders of magnitude, see, e.g., [6]. C. Circular motion

B. Radial infall The next situation which we consider is the redshift between the Earth–bound observerγ ˜ and another ob- Assume now that the observer γ is related to a satellite server on a circular geodesic orbit with radius r = R. in radial free fall, starting at some initial position r0 with The satellite’s motion is described by initial velocityr ˙ parallel to an r–line. During this radial 0 E L infall, light rays are emitted continuously and received by uµ = , 0, , 0 , (32) the Earth–bound observer. The motion of the infalling f(R) R2 satellite is described by where for circular geodesics we have µ p 2 R u = E/f(r), − E − f(r), 0, 0 , (27) E2 = f(R)2 , (33a) R − 3m mR2 where L = 0 holds since the angular momentum must L2 = . (33b) vanish for radial motion. The radial position as a func- R − 3m 6

0 3 2 ] ]

- 5 1

-1 - 10 0 -1 -2 -2

redshift z [ 10 -3 redshift z [ 10 -3 -4 -5

0 5 10 15 20 25 30 36 0 0.5r ⊕ 10 15 20 25 30 36 satellite height above Earth's surface[10 3km] satellite height above Earth's surface[10 3km]

FIG. 4. The redshift between an Earth–bound observer and FIG. 5. The redshift between an Earth–bound observer at an observer in radial infall at momentary height above the r⊕ and an observer on a circular geodesic with radius R. Earth’s surface. The motion starts with zero initial velocity from a geostationary radius. The total redshift contains transverse and parallel Doppler contributions as well as the “mislaunched” Galileo satellite 5 with a moderate eccen- gravitational effect. tricity of e = 0.15, iii) a highly elliptic orbit around the Earth with eccentricity e = 0.6, and iv) an elliptic orbit around a black hole that clearly shows perigee precession. The redshift is The redshift between the Earth–bound clock and the s p moving geodesic observer is ν f(r⊕) f(R) 1 + zcirc = = q ν ν˜ f(R) p 2 1 + zelliptic = 1 − 3m/R 1 + v⊕ ν˜ s p p 2 2 2 f(R) f(r⊕) E ± E − (L /r + 1)f(r) = = (1 + zstat) r (34) q 3m f(r) pf(r) 1 + v2 1 − ⊕ R E ± pE2 − (L2/r2 + 1)f(r) = (1 + zstat) , In Fig. 5, we show this redshift for circular geodesics with pf(r) different radii. (36) Note that there exist a special radius given by q 2 where E and L are the constants of motion along γ. They r = 1.5 r⊕ × 1 + v , for which the redshift vanishes ⊕ are related to the initial conditions by identically due to cancellation of Doppler and gravitational redshift contributions. 2 L = r0 ϕ˙ 0 (37a) A post–Newtonian expansion of the redshift result 2m L2 leads to 2 2 E =r ˙0 + 1 − 1 + 2 . (37b) r0 r ! 0 1 v2 v2 ⊕,SI sat We can also express L and E by a suitable defined semi– 1 + zcirc pN = 1 + 2 ∆U − + , (35) c 2 2 major axis and eccentricity of the orbit. The orbit has turning point rp (perigee) and ra (apogee). At these where we can clearly see the gravitational redshift re- turning points, dr/dτ = 0, and we can define eccentricity lated to potential differences ∆U/c2 between the clock e and semi-major axis a by on Earth and the clock on the circular orbit, as well as the transverse Doppler effect which is proportional to rp =: (1 − e)a , ra =: (1 + e)a . (38) 2 2 p v /c . Here, vsat = GM/r is the Newtonian velocity of a satellite on a circular Kepler orbit. Now, we can relate E and L to e and a by f(r ) − f(r ) L2 = p a , (39a) f(ra) f(rp) 2 − 2 D. Arbitrary geodesic motion ra rp L2 E2 = + 1 f(r ) . (39b) Now, we consider an observer γ that moves on an ar- r2 a bitrary bound orbit and evaluate the redshift w.r.t. the a Earth–bound observerγ ˜. We investigate in particular i) Inserting this result into Eq. (36) yields the redshift as a a nearly circular orbit around the Earth with a small ec- function of the momentary radial position, the eccentric- centricity, ii) the Schwarzschild analog of the orbit of the ity, and the semi–major axis. The radial position, for any 7

2 chosen combination of e and a, can only vary between rp where p := a(1 − e ) is the semilatus rectum and θ is and ra. the true anomaly. In the expansion (43), apart from We argue that, however, Eq. (36) might still be mis- the genuine gravitational redshift ∼ ∆U/c2, we recognize leading, because the redshift zelliptic is given as a func- parallel Doppler effects proportional to vsat k/c as well as tion of the momentary radial position r at the signal’s transverse Doppler terms proportional to v/c2 due to the emission. For an arbitrary elliptic orbit, all radial values circular motion of the Earth–bound clock as well as the between perigee and apogee are realized, but for a better transverse velocity of the satellite. It can easily be de- understanding we can insert the solution of the equation duced from the relations (45) and (46) for the parallel of motion to obtain the redshift as a function of proper velocity component that the parallel Doppler effect van- time or the azimuthal angle. This highlights in particular ishes at the turning points r = ra and r = rp as we expect the periodic character of the redshift signal. An analytic it to bee according to the special relativistic formulae. solution of the geodesic equation is As a possible application of our results, we consider m now four different elliptical bound orbits. These are i) an r(ϕ) = , (40) 2℘(ϕ − ϕ ) + 1/6 almost circular orbit around the Earth with a moderate in eccentricity e = 0.025, ii) a Schwarzschild version of the where ϕin is related to the initial conditions according to Galileo 5 orbit with an eccentricity of e = 0.15, iii) a highly elliptical orbit around the Earth with e = 0.6, and Z ∞ dz 1 m 1 ϕin = ϕ0 + , y0 = − . iv) a highly elliptical orbit with e = 0.6 and semi–major p 3 2 r 6 y0 4z − g2z − g3 0 axis of 100 m around a Schwarzschild black hole. The (41) results for the total as well as the gravitational redshift are shown in Figs. 6 – 9. The Weierstrass invariants g2 and g3 are determined by the constants of motion as follows: The gravitational redshift takes its largest value at the apogee since it is the largest distance and therefore the 1 m2 g = − , (42a) largest potential difference ∆U. In Eq. (43), we have 2 12 L2 shown that the gravitational redshift is, to lowest order, 1 1 m2 1 m2 sensitive to these potential differences. Figs. 6 – 8 show g = − − E2 − 1 . (42b) 2 216 12 L2 4 L2 how the profile of the gravitational redshift widens with increasing eccentricity. However, the gravitational red- For details on the analytic solution and possible appli- shift is 3–4 orders of magnitude smaller than the total cations, we refer the reader to the seminal paper by redshift due to the large Doppler contributions. Hence, Hagihara [24] and the work in Refs. [25–27]. In Ref. besides an accurate clock comparison, a precise knowl- [28] observables for bound orbits in more general axially edge of the satellite’s state vector is needed to recover symmetric spacetimes are considered, and in the present the gravitational redshift from experimental data. work, we add the exact description of the relativistic red- Investigating the peak–to–peak differences for the shift for the Schwarzschild spacetime as a possible observ- gravitational redshift in the three scenarios shown in able, see Eq. (36). Fig. 6 – 8, we observe the following relations: Inserting (40) into (36) yields the redshift as a function of the azimuthal angle as an alternative. orbital eccentricity peak–to–peak grav. redshift The post–Newtonian limit of Eq. (36) leads to

2 2 ! 0.025 ≈ 1 × 10−12 vsat k 1 v⊕,SI v (1 + z ) = 1 + + ∆U − + sat , −11 elliptic pN c c2 2 2 0.15 ≈ 5 × 10 0.6 ≈ 3 × 10−10 (43) where the satellites (Newtonian) orbital velocity is Fig. 9 shows the relation between the redshift and the s periapsis precession. We have plotted several orbits and 2 1 drawn vertical lines at multiples of 2π to guide the eye. v = GM − , (44) sat r a The perigee, i.e. the minimal radial position shifts after each orbit. At the same time, also the apoapsis precesses and the parallel velocity component turns out to be by the same angle and we see this shift phase–locked with r the maxima and minima of the gravitational redshift. GM (r − r )(r − r) v = a p . (45) sat k a r2 Using relations for Newtonian Kepler orbits, it can also IV. HIGHER ORDER MULTIPOLE EFFECTS be expressed in the well known form s GM So far, we have used the Schwarzschild spacetime, vsat k = e sin θ , (46) which possesses only a monopole moment, to calculate p the redshift between two observers. Within the theory of 8

GR, we can extend this framework to consider also the the framework of the redshift potential, see [22]. Insert- influence of higher order multipole moments on the grav- ing the expansion (48) for the Weyl metric function, we itational redshift. In the following, we will consider static finally obtain for the redshift observers in a general Weyl spacetime and, in particular, PN l+1 focus on the influence of the quadrupole moment on the exp l=0(−1) qlQl(˜x)Pl(˜y) gravitational redshift. 1 + zWeyl = . (53) exp PN (−1)l+1q Q (x)P (y) The Weyl class of spacetimes contains all static, ax- l=0 l l l isymmetric, and asymptotically flat solutions of Ein- This expression is valid for all Weyl spacetimes with mul- stein’s vacuum field equation [29]. The metric of a gen- tipole moments up to order N in the Newtonian limit, eral Weyl spacetime is and it is exact, i.e. no approximations are involved. For the choice q = 1 and q = 0 for all l > 0, we recover the µ ν 2ψ 2 2 −2ψ 2 2 2 0 l gµν dx dx = −e dt + m e (x − 1)(1 − y )dϕ Schwarzschild result of the previous section III. dx2 dy2 In Ref. [22] it is shown how the Newtonian limit of the + m2e−2ψe2γ (x2 − y2) + , (47) x2 − 1 1 − y2 metric function ψ is calculated. Based on these calcula- tions, we obtain here the post–Newtonian expression of where we use a time coordinate t, an azimuthal coordi- the redshift above, which is nate ϕ adapted to the symmetry, and spheroidal coordi- ∆U 1 + z = 1 + , (54) nates (x, y), see [30]. The metric function ψ is given by Weyl pN c2 the expansion where ∞ N X l+1 X Pl(cos ϑ) ψ(x, y) = (−1) ql Ql(x) Pl(y) , (48) U = −G N , (55) l rl+1 l=0 l=0 where Pl(y) are Legendre polynomials and the Ql(x) are which is the Newtonian gravitational potential of an ax- Legendre functions of the second kind as given in, e.g., isymmetric mass distribution with Newtonian multipole Ref. [31]. The parameters ql are related to the Newto- moments Nl. The Nl are then related to the ql via nian multipole moments of an axisymmetric gravitational l l! 2 l l+1 potential in the Newtonian limit, as shown in Refs. [30] Nl = (−1) ql (G/c ) M . (56) (2l + 1)!! and [22]. We can relate the spheroidal coordinates (x, y) to quasi–spherical (Schwarzschild–like) coordinates (r, ϑ) Eq. (54) demonstrates that also for a more general mul- by the transformation tipolar mass distribution, the post–Newtonian expression for the gravitational redshift between static observers is, x := r/m − 1 , y := cos ϑ . (49) to lowest order, determined by potential differences.

If only q0 6= 0 and all other ql for l > 1 vanish identically, the Schwarzschild spacetime is recovered. A. The quadrupole contribution Now, we consider two static observers with four– velocities u andu ˜, respectively, in a Weyl spacetime. For Besides the monopole, the next order multipole con- such static observers we have tribution to the gravitational redshift comes from the quadrupole. To treat such a conﬁguration in full GR, µ t µ t u = u , 0, 0, 0 , u˜ = u˜ , 0, 0, 0 . (50) we choose q0 = 1, q2 6= 0 and all other ql = 0. This choice yields the Erez-Rosen spacetime, where the met- The normalization of the four–velocity immediately ric potential is given by yields 2 x − 1 2 (3x − 1) t −ψ(x,y) t −ψ(˜x,y˜) 2ψ = log + q (3y − 1) u = e , u˜ = e . (51) x + 1 2 4 x − 1 3 According to the general equation (5), the redshift be- × log + x . (57) tween these two static observers is x + 1 2 This spacetime is a quadrupolar generalization of the ν ut eψ(˜x,y˜) 1 + zWeyl = = = , (52) Schwarzschild solution. We introduce the Schwarzschild– ν˜ u˜t eψ(x,y) like spherical coordinates by (49). Then, the exact gravitational redshift for two static observers in such a where (˜x, y˜) and (x, y) are the observers’ positions, re- quadrupolar spacetime becomes spectively. Note, however, that static observers in a Weyl spacetime are on surfaces of constant redshift po- s f(r⊕) h(r⊕, ϑ⊕) tential. Hence, the redshift (52) between any two of these 1 + zquadr = . (58) isochronometric surfaces can easily be calculated using f(r) h(r, ϑ) 9

One of the two observers shall be located on the sur- The post–Newtonian expansions of our results reveal face of the Earth, i.e. at (r⊕, ϑ⊕), and the second static the contributions to the total redshift due to the trans- observer is located in space at (r, ϑ). The function f(r) verse and parallel Doppler effects, as well as the genuine is the Schwarzschild metric function and h(r, ϑ) is given gravitational redshift, which is related to potential differ- by ences. The transverse Doppler contributions are due to the circular motion of the receiver clock on the Earth and 3 r 2 1 the perpendicular (w.r.t. the orbital trajectory) velocity h(r, ϑ) = exp q (3 cos2 ϑ − 1) × − 1 − 2 4 m 4 of the satellite that emits the electromagnetic signal for 2m 3 r the frequency comparison. The parallel Doppler effect × log 1 − + − 1 . (59) appears whenever the satellite has a non–vanishing radial r 2 m velocity and it vanishes at the turning points of bound The first term in (58) resembles the Schwarzschild orbits as shown by the respective equations Sec. III. result for a monopolar gravitational field, and the For the redshift between an observer on the surface of term h(r⊕, ϑ⊕)/h(r, ϑ) describes the modification of the rotating Earth and satellites in bound quasi–elliptical the gravitational redshift due to the general relativistic geodesic motion, we have considered three cases with dif- quadrupole. ferent eccentricities. We have shown how the profile of Calculating the post–Newtonian limit of the result (58) the gravitational redshift widens with increasing eccen- we obtain tricity and that Doppler effects on the redshift are usually a few orders of magnitude larger. Hence, the satellite’s 1 GM 3 cos2 ϑ − 1 ⊕ state vector must be known accurately to deduce the 1 + zquadr pN = 1 + 2 − − GN2 3 c r⊕ 2r gravitational redshift from experimental data in high– GM 3 cos2 ϑ − 1 precision frequency or clock comparison experiments. + GN2 3 r 2r To analyze the influence of relativistic higher order ∆U = 1 + quadr , (60) multipoles, we have derived an exact expression for the c2 gravitational redshift for static observers in a general Weyl spacetime. In particular, we have shown how the where the value of q is chosen to be related to the Earth’s 2 relativistic quadrupole modifies the Schwarzschild result. Newtonian quadrupole The Newtonian limit of our result for the gravitational 15 N redshift was related to potential differences of axisym- q = 2 . (61) 2 2 Mm2 metric Newtonian gravitational potentials in lowest order. Instead of N2, sometimes the dimensionless parameter In our future work, we will extend our framework to N2 cover also non–radial signal transmission and gravito- J2 = 2 (62) a⊕M magnetic contributions to the redshift and clock comparison by considering rotating spacetimes, such as the is used. Here, a⊕ is the Earth’s mean radius. Kerr spacetime, and moving observers in these geome- For the Earth, the value of J2 is approximately tries. Furthermore, we will analytically and without ap- −3 J2 ≈ 1.0826 × 10 . Hence, the quadrupolar contribu- proximations treat the influence of higher order relativis- tion to the Newtonian gravitational potential is three or- tic multipole moments on the Doppler and gravitational ders of magnitude smaller than the monopolar contribu- redshift by considering for instance Weyl spacetimes and tion due to the total mass M. Our result in the post– Quevedo–Mashhoon spacetimes and the mutual redshift Newtonian limit is in agreement with the well–known also for moving observers. These axisymmetric static equations for the gravitational redshift beyond spheri- spacetimes possess well–defined Newtonian limits, see cally symmetric gravitational fields, see, e.g., Ref. [14]. Sec. IV, and this allows also to recover the influence of multipole moments in the post–Newtonian framework. It has to be analyzed how our exact results for the rela- V. CONCLUSION tivistic redshift can contribute to future satellite experiments and data analysis. However, for high–precision Within the framework of General Relativity, we have measurements, environmental effects on the satellite or- derived an exact analytic expression for the redshift be- bits need to be taken into account, and the satellite’s tween an Earth–bound receiver clock and arbitrary ob- motion is no longer described by unperturbed Kepler– servers in space that emit radial light signals for the fre- like or Schwarzschild geodesics. Even though a first or- quency comparison in Schwarzschild spacetime. Further- der post–Newtonian treatment of the relativistic redshift more, we have specified the results to the case of a static might be sufficient in some situation, contemporary and observer hovering at a constant spatial position, geodesic future measurements should be accompanied with the observers in radial free fall, in circular orbit, as well as in best available theoretical framework in full GR to ensure general bound elliptical motion. an undoubtedly correct interpretation. 10

6 4.50 semi-major axis 27977600 m 4 eccentricity 0.025 4.45

2 ⊕

4.40 ⊕ 0 Earth y / r

4.35

-2 orbital radius r [ s ]/ r

-4 4.30

-6 -6 -4 -2 0 2 4 6 0 T/2 T

x/r ⊕ proper time s

] -5.34 2 - 10 ]

- 7 1 -5.36 0

-1 -5.38 total redshift z [ 10

-2 pure gravitational redshift z [ 10 -5.40 -3

0 T/2 T 0 T/2 T proper time s proper time s

FIG. 6. Redshift between an almost circular orbit with eccentricity e = 0.025 and semi major axis a = 27977600 m and an observer on the surface of the rotating Earth. We show the orbit as well as the radial coordinate r over one full orbital period (upper row). In the bottom row, we show the total and gravitational redshift, respectively. The total redshift contains besides the gravitational contribution, transverse and parallel Doppler terms. The peak–to–peak diﬀerence in the gravitational redshift is about 1 × 10−12.

ACKNOWLEDGMENTS

This work was supported by the Deutsche Forschungs- gemeinschaft (DFG) through the Collaborative Research Center (SFB) 1128 “geo-Q” and the Research Training Group 1620 “Models of Gravity”. We also acknowledge support by the German Space Agency DLR and the RE- LAGAL study with funds provided by the Federal Min- istry of Economics and Technology (BMWi) under grant number DLR 50WM1547. 11

semi-major axis 27977600 m 5.0 4 eccentricity 0.15 4.8 ⊕ 2 4.6 ⊕ 0 Earth 4.4 y / r

4.2

-2 orbital radius r [ s ]/ r

4.0 -4 3.8

-6 -6 -4 -2 0 2 4 6 0 T/2 T

x/r ⊕ proper time s

2 -5.1 ] - 10 1 -5.2 ] - 6

-5.3 0

-5.4 total redshift z [ 10 -1

pure gravitational redshift- z [ 10 5.5

-2 -5.6 0 T/2 T 0 T/2 T proper time s proper time s

FIG. 7. Redshift between the Schwarzschild version of the Galileo 5 orbit with an orbital eccentricity e = 0.15 and semi major axis a = 27977600 m and an observer on the surface of the rotating Earth. We show the orbit as well as the radial coordinate r over one full orbital period (upper row). In the bottom row, we show the total and gravitational redshift, respectively. The total redshift contains besides the gravitational contribution, transverse and parallel Doppler terms. The peak–to–peak diﬀerence in the gravitational redshift is about 5 × 10−11. 12

7 semi-major axis 27977600 m eccentricity 0.6 5 6 ⊕ 5 ⊕ 0 Earth y / r 4 orbital radius r [ s ]/ r 3 -5

-5 0 5 0 T/2 T

x/r ⊕ proper time s

10 -3.0

] -3.5 - 10 5 ]

- 6 -4.0

0 -4.5

-5.0 total redshift z [ 10 -5

pure gravitational redshift- z [ 10 5.5

-10 -6.0 0 T/2 T 0 T/2 T proper time s proper time s

FIG. 8. Redshift between a highly elliptical orbit with eccentricity e = 0.6 and semi major axis a = 27977600 m and an observer on the surface of the rotating Earth. We show the orbit as well as the radial coordinate r over one full orbital period (upper row). In the bottom row, we show the total and gravitational redshift, respectively. The total redshift contains besides the gravitational contribution, transverse and parallel Doppler terms. The peak–to–peak diﬀerence in the gravitational redshift is about 3 × 10−10. 13

160 semi-major axis 100m eccentricity 0.6 140 100

120 50

0 BH 100 y / m

-50 80 orbital radius r [ s ]/ m

-100 60

40 -100 -50 0 50 100 0 2π 4π 6π 8π 10π x/m azimuthal angle

0.05

-0.030

0.00

-0.035 total redshift z -0.05 -0.040 pure gravitational redshift z

-0.10 -0.045 0 2π 4π 6π 8π 10π 0 2π 4π 6π 8π 10π azimuthal angle azimuthal angle

FIG. 9. Redshift between a highly elliptical orbit with eccentricity e = 0.6 and semi major 100 m around a Schwarzschild black hole and static observers at r = 20 m. We show the orbit as well as the radial coordinate r over one full orbital period (upper row). The black circle in the left plot indicates the position of the static observers that receive the signals. In the bottom row, we show the total and gravitational redshift, respectively. The total redshift contains besides the gravitational contribution, transverse and parallel Doppler terms. 14

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