Velocity effects on second order contributions to gravitational frequency shift by a moving Kerr–Newman black hole

The significance of the radial velocity effects on the leading-order gravitational frequency shift in some special cases was reported recently. This paper investigates the gravitational shift of frequency of light propagating in the equatorial plane of a radially moving Kerr–Newman black hole up to the second post-Minkowskian order, and discusses the radial velocity effects on the second-order contributions to the frequency shift. It is found that a new radial velocity effect appears in the second-order Schwarzschild contribution to the frequency shift, in contrast to no radial velocity effect in the first-order contribution, when both the emission and reception events are far away from the lens. Velocity effects on the gravitational frequency shifts induced respectively by the lens’s electrical charge and spin are also analyzed.


Introduction
The translational motion of a gravitational system makes a difference on the propagation of electromagnetic signals, which is the so-called velocity effect [1,2]. The issue of velocity effects on the gravitational frequency shift of light was discussed in many papers [2][3][4][5][6][7][8][9][10][11]. Birkinshaw and Gull [3] studied the transversal velocity effect of cluster of galaxies and showed that the gravitational frequency shift of light is proportional to the low transverse lens velocity and the deflection angle of light. Since this pioneering work, the conclusion that the transversal velocity effect on the gravitational frequency shift is dominant relative to that of the radial lens motion has been well established in the full relativistic framework [2,6,7].
Recently, the radial-velocity effects on the leading-order gravitational frequency shift induced by a constantly mova e-mail: hgs@usc.edu.cn b e-mail: lwb@usc.edu.cn ing Schwarzschild black hole were re-examined, and these effects are found to be necessary for the scenarios when the light emitter or the receiver is near to the gravitational lens [12]. As shown in Ref. [12], these velocity effects under these circumstances were, for example, of the order of magnitude of 10 −9 , which is larger than the observational precision of current detectors. For example, the precision of the Cassini for measuring gravitational frequency shift is about ∼ 10 −14 [13][14][15]. The Cassini is a sophisticated spacecraft orbiting the ringed planet and its main mission is to explore the Saturnian system, including measuring the plasma density, magnetic field, temperatures and atmospheric composition in Saturn's atmosphere, as well as studying its largest moon Titan. It bases on the Doppler shift of light signals to and from the spacecraft to determine its speed and the signal timing to measure its distance. When the spacecraft passes behind a massive central body (e.g. the Sun) on its cruise to Saturn, whether the body moves or not, these techniques can be also used to detect frequency shifts caused by gravity with a high accuracy. Highly precise measurements of these velocity effects are thus indeed possible by missions such as the Cassini. Considering the rapid progresses made in the gravitational-frequency-shift measurements (such as the ground-based [16][17][18][19][20][21] and satellite-based [22][23][24] measurements), we expect that the velocity effects on the secondorder contributions to the gravitational frequency shift might also be detectable in future. Thus, a full theoretical treatment of the gravitational shift of frequency of light due to a radially moving lens in the second post-Minkowskian (PM) approximation is necessary.
In this work, we extend the method of Ref. [12] to deriving the gravitational frequency shift of light propagating in the equatorial plane of a radially moving Kerr-Newman (KN) black hole, and concentrate on the radial-velocity effects on the second-order Schwarzschild, Kerr, and charge-induced contributions to the frequency shift. We find that for both the emission and reception events far away from the lens, the radial velocity effect makes a significant correction to the second-order Schwarzschild frequency shift, although this effect on the first-order one can be usually neglected.
The structure of this article is as follows. In Sect. 2, we adopt the weak-field metric of the moving KN source to calculate the equatorial frequency shift up to the 2PM order. Section 3 gives the discussions of the velocity effects on the second-order contributions to the frequency shift. Section 4 presents a summary. Our discussions are restricted in the weak-field, small-angle, and thin lens approximation.
Throughout the paper, Latin indices run from 1 to 3, and natural units in which G = c = 1 are used.
2 Gravitational frequency shift due to a radially moving KN source 2.1 The weak-field metric for a radially moving Kerr-Newman black hole Let {e 1 , e 2 , e 3 } denote the orthonormal basis of a 3D Cartesian coordinate system. The rest frames of the observer and the lens are denoted by (t, x, y, z) and (X 0 , X 1 , X 2 , X 3 ), respectively. The angular momentum vector J of the lens is assumed to be J e 3 . The harmonic metric of a moving KN black hole with a constant radial velocity v = ve 1 up to the 2PM order reads [25]: where δ i j denotes Kronecker delta, γ = (1 − v 2 ) − 1 2 is the Lorentz factor, and , and a ≡ J/M are the rest mass, the electrical charge, and the angular momentum per mass of the lens, respectively.
, with i jk being the Levi-Civita symbol. The relation M 2 ≥ a 2 + Q 2 is assumed to avoid the naked sin-gularity of the black hole. The coordinates X 0 , X 1 , X 2 , and X 3 are related to t, x, y, and z by the Lorentz transformation 2.2 Moving-KN frequency shift up to the 2PM order For simplicity, we consider the gravitational frequency shift of light which is restricted in the equatorial plane (z = ∂ ∂z = 0) of the moving Kerr-Newman black hole, as shown in Fig. 1. b is the impact parameter. The spatial coordinates of the light emitter A and the receiver B are denoted in the observer's rest frame by (x A , y A , 0) and (x B , y B , 0), respectively, with y A < 0, x A < 0, and x B > 0. The initial velocity of a photon, parallel to the lens's velocity v, is assumed to be w| x→−∞ (= e 1 ). The locations of A and B are denoted in the lens's rest frame by (X A , Y A , 0) and (X B , Y B , 0), respectively.
The frequency shift of a photon between emission and observation is defined by [6,7,[27][28][29] Here, τ A and τ B are the proper times of the light emitter and the receiver, respectively, and t A and t B denote respectively their coordinate times. For the case of the light emitter and the receiver being static (v A = v B = 0) in the observer's rest frame, the explicit forms of dτ A dt A and dt B dτ B up to the 2PM order are given as follows: The analytical form of dt A dt B up to the 2PM order can be obtained from the following gravitational time delay induced by the moving KN black hole [26] source plane lens plane observer plane with where can be calculated from Eq. (12) as follows: Moreover, the 1PM form of X 2 is given by [26] which yields The substitution of Eqs. (9), (10) and (13)- (18) into Eq. (8) yields the explicit form of the gravitational frequency shift up to the 2PM order caused by the moving KN source: Actually, Eq. (19) can also be expressed in terms of the quantities in the observer's rest frame (t, x, y, z). First, we adopt the following forms of X A and X B up to the 1PM order [26] with Substituting Eqs. (20), (21) into Eq. (19), up to the 2PM order, we obtain If all of the second-order contributions are dropped, Eq. (23) reduces to the first-order frequency shift caused by a radially moving Schwarzschild black hole [12] Δν which is consistent with the result obtained by the method of the Liénard-Wiechert gravitational potential [7], as shown in Appendix A. For the case of no translational motion of the lens (v = 0), Eq. (23) reduces to the second-order KN frequency shift In addition, it can be seen from Eq. (23) that the gravitational frequency shift induced by the charge Q can be written as

Radial velocity effects on the second-order contributions to the frequency shift
Since the radial velocity effect on the leading-order gravitational frequency shift has been discussed [12], we now focus on the radial velocity effect on the second-order contributions to the frequency shift.

Analytical forms of the radial-velocity effects
The comparison between Eqs. (23) and (25) results in the explicit forms of the radial-velocity effects as follows: where Δ M 2 (v) and Δ Q (v) denote the velocity effects on the second-order Schwarzschild and charge-induced contributions to the frequency shift, respectively.

Magnitudes and possible detection of the radial-velocity effects
In this subsection, we discuss the magnitudes of the radial-velocity effects on the second-order contributions to the frequency shift, and analyze the possibilities of their astronomical detection

General cases
We first discuss the magnitudes of the radial-velocity effects for arbitrary radial velocity v of the lens. According to Fig. 2c or d, we can see that the magnitudes of Δ a (v) may be larger than the Cassini's precision only when the lens has a relativistic radial motion. For instance, |Δ a (v)| will exceed 1.0 × 10 −14 when v > 0.38 for the case of x A = −10b and x B = 1.0 × 10 5 b. The possibility of detecting Δ a (v) via current projects is thus rather small. In addition, it can also be seen from Fig. 2e or f that there is no possibility of detection of Δ Q (v) by current detectors, since its magnitudes are much smaller than the Cassini's precision. It is only by using the ultrahigh accuracy techniques in future that we may observe these two correctional effects.
However, it should be pointed out that our discussions depend on the choice of the intrinsic parameters of the central body, which implies these velocity effects will be more evident and a easier detection of them if M, a, and Q become larger.
In addition, we emphasize that the requirements for their realistic detections are more demanding. For instance, the radial velocity of the Sun during the Cassini experiment in 2002 is not larger than 0.0001 in the geocentric frame and 5×10 −8 in the barycentric frame of the solar system [15,30,31]. The corresponding precision of the measurements for Δ M 2 (v) should be at least better than 8.0×10 −16 in the geocentric frame and 3.2 × 10 −19 in the barycentric frame, for the case of x A = −10b, x B = 1.0 × 10 5 b and b = 1.0 × 10 6 M (as an example). Therefore, if we wish to detect conservatively the radial velocity effects on the second-order contributions to the frequency shift in future actual measurements, the central body should move at a higher radial velocity (e.g. |v| > 1.26 × 10 −3 , for the case of x A = −10b, x B = 1.0 × 10 5 b and b = 1.0 × 10 6 M), or a better precision for measurements is required.
In contrast with Δ M 2 (v), Δ a (v), and Δ Q (v), Fig. 3 gives the magnitudes of the radial-velocity effect Δ M (v) and the transversalvelocity effect Δ M (v T ) on the leading-order gravitational frequency shift of light, which have been given analytically in Ref. [12] as follows: where v T is the magnitude of the transversal velocity v = v T e 2 of a moving Schwarzschild source along y-axis,

The low-velocity limit
We now consider the radial motion effects for four special scenarios in the low-velocity approximation, under which the second-and higher-order terms in velocity are omitted and Eqs.
where the terms with the factor v/x i A or v/x i B (i = 1, 2, 4) have been dropped since they are much smaller than the term with the factor v/b 2 .
It can be seen that under this scenario, a radial velocity effect (i.e., 8M 2 v b 2 ) on the second-order Schwarzschild frequency shift appears, although the radial velocity effect on the first-order one can be indeed neglected [2,6,7]. To some extent, this effect is similar to the famous formula Mv ⊥ b for the transversal velocity effect on the first-order frequency shift, with v ⊥ being the transversal lens velocity. For the case of M/b = 0.0001 (weak-field), the magnitudes of this effect are 8.0 × 10 −11 and 8.0 × 10 −13 for v = 0.001 and 0.00001, respectively. Therefore, it is possible to detect this new effect, since its magnitude may still be comparable to the precision (∼ 10 −14 ) of the Cassini in the weak-field and low-velocity limit.
where the terms with the factor v/x i B (i = 1, 2, 4) have been dropped. (c) The scenario for x A −b and x B ∼ b When the light emitter is far away from the lens and the observer is relative close to it (x B ∼ b), Eqs. (27)-(29) then reduce to where the terms with the factor v/x i A (i = 1, 2, 4) have been dropped.

Summary
In this paper we have calculated the gravitational frequency shift up to the 2PM order for light propagating in the equatorial plane of a radially moving Kerr-Newman source. It is found that for the astronomical scenario where both the light emitter and observer are far away from the lens, a new radial velocity effect 8M 2 v b 2 appears in the second-order Schwarzschild contribution to the frequency shift, in contrast to no radial-velocity effect in the first-order frequency shift. The velocity effects on the gravitational frequency shifts induced respectively by the lens' electrical charge and spin are also analyzed. It might be possible to detect the radial velocity effects on the second-order contributions to the frequency shift by high-accuracy measurements in future.

Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors' comment: This manuscript has no associated data.] Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Funded by SCOAP 3 .

Appendix A: Comparison between our formulation with that obtained via Liénard-Wiechert representation
In this appendix, we compare Eq. (24) with the analytical form of the first-order gravitational frequency shift obtained from the retarded Liénard-Wiechert gravitational potential [7]. For comparison, we consider a point mass M moving with a constant radial velocity v = ve 1 in the observer's rest frame (t, x, y, z), and follow the notations used in Ref. [26] for display convenience. According to the results presented in [7] (t A , x A ) and the receiver B (t B , x B ) being non-moving (v A = v B = 0) are written as follows: (A.12) Here, the two retarded times s A and s B are related to the emission time t A and the receiver time t B by the light-cone equations (A.14) (A. 24) Up to the 1PM order, Eq. (A.24) yields 25) which is the same as that given in Eq. (24).