Second-order time delay by a radially moving Kerr-Newman black hole
SSecond-order time delay by a radially moving Kerr-Newman black hole
Guansheng He and Wenbin Lin ∗ School of Physical Science and Technology, Southwest Jiaotong University,Chengdu 610031, People’s Republic of China
Abstract
We derive the analytical time delay of light propagating in the equatorial plane and parallel to the velocityof a moving Kerr-Newman black hole up to the second post-Minkowskian order via integrating the nullgeodesic equations. The velocity effects are expressed by a very compact form. We then concentrate onanalyzing the magnitudes of the correctional effects on the second-order contributions to the delay anddiscuss their possible detection. Our result in the first post-Minkowskian approximation is in agreementwith Kopeikin and Sch¨afer’s formulation which is based on the retarded Li´enard-Wiechert potential.
PACS numbers: ∗ [email protected] a r X i v : . [ g r- q c ] J u l . INTRODUCTION The time dependence of a background field caused by the translational motion of a gravitationalsource usually exerts an influence on the propagation of electromagnetic waves. This kind of kine-matical effect (also called the velocity effect [1]) on the gravitational time delay of light has beeninvestigated in detail in the last two decades [2–11]. In particular, Kopeikin and Sch¨afer [2] pio-neered the Lorentz-covariant theory for light propagating in the gravitational field of an ensembleof arbitrarily moving bodies, in which the generalized form of the Shapiro time delay [12, 13] wasobtained in the first post-Minkowskian (1PM) approximation. Their calculations were based onthe Li´enard-Wiechert gravitational potential and later extended in Ref. [3] to investigate the spin-dependent gravitomagnetic effects. Sereno [4, 6] employed Fermat’s principle [14] to study thegravitational lensing caused by a slowly moving, spinning body in the framework of the standardlens theory, including the kinematically correctional effects on the light delay. Recently, the timetransfer functions proposed by Teyssandier and Le Poncin-Lafitte [15] were applied to calculatethe observable relativistic effects containing the Shapiro effect in the field of moving axisymmetricbodies [10]. This approach was also used to deal with the light delay due to a moving gravitationalsource with a low velocity and arbitrary multipoles [11].These previous surveys of the velocity corrections were mainly aimed at the first-order grav-itational signals delay. The magnitudes of these correctional effects are so relatively large thatthey are very likely to be detected, whether the motion of the gravitational source is relativisticor not. Not to be forgotten, the nonrelativistic velocity effects (appearing as extrinsic gravitomag-netic effects [7, 16]) were confirmed by the Jovian deflection experiment in 2002 [5, 7, 17], withan accuracy of . As is known, nowadays techniques in the Shapiro delay measurements havemade rapid progresses and achieved a high precision at the picosecond ( ps ∼ − s ) level oreven better. For example, the delay precision of the next generation of the Very-Long-BaselineInterferometry (VLBI) system was proposed to be ps [18, 19]. One can expect that the kinemat-ical effects on the second-order time delay might also be detectable and, therefore, deserve ourattentions. It requires full theoretical treatment of the gravitational retardation effect induced by amoving lens in the second post-Minkowskian (2PM) approximation.In the present paper, we investigate the velocity effects on the second-order gravitational delayof light propagating in the equatorial plane of a moving Kerr-Newman (KN) black hole with aconstant radial velocity, which serves as a natural extension of our previous result [20]. We restrict2ur discussions to the weak-field, small-angle, and thin-lens approximation. For the convenienceof the computations, we will define the impact parameter by assuming that light signals come frominfinity with an initial velocity being parallel to the black hole’s velocity.This paper is organized as follows. In Sec. II, we first review the weak-field metric of themoving KN source, and then derive the explicit time delay up to the 2PM order. Section III isdevoted to estimating the magnitudes of the velocity effects on the delay for three typical cases ofthe lens’ mass. In Sec. IV, we discuss the possibility of detecting the correctional effects on thesecond-order contributions to the delay. The summary is given in Sec. V. In what follows, we usenatural units in which G = c = 1 . II. SECOND-ORDER MOVING KERR-NEWMAN TIME DELAYA. The 2PM metric for a moving KN source with a constant radial velocity
The second post-Minkowskian metric of a radially moving Kerr-Newman black hole can beobtained from the harmonic Kerr-Newman metric [21] via a Lorentz boost transformation. Weassume { e , e , e } to be the orthonormal basis of a three-dimensional Cartesian coordinatesystem. Let ( t, x, y, z ) and ( X , X , X , X ) denote the rest frame of the background and thecomoving frame for the barycenter of the gravitational source, respectively. The 2PM harmonicmetric of a moving KN black hole with a constant radial velocity v = v e can be written as [22] g = −
1+ 2(1+ v ) γ MR − M + γ ( M + Q ) R − vγ aM X R + v γ ( M − Q ) X R + O ( G ) , (1) g i = γζ i − vγ (cid:18) MR − M + Q R (cid:19) δ i − vγ ( M − Q ) X [ X i +( γ − X δ i ] R + ( γ − γ + v γ ) × aM X δ i R + O ( G ) , (2) g ij = (cid:18) MR (cid:19) δ ij + v γ (cid:18) MR − M + Q R (cid:19) δ i δ j − v γ (cid:20) ζ i δ j + ζ j δ i + 4( γ − aM X δ i δ j R (cid:21) + ( M − Q ) [ X i + ( γ − X δ i ] [ X j + ( γ − X δ j ] R + O ( G ) , (3)where i and j take values among the set { , , } , δ ij denotes the Kronecker delta, and γ =(1 − v ) − is the Lorentz factor. M , Q , and J (= J e ) are the rest mass, electrical charge, and an-gular momentum vector of the gravitational source, respectively. Φ ≡ − MR represents Newtoniangravitational potential, with X + X R + a + X R = 1 and X · d X ≡ X dX + X dX + X dX . a ≡ JM is theangular momentum per mass, and ζ ≡ aMR ( X × e ) = ( ζ , ζ , . The relation M ≥ a + Q oving KN black holeb J v × Oz unperturbed pathperturbed pathA H x A , y A , 0 L B H x B , y B , 0 L x A x B source plane lens plane observer plane xy FIG. 1. Schematic model for light propagating in the gravitational field of the moving KN black hole. Thegravitational deflection is greatly exaggerated to distinguish the blue line (perturbed path) from the dashedhorizontal line (unperturbed path). Light is supposed to take the prograde motion relative to the rotation J of the gravitational source. is assumed to avoid naked singularity for the black hole. Notice that the coordinates X , X , X ,and X are related to t, x, y , and z by the Lorentz transformation as follow X = γ ( t − vx ) , (4) X = γ ( x − vt ) , (5) X = y , (6) X = z . (7) B. Second-order time delay caused by the moving KN black hole
We consider the gravitational time delay of light caused by the moving Kerr-Newman blackhole. For simplicity, light signals are assumed to propagate in the equatorial plane of the black4ole ( z = ∂∂z = 0) . In contrast to the gravitational deflection case where the light emitter andreceiver can be set at infinity, for the time delay, the emitter and receiver cannot be located atinfinity, otherwise the time delay will become infinity. Hence, we set the emitter and receiver tobe located at the points A and B , respectively, both of which are far away from the lens.The schematic model for light propagation is shown in Fig. 1. The spatial coordinates of thelight emitter (denoted by A on the source plane) and receiver (denoted by B on the observer plane)are assumed to be ( x A , y A , and ( x B , y B , , respectively, in the background’s rest frame, where y A < , x A < and x B > . b denotes the impact parameter which is strictly defined as follows.Let the blue line represent the propagation path of a photon coming from p = −∞ with the initialvelocity w | p →−∞ (= e ) being parallel to the central mass’s velocity v , where p denotes the affineparameter of the trajectory [23, 24]. Then, the impact parameter is defined via the y coordinateof a photon as b ≡ − y | p →−∞ , which denotes the geometrical distance between the x axis andunperturbed path of light. This definition is convenient for the Cartesian coordinate system, andit is a little bit different from the definition by the conservation of the angular momentum of aphoton [25] since the black hole is not static. Notice that the locations of A and B are denoted by ( X A , Y A , and ( X B , Y B , in the comoving frame, respectively.The general form of the null curve is given in the background’s rest frame as ds = g µν dx µ dx ν , (8)where the indices µ and ν run over the values , , , . For light propagating in the equatorialplane, Eq. (8) is reduced to g dt + g dx + g dy + 2 g dtdx + 2 g dtdy + 2 g dxdy , (9)which results in dtdx = − M + √ M − N , (10)with M = 2 (cid:18) g g + g g dydx (cid:19) , (11) N = g g + 2 g g dydx + g g (cid:18) dydx (cid:19) . (12)Here, we have ignored the other solution for its nonphysical property, since N = − O ( G ) < inthe weak-field and small-angle approximation. Note that dydx is related to the gravitational deflectionangle α of light by α = arctan dydx (cid:12)(cid:12) p → + ∞ p →−∞ = arctan dydx (cid:12)(cid:12) x → + ∞ x →−∞ .5e then perform an indefinite integral over x for Eq. (10) and get t = (cid:90) (cid:115) − g g + g + (cid:18) dydx (cid:19) − g g dx , (13)where the third- and higher-order terms (e.g., g g dydx ) have been neglected for the calculations oftime delay up to the second order. Notice that the terms with the factor aMR or Q R (cid:16) ≤ M R (cid:17) inEqs. (1) - (3) are regarded as second-order terms because of the assumption M ≥ a + Q [21,26, 27]. This is a little bit different from the definition of the second-order terms in Ref. [28],where these terms are regarded as first-order terms which are of order O ( G ) .In order to obtain the analytical coordinate time t , we only need to calculate the explicit formof dydx to the first order. We begin with the 1PM equations of motion for light in the gravitationalfield of a moving Schwarzschild black hole with the radial velocity v [29] t + (cid:2) v ( v −
3) ˙ x + 2(1 + v ) ˙ t ˙ x − v (1 + v ) ˙ t (cid:3) γ M X R + O ( G ) , (14) x + (cid:2) (1 − v ) ˙ t + 2 v (1 + v ) ˙ t ˙ x − (1 + v ) ˙ x (cid:3) γ M X R + O ( G ) , (15) y + (cid:2) (1 + v ) ˙ t − v ˙ t ˙ x + (1 + v ) ˙ x (cid:3) γ M X R + O ( G ) , (16)where a dot denotes the derivative with respect to p which is assumed to be x to calculate thefirst-order form of dydx , as done in Ref. [23].With the help of the boundary conditions ˙ t | x →−∞ = 1 and ˙ x | x →−∞ = 1 , we can obtain thezero-order values for ˙ t and ˙ x from Eqs. (14) and (15) as follows: ˙ t = 1 + O ( G ) , (17) ˙ x = 1 + O ( G ) . (18)We then substitute Eqs. (17) and (18) into Eq. (16), and integrate Eq. (16) over x to get theanalytical form for ˙ y , dydx = 2(1 − v ) γMb (cid:32) X (cid:112) X + b (cid:33) + O ( G ) , (19)where the zero-order parameter transformation dX = γ ( dx − vdt ) = (1 − v ) γdx [23] and theboundary conditions (in the comoving frame) ˙ y | X →−∞ = 0 and y | X →−∞ = − b have been used.Notice that in the limit X → + ∞ , the first-order moving Schwarzschild deflection angle can beobtained from Eq. (19) as α = − v ) γMb [23]. 6lugging Eqs. (1) and (3) and (19) into Eq. (13), we have t = (cid:90) (cid:40) (cid:34) v ) γ MR + M − v γ ( M + Q ) R − v γ a M yR + γ ( M − Q ) X R − v ) γ MR + M + γ ( M + Q ) R + v γ a M yR − v γ ( M − Q ) X R + 16 v γ M R + 4(1 − v ) γ M b (cid:32) X (cid:112) X + b (cid:33) (cid:35) − vγ (cid:104) MR − M + Q R + ( M − Q ) X R (cid:105) − v ) γ aMyR − v ) γ MR (cid:41) dx = (cid:90) (cid:40)(cid:34)
1+ 4(1+ v ) γ MR + 8(1+ v ) γ M +16 v γ M − (1+ v ) γ ( M + Q ) R − vγ aM yR + (1 + v ) γ ( M − Q ) X R + 4 (1 − v ) γ M b (cid:32) X (cid:112) X + b (cid:33) (cid:35) − vγ ( M − Q ) X R − v γ MR − v (1 + v ) γ M − v γ ( M + Q ) R + 2 (1 + v ) γ a M yR (cid:41) dx = (cid:90) (cid:40) − v ) γ MR + (1 − v ) [ (3 − v ) M − (1+ v ) Q ]2(1 + v ) R + (1 − v ) γ ( M − Q ) X R + 2(1 − v ) γ M b (cid:32) X (cid:112) X + b (cid:33) + 2 ( 1 − v ) γ a M yR (cid:41) dx , (20)where the third- and higher-order terms have been ignored.To get the explicit form of y up to the 1PM order on the right-hand side of Eq. (20), we integrateEq. (19) over x , y = − b − M (cid:16)(cid:112) X + b + X (cid:17) b + O ( G ) , (cid:32) M ( (cid:112) X + b + X ) b (cid:28) (cid:33) , (21)where the zero-order approximation dX = (1 − v ) γdx and the boundary condition y | X →−∞ = − b have been used. Substituting Eq. (21) into Eq. (20), we can obtain t = (cid:90) (cid:40)
1+ 2 (1 − v ) γ M (cid:112) X + b (cid:114) − M (cid:16)(cid:112) X + b + X (cid:17) / ( X + b ) + (1 − v ) [(3 − v ) M − (1+ v ) Q ]2 ( 1 + v ) ( X + b )+ 2(1 − v ) γ M b (cid:32) X (cid:112) X + b (cid:33) + (1 − v ) γ ( M − Q ) X X + b ) + 2(1 − v ) γ aM y ( X + b ) (cid:41) dx = x + (1 − v ) γ (cid:90) (cid:34) M (cid:112) X + b + 4 M (cid:16)(cid:112) X + b + X (cid:17) ( X + b ) + (3 − v ) M v − Q X + b ) + ( M − Q ) X X + b ) + 2 M b (cid:32) X (cid:112) X + b (cid:33) − aM b ( X + b ) (cid:35) dx , (22)7ith the third- and higher-order terms being dropped in the derivation.In order to integrate the second part on the right-hand side of Eq. (22) more conveniently, weperform a coordinate transformation dX = γ (1 − v ˙ t/ ˙ x ) dx which is to be calculated up to the firstpost-Minkowskian order. We substitute Eqs. (17) and (18) into Eqs. (14) and (15), and integratethe latter over x to obtain the explicit forms of ˙ t and ˙ x up to the 1PM order as follows: ˙ t = 1 + 2(1 − v ) γ M (cid:112) X + b + O ( G ) , (23) ˙ x = 1 + 2 v (1 − v ) γ M (cid:112) X + b + O ( G ) . (24)Thus, we can get dX = γ (cid:34) − v (cid:32) − v ) γ M (cid:112) X + b (cid:33) + O ( G ) (cid:35) dx = (1 − v ) γ (cid:34) − v (1 − v ) γ M (cid:112) X + b + O ( G ) (cid:35) dx . (25)Notice that up to the zero order Eq. (25) reduces to dX = (1 − v ) γdx , which is enough to dealwith the 1PM gravitational deflection [23]. Plugging Eq. (25) into Eq. (22), we have t = x + (1 − v ) γ (cid:90) (cid:34) M (cid:112) X + b + 2 M b (cid:32) X (cid:112) X + b (cid:33) + 3 M − Q X + b ) + 4 M (cid:16)(cid:112) X + b + X (cid:17) ( X + b ) + ( M − Q ) X X + b ) − aM b ( X + b ) (cid:35) dX = x + (1 − v ) γ (cid:34) M ln (cid:18)(cid:113) X + b + X (cid:19) + 4 M (cid:16)(cid:112) X + b + X (cid:17) b + 3(5 M − Q )4 b ArcT an X b − M (cid:112) X + b − ( M − Q ) X X + b ) − aM X b (cid:112) X + b (cid:35) + C , (26)where C denotes the integral constant and the third- and higher-order terms have been dropped.Finally, the explicit form of the time delay up to the second order for light propagating fromthe light emitter A to the receiver B can be calculated from Eq. (26) as follows: t ( B, A ) = ( x B − x A )+(1 − v ) γ (cid:34) M ln (cid:32)(cid:112) X B + b + X B (cid:112) X A + b + X A (cid:33) +4 M (cid:32) (cid:112) X A + b − (cid:112) X B + b (cid:33) + 4 M b (cid:18) X B − X A + (cid:113) X B + b − (cid:113) X A + b (cid:19) + M − Q (cid:18) X A X A + b − X B X B + b (cid:19) + 15 M − Q b (cid:18) arctan X B b − arctan X A b (cid:19) + 2 aMb (cid:32) X A (cid:112) X A + b − X B (cid:112) X B + b (cid:33)(cid:35) , (27)where X A = γ ( x A − vt A ) and X B = γ ( x B − vt B ) .8 . Discussion of the result The first term on the right-hand side of Eq. (27), which is independent of the gravitationalsource, represents the geometrical time for light traveling in a straight line. In the first post-Minkowskian approximation, Eq. (27) reduces to t ( B, A ) = ( x B − x A ) + 2(1 − v ) γM ln (cid:32) (cid:112) X B + b + X B (cid:112) X A + b + X A (cid:33) . (28)For a nonmoving Kerr-Newman source, Eq. (27) can be simplified to [20] t ( B, A ) = ( x B − x A )+2 M ln (cid:32) (cid:112) x B + b + x B (cid:112) x A + b + x A (cid:33) + 8 M x B b + 15 πM b − aMb − πQ b , (29)where the assumptions x A (cid:28) − b and x B (cid:29) b have been considered, and the second-order termswith the factor x A or x B have been dropped since they are found to be much smaller than thosecontaining the factor b .Provided the angular momentum and electrical charge are dropped from the gravitational source ( a = Q = 0) , Eq. (27) is reduced to the second-order moving Schwarzschild time delay, whichreads t ( B, A ) = ( x B − x A )+(1 − v ) γ (cid:34) M ln (cid:32)(cid:112) X B + b + X B (cid:112) X A + b + X A (cid:33) +4 M (cid:32) (cid:112) X A + b − (cid:112) X B + b (cid:33) + 4 M b (cid:18) X B − X A + (cid:113) X B + b − (cid:113) X A + b (cid:19) + 15 M b (cid:18) arctan X B b − arctan X A b (cid:19) + M (cid:18) X A X A + b − X B X B + b (cid:19)(cid:21) . (30)In regard to Eq. (27), we emphasize two points. First, the analytical 1PM forms for X A and X B in Eq. (27) can be determined by the iteration technique as follows:We adopt an alternative method to determine the integral constant C in Eq. (26) by imposing t = t A and x = x A . Without loss of generality, we set t A = x A + O ( G ) and obtain X A = (1 − v ) γx A + O ( G ) , (31) C = − − v ) γM ln (cid:18)(cid:113) X A + b + X A (cid:19) + O ( G )= − − v ) γM ln (cid:20)(cid:113) (1 − v ) γ x A + b + (1 − v ) γx A (cid:21) + O ( G ) . (32)In addition, Eq. (26) up to the 0PM order leads to t B = x B + O ( G ) . (33)9ubstituting Eqs. (32) and (33) into Eq. (26), up to the 1PM order, we have t B = x B + 2(1 − v ) γM ln (cid:34) (cid:112) (1 − v ) γ x B + b + (1 − v ) γx B (cid:112) (1 − v ) γ x A + b + (1 − v ) γx A (cid:35) + O ( G ) , (34)where X B = γ ( x B − vt B ) = (1 − v ) γx B + O ( G ) has been used. From Eq. (34), the 1PM form of X B can be expressed as X B = (1 − v ) γx B − vM v ln (cid:34) (cid:112) (1 − v ) γ x B + b + (1 − v ) γx B (cid:112) (1 − v ) γ x A + b + (1 − v ) γx A (cid:35) + O ( G ) . (35)Considering Eqs. (31) and (35), we can express Eq. (27) by the quantities in the background’srest frame ( t, x, y, z ) as follows t ( B, A ) = ( x B − x A ) + (1 − v ) γ (cid:40) M ln (cid:34) (cid:112) (1 − v ) γ x B + b + (1 − v ) γ x B (cid:112) (1 − v ) γ x A + b + (1 − v ) γ x A (cid:35) + 4 M b × (cid:20)(cid:113) (1 − v ) γ x B + b − (cid:113) (1 − v ) γ x A + b + (1 − v ) γ ( x B − x A ) (cid:21) + 15 M − Q b × (cid:20) arctan (1 − v ) γx B b − arctan (1 − v ) γx A b (cid:21) − aMb (cid:34) (1 − v ) γx B (cid:112) (1 − v ) γ x B + b − (1 − v ) γx A (cid:112) (1 − v ) γ x A + b (cid:35) − M (cid:34) (cid:112) (1 − v ) γ x B + b − (cid:112) (1 − v ) γ x A + b (cid:35) − M − Q (cid:20) (1 − v ) γ x B (1 − v ) γ x B + b − (1 − v ) γ x A (1 − v ) γ x A + b (cid:21) − v (1 − v ) γ M (cid:112) (1 − v ) γ x B + b ln (cid:34) (cid:112) (1 − v ) γ x B + b + (1 − v ) γx B (cid:112) (1 − v ) γ x A + b + (1 − v ) γx A (cid:35)(cid:41) , (36)where two series expansions have been performed and the third- and higher-order terms have beendropped. Notice that Eq. (36) is valid for both nonrelativistic and relativistic (such as v = 0 . )motions of the gravitational source. In the limit | x A | (cid:29) b and x B (cid:29) b , Eq. (36) is reduced to t ( B, A ) = ( x B − x A ) + (1 − v ) γ (cid:40) M ln (cid:34) (cid:112) (1 − v ) γ x B + b +(1 − v ) γx B (cid:112) (1 − v ) γ x A + b +(1 − v ) γx A (cid:35) + 8 (1 − v ) γM x B b + 15 πM b − aMb − π Q b (cid:27) , (37)where the second-order terms with the factor x A or x B have been dropped for the same reasonmentioned above.Second, for using Eq. (27), we can replace the impact parameter b by the coordinates x A and y A to express the time delay up to the 2PM order. From Eqs. (21) and (31), we obtain the explicit10orm of b up to the 1PM order by the iteration technique as b = − y A (cid:32) M (cid:112) X A + y A − X A (cid:33) + O ( G )= − y A (cid:34) M (cid:112) (1 − v ) γ x A + y A − (1 − v ) γx A (cid:35) + O ( G ) . (38)Plugging Eq. (38) into Eq. (36), up to the 2PM order, we can rewrite Eq. (36) as follow: t ( B, A ) = ( x B − x A ) + (1 − v ) γ (cid:40) M ln (cid:34) (cid:112) (1 − v ) γ x B + y A + (1 − v ) γ x B (cid:112) (1 − v ) γ x A + y A + (1 − v ) γ x A (cid:35) + 4 M y A × (cid:20)(cid:113) (1 − v ) γ x B + y A − (cid:113) (1 − v ) γ x A + y A + (1 − v ) γ ( x B − x A ) (cid:21) + 15 M − Q y A × (cid:20) arctan (1 − v ) γx B y A − arctan (1 − v ) γx A y A (cid:21) + 2 aMy A (cid:34) (1 − v ) γ x B (cid:112) (1 − v ) γ x B + y A − (1 − v ) γ x A (cid:112) (1 − v ) γ x A + y A (cid:35) − M (cid:34) (cid:112) (1 − v ) γ x B + y A − (cid:112) (1 − v ) γ x A + y A (cid:35) − M − Q (cid:20) (1 − v ) γ x B (1 − v ) γ x B + y A − (1 − v ) γ x A (1 − v ) γ x A + y A (cid:21) − v (1 − v ) γ M (cid:112) (1 − v ) γ x B + y A ln (cid:34) (cid:112) (1 − v ) γ x B + y A + (1 − v ) γx B (cid:112) (1 − v ) γ x A + y A + (1 − v ) γx A (cid:35) + 4 M (cid:112) (1 − v ) γ x B + y A (cid:112) (1 − v ) γ x B + y A − (1 − v ) γx B (cid:112) (1 − v ) γ x A + y A − (1 − v ) γx A − M (cid:112) (1 − v ) γ x A + y A (cid:41) . (39)Correspondingly, for the case of | x A | (cid:29) | y A | and x B (cid:29) | y A | , Eq. (39) can be simplified to t ( B, A ) = ( x B − x A ) + (1 − v ) γ (cid:26) M ln (cid:20) − − v ) γ x A x B y A (cid:21) + 8 (1 − v ) γM x B y A − πM y A + 4 aMy A + 3 π Q y A (cid:27) , (40)where the the second-order terms with the factor x A or x B have been ignored. III. MAGNITUDES OF THE KINEMATICALLY CORRECTIONAL EFFECTS ON THE DELAY
In this section we analyze the magnitudes of the velocity effects on the time delay. We considernonrelativistic as well as relativistic cases for the motion of the gravitational source, since thereare some celestial bodies moving with a high radial velocity [30–32]. For illustration, we takeEq. (40) as an example. In order to evaluate their magnitudes, we follow the notations in Ref. [29]11 ∆ F M ( v ) ∆ SM − ( v ) ∆ SM − ( v ) ∆ a ( v ) ∆ Q ( v ) µs µs ns − ps − f s µs ns ps − ps (cid:63) µs ns ps − f s (cid:63) ns ps f s (cid:63) (cid:63) ps f s (cid:63) (cid:63) (cid:63) TABLE I. The magnitudes of the kinematical corrections to the light delay for various v , with the lens’s mass M being M (cid:12) . Hereafter, our attention is concentrated on the absolute values ( ≥ of these magnitudes,and the star “ (cid:63) ” denotes the absolute value which is less than f s . We present the case with non-zeroelectrical charge mainly for illustration. and present the general forms of the velocity-induced correctional effects in Eq. (40) as follows ∆ F M ( v ) = 2 M (cid:40) ln (cid:18) − x A x B y A (cid:19) − (cid:114) − v v ln (cid:20) − − v ) x A x B (1 + v ) y A (cid:21)(cid:41) , (41) ∆ SM − ( v ) = 16 v M x B (1 + v ) y A , (42) ∆ SM − ( v ) = − (cid:32) − (cid:114) − v v (cid:33) π M y A , (43) ∆ a ( v ) = (cid:32) − (cid:114) − v v (cid:33) a My A , (44) ∆ Q ( v ) = (cid:32) − (cid:114) − v v (cid:33) π Q y A , (45)where ∆ F M ( v ) , ∆ a ( v ) , and ∆ Q ( v ) denote the velocity corrections to the first-order Schwarzschild,second-order Kerr, and charge-induced terms on the right-hand side of Eq. (29), respectively. ∆ SM − ( v ) and ∆ SM − ( v ) represent the velocity corrections to the larger and smaller second-orderSchwarzschild contributions to the delay in Eq. (29), respectively.As an example, the related parameters are preset as follows. We set y A = − . × M to guarantee a weak field. x B and | x A | are much larger than | y A | and set to be x B = − x A =1 . × M (= − y A ) , a = 0 . M , and Q = 0 . M . Notice that for certain parameters y A , x A , x B , a , and Q , the correctional effects defined in Eqs. (41) - (45) are not only dependent on the radialvelocity v but also proportional to the mass M of the gravitational source. Moreover, it is generallybelieved that there are three classifications for black holes by their masses in our Universe, i.e.,12 ∆ F M ( v ) ∆ SM − ( v ) ∆ SM − ( v ) ∆ a ( v ) ∆ Q ( v ) s µs ns − ns − ps s µs ns − ns − ps µs ns ps − ps − f s µs ns ps − f s (cid:63) ns ps f s (cid:63) (cid:63) TABLE II. The magnitudes of the kinematical corrections for the lens with an intermediate mass M =1000 M (cid:12) . v ∆ F M ( v ) ∆ SM − ( v ) ∆ SM − ( v ) ∆ a ( v ) ∆ Q ( v ) s s µs − µs − ns s s µs − µs − ns s µs µs − ns − ps s µs ns − ps − f s µs ns ps − ps (cid:63) TABLE III. The magnitudes of the kinematical corrections for the lens with a supermassive mass M =4 . × M (cid:12) . stellar-mass ( M ∼ − M (cid:12) ), intermediate-mass ( M ∼ − M (cid:12) ), and supermassive ( M ∼ − M (cid:12) ) black holes [33, 34]. Therefore, we typically assume that the rest mass M of themoving KN black hole to be , , and . × M (cid:12) in Tabs. I - III, respectively, to show themagnitudes of these correctional effects for various velocities of the lens, with M (cid:12) (= 1 . km ) being the mass of the Sun. IV. POSSIBLE APPLICATIONS AND DETECTION OF THE VELOCITY EFFECTS ON THESECOND-ORDER DELAY
Velocity effects on the second-order gravitational delay may influence on the high-precisionmeasurements of some crucial parameters, such as the post-Newtonian parameters [7–9], andHubble’s constant when measured via the time delay between two lensed images [35–38]. Forexample, the relation between the time delay and the post-Newtonian parameters in the time-dependent gravitational field has been given in Ref. [9] (see Eq. (71) therein). The velocity effects13n the static time delay make the observed numerical values of these post-Newtonian parametersbiased. Therefore, it is worthwhile to consider their possible detection.Based on the results shown in Tabs. I - III, we analyze the possibility of detecting the velocityeffects qualitatively. For the convenience of our discussion, we fix the parameters x A , x B , and y A as given above. We focus on the velocity effects on the second-order contributions to the delay,since the correctional effect on the first-order delay has been studied in detail.We first consider ∆ SM − ( v ) . For a supermassive black hole with mass M = 4 . × M (cid:12) ,Tab. III shows that the kinematically correctional effect ( > ns ) on the larger second-orderSchwarzschild contribution to the delay is much larger than the accuracy ( ∼ ps ) of today’s high-precision techniques, for both relativistic and nonrelativistic motions of the black hole. For exam-ple, ∆ SM − ( v ) is about . ns when this black hole moves at an extremely low radial velocity v = 30 m/s . Thus, the possibility of detecting the correctional effect ∆ SM − ( v ) might be verylarge for moving supermassive black holes. This conclusion also holds for moving black holeswith an intermediate mass, since the magnitude of ∆ SM − ( v ) in the case of M = 1000 M (cid:12) islarger than ps for almost all of the range of the radial velocity v , as presented in Tab. II. Even fora stellar-mass black hole moving with a low radial velocity, we find there exists the possibility todetect ∆ SM − ( v ) . For instance, ∆ SM − ( v ) still exceeds ps for the case of v = 3 km/s in Tab. I.We then discuss the correctional effect on the smaller second-order Schwarzschild contributionto the delay, i.e., ∆ SM − ( v ) . From Tab. III, we notice that ∆ SM − ( v ) can be larger than ps fora supermassive black hole ( M = 4 . × M (cid:12) ) moving at a very low velocity. Thus, it is stillvery likely to detect ∆ SM − ( v ) for moving supermassive black holes. When the mass of the lensdecreases from the supermassive class to the stellar class, the motion of the gravitational source hasto change from a nonrelativistic case to a relativistic case, for the possible detection of ∆ SM − ( v ) .Rotating black holes such as the Sagittarius A ∗ (a supermassive black hole) in the Galacticcenter [39] are very common in our universe. It is also necessary to take the velocity effect onthe Kerr delay into account. Tabs. I and II indicate that it is possible to detect ∆ a ( v ) only whenthe lens’ motion tends to be relativistic for stellar-mass and intermediate-mass black holes. Incontrast to these relatively small black holes, we might observe ∆ a ( v ) for supermassive blackholes with a nonrelativistic radial velocity. For example, ∆ a ( v ) can largely exceed ps for thecase of M = 4 . × M (cid:12) and v = 3 km/s .With respect to the correctional effect on the charge-induced contribution to the delay, namely ∆ Q ( v ) , we have to conclude that there may not be any chance for its possible detection via today’s14echniques. As shown in Tab. III, even for a supermassive black hole with M = 4 . × M (cid:12) and v = 300 km/s , ∆ Q ( v ) is still a little larger than ps . In addition, the original charge ofa black hole in the Universe might have been neutralized or become very small in most cases,which makes the detection more difficult. However, recently, some new techniques to detect tinytime delays of light in the framework of special relativity have been proposed [40, 41], with anunprecedented ultimate precision limit which is much less than f s . There is the possibility thatthey are developed into the astronomical domain for the measurements of Shapiro delays, and ∆ Q ( v ) might also be observed at that time. V. SUMMARY
In this work we calculate the second-order gravitational time delay of light propagating inthe equatorial plane and parallel to the velocity of a constantly moving Kerr-Newman black hole,based on the 2PM harmonic metric. With respect to the velocity effects, we find that the relativisticcorrectional factor (1 − v ) γ applies not only to the first-order term but also to all the second-orderterms in the time delay. We also analyze the magnitudes of the correctional effects on the second-order contributions to the delay and their possible detections. We conclude that it is likely todetect the velocity effects on the second-order Schwarzschild and Kerr contributions to the delayby today’s high-accuracy techniques such as the VLBI. Our result and conclusions might be usefulin future astronomical observations. ACKNOWLEDGEMENTS
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