Second Post-Minkowskian Metric for a Moving Kerr Black Hole
aa r X i v : . [ g r- q c ] J u l Second Post-Minkowskian Metric for a Moving Kerr Black Hole
Guansheng He, Chunhua Jiang, and Wenbin Lin ∗ School of Physical Science and Technology,Southwest Jiaotong University, Chengdu 610031, China (Dated: July 28, 2020)
Abstract
The harmonic metric for a moving Kerr black hole is presented in the second post-Minkowskianapproximation. It is further demonstrated that the obtained metric is consistent with the Li´enard-Wiechert gravitational potential for a moving and spinning source with an arbitrary constantvelocity. Based on the metric, we also give the post-Newtonian equations of motion for photon andmassive test particle in the time-dependent gravitational field.
PACS numbers: 04.20-q, 04.25.Nx, 04.70.Bw, 95.30.Sf ∗ Email: [email protected]. . INTRODUCTION The time-dependent gravitational field motivated by moving source system is an attrac-tive subject. Several groups have addressed the corrections due to the motion of source(s)to the propagation of a test particle theoretically. In order to study light propagation inthe gravitational field of arbitrarily moving N-body system, Kopeikin and Sch¨afer [1] ob-tained their explicit first post-Minkowskian Li´enard-Wiechert gravitational potential. Thisretarded solution of field equations was validated via using a Lorentz boost to the equa-tions of light propagation in the static field of source [2]. The numerical simulations forlight deflection by moving gravitational bodies were made in Ref. [3], where the two authorsconsidered the equations of light trajectory in both post-Newtonian and post-Minkowskianapproximation, respectively. The new technique of integration in Ref. [1] was also used toinvestigate the gravito-magnetism for light deflection in the weak field of moving and rotat-ing bodies by Kopeikin and Mashhoon [4]. According to the generalized idea of Fermat’sprinciple, Sereno [5, 6] considered the velocity-effect corrections of spinning source with lowtranslational speed to gravitational lensing of light. Then, Sereno [7] evaluated both theeffects of peculiar motion and rotation of galaxy clusters on the properties of gravitationallensing. Bonvin [8] discussed the weak-field perturbation of peculiar motion of galaxieson the light propagation, and showed that it obviously affected the magnification and therelation between cosmic shear and convergence. Wucknitz and Sperhake [9] studied notonly the light’s gravitational deflection, but also a massive test particle’s deflection, in thetime-dependent field of a uniformly moving source.On the other hand, the motion effects associated with observational experiments werealso investigated extensively. The effects of the solar barycentric motion to post-Newtonianparameter γ in Cassini spacecraft experiment were analyzed under Lorentz transformation(see, e.g. Refs. [10–12]). Kopeikin and Makarov [13] calculated the planetary monopolar,dipolar and quadrupole deflection of light by moving source in solar system and discussedthe possibilities for the actual observation of these effects. The authors of Refs. [14, 15]simplified the quadrupole formula given in Ref. [13] to examine the general relativity inexploration of Gaia mission [16]. There are also increasing interests for measuring propermotion of stars and other deflection sources [17–22].When considering motion effects of gravitational source in theoretical calculations for2igh-order gravitational deflection and the equations of motion, usually we need to includethe high-order effects produced by the source’s mass. The high-order velocity effects can becalculated only when the metric of this kind of source is known.In this paper we derive the second post-Minkowskian harmonic metric of a moving Kerrblack hole with an arbitrary constant speed, and calculate the equations of motion of photonand massive test particle. We demonstrate the consistency of the solution obtained viaLorentz transformation with the retarded Li´enard-Wiechert potential [1] for the first post-Minkowskian limit.Throughout the units where G = c = 1 are used. In addition, as done in Ref. [9], we usethe primed coordinates to distinguish the rest frame of the gravitational source ( t ′ , x ′ , y ′ , z ′ )from that of the observer ( t, x, y, z ) (unprimed coordinates). The observer is assumed tobe static relative to the background. II. SECOND POST-MINKOWSKIAN HARMONIC METRIC FOR A MOVINGKERR BLACK HOLE
Let e i ( i = 1 , ,
3) denote the unit vector of three-dimensional rectangular coordinatesystem. We assume m and J (= J e ) stand for the mass and angular momentum of a Kerrblack hole, respectively. Then the harmonic metric of Kerr black hole in the center of mass’srest frame can be expressed as [23, 24]: ds = − dX + R ( R + m ) + a X (cid:0) R + a R X (cid:1) (cid:16) X · d X + a R X dX (cid:17) R + a − m + X R (cid:16) X · d X − R X dX (cid:17) R − X + ( R + m ) + a R − X R m a (cid:0) R − X (cid:1)(cid:16) X · d X + a R X dX (cid:17) ( R + a − m )( R + a )( R + a X ) + R ( X dX − X dX ) R + a + 2 m ( R + m )( R + m ) + a R X Rm a (cid:0) R − X (cid:1)(cid:16) X · d X + a R X dX (cid:17) ( R + a − m )( R + a ) ( R + a X ) + a ( X dX − X dX ) R + a + dX , (1) where X · d X ≡ X dX + X dX + X dX and a ≡ Jm is the angular momentum per mass. R is related to X , X and X via the relation X + X R + a + X R = 1. Notice that here X µ denotes the 4-dimensional coordinates x ′ µ = ( t ′ , x ′ , y ′ , z ′ ) for display convenience. Thereare many efforts contributed to derive the Kerr metric in harmonic coordinates, and Eq. (1)has been compared with the results of other researchers in Ref. [23]. If we drop all Kerr-3elated terms ( a = 0), this equation will reduce to the Schwarzschild metric in the harmoniccoordinates [25, 26].In the limit of weak field, the metric can be reduced to g = − − − , (2) g i = ζ i , (3) g ij = (1 − Φ) δ ij + Φ X i X j R , (4)where the order 1 /R is kept. Φ ≡ − mR is Newtonian gravitational potential, and for given X , X , X , R = s X + X + X − a + p ( X + X + X − a ) + 4 a X , (5)which reduces to R = p X + X + X when the angular momentum vanishes ( a = 0). ζ ≡ amR ( X × e ) denotes the vector potential due to the rotation of Kerr black hole. Thisapproximate metric is in accord with the results of multipole expansions [27], as shown inAppendix A.Using Lorentz transformation, one can obtain the metric for an arbitrarily moving Kerrblack hole from Eqs. (2) - (4). Here we denote the translational velocity in arbitrary directionof this black hole as v = v e + v e + v e . The general Lorentz transformation between( t ′ , x ′ , y ′ , z ′ ) and the coordinate frame of the background ( t, x, y, z ) can be written as x ′ α = Λ αβ x β , (6)and Λ = γ , (7)Λ i = Λ i = − v i γ , (8)Λ ij = δ ij + v i v j γ − v , (9)where γ = (1 − v ) − is Lorentz factor and v = v = v + v + v . Following the definitionof covariant metric tensor g µν = g ′ ρσ Λ ρµ Λ σν , we can write down the second post-Minkowskian4etric of the moving Kerr black hole: g = − − γ (cid:0) v (cid:1) Φ − (cid:0) γ (cid:1) Φ − γ ( v · ζ ) + γ Φ R ( v · X ) , (10) g i = v i γ (cid:0) (cid:1) + γ (cid:20) ζ i + (cid:18) γ − v + γ (cid:19) ( v · ζ ) v i (cid:21) − γ Φ R (cid:20) v i ( γ − v ( v · X ) + X i ( v · X ) (cid:21) , (11) g ij = (1 − Φ) δ ij − v i v j γ (cid:0)
4Φ + Φ (cid:1) − γ (cid:20) ζ i v j + ζ j v i + 2( γ − v ( v · ζ ) v i v j (cid:21) + Φ R (cid:20) X i X j + v i v j ( γ − v ( v · X ) + ( γ −
1) ( v i X j + v j X i ) v ( v · X ) (cid:21) . (12) For simplicity, here we formulate the metric in terms of the old coordinates, which arerelated to the new coordinates via x ′ α = Λ αβ x β as follows: x ′ = X = x + v (cid:20) γ − v ( v · x ) − γt (cid:21) , (13) y ′ = X = y + v (cid:20) γ − v ( v · x ) − γt (cid:21) , (14) z ′ = X = z + v (cid:20) γ − v ( v · x ) − γt (cid:21) . (15)In terms of the new coordinates, R , ζ , v · X and v · ζ can also be written as R = 1 √ n x + y + z − t + γ ( v · x − t ) − a + r(cid:2) x + y + z − t + γ ( v · x − t ) − a (cid:3) +4 a (cid:8) z + v (cid:2) γ − v ( v · x ) − γt (cid:3)(cid:9) o , (16) ζ = 2 ma n y e − x e + ( v e − v e ) h γ − v ( v · x ) − γt io R , (17) v · X = γ (cid:0) v · x − v t (cid:1) , (18) v · ζ = 2 ma ( v y − v x ) R . (19) Eqs. (10) - (12) extend the metric presented in Ref. [28] to arbitrary direction of translationalvelocity of gravitational source. In the limit of low velocity ( v → g = − − (cid:0) v (cid:1) Φ − , (20) g i = 4 v i Φ + ζ i , (21) g ij = (1 − δ ij , (22)where Φ = − m/ p x + y + z − t + γ ( v · x − t ) up to the first post-Newtonian order.5 II. COMPATIBILITY BETWEEN LORENTZ TRANSFORMATION SOLUTIONAND RETARDED LI ´ENARD-WIECHERT POTENTIAL
Considering the characteristic of the linear gravitational perturbation theory, here welimit to discussing the first post-Minkowskian consistency with the solution of directly solv-ing the linear field equations.
A. Comparison with Mass-induced Li´enard-Wiechert Potential of a MovingMonopole
1. Retarded Li´enard-Wiechert Gravitational Potential
For a massive point-like particle moving with the arbitrary constant speed v mentionedabove, the corresponding energy-momentum tensor in the background’s rest frame can bewritten as [25] T ( x , t ) = γmδ ( x − x ( t )) , (23) T i ( x , t ) = v i γmδ ( x − x ( t )) , (24) T ij ( x , t ) = v i v j γmδ ( x − x ( t )) , (25)where i, j = 1 , m denotes the static mass of the particle.In the linear perturbation theory, the metric tensor g µν takes the form of g µν ( x , t ) = η µν + h µν ( x , t ) , (26)where η µν = ( − , , ,
1) denotes the Minkowski metric and the perturbation h µν ( x , t )related to Eqs. (23) - (25) in the first post-Minkowskian approximation is [1] h µν ( x , t ) = h µνM ( x , t ) = 4 (cid:0) T µν − η µν T λλ (cid:1) r ( s ) − v ( s ) · r ( s ) . (27)Here s = s ( t, x ) represents the retarded time, and the symbol M denotes the case that theperturbation here only depends on the mass of the source. Explicitly, we have h ( x , t ) = 2(1 + v ) γmr ( s ) − v ( s ) · r ( s ) , (28) h i ( x , t ) = − v i γmr ( s ) − v ( s ) · r ( s ) , (29) h ij ( x , t ) = 4 v i v j γm + mγ δ ij r ( s ) − v ( s ) · r ( s ) . (30)6otice that the retarded denominator r ( s ) − v ( s ) · r ( s ) for the uniform motion of gravitationalsource can be also expressed as [29] r ( s ) − v ( s ) · r ( s ) ≡ h r − v · r i = r p − v sin θ , (31)where θ is the angle between the present position vector r and the translational velocityvector v , r = | r | and the angle brackets denote the retardation symbol. Hence, Eqs. (28)- (30) become h ( x , t ) = 2(1 + v ) γmr p − v sin θ , (32) h i ( x , t ) = − v i γmr p − v sin θ , (33) h ij ( x , t ) = 4 v i v j γm + mγ δ ij r p − v sin θ . (34)Therefore, Eq. (26) has been expressed in terms of the present time-dependent metric viaquantities of present time for comparison.
2. First Post-Minkowskian Metric via Coordinate Transformation
Now we turn our attention to the result to be analyzed based on Lorentz transformation.Here we don’t distinguish the gravitational field of a Schwarzschild black hole from thatof one static point mass m , as done in Ref. [30]. Thus, according to Eqs. (10) - (12), thefirst post-Minkowskian time-dependent metric of a moving point mass with the arbitraryconstant speed v is g = − − v ) γ Φ , (35) g i = 4 v i γ Φ , (36) g ij = (1 − δ ij − v i v j γ Φ . (37)Notice that Φ in Eqs. (35) - (37) is reduced to Φ = − m/ p X + X + X in the first post-Minkowskian limit and that the metric here is denoted by quantities of co-moving frame X i ≡ ( X , X , X , X ) of the translationally moving Schwarzschild black hole.For the convenience of comparison, we need to express Φ in terms of quantities of thebackground’s rest frame. Using Lorentz transformation, Eq. (6) between the co-moving7 H v t, v t, v t L P H x,y,z L O z r × r × v Ó Θj xy FIG. 1. Sketch map of the geometrical relations between retarded position vector r and presentposition vector r for the moving Schwarzschild black hole. P ( x, y, z ) and S denote the field pointand source point, respectively. We locate the source S at the origin O when t = 0 so that the sourceis located at the position ( v t, v t, v t ) at the present time t ( > θ denotes the angle betweenthe present position vector r and velocity vector v . ϕ denotes the angle between the retardedposition vector r and v . frame and background’s rest frame, we obtain R in Φ as the function of the present distance r between field point P and source point S as follow R = q(cid:0) Λ j x j (cid:1) + (cid:0) Λ j x j (cid:1) + (cid:0) Λ j x j (cid:1) = γ q r − [ v r − ( v · x ) ] , (38)where r ≡ | r | = p x + y + z denotes the distance between field point and the origin O of coordinate frame ( t, x, y, z ). Generally, the geometrical relations of the related quantitiesare showed in Fig. 1. For the velocity v in arbitrary direction, the relation below holds: r sin θ = r sin ϕ . (39)8herefore, Eq. (38) is simplified to R = γr p − v sin θ . (40)We rewrite Eqs. (35) - (37) as follows: g = − v ) γmr p − v sin θ , (41) g i = − v i γmr p − v sin θ , (42) g ij = δ ij + 4 v i v j γm + mγ δ ij r p − v sin θ , (43)which is the same as the result by solving linearized field equation. For the special case thatthe point mass moves along positive x-axis, namely v = v e , the exact solution [30] via com-mon Lorentz transformation is certainly consistent with the Li´enard-Wiechert GravitationalPotential. In the limit of low velocity (within post-Newtonian order), the gravielectric andgravimagnetic potentials from Eqs. (41) - (42) are consistent with that via solving gravita-tional wave equation in analogy to its electromagnetic counterpart [31]. B. Comparison with Spin-induced Li´enard-Wiechert Potential of a Moving Spin-ning Point Mass
In this section, we compare the spin-induced Li´enard-Wiechert potential (see Eq. (16) inRef. [4]) with the spin-induced terms in Eqs. (10) - (12) g S ∼ − γ ( v · ζ ) , (44) g S i ∼ γ (cid:20) ζ i + (cid:18) γ − v + γ (cid:19) ( v · ζ ) v i (cid:21) , (45) g Sij ∼ − γ (cid:20) ζ i v j + ζ j v i + 2( γ − v ( v · ζ ) v i v j (cid:21) . (46)According to the results presented in Ref. [4], the spin-induced Li´enard-Wiechert potentialreads h αβS = 4 γ r ξ S ξ ( α u β ) [ r ( s ) − v · r ( s )] , (47)which is determined by the energy-momentum tensor generated by the spin of the movingpoint mass. Here the symbol S denotes the spin-dependence, in contrast with the mass-dependence ( M ) mentioned above. 9he detailed comparison of Eqs. (44) - (46) with Eq. (47) is performed in Appendix B,explicitly. We conclude that the spin-induced terms in our metric are also consistent withthe retarded Li´enard-Wiechert potential solution, except for an additional factor 1 /γ whichis missing in Eq. (15) in Ref. [4] and has also been noticed in previous literatures [32, 33].Since the linear metric perturbation h αβ for a moving spinning gravitational source canbe divided into two parts, namely the mass-dependent part h αβM and the spin-induced part h αβS in the first post-Minkowskian approximation [4], the compatibility demonstrated aboveis helpful to verify the solution obtained by the method of Lorentz transformation. IV. EQUATIONS OF MOTION OF PHOTON AND MASSIVE TEST PARTICLEIN POST-NEWTONIAN APPROXIMATION
Based on Eqs. (10) - (12), we can derive the post-Newtonian equations of motion for testparticles via calculating Christoffel symbols and then substituting them into the geodesicequations. We consider a massive test particle or a photon in the weak gravitational field ofthe moving Kerr black hole. Eqs. (10) - (12) in the low-velocity limit reads g = − − v )Φ − − v · ζ + O ( v ) , (48) g i = 4 v i Φ + ζ i + O ( v ) , (49) g ij = (1 − δ ij + O ( v ) , (50)where v denotes typical velocity of a non-relativistic system in the post-Newtonian approx-imation.Up to order of v /r ( r denotes typical separation of a system of particles), the equationof motion for a massive test particle can be written as d u dt = −∇ (cid:0) Φ+2 v Φ+2Φ + v · ζ (cid:1) − ∂ ξ ∂t + u × ( ∇× ξ )+3 u ∂ Φ ∂t +4 u ( u ·∇ ) Φ − u ∇ Φ , (51)where ξ = 4 v Φ + ζ . The equation of motion for photon, up to order v , are obtained as d u dt = − (1 + u ) ∇ Φ + 4(1 − v · u ) u ( u · ∇ ) Φ + u × [ ∇ × (4 v Φ)] + (3 − u ) u ∂ Φ ∂t . (52)One can notice that Eqs. (51) - (52) are fully consistent with the first post-Newtonian equationsof motion given in Ref. [25]. onsidering the fact that the angular momentum per mass a is less than mass m , we can seethat the effects of the terms with Φ may be larger than that of the term containing ζ . When v = 0, we have ξ = ζ , and Eqs. (51) and (52) reduce to describe the post-Newtonian equations ofmotion of test particles in the gravitational field of Kerr black hole [23]. When ζ = 0, Eqs. (51)and (52) reduce to the post-Newtonian equations of motion of test particles in field of a movingSchwarzschild black hole [9, 30]. V. CONCLUSION
We have applied a general Lorentz transformation to the harmonic Kerr metric, and obtainedthe second post-Minkowskian metric for a moving Kerr black hole with an arbitrary constant speed.The perturbation of the trajectory of test particles due to the terms with Φ in this metric maybe larger than that of the term containing the source’s spin a . We have also illustrated that themetric obtained via Lorentz transformation is in agreement with the Li´enard-Wiechert solution fora moving and spinning point mass in the first post-Minkowskian approximation, not limiting tolow velocity. Furthermore, the resulting metric via Lorentz transformation doesn’t depend on thechoice of spin supplementary conditions while the Li´enard-Wiechert gravitational potential is basedon it. As an application, we calculated the post-Newtonian equations of motion for the massive testparticle and photon based on the metric, which can also be used to investigate the second-ordergravitational deflection of light as well as massive test particles, in the field of a moving Kerr blackhole. ACKNOWLEDGMENTS
This work was supported in part by the Program for New Century Excellent Talents in Univer-sity (No. NCET-10-0702), the National Basic Research Program of China (973 Program) GrantNo. 2013CB328904, and the Ph.D. Programs Foundation of Ministry of Education of China (No.20110184110016), as well as the Fundamental Research Funds for the Central Universities. ppendix A: Comparison between Eqs. (2) - (4) with the multipole expansions The multipole expansions for static field in Ref. [27] (see, Eqs. (8.13) and (10.6)) are given as g = − gr − g r + O (1 /r ) , (A1) g j = − ǫ jpq S p n q r + O (1 /r ) , (A2) g ij = δ ij (cid:18) gr (cid:19) + g r ( δ ij + n i n j ) + O (1 /r ) , (A3)where n i = x i r and ǫ ijk is the completely antisymmetric Minkowskian tensor with ǫ = +1. g (= M = m ) and S i denote the constant mass and angular momentum of the source, respectively.For the case of J = ma e , we rewrite the component g i as g = − ǫ pq S p n q r = − ǫ S n + ǫ S n ) r = 2 max r , (A4) g = − ǫ pq S p n q r = − ǫ S n + ǫ S n ) r = − max r , (A5) g = − ǫ pq S p n q r = 0 . (A6)It can be viewed obviously that Eqs. (A1) - (A6) are consistent with Eqs. (2) - (4). ppendix B: Comparison with the spin-induced Li´enard-Wiechert potential Considering the transformation Eqs. (6) - (9) and Eq. (40), we present the spin-induced terms g Sµν in Eqs. (10) - (12) explicitly in terms of the quantities of the background’s rest frame as g S ∼ ma ( v x − v y ) γ (cid:16) r p − v sin θ (cid:17) , (B1) g S ∼ ma h y − v t + v ( v y − v x ) + γv ( v z − v y ) γ +1 i γ (cid:16) r p − v sin θ (cid:17) , (B2) g S ∼ ma h − x + v t + v ( v y − v x ) − γv ( v z − v x ) γ +1 i γ (cid:16) r p − v sin θ (cid:17) , (B3) g S ∼ mav ( v y − v x ) (2 γ + 1) γ ( γ + 1) (cid:16) r p − v sin θ (cid:17) , (B4) g S ∼ − mav h y − v t + γv ( v z − v y ) γ +1 i γ (cid:16) r p − v sin θ (cid:17) , (B5) g S ∼ mav h x − v t + γv ( v z − v x ) γ +1 i γ (cid:16) r p − v sin θ (cid:17) , (B6) g S ∼ − mav ( v y − v x )( γ + 1) (cid:16) r p − v sin θ (cid:17) , (B7) g S ∼ ma h(cid:16) t − γv zγ +1 (cid:17) ( v − v ) + ( v x − v y ) (cid:16) − γv γ +1 (cid:17)i γ (cid:16) r p − v sin θ (cid:17) , (B8) g S ∼ mav n v t − y − γγ +1 (cid:2) v ( v z − v x ) + y ( v − v ) (cid:3)o γ (cid:16) r p − v sin θ (cid:17) , (B9) g S ∼ mav n x − v t + γγ +1 (cid:2) v ( v z − v y ) + x ( v − v ) (cid:3)o γ (cid:16) r p − v sin θ (cid:17) . (B10)On the other hand, substituting Eqs. (D1) - (D2) in Ref. [4] and Eq. (31) into Eq. (47), wewrite the explicit components of the perturbation with J = ma e as follow: h S = 4 ma ( x v − x v ) (cid:16) r p − v sin θ (cid:17) , (B11) S i = − n r ( v × J ) i + ma ( x v − x v ) v i +( x ǫ ik + x ǫ ik + x ǫ ik ) h J k − v k γγ +1 ( v · J ) io(cid:16) r p − v sin θ (cid:17) , (B12) h Sij = 2 n ( x ǫ ik + x ǫ ik + x ǫ ik ) J k v j +( x ǫ jk + x ǫ jk + x ǫ jk ) J k v i − γγ + 1 × ( v · J ) [( x ǫ ik + x ǫ ik + x ǫ ik ) v k v j + ( x ǫ jk + x ǫ jk + x ǫ jk ) v k v i ]+ r h ( v × J ) i v j + ( v × J ) j v i i o / (cid:16) r p − v sin θ (cid:17) , (B13)where i, j, k = 1 , r α = ( − r, r ) = ( − t, x , x , x ) and the symbol of spin-dependence S has been shifted up for display convenience. Namely, h S = 4 ma ( v x − v x ) (cid:16) r p − v sin θ (cid:17) , (B14) h S = 2 ma h x − v t + v ( v x − v x ) + γv ( v x − v x ) γ +1 i(cid:16) r p − v sin θ (cid:17) , (B15) h S = 2 ma h − x + v t + v ( v x − v x ) − γv ( v x − v x ) γ +1 i(cid:16) r p − v sin θ (cid:17) , (B16) h S = 2 mav ( v x − v x ) (2 γ + 1)( γ + 1) (cid:16) r p − v sin θ (cid:17) , (B17) h S = 4 mav h v t − x + γv ( v x − v x ) γ +1 i(cid:16) r p − v sin θ (cid:17) , (B18) h S = 4 mav h x − v t + γv ( v x − v x ) γ +1 i(cid:16) r p − v sin θ (cid:17) , (B19) h S = 4 mav ( v x − v x ) γ ( γ + 1) (cid:16) r p − v sin θ (cid:17) , (B20) h S = 2 ma h(cid:16) t − γv x γ +1 (cid:17) ( v − v ) + ( v x − v x ) (cid:16) − γv γ +1 (cid:17)i(cid:16) r p − v sin θ (cid:17) , (B21) h S = 2 mav n v t − x − γγ +1 (cid:2) v ( v x − v x ) + x ( v − v ) (cid:3)o(cid:16) r p − v sin θ (cid:17) , (B22) h S = 2 mav n x − v t + γγ +1 (cid:2) v ( v x − v x ) + x ( v − v ) (cid:3)o(cid:16) r p − v sin θ (cid:17) . 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