Radio pulsars emit highly polarized radiation whose polarization properties are commonly described by the Rotating Vector Model (RVM). In this work, we investigate the effects of gravitational light bending and parallel transport on the polarization profile of a pulsar in a hypothetical binary system with a black hole (BH) companion. We combine the RVM with numerical integration of null geodesics to calculate the polarization direction along the photon trajectory and model the orbital-phase-dependent variation of the observed polarization position angle (PPA). We consider both non-spinning and spinning black hole companions and include the evolution of the orbital geometry due to relativistic spin-orbit coupling. For a non-spinning black hole, the change in the observed polarization direction arises from the geometric shift associated with gravitational light bending. For a spinning black hole, frame dragging introduces an additional geometric shift, while parallel transport produces an additional polarization rotation measured relative to our chosen reference construction. We investigate the dependence of these polarization changes on the binary parameters and the orientation of the pulsar spin. We find that gravitational light bending can significantly distort the characteristic S-shaped PPA profile, particularly near superior conjunction, and can reverse its overall sense of variation for the supermassive black hole system considered here. Relativistic spin-orbit evolution also affects the orbital-phase dependence of the polarization profile, particularly in the supermassive black hole system. These results show that relativistic photon propagation can modify both the shape and the phase dependence of pulsar polarization profiles and may provide a useful probe of the geometry of pulsar–BH binaries when such systems are discovered.
Introduction
Radio pulsars are rapidly rotating neutron stars that emit highly polarized radio radiation [1 ] . The polarization position angle (PPA) across the pulse profile provides information about the geometry of the pulsar emission region and the orientations of its magnetic and rotation axes [2 ] . The observed PPA is commonly described by the Rotating Vector Model (RVM), which predicts a characteristic S-shaped variation of the polarization angle across the pulse [3 , 4 ] .
The exceptional timing precision of pulsars, particularly millisecond pulsars, makes them powerful probes of a wide range of astrophysical phenomena [5 ] . Their stable rotational periods enable stringent tests of general relativity [6 , 7 ] , studies of the interstellar medium [8 ] , and searches for low-frequency gravitational waves [9 , 10 ] . Pulsar-BH binaries are of particular interest because their timing and propagation signals can provide information about both the pulsar and the gravitational field of the black hole companion [11 ] . Intermediate-mass black holes (IMBHs) are also expected to occur in globular clusters, although their existence has not yet been firmly established observationally [12 , 13 ] . Pulsar-BH systems in such environments are therefore of particular interest for searches for these otherwise difficult-to-detect black holes. The increasing sensitivity of modern radio telescopes can improve the prospects for finding such systems by enabling pulsar surveys to be carried out with shorter integration times [14 ] .
When a pulsar is part of a binary system, the propagation of its radiation can be significantly affected by the gravitational field of the companion [15 , 16 , 17 , 18 ] . Depending on the nature and geometry of the binary, the observed pulsar signal can be affected by several propagation effects. These include the Römer delay associated with the orbital motion of the pulsar, the Shapiro delay produced by the curvature of spacetime, and the geometric propagation delay [19 ] . In addition, gravitational bending of the light ray introduces a bending delay that depends on the geometry of the pulsar-companion system [15 , 16 ] . These relativistic propagation effects have been studied extensively in the context of pulsar-BH binaries and can be investigated through precise pulsar timing [20 ] .
The effect of gravitational light bending becomes particularly important when the pulsar is observed close to superior conjunction, where the line of sight passes close to the companion [21 ] . Light bending changes both the direction and the path of the light ray between the pulsar and the observer. Previous studies have investigated the latitudinal and longitudinal displacements of the observed signal and the corresponding bending delay in pulsar-BH systems [21 , 22 ] . The change in the direction of the light ray also affects the observed polarization of the pulsar radiation. Since the polarization direction predicted by the RVM depends on the emission direction, gravitational light bending produces an apparent geometric shift in the observed polarization angle.
In addition to this geometric effect, the polarization vector is parallel transported along the null geodesic followed by the radiation in curved spacetime [23 , 24 ] . The propagation of the polarization vector can therefore introduce an additional change in the observed polarization direction [25 , 26 ] . For a spinning black hole, frame dragging also introduces an additional geometric shift in the observed polarization direction. Thus, the polarization changes considered in this work include the geometric contribution associated with gravitational light bending, the additional geometric contribution due to frame dragging, and the polarization rotation associated with parallel transport. Separating these contributions is important for understanding the relativistic modification of the pulsar polarization profile. Although the contributions associated with black hole spin are smaller than the leading geometric shift for the systems considered here, they may become relevant in sufficiently precise polarization measurements. Gravitational light bending can therefore modify the characteristic S-shaped PPA profile predicted by the RVM.
In this work, we study the modification of pulsar polarization caused by relativistic photon propagation in a binary system containing a black hole companion. We combine the RVM description of pulsar emission with a general relativistic calculation of the photon trajectory and the parallel transport of the polarization vector along the null geodesic. This allows us to follow the polarization from the emission point to a distant observer and to quantify the geometric shift and the polarization rotation associated with parallel transport relative to our chosen reference construction. We consider both non-spinning and spinning black hole companions and examine how the resulting polarization changes depend on the binary parameters and the orientation of the pulsar spin. We also investigate the effect of relativistic spin-orbit evolution on the orbital-phase dependence of the polarization profile.
The paper is organized as follows. In Section II, we describe the pulsar emission geometry and the polarization direction within the RVM. In Section III, we introduce the spin-orbit dynamics and describe the binary geometry and the coordinate frames used in our calculation. In Section IV, we define the different contributions to the polarization change and describe how they are quantified. In Section V, we present the full general relativistic formalism used to calculate the null geodesic and parallel transport the polarization vector along the photon trajectory. In Section VI, we present the numerical results for two representative systems: an intermediate-mass black hole companion with a mass of 10 5 M ⊙ 10^{5}\,M_{\odot} 1 0 5 M ⊙
Polarization model of an Isolated Pulsar
Pulsar Emission and Polarization Beam Geometry
The classic “lighthouse model” of pulsar emission, proposed by Radhakrishnan and Cooke [3 ] , provides the standard geometrical framework for understanding the polarization properties of radio pulsars. In this model, a pulsar is treated as a rotating magnetic dipole, and relativistic charged particles are assumed to stream along the open magnetic field lines originating from the magnetic poles [27 , 28 , 29 , 30 , 31 , 32 ] . Due to the curvature of these magnetic field lines, the particles emit highly beamed radiation along the local tangent to the magnetic field [33 , 34 , 35 ] . This mechanism is known as curvature radiation. The emitted radiation is highly polarized, with its polarization direction determined by the acceleration of the radiating particles [36 ] .
In Fig. 1, we illustrate the geometric configuration of the pulsar emission and polarization beam. We define a Cartesian reference frame X I Y I Z I X_IY_IZ_I X I Y I Z I I I I Z I Z_I Z I N ^ I \hat{N}_I N ^ I X I Z I X_IZ_I X I Z I ζ N \zeta_N ζ N m ^ I \hat{m}_I m ^ I α \alpha α t = 0 t = 0 t = 0 m ^ I \hat{m}_I m ^ I X I Z I X_IZ_I X I Z I Φ m ( t ) = 2 π t / P s \Phi_m(t) = 2\pi t/P_s Φ m ( t ) = 2 π t / P s P s P_s P s Φ m \Phi_m Φ m X I X_I X I X I Y I X_IY_I X I Y I m ^ I \hat{m}_I m ^ I Γ \Gamma Γ β = ζ N − α \beta= \zeta_N - \alpha β = ζ N − α [37 ] .
As the charged particles stream along the open magnetic field lines, they emit radio radiation along the tangent to the field lines. The characteristic altitude at which the radio emission originates depends on the observing frequency ν \nu ν [38 , 39 ] , the radius-to-frequency mapping for the emission height h ν , e m h_{\nu,\mathrm{em}} h ν , em
h ν , e m = 400 k m ( ν 10 9 H z ) − 0.26 ( P ˙ s 10 − 15 s s − 1 ) 0.07 ( P s s ) 0.30 . (1) h_{\nu,\mathrm{em}}=400\ \mathrm{km}\left(\frac{\nu}{10^9\ \mathrm{Hz}}\right)^{-0.26}\left(\frac{\dot{P}_{\mathrm{s}}}{10^{-15}\ \mathrm{s}\ \mathrm{s}^{-1}}\right)^{0.07}\left(\frac{P_{\mathrm{s}}}{\mathrm{s}}\right)^{0.30}. \tag*{(1)} h ν , em = 400 km ( 1 0 9 Hz ν ) − 0.26 ( 1 0 − 15 s s − 1 P ˙ s ) 0.07 ( s P s ) 0.30 . (1) where P ˙ s \dot{P}_{\mathrm{s}} P ˙ s
In Figure 1 , two light-ray directions n ^ 1 \hat{n}_1 n ^ 1 n ^ 2 \hat{n}_2 n ^ 2 B 1 B_1 B 1 B 2 B_2 B 2 n ^ 1 \hat{n}_1 n ^ 1 n ^ 2 \hat{n}_2 n ^ 2 h ν , e m h_{\nu,\mathrm{em}} h ν , em [40 ] . In this simple dipolar emission model, the physical conal beam is centered on the magnetic axis [41 , 42 ] . Since the emission height is much smaller than the distance to the observer, the geometrical representation of the conal beam can be translated along m ^ I \hat{m}_{I} m ^ I I I I
Figure 1. Schematic diagram of the pulsar emission and polarization beam geometry. The meaning of the various vectors, coordinate frames, and angles is explained in the text.
In Figure 1 , the physical conal beam generated at the emission height h ν , e m h_{\nu,\mathrm{em}} h ν , em I I I W ν W_{\nu} W ν W 1.4 W_{1.4} W 1.4 [21 , 38 , 39 , 43 ] :
W 1.4 = 5.83 ∘ ( P s s ) − 0.5 . (2) W_{1.4}=5.83^\circ\left(\frac{P_{\mathrm{s}}}{\mathrm{s}}\right)^{-0.5}. \tag*{(2)} W 1.4 = 5.8 3 ∘ ( s P s ) − 0.5 . (2) The gravitational transport of the polarization vector considered in the present work does not depend explicitly on the choice of the half-opening angle. The value of W ν W_{\nu} W ν
In this lighthouse model of pulsar emission, the emitted radiation is highly polarized and the polarization direction is related to the acceleration vector of the radiating particles [36 ] . In Figure 1 , the polarization vectors corresponding to the propagation directions n ^ 1 \hat{n}_1 n ^ 1 n ^ 2 \hat{n}_2 n ^ 2 f ⃗ 1 \vec{f}_1 f 1 f ⃗ 2 \vec{f}_2 f 2 I I I a ⃗ \vec{a} a [44 ] :
a ⃗ = c ∂ n ^ ∂ t + c ( v ⃗ ⋅ ∇ ) n ^ + c Ω ⃗ p × n ^ + Ω ⃗ p × ( Ω ⃗ p × r ⃗ 0 ) , (3) \vec{a}=c\frac{\partial\hat{n}}{\partial t}+c(\vec{v}\cdot\nabla)\hat{n}+c\vec{\Omega}_{p}\times\hat{n}+\vec{\Omega}_{p}\times\left(\vec{\Omega}_{p}\times\vec{r}_{0}\right), \tag*{(3)} a = c ∂ t ∂ n ^ + c ( v ⋅ ∇ ) n ^ + c Ω p × n ^ + Ω p × ( Ω p × r 0 ) , (3) where n ^ \hat{n} n ^ Ω ⃗ p \vec{\Omega}_{p} Ω p v ⃗ \vec{v} v r ⃗ 0 \vec{r}_{0} r 0 ∣ r ⃗ 0 ∣ = h ν , e m \lvert\vec{r}_{0}\rvert=h_{\nu,\mathrm{em}} ∣ r 0 ∣ = h ν , em
The third and fourth terms on the right-hand side of Eq. (3) represent the Coriolis and centripetal accelerations associated with the rotation of the pulsar, respectively. In the low-altitude limit, where the emission height is much smaller than the light-cylinder radius (h ν , e m ≪ r L C h_{\nu,\mathrm{em}}\ll r_{\mathrm{LC}} h ν , em ≪ r LC
The standard Rotating Vector Model (RVM) proposed by Radhakrishnan and Cooke [3 ] provides an idealized description of the variation of the polarization direction based on the geometry of a rotating dipolar magnetic field. To establish a simple baseline model for the present work, we restrict our analysis to the spatial curvature of the magnetic field lines and neglect the explicit time-dependent term. Consequently, Eq. (3) reduces to
a ⃗ ≈ c ( v ⃗ ⋅ ∇ ) n ^ . (4) \vec{a}\approx c(\vec{v}\cdot\nabla)\hat{n}. \tag*{(4)} a ≈ c ( v ⋅ ∇ ) n ^ . (4) Since the emitting particles are ultra-relativistic, we further consider v ⃗ ≈ c n ^ \vec{v}\approx c\hat{n} v ≈ c n ^
Construction of the initial direction of the polarization vector from RVM geometry
To evaluate the curvature acceleration given by Eq. (4) , we introduce a corotating Cartesian reference frame X m Y m Z m X_mY_mZ_m X m Y m Z m m m m Z m Z_m Z m m ^ I \hat{m}_{I} m ^ I Z m Z_m Z m Z I Z_I Z I α \alpha α X m X_m X m m m m I I I X m X_m X m X I Y I X_IY_I X I Y I Φ m \Phi_m Φ m X I X_I X I Φ m = ∣ Ω ⃗ p ∣ t \Phi_m=\lvert\vec{\Omega}_{p}\rvert t Φ m = ∣ Ω p ∣ t t t t X I Z I X_IZ_I X I Z I [37 ] , an arbitrary vector V ⃗ m \vec{V}_m V m m m m I I I
V ⃗ I = R Z ( Φ m ) R Y ( α ) V ⃗ m , (5) \vec{V}_{I}=R_Z(\Phi_m)R_Y(\alpha)\vec{V}_m, \tag*{(5)} V I = R Z ( Φ m ) R Y ( α ) V m , (5) where R Z ( θ ) R_Z(\theta) R Z ( θ ) R Y ( θ ) R_Y(\theta) R Y ( θ ) Z Z Z Y Y Y θ \theta θ I I I ζ L \zeta_L ζ L Φ L \Phi_L Φ L ζ L \zeta_L ζ L Φ L \Phi_L Φ L X I X_I X I X I Y I X_IY_I X I Y I [21 ] . The unit vector along the direction of the light ray can therefore be written as
n ^ I = [ sin ζ L cos Φ L , sin ζ L sin Φ L , cos ζ L ] . (6) \hat{n}_I = [\sin\zeta_L \cos\Phi_L, \sin\zeta_L \sin\Phi_L, \cos\zeta_L]. \tag*{(6)} n ^ I = [ sin ζ L cos Φ L , sin ζ L sin Φ L , cos ζ L ] . (6) The corresponding direction in the corotating m m m
n ^ m = R Y ( − α ) R Z ( − Φ m ) n ^ I . (7) \hat{n}_m = R_Y(-\alpha)R_Z(-\Phi_m)\hat{n}_I. \tag*{(7)} n ^ m = R Y ( − α ) R Z ( − Φ m ) n ^ I . (7) Assuming a pure static dipolar magnetic-field configuration, the unit tangent vector to a magnetic field line at the emission point can be written as
n ^ m = 1 N ( 2 cos θ m r ^ m + sin θ m θ ^ m ) , (8) \hat{n}_m = \frac{1}{\mathcal{N}}\left(2\cos\theta_m\hat{r}_m+\sin\theta_m\hat{\theta}_m\right), \tag*{(8)} n ^ m = N 1 ( 2 cos θ m r ^ m + sin θ m θ ^ m ) , (8) where r ^ m \hat{r}_m r ^ m θ ^ m \hat{\theta}_m θ ^ m θ m \theta_m θ m m m m
N = 1 + 3 cos 2 θ m . (9) \mathcal{N}=\sqrt{1+3\cos^2\theta_m}. \tag*{(9)} N = 1 + 3 cos 2 θ m . (9) The unit basis vectors can be written in terms of the Cartesian basis vectors of the m m m
r ^ m = sin θ m cos ϕ m X ^ m + sin θ m sin ϕ m Y ^ m + cos θ m Z ^ m , (10) \hat{r}_m=\sin\theta_m\cos\phi_m\hat{X}_m+\sin\theta_m\sin\phi_m\hat{Y}_m+\cos\theta_m\hat{Z}_m, \tag*{(10)} r ^ m = sin θ m cos ϕ m X ^ m + sin θ m sin ϕ m Y ^ m + cos θ m Z ^ m , (10) θ ^ m = cos θ m cos ϕ m X ^ m + cos θ m sin ϕ m Y ^ m − sin θ m Z ^ m . (11) \hat{\theta}_m=\cos\theta_m\cos\phi_m\hat{X}_m+\cos\theta_m\sin\phi_m\hat{Y}_m-\sin\theta_m\hat{Z}_m. \tag*{(11)} θ ^ m = cos θ m cos ϕ m X ^ m + cos θ m sin ϕ m Y ^ m − sin θ m Z ^ m . (11) Equations (7) and (8) describe the same light-ray direction in the m m m ( θ m , ϕ m ) (\theta_m,\phi_m) ( θ m , ϕ m )
We next calculate the curvature acceleration at this emission point. Using the ultra-relativistic approximation v ⃗ ≈ c n ^ m \vec{v}\approx c\hat{n}_m v ≈ c n ^ m m m m
a ⃗ m = c ( v ⃗ ⋅ ∇ ) n ^ m = c 2 ( n ^ m ⋅ ∇ ) n ^ m . (12) \vec{a}_m=c(\vec{v}\cdot\nabla)\hat{n}_m=c^2(\hat{n}_m\cdot\nabla)\hat{n}_m. \tag*{(12)} a m = c ( v ⋅ ∇ ) n ^ m = c 2 ( n ^ m ⋅ ∇ ) n ^ m . (12) By evaluating the spatial derivatives of the dipolar field configuration given in Eq. (8), at the radio emission height ∣ r ⃗ 0 ∣ = h ν , e m |\vec{r}_0|=h_{\nu,\mathrm{em}} ∣ r 0 ∣ = h ν , em
a ⃗ m = c 2 sin θ m N h ν , e m [ − ( 2 sin θ m N 3 + sin θ m N ) r ^ m + ( 4 cos θ m N 3 + 2 cos θ m N ) θ ^ m ] . (13) \vec{a}_m=\frac{c^2\sin\theta_m}{\mathcal{N}h_{\nu,\mathrm{em}}}\left[-\left(\frac{2\sin\theta_m}{\mathcal{N}^3}+\frac{\sin\theta_m}{\mathcal{N}}\right)\hat{r}_m+\left(\frac{4\cos\theta_m}{\mathcal{N}^3}+\frac{2\cos\theta_m}{\mathcal{N}}\right)\hat{\theta}_m\right]. \tag*{(13)} a m = N h ν , em c 2 sin θ m [ − ( N 3 2 sin θ m + N sin θ m ) r ^ m + ( N 3 4 cos θ m + N 2 cos θ m ) θ ^ m ] . (13) Using the expressions for the spherical basis vectors given in Eq. (10), the acceleration vector can be expressed in terms of the Cartesian basis vectors of the m m m
The resulting acceleration vector is then transformed into the pulsar’s inertial frame using Eq. (5):
a ⃗ I = R Z ( Φ m ) R Y ( α ) a ⃗ m . (14) \vec{a}_I=R_Z(\Phi_m)R_Y(\alpha)\vec{a}_m. \tag*{(14)} a I = R Z ( Φ m ) R Y ( α ) a m . (14) In the present simplified model, the polarization direction of the emitted light ray is taken to be along the acceleration vector of the radiating particles. The corresponding unit polarization vector in the I I I
f ^ I = a ⃗ I ∣ a ⃗ I ∣ . (15) \hat{f}_I=\frac{\vec{a}_I}{|\vec{a}_I|}. \tag*{(15)} f ^ I = ∣ a I ∣ a I . (15) It is worth noting that along the exact magnetic axis (θ m = 0 \theta_m=0 θ m = 0 θ m > 0 \theta_m>0 θ m > 0
Binary Pulsar Setup
Orbital Geometry and Coordinate Systems
To describe the geometry of a pulsar in a binary system, we introduce several Cartesian coordinate systems. These frames are related to each other by rotations and parallel translations and are used to describe the orbital motion, the orientation of the pulsar spin axis, and the direction of the emitted light rays. These frames are described in detail in [37 ] .
We first define the X I ′ Y I ′ Z I ′ X'_IY'_IZ'_I X I ′ Y I ′ Z I ′ X I Y I Z I X_IY_IZ_I X I Y I Z I X I ′ ∥ X I X'_I\parallel X_I X I ′ ∥ X I Y I ′ ∥ Y I Y'_I\parallel Y_I Y I ′ ∥ Y I Z I ′ ∥ Z I Z'_I\parallel Z_I Z I ′ ∥ Z I S ^ p \hat{S}_p S ^ p Z I ′ Z'_I Z I ′
We next introduce the observer-oriented X S Y S Z S X_SY_SZ_S X S Y S Z S s s s Z S Z_S Z S X S Y S X_SY_S X S Y S X S X_S X S
The orbital motion of the binary is described in the X b Y b Z b X_bY_bZ_b X b Y b Z b b b b X b Y b X_bY_b X b Y b X b X_b X b Z b Z_b Z b ι \iota ι Z S Z_S Z S Z b Z_b Z b ω \omega ω A T A_T A T A T + ω A_T+\omega A T + ω
Figure 2 illustrates the relative orientation of these coordinate systems. The plane of the sky, represented by the X s Y s X_{\mathrm{s}}Y_{\mathrm{s}} X s Y s X b Y b X_{\mathrm{b}}Y_{\mathrm{b}} X b Y b X I ′ Y I ′ X'_{\mathrm{I}}Y'_{\mathrm{I}} X I ′ Y I ′
For convenience, we also introduce two additional coordinate systems centered on the two members of the binary. The X p Y p Z p X_{\mathrm{p}}Y_{\mathrm{p}}Z_{\mathrm{p}} X p Y p Z p p p p X c Y c Z c X_{\mathrm{c}}Y_{\mathrm{c}}Z_{\mathrm{c}} X c Y c Z c c c c b b b b b b b b b p p p c c c
For a rotating companion, we additionally define a companion-fixed X b h Y b h Z b h X_{\mathrm{bh}}Y_{\mathrm{bh}}Z_{\mathrm{bh}} X bh Y bh Z bh Z b h Z_{\mathrm{bh}} Z bh S ^ b h \hat{\mathbf{S}}_{\mathrm{bh}} S ^ bh Y b h Y_{\mathrm{bh}} Y bh X b h ′ Y b h ′ Z b h ′ X'_{\mathrm{bh}}Y'_{\mathrm{bh}}Z'_{\mathrm{bh}} X bh ′ Y bh ′ Z bh ′ c c c Z c Z_{\mathrm{c}} Z c Z b h Z_{\mathrm{bh}} Z bh
For a pulsar-BH binary, the mass of the black hole is expected to be much larger than that of the pulsar. Consequently, the barycenter lies close to the center of the black hole and the b b b c c c
The orientation of the pulsar spin axis with respect to the observer is specified by two angles, λ p \lambda_{\mathrm{p}} λ p η p \eta_{\mathrm{p}} η p λ p \lambda_{\mathrm{p}} λ p Z s Z_{\mathrm{s}} Z s η p \eta_{\mathrm{p}} η p X s X_{\mathrm{s}} X s λ b h \lambda_{\mathrm{bh}} λ bh η b h \eta_{\mathrm{bh}} η bh η b h \eta_{\mathrm{bh}} η bh Y b h Y_{\mathrm{bh}} Y bh
The position of the pulsar in the b b b r ⃗ b \vec{r}_{\mathrm{b}} r b r ⃗ ^ b \hat{\vec{r}}_{\mathrm{b}} r ^ b
∣ r ⃗ c ∣ = ∣ r ⃗ b h ∣ = ( 1 + M p M c ) ∣ r ⃗ b ∣ , (16) |\vec{r}_{\mathrm{c}}|=|\vec{r}_{\mathrm{bh}}|=\left(1+\frac{M_{\mathrm{p}}}{M_{\mathrm{c}}}\right)|\vec{r}_{\mathrm{b}}|, \tag*{(16)} ∣ r c ∣ = ∣ r bh ∣ = ( 1 + M c M p ) ∣ r b ∣ , (16) where M p M_{\mathrm{p}} M p M c M_{\mathrm{c}} M c b b b c c c r ⃗ ^ c = r ⃗ ^ b \hat{\vec{r}}_{\mathrm{c}}=\hat{\vec{r}}_{\mathrm{b}} r ^ c = r ^ b
The position vector in the companion spin-aligned frame is obtained through the following sequence of rotations:
r ⃗ b h = R y ( − λ b h ) R z ( − η b h ) R x ( i ) R z ( ω ) r ⃗ c , (17) \vec{r}_{\mathrm{bh}}=R_y(-\lambda_{\mathrm{bh}})R_z(-\eta_{\mathrm{bh}})R_x(i)R_z(\omega)\vec{r}_{\mathrm{c}}, \tag*{(17)} r bh = R y ( − λ bh ) R z ( − η bh ) R x ( i ) R z ( ω ) r c , (17) while its representation in the sky frame is
r ⃗ s = R x ( i ) R z ( ω ) r ⃗ b . (18) \vec{r}_{\mathrm{s}}=R_x(i)R_z(\omega)\vec{r}_{\mathrm{b}}. \tag*{(18)} r s = R x ( i ) R z ( ω ) r b . (18) The unit vectors required in the subsequent calculation can be written directly in terms of the corresponding angular coordinates as
S ^ b h = [ cos η b h sin λ b h , sin η b h sin λ b h , cos λ b h ] , (19) \hat{\mathbf{S}}_{\mathrm{bh}}=[\cos\eta_{\mathrm{bh}}\sin\lambda_{\mathrm{bh}},\sin\eta_{\mathrm{bh}}\sin\lambda_{\mathrm{bh}},\cos\lambda_{\mathrm{bh}}], \tag*{(19)} S ^ bh = [ cos η bh sin λ bh , sin η bh sin λ bh , cos λ bh ] , (19) r ⃗ ^ b = [ cos A T , sin A T , 0 ] , (20) \hat{\vec{r}}_{\mathrm{b}}=[\cos A_T,\sin A_T,0], \tag*{(20)} r ^ b = [ cos A T , sin A T , 0 ] , (20) and
N ^ s = [ 0 , 0 , − 1 ] . (21) \hat{\mathbf{N}}_{\mathrm{s}}=[0,0,-1]. \tag*{(21)} N ^ s = [ 0 , 0 , − 1 ] . (21) We now introduce a ray-dependent reference frame, referred to as the T T T X T Y T Z T X_TY_TZ_T X T Y T Z T Y T Y_T Y T r ⃗ ^ b \hat{\vec{r}}_{\mathrm{b}} r ^ b Z T Z_T Z T r ⃗ ^ b × n ^ b \hat{\vec{r}}_{\mathrm{b}}\times\hat{\mathbf{n}}_{\mathrm{b}} r ^ b × n ^ b n ^ b \hat{\mathbf{n}}_{\mathrm{b}} n ^ b b b b
The initial propagation direction of a light ray is first obtained in the pulsar’s inertial frame from the emission geometry discussed in Section II. It is then transformed into the b b b
n ^ p = R z ( − ω ) R x ( − i ) R z ( η p ) R y ( λ p ) n ^ i . (22) \hat{\mathbf{n}}_{\mathrm{p}}=R_z(-\omega)R_x(-i)R_z(\eta_{\mathrm{p}})R_y(\lambda_{\mathrm{p}})\hat{\mathbf{n}}_{\mathrm{i}}. \tag*{(22)} n ^ p = R z ( − ω ) R x ( − i ) R z ( η p ) R y ( λ p ) n ^ i . (22) With the above choice of the T T T Y T Y_T Y T
The angle between the pulsar spin axis and the orbital angular momentum is denoted by ξ p \xi_{\mathrm{p}} ξ p
ξ p = cos − 1 ( cos λ p cos i − sin λ p sin η p sin i ) . (23) \xi_{\mathrm{p}}=\cos^{-1}(\cos\lambda_{\mathrm{p}}\cos i-\sin\lambda_{\mathrm{p}}\sin\eta_{\mathrm{p}}\sin i). \tag*{(23)} ξ p = cos − 1 ( cos λ p cos i − sin λ p sin η p sin i ) . (23) Orbital and Spin Evolution
The orientation of the pulsar and the companion does not necessarily remain fixed during the evolution of a relativistic binary. In addition to the secular evolution of the orbital parameters due to gravitational-wave emission, relativistic spin-orbit coupling, and possibly spin-spin interaction, can produce precession of the spin axes and changes in the orbital orientation. These effects can therefore modify the geometry of the pulsar beam and, consequently, the direction of the emitted radiation.
Figure 2. The orbital geometry of a pulsar (marked by a ⋆ \star ⋆ X s Y s X_{\mathrm{s}}Y_{\mathrm{s}} X s Y s X b Y b X_{\mathrm{b}}Y_{\mathrm{b}} X b Y b X I ′ Y I ′ X'_{\mathrm{I}}Y'_{\mathrm{I}} X I ′ Y I ′ S ^ p \hat{S}_{\mathrm{p}} S ^ p S ^ b h \hat{S}_{\mathrm{bh}} S ^ bh ω \omega ω A T A_T A T i i i λ p \lambda_{\mathrm{p}} λ p η p \eta_{\mathrm{p}} η p λ b h \lambda_{\mathrm{bh}} λ bh η b h \eta_{\mathrm{bh}} η bh [22 ] .
For the orbital evolution, we consider the secular changes in the orbital period and eccentricity caused by gravitational-wave emission. The corresponding expressions for P ˙ b \dot{P}_{\mathrm{b}} P ˙ b e ˙ \dot{e} e ˙ [19 , 47 ] . Since the present calculation tracks the binary over a finite interval, these secular changes are implemented by updating the orbital parameters with time.
The spin evolution is treated separately. In the present work, we consider systems in which the spin axis of the black hole is aligned with the orbital angular momentum. Under this assumption, the black-hole spin does not undergo an appreciable change in orientation. However, the pulsar spin can precess around the orbital angular momentum due to relativistic spin-orbit coupling.
The evolution of the pulsar spin vector can be written as
d S ⃗ p d t = Ω ⃗ S p × S ⃗ p , (24) \frac{d\vec{S}_{\mathrm{p}}}{dt}=\vec{\Omega}_{S_{\mathrm{p}}}\times\vec{S}_{\mathrm{p}}, \tag*{(24)} d t d S p = Ω S p × S p , (24) where S ⃗ p \vec{S}_{\mathrm{p}} S p Ω ⃗ S p \vec{\Omega}_{S_{\mathrm{p}}} Ω S p
Ω ⃗ S p = Ω ⃗ 1 P N p + Ω ⃗ L T p , (25) \vec{\Omega}_{S_{\mathrm{p}}}=\vec{\Omega}_{1\mathrm{PN}_{\mathrm{p}}}+\vec{\Omega}_{\mathrm{LT}_{\mathrm{p}}}, \tag*{(25)} Ω S p = Ω 1 PN p + Ω LT p , (25) where the first term represents the standard relativistic spin precession and the second term describes the Lense-Thirring contribution.
Following [48 ] , the leading spin-precession term is written as
Ω ⃗ 1 P N p = A 1 P N p L ^ , (26) \vec{\Omega}_{1\mathrm{PN}_{\mathrm{p}}}=A_{1\mathrm{PN}_{\mathrm{p}}\!}\hat{L}, \tag*{(26)} Ω 1 PN p = A 1 PN p L ^ , (26) where L ^ \hat{L} L ^
A 1 P N p = ( G c 3 ) 2 / 3 σ b 5 / 3 1 − e 2 M c ( 4 M p + 3 M c ) 2 ( M p + M c ) 4 / 3 . (27) A_{1\mathrm{PN}_{\mathrm{p}}}=\left(\frac{G}{c^3}\right)^{2/3}\frac{\sigma_{\mathrm{b}}^{5/3}}{1-e^2}\frac{M_{\mathrm{c}}(4M_{\mathrm{p}}+3M_{\mathrm{c}})}{2(M_{\mathrm{p}}+M_{\mathrm{c}})^{4/3}}. \tag*{(27)} A 1 PN p = ( c 3 G ) 2/3 1 − e 2 σ b 5/3 2 ( M p + M c ) 4/3 M c ( 4 M p + 3 M c ) . (27) Here, e e e
σ b = 2 π P b . (28) \sigma_{\mathrm{b}}=\frac{2\pi}{P_{\mathrm{b}}}. \tag*{(28)} σ b = P b 2 π . (28) The Lense-Thirring contribution to the pulsar spin precession is given by
Ω ⃗ L T p = A L T p [ S ^ b h − 3 ( L ^ ⋅ S ^ b h ) L ^ ] . (29) \vec{\Omega}_{\mathrm{LT}_{\mathrm{p}}}=A_{\mathrm{LT}_{\mathrm{p}}}\left[\hat{S}_{\mathrm{bh}}-3(\hat{L}\cdot\hat{S}_{\mathrm{bh}})\hat{L}\right]. \tag*{(29)} Ω LT p = A LT p [ S ^ bh − 3 ( L ^ ⋅ S ^ bh ) L ^ ] . (29) Under the assumption adopted here that S ^ b h ∥ L ^ \hat{S}_{\mathrm{bh}}\parallel\hat{L} S ^ bh ∥ L ^
Ω ⃗ L T p = − 2 A L T p L ^ , (30) \vec{\Omega}_{\mathrm{LT}_{\mathrm{p}}}=-2A_{\mathrm{LT}_{\mathrm{p}}}\hat{L}, \tag*{(30)} Ω LT p = − 2 A LT p L ^ , (30) where
A L T p = G c 3 a ~ σ b 2 ( 1 − e 2 ) 3 / 2 M c 2 2 ( M p + M c ) . (31) A_{\mathrm{LT}_{\mathrm{p}}}=\frac{G}{c^3}\frac{\tilde{a}\sigma_{\mathrm{b}}^2}{(1-e^2)^{3/2}}\frac{M_{\mathrm{c}}^2}{2(M_{\mathrm{p}}+M_{\mathrm{c}})}. \tag*{(31)} A LT p = c 3 G ( 1 − e 2 ) 3/2 a ~ σ b 2 2 ( M p + M c ) M c 2 . (31) The dimensionless parameter a ~ \tilde{a} a ~
S b h = a ~ G M c 2 c . (32) S_{\mathrm{bh}}=\tilde{a}\frac{GM_{\mathrm{c}}^{2}}{c}. \tag*{(32)} S bh = a ~ c G M c 2 . (32) The positive and negative values of a ~ \tilde{a} a ~ ∣ a ~ ∣ ≤ 1 |\tilde{a}|\leq1 ∣ a ~ ∣ ≤ 1
The spin-orbit coupling also contributes to the secular advance of the periastron. We write the total rate of periastron advance as
ω ˙ = ω ˙ P N + ω ˙ L T , p + ω ˙ L T , c , (33) \dot{\omega}=\dot{\omega}_{\mathrm{PN}}+\dot{\omega}_{\mathrm{LT,p}}+\dot{\omega}_{\mathrm{LT,c}}, \tag*{(33)} ω ˙ = ω ˙ PN + ω ˙ LT , p + ω ˙ LT , c , (33) where the first term denotes non-spinning post-Newtonian and the remaining terms arise from the spin-orbit coupling of the pulsar and the companion.
The post-Newtonian contribution is written as
ω ˙ P N = 3 β 0 2 σ b 1 − e 2 [ 1 + β 0 2 f 0 ] , (34) \dot{\omega}_{\mathrm{PN}}=\frac{3\beta_{0}^{2}\sigma_{\mathrm{b}}}{1-e^{2}}\left[1+\beta_{0}^{2}f_{0}\right], \tag*{(34)} ω ˙ PN = 1 − e 2 3 β 0 2 σ b [ 1 + β 0 2 f 0 ] , (34) with
β 0 = ( G M T σ b ) 1 / 3 c , (35) \beta_{0}=\frac{(GM_{\mathrm{T}}\sigma_{\mathrm{b}})^{1/3}}{c}, \tag*{(35)} β 0 = c ( G M T σ b ) 1/3 , (35) where
M T = M p + M c . (36) M_{\mathrm{T}}=M_{\mathrm{p}}+M_{\mathrm{c}}. \tag*{(36)} M T = M p + M c . (36) The mass-ratio factor f 0 f_{0} f 0
f 0 = 1 1 − e 2 ( 39 4 X p 2 + 27 4 X c 2 + 15 X p X c ) − ( 13 4 X p 2 + 1 4 X c 2 + 13 3 X p X c ) , (37) f_{0}=\frac{1}{1-e^{2}}\left(\frac{39}{4}\mathcal{X}_{\mathrm{p}}^{2}+\frac{27}{4}\mathcal{X}_{\mathrm{c}}^{2}+15\mathcal{X}_{\mathrm{p}}\mathcal{X}_{\mathrm{c}}\right)-\left(\frac{13}{4}\mathcal{X}_{\mathrm{p}}^{2}+\frac{1}{4}\mathcal{X}_{\mathrm{c}}^{2}+\frac{13}{3}\mathcal{X}_{\mathrm{p}}\mathcal{X}_{\mathrm{c}}\right), \tag*{(37)} f 0 = 1 − e 2 1 ( 4 39 X p 2 + 4 27 X c 2 + 15 X p X c ) − ( 4 13 X p 2 + 4 1 X c 2 + 3 13 X p X c ) , (37) where
X p = M p M T , X c = M c M T . (38) \mathcal{X}_{\mathrm{p}}=\frac{M_{\mathrm{p}}}{M_{\mathrm{T}}},\qquad\mathcal{X}_{\mathrm{c}}=\frac{M_{\mathrm{c}}}{M_{\mathrm{T}}}. \tag*{(38)} X p = M T M p , X c = M T M c . (38) The spin-orbit contributions to the periastron advance are as follows:
ω ˙ L T , p = − 3 β 0 3 σ b 1 − e 2 [ β s p g s p ] , (39) \dot{\omega}_{\mathrm{LT,p}}=-\frac{3\beta_{0}^{3}\sigma_{\mathrm{b}}}{1-e^{2}}\left[\beta_{\mathrm{sp}}g_{\mathrm{sp}}\right], \tag*{(39)} ω ˙ LT , p = − 1 − e 2 3 β 0 3 σ b [ β sp g sp ] , (39) and
ω ˙ L T , c = − 3 β 0 3 σ b 1 − e 2 [ a ~ g s c ] . (40) \dot{\omega}_{\mathrm{LT,c}}=-\frac{3\beta_{0}^{3}\sigma_{\mathrm{b}}}{1-e^{2}}\left[\tilde{a}g_{\mathrm{sc}}\right]. \tag*{(40)} ω ˙ LT , c = − 1 − e 2 3 β 0 3 σ b [ a ~ g sc ] . (40) The dimensionless spin parameter of the pulsar is
β s p = c S p G M p 2 = 2 π c I p G M p 2 P s , (41) \beta_{\mathrm{sp}}=\frac{cS_{\mathrm{p}}}{GM_{\mathrm{p}}^{2}}=\frac{2\pi cI_{\mathrm{p}}}{GM_{\mathrm{p}}^{2}P_{\mathrm{s}}}, \tag*{(41)} β sp = G M p 2 c S p = G M p 2 P s 2 π c I p , (41) where
S p = 2 π I p P s (42) S_{\mathrm{p}}=\frac{2\pi I_{\mathrm{p}}}{P_{\mathrm{s}}} \tag*{(42)} S p = P s 2 π I p (42) is the magnitude of the pulsar’s spin angular momentum. Here, I p I_{\mathrm{p}} I p 1.8 M ⊙ 1.8\,M_{\odot} 1.8 M ⊙ I p = 1.8 × 10 45 g c m 2 I_{\mathrm{p}}=1.8\times10^{45}\,\mathrm{g\,cm}^{2} I p = 1.8 × 1 0 45 g cm 2
Following [50 ] , the geometrical factor associated with the pulsar spin is
g s p = X p ( 4 X p + 3 X c ) 6 ( 1 − e 2 ) 1 / 2 sin 2 i [ ( 3 sin 2 i − 1 ) cos ξ p + cos i cos λ p ] . (43) g_{\mathrm{sp}}=\frac{\mathcal{X}_{\mathrm{p}}(4\mathcal{X}_{\mathrm{p}}+3\mathcal{X}_{\mathrm{c}})}{6(1-e^{2})^{1/2}\sin^{2}i}\left[(3\sin^{2}i-1)\cos\xi_{\mathrm{p}}+\cos i\cos\lambda_{\mathrm{p}}\right]. \tag*{(43)} g sp = 6 ( 1 − e 2 ) 1/2 sin 2 i X p ( 4 X p + 3 X c ) [ ( 3 sin 2 i − 1 ) cos ξ p + cos i cos λ p ] . (43) The angle ξ p \xi_{\mathrm{p}} ξ p
For the companion, the alignment S ^ b h ∥ L ^ \hat{\mathbf{S}}_{\mathrm{bh}}\parallel\hat{\mathbf{L}} S ^ bh ∥ L ^
g s c = X c ( 4 X c + 3 X p ) 3 ( 1 − e 2 ) 1 / 2 . (44) g_{\mathrm{sc}}=\frac{\mathcal{X}_{\mathrm{c}}(4\mathcal{X}_{\mathrm{c}}+3\mathcal{X}_{\mathrm{p}})}{3(1-e^{2})^{1/2}}. \tag*{(44)} g sc = 3 ( 1 − e 2 ) 1/2 X c ( 4 X c + 3 X p ) . (44) The Lense-Thirring interaction can also modify the orbital inclination. Since the black-hole spin is assumed to be parallel to the orbital angular momentum, its contribution to the evolution of i i i [51 ] as
d i d t = σ b 2 2 ( M p + M c ) c 2 ∑ j = p , b h S ^ j ⋅ Υ ^ = π I p σ b 2 ( M p + M c ) c 2 S ^ p ⋅ Υ ^ , (45) \frac{di}{dt}=\frac{\sigma_{\mathrm{b}}^{2}}{2(M_{\mathrm{p}}+M_{\mathrm{c}})c^{2}}\sum_{j=\mathrm{p,bh}}\hat{\mathbf{S}}_{j}\cdot\hat{\boldsymbol{\Upsilon}}=\frac{\pi I_{\mathrm{p}}\sigma_{\mathrm{b}}^{2}}{(M_{\mathrm{p}}+M_{\mathrm{c}})c^{2}}\hat{\mathbf{S}}_{\mathrm{p}}\cdot\hat{\boldsymbol{\Upsilon}}, \tag*{(45)} d t d i = 2 ( M p + M c ) c 2 σ b 2 j = p , bh ∑ S ^ j ⋅ Υ ^ = ( M p + M c ) c 2 π I p σ b 2 S ^ p ⋅ Υ ^ , (45) where Υ ^ \hat{\boldsymbol{\Upsilon}} Υ ^ i i i
The evolution of the pulsar spin modifies its orientation with respect to the observer. In particular, the angles λ p \lambda_{\mathrm{p}} λ p η p \eta_{\mathrm{p}} η p (23) , these changes also modify ξ p \xi_{\mathrm{p}} ξ p
For sufficiently compact binaries, higher-order spin and orbital coupling terms may become important [52 , 53 , 54 ] . Such effects are particularly relevant for systems containing a millisecond pulsar orbiting a supermassive black hole.
Polarization Rotation Induced by Gravitational Bending
The light rays emitted by the pulsar propagate through the curved spacetime generated by the companion. The curvature of spacetime affects both the propagation time and the direction of the light rays. These effects become most significant near superior conjunction, where the pulsar lies closest to the line of sight to the companion. For an edge-on orbit, this configuration corresponds to i = 90 ∘ i=90^\circ i = 9 0 ∘ A T + ω = 90 ∘ A_{\mathrm{T}}+\omega=90^\circ A T + ω = 9 0 ∘ [15 , 16 , 55 , 56 ] . These delays arise because the latitude and longitude of a light ray reaching the observer after gravitational bending differ from those of the corresponding ray in the absence of gravitational bending [21 ] .
According to the Rotating Vector Model (RVM), the polarization direction of the radiation emitted by the pulsar depends on the direction of the emitted light ray relative to the magnetic axis [57 ] . Gravitational bending changes the emission direction required for the ray to reach the observer, and consequently, the polarization direction. In addition to this geometric effect, the polarization vector is parallel transported along the null geodesic in curved spacetime. Since the initial light ray is not directed toward the observer, its polarization vector at the emission point does not, in general, lie in the observer’s sky plane, whereas the final polarization vector does. Therefore, the initial and final polarization vectors cannot be directly compared to separate the effect of parallel transport. To distinguish the geometric contribution from the additional polarization rotation associated with propagation in curved spacetime, we construct a reference final polarization vector by connecting the initial and final ray directions through a sequence of rotations. The difference between the reference final polarization direction and the direction obtained by parallel transport defines the polarization transport rotation relative to our chosen reference construction.
To quantify these effects, we consider the polarization vectors in the X s Y s X_sY_s X s Y s f ^ s k y , s \hat{\mathbf{f}}_{\mathrm{sky},s} f ^ sky , s Φ m = 0 \Phi_m = 0 Φ m = 0 X s Y s X_sY_s X s Y s f ^ s k y , s ′ \hat{\mathbf{f}}'_{\mathrm{sky},s} f ^ sky , s ′ f ^ s k y , f ′ \hat{\mathbf{f}}'_{\mathrm{sky},f} f ^ sky , f ′ 1
The reference vector f ^ s k y , f ′ \hat{\mathbf{f}}'_{\mathrm{sky},f} f ^ sky , f ′ f ^ s k y , f \hat{\mathbf{f}}_{\mathrm{sky},f} f ^ sky , f § 2.3 B. The geometry and the polarization vectors used in these definitions are illustrated schematically in Figure 3 .
Figure 3. Schematic diagram illustrating the geometry and the polarization vectors used in this work. The pulsar is marked by a star, and the companion black hole is denoted by BH. The red plane represents the observer’s sky plane. The fiducial unbent light ray is shown by the dashed line, with its polarization vector f ^ s k y , s \hat{\mathbf{f}}_{\mathrm{sky,s}} f ^ sky , s n ^ s \hat{\mathbf{n}}_{\mathrm{s}} n ^ s f ^ s k y , s ′ \hat{\mathbf{f}}'_{\mathrm{sky,s}} f ^ sky , s ′ n ^ s \hat{\mathbf{n}}_{\mathrm{s}} n ^ s f ^ s k y , s ′ \hat{\mathbf{f}}'_{\mathrm{sky,s}} f ^ sky , s ′ n ^ f \hat{\mathbf{n}}_{\mathrm{f}} n ^ f f ^ s k y , f ′ \hat{\mathbf{f}}'_{\mathrm{sky,f}} f ^ sky , f ′ f ^ s k y , f \hat{\mathbf{f}}_{\mathrm{sky,f}} f ^ sky , f
We define the polarization position angle (PPA) of the fiducial unbent ray as
ψ s = atan2 ( f ^ s k y , s y , f ^ s k y , s x ) , (46) \psi_s = \operatorname{atan2}\left(\hat{f}_{\mathrm{sky},s_y},\hat{f}_{\mathrm{sky},s_x}\right), \tag*{(46)} ψ s = atan2 ( f ^ sky , s y , f ^ sky , s x ) , (46) where atan2 ( y , x ) \operatorname{atan2}(y,x) atan2 ( y , x )
ψ f ′ = atan2 ( f ^ s k y , f y ′ , f ^ s k y , f x ′ ) . (47) \psi'_f = \operatorname{atan2}\left(\hat{f}'_{\mathrm{sky},f_y},\hat{f}'_{\mathrm{sky},f_x}\right). \tag*{(47)} ψ f ′ = atan2 ( f ^ sky , f y ′ , f ^ sky , f x ′ ) . (47) The difference between these two angles characterizes the geometric polarization shift associated with the change in the emission geometry caused by gravitational bending. We define this quantity as
Δ ψ g e o = ψ s − ψ f ′ . (48) \Delta\psi_{\mathrm{geo}} = \psi_s - \psi'_f. \tag*{(48)} Δ ψ geo = ψ s − ψ f ′ . (48) Here, the reference construction provides the basis for determining the change associated with the different emission geometry, without including the effect of polarization transport in curved spacetime.
For the bent ray, the final polarization vector obtained by parallel transport in curved spacetime is denoted by f ^ s k y , f \hat{\mathbf{f}}_{\mathrm{sky},f} f ^ sky , f
ψ f = atan2 ( f ^ s k y , f y , f ^ s k y , f x ) . (49) \psi_f = \operatorname{atan2}\left(\hat{f}_{\mathrm{sky},f_y},\hat{f}_{\mathrm{sky},f_x}\right). \tag*{(49)} ψ f = atan2 ( f ^ sky , f y , f ^ sky , f x ) . (49) We define the total change in the polarization direction between the fiducial unbent ray and the final bent ray as the total relativistic polarization rotation , Δ ψ t o t \Delta\psi_{\mathrm{tot}} Δ ψ tot
Δ ψ t o t = ψ s − ψ f . (50) \Delta\psi_{\mathrm{tot}} = \psi_s - \psi_f. \tag*{(50)} Δ ψ tot = ψ s − ψ f . (50) The total change includes the geometric contribution and the additional change associated with polarization transport in curved spacetime. We therefore define the contribution associated with curved-spacetime transport as
Δ ψ p t = Δ ψ t o t − Δ ψ g e o . (51) \Delta\psi_{\mathrm{pt}} = \Delta\psi_{\mathrm{tot}} - \Delta\psi_{\mathrm{geo}}. \tag*{(51)} Δ ψ pt = Δ ψ tot − Δ ψ geo . (51) Thus, Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt [59 , 60 , 61 ] . However, polarization rotation in a gravitational field depends on the choice of the local polarization basis, and a geometric contribution associated with the reference frame can also be present [58 ] . Although our reference construction explicitly preserves the polarization components parallel and perpendicular to the corresponding propagation planes, we therefore do not interpret Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt [58 ] . For a non-rotating black hole companion, Δ ψ p t = 0 \Delta\psi_{\mathrm{pt}}=0 Δ ψ pt = 0
The spin of the companion also affects the propagation of light rays through the frame-dragging effect, thereby modifying their directions [62 ] . Light rays propagating in directions co-rotating with the companion spin experience a different amount of bending from those propagating in counter-rotating directions. This difference modifies both the latitudinal and longitudinal bending delays, giving rise to the frame-dragging bending delays [22 ] . Consequently, the companion spin can also modify the observed polarization direction.
To quantify the spin-induced change in the geometric polarization shift, we compare the results for a spinning companion with those for a non-spinning companion. The latter corresponds to α ~ = 0 \tilde{\alpha}=0 α ~ = 0 α ~ \tilde{\alpha} α ~ frame-dragging geometric polarization shift , Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo
Δ ψ F D , g e o = ( Δ ψ g e o ) α ~ − ( Δ ψ g e o ) 0 (52) \Delta\psi_{\mathrm{FD,geo}} = \left(\Delta\psi_{\mathrm{geo}}\right)_{\tilde{\alpha}} - \left(\Delta\psi_{\mathrm{geo}}\right)_0 \tag*{(52)} Δ ψ FD , geo = ( Δ ψ geo ) α ~ − ( Δ ψ geo ) 0 (52) where ( Δ ψ g e o ′ ) α ~ \left(\Delta\psi'_{\mathrm{geo}}\right)_{\tilde{\alpha}} ( Δ ψ geo ′ ) α ~ ( Δ ψ g e o ) 0 \left(\Delta\psi_{\mathrm{geo}}\right)_0 ( Δ ψ geo ) 0
Relativistic Light Propagation and Polarization Transport
Relativistic Null Geodesics and Lensing
To describe the propagation of the radiation emitted by the pulsar in the gravitational field of the companion, we solve the null geodesic equations in the Kerr spacetime. The gravitational contribution of the pulsar to the spacetime is neglected as we are assuming a significantly more massive companion. We use Boyer–Lindquist coordinates x μ x^\mu x μ x 0 = c t x^0=ct x 0 = c t x 1 = r x^1=r x 1 = r x 2 = θ x^2=\theta x 2 = θ x 3 = ϕ x^3=\phi x 3 = ϕ
Following [63 , 64 , 65 , 66 ] , the equations governing the null geodesics in the Kerr spacetime can be written in terms of separated radial and latitudinal integrals as
I r = G θ , (53) I_r = G_\theta, \tag*{(53)} I r = G θ , (53) and
ϕ f − ϕ s = I ϕ + D λ G ϕ . (54) \phi_f-\phi_s = I_\phi+ D_\lambda G_\phi. \tag*{(54)} ϕ f − ϕ s = I ϕ + D λ G ϕ . (54) The radial and latitudinal effective potentials are given by
R ( r ) = r 4 + [ ( a ~ G M c c 2 ) 2 − D λ 2 − D q ] r 2 + 2 ( G M c c 2 ) [ D q + { D λ − ( a ~ G M c c 2 ) 2 } ] r − ( a ~ G M c c 2 ) 2 ( D q − M c 2 c 4 E 2 ) r 2 Δ , (55) \begin{aligned}
\mathcal{R}(r)={}&r^4+\left[\left(\tilde{a}\frac{GM_c}{c^2}\right)^2-D_\lambda^2-D_q\right]r^2\\
&+2\left(\frac{GM_c}{c^2}\right)\left[D_q+\left\{D_\lambda-\left(\tilde{a}\frac{GM_c}{c^2}\right)^2\right\}\right]r\\
&-\left(\tilde{a}\frac{GM_c}{c^2}\right)^2\left(D_q-\frac{M_c^2c^4}{E^2}\right)r^2\Delta, \tag*{(55)}
\end{aligned} R ( r ) = r 4 + [ ( a ~ c 2 G M c ) 2 − D λ 2 − D q ] r 2 + 2 ( c 2 G M c ) [ D q + { D λ − ( a ~ c 2 G M c ) 2 } ] r − ( a ~ c 2 G M c ) 2 ( D q − E 2 M c 2 c 4 ) r 2 Δ , (55) Θ ( θ ) = D q + ( a ~ G M c c 2 ) 2 cos 2 θ − D λ 2 cot 2 θ − ( M c 2 c 4 E 2 ) ( a ~ G M c c 2 ) 2 cos 2 θ , (56) \Theta(\theta)=D_q+\left(\tilde{a}\frac{GM_c}{c^2}\right)^2\cos^2\theta-D_\lambda^2\cot^2\theta-\left(\frac{M_c^2c^4}{E^2}\right)\left(\tilde{a}\frac{GM_c}{c^2}\right)^2\cos^2\theta, \tag*{(56)} Θ ( θ ) = D q + ( a ~ c 2 G M c ) 2 cos 2 θ − D λ 2 cot 2 θ − ( E 2 M c 2 c 4 ) ( a ~ c 2 G M c ) 2 cos 2 θ , (56) I r = ∫ r ∞ d r sign ( p r ) R ( r ) , (57) I_r=\int_{r}^{\infty}\frac{dr}{\operatorname{sign}(p_r)\sqrt{\mathcal{R}(r)}}, \tag*{(57)} I r = ∫ r ∞ sign ( p r ) R ( r ) d r , (57) I ϕ = ∫ r ∞ ( a ~ G M c c 2 ) [ ( 2 G M c / c 2 ) r − D λ ( a ~ G M c c 2 ) ] sign ( p r ) R ( r ) Δ d r , (58) I_\phi=\int_{r}^{\infty}\frac{\left(\tilde{a}\frac{GM_c}{c^2}\right)\left[(2GM_c/c^2)r-D_\lambda\left(\tilde{a}\frac{GM_c}{c^2}\right)\right]}{\operatorname{sign}(p_r)\sqrt{\mathcal{R}(r)}\Delta}\,dr, \tag*{(58)} I ϕ = ∫ r ∞ sign ( p r ) R ( r ) Δ ( a ~ c 2 G M c ) [ ( 2 G M c / c 2 ) r − D λ ( a ~ c 2 G M c ) ] d r , (58) G θ = ∫ θ π / 2 d θ sign ( p θ ) Θ ( θ ) , (59) G_\theta=\int_{\theta}^{\pi/2}\frac{d\theta}{\operatorname{sign}(p_\theta)\sqrt{\Theta(\theta)}}, \tag*{(59)} G θ = ∫ θ π /2 sign ( p θ ) Θ ( θ ) d θ , (59) G ϕ = ∫ θ π / 2 csc 2 θ sign ( p θ ) Θ ( θ ) d θ , (60) G_\phi=\int_{\theta}^{\pi/2}\frac{\csc^2\theta}{\operatorname{sign}(p_\theta)\sqrt{\Theta(\theta)}}\,d\theta, \tag*{(60)} G ϕ = ∫ θ π /2 sign ( p θ ) Θ ( θ ) csc 2 θ d θ , (60) D λ = c L z E , (61) D_\lambda=\frac{cL_z}{E}, \tag*{(61)} D λ = E c L z , (61) D q = L E 2 . (62) D_q=\frac{\mathcal{L}}{E^2}. \tag*{(62)} D q = E 2 L . (62) Here, E E E L z L_z L z L \mathcal{L} L D λ D_\lambda D λ D q D_q D q M c M_c M c a ~ \tilde{a} a ~
S b h = a ~ G M c 2 c . (63) S_{\rm bh}=\tilde{a}\frac{GM_c^2}{c}. \tag*{(63)} S bh = a ~ c G M c 2 . (63) The photon four-momentum is denoted by p μ = ( p t , p r , p θ , p ϕ ) p_\mu=(p_t,p_r,p_\theta,p_\phi) p μ = ( p t , p r , p θ , p ϕ )
The metric functions appearing in the above equations are
Δ = r 2 − 2 G M c c 2 r + ( a ~ G M c c 2 ) 2 , (64) \Delta=r^2-2\frac{GM_c}{c^2}r+\left(\tilde{a}\frac{GM_c}{c^2}\right)^2, \tag*{(64)} Δ = r 2 − 2 c 2 G M c r + ( a ~ c 2 G M c ) 2 , (64) ρ 2 = r 2 + ( a ~ G M c c 2 ) 2 cos 2 θ , (65) \rho^2=r^2+\left(\tilde{a}\frac{GM_c}{c^2}\right)^2\cos^2\theta, \tag*{(65)} ρ 2 = r 2 + ( a ~ c 2 G M c ) 2 cos 2 θ , (65) Σ 2 = [ r 2 + ( a ~ G M c c 2 ) 2 ] 2 − ( a ~ G M c c 2 ) 2 Δ sin 2 θ , (66) \Sigma^2=\left[r^2+\left(\tilde{a}\frac{GM_c}{c^2}\right)^2\right]^2-\left(\tilde{a}\frac{GM_c}{c^2}\right)^2\Delta\sin^2\theta, \tag*{(66)} Σ 2 = [ r 2 + ( a ~ c 2 G M c ) 2 ] 2 − ( a ~ c 2 G M c ) 2 Δ sin 2 θ , (66) and
Ω = 2 ( G M c c 2 ) r ( a ~ G M c c 2 ) Σ 2 . (67) \Omega=\frac{2\left(\frac{GM_c}{c^2}\right)r\left(\tilde{a}\frac{GM_c}{c^2}\right)}{\Sigma^2}. \tag*{(67)} Ω = Σ 2 2 ( c 2 G M c ) r ( a ~ c 2 G M c ) . (67) The constants of motion are determined from the initial conditions of the light ray at the emission point. As described in Section III A, the initial direction of the ray is specified in the T-frame. In this frame, p θ T , s = 0 p_{\theta T,s}=0 p θ T , s = 0 χ \chi χ r ⃗ b \vec{r}_b r b n ^ b \hat{n}_b n ^ b
p ϕ T , s p t T , s = − sin χ . (68) \frac{p_{\phi T,s}}{p_{tT,s}}=-\sin\chi. \tag*{(68)} p tT , s p ϕT , s = − sin χ . (68) We define the normalized angular momentum vector at the source as
L ~ ⃗ T , s = L ~ ⃗ T , s ∣ r ⃗ b ∣ p t T , s . (69) \vec{\tilde{L}}_{T,s}=\frac{\vec{\tilde{L}}_{T,s}}{|\vec{r}_b|p_{tT,s}}. \tag*{(69)} L ~ T , s = ∣ r b ∣ p tT , s L ~ T , s . (69) Its Cartesian components in the T-frame are
L ~ X T , s = − [ p ϕ T , s p t T , s cos θ T , s cos ϕ T , s + p θ T , s p t T , s sin ϕ T , s ] , (70) \tilde{L}_{XT,s}=-\left[\frac{p_{\phi T,s}}{p_{tT,s}}\cos\theta_{T,s}\cos\phi_{T,s}+\frac{p_{\theta T,s}}{p_{tT,s}}\sin\phi_{T,s}\right], \tag*{(70)} L ~ X T , s = − [ p tT , s p ϕT , s cos θ T , s cos ϕ T , s + p tT , s p θ T , s sin ϕ T , s ] , (70) L ~ Y T , s = − [ p ϕ T , s p t T , s cos θ T , s sin ϕ T , s − p θ T , s p t T , s cos ϕ T , s ] , (71) \tilde{L}_{YT,s}=-\left[\frac{p_{\phi T,s}}{p_{tT,s}}\cos\theta_{T,s}\sin\phi_{T,s}-\frac{p_{\theta T,s}}{p_{tT,s}}\cos\phi_{T,s}\right], \tag*{(71)} L ~ Y T , s = − [ p tT , s p ϕT , s cos θ T , s sin ϕ T , s − p tT , s p θ T , s cos ϕ T , s ] , (71) L ~ Z T , s = p ϕ T , s p t T , s sin θ T , s . (72) \tilde{L}_{ZT,s}=\frac{p_{\phi T,s}}{p_{tT,s}}\sin\theta_{T,s}. \tag*{(72)} L ~ Z T , s = p tT , s p ϕT , s sin θ T , s . (72) At the emission point, the construction of the T-frame gives θ T , s = 90 ∘ \theta_{T,s}=90^\circ θ T , s = 9 0 ∘ ϕ T , s = 90 ∘ \phi_{T,s}=90^\circ ϕ T , s = 9 0 ∘
L ~ ⃗ b h ′ , s = R y ( − λ b h ) R z ( − η b h ) R x ( i ) R z ( ω ) R z ( ϕ T b ) R y ( ϑ T b ) L ~ ⃗ T , s . (73) \vec{\tilde{L}}_{\rm bh',s}=R_y(-\lambda_{\rm bh})R_z(-\eta_{\rm bh})R_x(i)R_z(\omega)R_z(\phi_{Tb})R_y(\vartheta_{Tb})\vec{\tilde{L}}_{T,s}. \tag*{(73)} L ~ b h ′ , s = R y ( − λ bh ) R z ( − η bh ) R x ( i ) R z ( ω ) R z ( ϕ T b ) R y ( ϑ T b ) L ~ T , s . (73) where ϑ T b \vartheta_{\mathrm{Tb}} ϑ Tb Z T Z_{\mathrm{T}} Z T Z b Z_{\mathrm{b}} Z b φ T b \varphi_{\mathrm{Tb}} φ Tb Y T Y_{\mathrm{T}} Y T Y b Y_{\mathrm{b}} Y b [21 ] . Since the bh- and bh′-frames are parallel,
L ~ b h , s = L ~ b h ′ , s . (74) \widetilde{\mathbf{L}}_{\mathrm{bh},s}=\widetilde{\mathbf{L}}_{\mathrm{bh}',s}. \tag*{(74)} L bh , s = L bh ′ , s . (74) The components of the angular momentum vector determine the initial momentum ratios in the companion frame:
p θ b h , s p t b h , s = L ~ Y b h , s cos ϕ b h , s − L ~ X b h , s sin ϕ b h , s , (75) \frac{p_{\theta_{\mathrm{bh},s}}}{p_{t_{\mathrm{bh},s}}}
=\widetilde{L}_{Y_{\mathrm{bh},s}}\cos\phi_{\mathrm{bh},s}
-\widetilde{L}_{X_{\mathrm{bh},s}}\sin\phi_{\mathrm{bh},s}, \tag*{(75)} p t bh , s p θ bh , s = L Y bh , s cos ϕ bh , s − L X bh , s sin ϕ bh , s , (75) p ϕ b h , s p t b h , s = L ~ Z b h , s sin θ b h , s . (76) \frac{p_{\phi_{\mathrm{bh},s}}}{p_{t_{\mathrm{bh},s}}}
=\frac{\widetilde{L}_{Z_{\mathrm{bh},s}}}{\sin\theta_{\mathrm{bh},s}}. \tag*{(76)} p t bh , s p ϕ bh , s = sin θ bh , s L Z bh , s . (76) Because the transformation between the T- and bh-frames consists only of spatial rotations, the temporal component of the photon momentum remains unchanged:
p t b h , s = p t T , s . (77) p_{t_{\mathrm{bh},s}}=p_{t_{\mathrm{T},s}}. \tag*{(77)} p t bh , s = p t T , s . (77) Following [67 ] , the constants of motion can then be calculated from the initial momentum components:
D λ = sin θ b h ( p ϕ b h / p t b h ) − Δ ρ 2 / Σ 2 + Ω sin θ b h ( p ϕ b h / p t b h ) , (78) D_{\lambda}=
\frac{\sin\theta_{\mathrm{bh}}\left(p_{\phi_{\mathrm{bh}}}/p_{t_{\mathrm{bh}}}\right)}
{-\sqrt{\Delta}\rho^{2}/\Sigma^{2}+\Omega\sin\theta_{\mathrm{bh}}\left(p_{\phi_{\mathrm{bh}}}/p_{t_{\mathrm{bh}}}\right)}, \tag*{(78)} D λ = − Δ ρ 2 / Σ 2 + Ω sin θ bh ( p ϕ bh / p t bh ) sin θ bh ( p ϕ bh / p t bh ) , (78) D q = [ D λ 2 sin 2 θ b h − ( a G M c c 2 ) 2 ] cos 2 θ b h + [ p θ b h p t b h ( 1 − D λ Ω ) ] 2 Σ 2 Δ . (79) D_q=
\left[
\frac{D_{\lambda}^{2}}{\sin^{2}\theta_{\mathrm{bh}}}
-\left(\frac{aGM_{\mathrm{c}}}{c^{2}}\right)^{2}
\right]\cos^{2}\theta_{\mathrm{bh}}
+\left[
\frac{p_{\theta_{\mathrm{bh}}}}{p_{t_{\mathrm{bh}}}}(1-D_{\lambda}\Omega)
\right]^{2}\frac{\Sigma^{2}}{\Delta}. \tag*{(79)} D q = [ sin 2 θ bh D λ 2 − ( c 2 a G M c ) 2 ] cos 2 θ bh + [ p t bh p θ bh ( 1 − D λ Ω ) ] 2 Δ Σ 2 . (79) Evaluating these quantities at the emission point fixes the constants D λ D_{\lambda} D λ D q D_q D q [68 ] and [66 ] to determine the trajectory of the ray and its direction at asymptotic infinity. The final angular coordinates θ b h , f \theta_{\mathrm{bh},f} θ bh , f ϕ b h , f \phi_{\mathrm{bh},f} ϕ bh , f
n ^ b h , f = [ cos ϕ b h , f sin θ b h , f , sin ϕ b h , f sin θ b h , f , cos θ b h , f ] . (80) \widehat{\mathbf{n}}_{\mathrm{bh},f}
=[\cos\phi_{\mathrm{bh},f}\sin\theta_{\mathrm{bh},f},
\sin\phi_{\mathrm{bh},f}\sin\theta_{\mathrm{bh},f},
\cos\theta_{\mathrm{bh},f}]. \tag*{(80)} n bh , f = [ cos ϕ bh , f sin θ bh , f , sin ϕ bh , f sin θ bh , f , cos θ bh , f ] . (80) The final propagation direction is transformed from the companion frame to the pulsar inertial frame according to
n ^ I , f = R y ( − λ p ) R z ( − η p ) R z ( η b h ) R y ( λ b h ) n ^ b h , f . (81) \widehat{\mathbf{n}}_{\mathrm{I},f}
=R_y(-\lambda_p)R_z(-\eta_p)R_z(\eta_{\mathrm{bh}})R_y(\lambda_{\mathrm{bh}})
\widehat{\mathbf{n}}_{\mathrm{bh},f}. \tag*{(81)} n I , f = R y ( − λ p ) R z ( − η p ) R z ( η bh ) R y ( λ bh ) n bh , f . (81) Parallel Transport of the Polarization Vector
The gravitationally bent ray changes not only its propagation direction but also the polarization direction measured by the observer. For each light ray, the initial polarization vector is determined from the emission geometry described in Section II. We denote this vector in the pulsar inertial frame by f ^ I , s \widehat{\mathbf{f}}_{\mathrm{I},s} f I , s f ^ I , s \widehat{\mathbf{f}}_{\mathrm{I},s} f I , s T T T
f ^ T , s = R y ( − ϑ T b ) R z ( − φ T b ) R z ( − ω ) R x ( − i ) × R z ( η p ) R y ( λ p ) f ^ I , s . (82) \widehat{\mathbf{f}}_{\mathrm{T},s}
=R_y(-\vartheta_{\mathrm{Tb}})R_z(-\varphi_{\mathrm{Tb}})R_z(-\omega)R_x(-i)
\times R_z(\eta_p)R_y(\lambda_p)\widehat{\mathbf{f}}_{\mathrm{I},s}. \tag*{(82)} f T , s = R y ( − ϑ Tb ) R z ( − φ Tb ) R z ( − ω ) R x ( − i ) × R z ( η p ) R y ( λ p ) f I , s . (82) Let n ^ T , s \widehat{\mathbf{n}}_{\mathrm{T},s} n T , s n ^ T , f \widehat{\mathbf{n}}_{\mathrm{T},f} n T , f T T T ( θ T , s , φ T , s ) (\theta_{\mathrm{T},s},\varphi_{\mathrm{T},s}) ( θ T , s , φ T , s ) ( θ T , f , φ T , f ) (\theta_{\mathrm{T},f},\varphi_{\mathrm{T},f}) ( θ T , f , φ T , f ) X T Y T X_{\mathrm{T}}Y_{\mathrm{T}} X T Y T θ T , s = θ T , f = 90 ∘ \theta_{\mathrm{T},s}=\theta_{\mathrm{T},f}=90^\circ θ T , s = θ T , f = 9 0 ∘
f ^ T , f = R z ( φ T , f ) R y ( θ T , f ) R y ( − θ T , s ) R z ( − φ T , s ) f ^ T , s . (83) \widehat{\mathbf{f}}_{\mathrm{T},f}
=R_z(\varphi_{\mathrm{T},f})R_y(\theta_{\mathrm{T},f})R_y(-\theta_{\mathrm{T},s})R_z(-\varphi_{\mathrm{T},s})\widehat{\mathbf{f}}_{\mathrm{T},s}. \tag*{(83)} f T , f = R z ( φ T , f ) R y ( θ T , f ) R y ( − θ T , s ) R z ( − φ T , s ) f T , s . (83) This transformation connects the initial and final ray directions while preserving the polarization components relative to the corresponding propagation planes. In particular, we define the initial plane of propagation using r ^ T × n ^ T , s \widehat{\mathbf{r}}_{\mathrm{T}}\times\widehat{\mathbf{n}}_{\mathrm{T},s} r T × n T , s n ^ T , f × r ^ T \widehat{\mathbf{n}}_{\mathrm{T},f}\times\widehat{\mathbf{r}}_{\mathrm{T}} n T , f × r T r ^ T \widehat{\mathbf{r}}_{\mathrm{T}} r T T T T f ^ T , s \widehat{\mathbf{f}}_{\mathrm{T},s} f T , s f ^ T , f \widehat{\mathbf{f}}_{\mathrm{T},f} f T , f (83) provides the flat-spacetime reference construction used to determine the geometric polarization shift. It does not involve an explicit integration of the parallel-transport equation along the photon trajectory.
The reference polarization vector f ^ T , f \widehat{\mathbf{f}}_{\mathrm{T},f} f T , f
We next calculate the parallel transport of the polarization vector along the null geodesic in the curved spacetime of the companion. For this purpose, the initial polarization vector f ^ I , s \widehat{\mathbf{f}}_{\mathrm{I},s} f I , s
f ^ b h , s = R y ( − λ b h ) R z ( − η b h ) R z ( η p ) R y ( λ p ) f ^ I , s . (84) \widehat{\mathbf{f}}_{\mathrm{bh},s}
=R_y(-\lambda_{\mathrm{bh}})R_z(-\eta_{\mathrm{bh}})R_z(\eta_p)R_y(\lambda_p)\widehat{\mathbf{f}}_{\mathrm{I},s}. \tag*{(84)} f bh , s = R y ( − λ bh ) R z ( − η bh ) R z ( η p ) R y ( λ p ) f I , s . (84) The polarization vector is then expressed in the spherical coordinate basis associated with the companion frame. The Cartesian components of the polarization vector in the companion frame, denoted by f b h , s x f_{\mathrm{bh},s_x} f bh , s x f b h , s y f_{\mathrm{bh},s_y} f bh , s y f b h , s z f_{\mathrm{bh},s_z} f bh , s z
f ( b h , s ) r = f b h , s x cos ϕ b h , s sin θ b h , s + f b h , s y sin ϕ b h , s sin θ b h , s + f b h , s z cos θ b h , s , (85) f_{(\mathrm{bh},s)r}
=f_{\mathrm{bh},s_x}\cos\phi_{\mathrm{bh},s}\sin\theta_{\mathrm{bh},s}
+f_{\mathrm{bh},s_y}\sin\phi_{\mathrm{bh},s}\sin\theta_{\mathrm{bh},s}
+f_{\mathrm{bh},s_z}\cos\theta_{\mathrm{bh},s}, \tag*{(85)} f ( bh , s ) r = f bh , s x cos ϕ bh , s sin θ bh , s + f bh , s y sin ϕ bh , s sin θ bh , s + f bh , s z cos θ bh , s , (85) f ( b h , s ) θ = f b h , s x cos ϕ b h , s cos θ b h , s + f b h , s y sin ϕ b h , s cos θ b h , s − f b h , s z sin θ b h , s . (86) f_{(\mathrm{bh},s)\theta}
=f_{\mathrm{bh},s_x}\cos\phi_{\mathrm{bh},s}\cos\theta_{\mathrm{bh},s}
+f_{\mathrm{bh},s_y}\sin\phi_{\mathrm{bh},s}\cos\theta_{\mathrm{bh},s}
-f_{\mathrm{bh},s_z}\sin\theta_{\mathrm{bh},s}. \tag*{(86)} f ( bh , s ) θ = f bh , s x cos ϕ bh , s cos θ bh , s + f bh , s y sin ϕ bh , s cos θ bh , s − f bh , s z sin θ bh , s . (86) and
f ( b h , s ) ϕ = − f b h , s x sin ϕ b h , s + f b h , s y cos ϕ b h , s , (87) f_{(\mathrm{bh,s})\phi}=-f_{\mathrm{bh,s}x}\sin\phi_{\mathrm{bh,s}}+f_{\mathrm{bh,s}y}\cos\phi_{\mathrm{bh,s}}, \tag*{(87)} f ( bh , s ) ϕ = − f bh , s x sin ϕ bh , s + f bh , s y cos ϕ bh , s , (87) where θ b h , s \theta_{\mathrm{bh,s}} θ bh , s ϕ b h , s \phi_{\mathrm{bh,s}} ϕ bh , s
At the emission point, the polarization direction is specified in the local spatial frame of the companion. We therefore choose the corresponding polarization four-vector to have a vanishing temporal component [25 ] ,
f ( b h , s ) μ = ( 0 , f ( b h , s ) r , f ( b h , s ) θ , f ( b h , s ) ϕ ) . (88) f_{(\mathrm{bh,s})\mu}=\left(0,f_{(\mathrm{bh,s})r},f_{(\mathrm{bh,s})\theta},f_{(\mathrm{bh,s})\phi}\right). \tag*{(88)} f ( bh , s ) μ = ( 0 , f ( bh , s ) r , f ( bh , s ) θ , f ( bh , s ) ϕ ) . (88) This choice represents the polarization as a purely spatial vector in the local companion frame. The initial polarization vector is required to be transverse to the propagation direction, and we explicitly verify that
f ^ b h , s ⋅ n ^ b h , s = 0. (89) \hat{f}_{\mathrm{bh,s}}\cdot\hat{n}_{\mathrm{bh,s}}=0. \tag*{(89)} f ^ bh , s ⋅ n ^ bh , s = 0. (89) The initial polarization four-vector is transformed from the companion frame to the Boyer–Lindquist coordinate basis using the transformation matrix E ~ ν μ \widetilde{E}^{\mu}_{\ \nu} E ν μ
f ( B L , s ) ν = E ~ ν μ f ( b h , s ) μ . (90) f_{(\mathrm{BL,s})\nu}=\widetilde{E}^{\mu}_{\ \nu}f_{(\mathrm{bh,s})\mu}. \tag*{(90)} f ( BL , s ) ν = E ν μ f ( bh , s ) μ . (90) The initial polarization four-vector also satisfies
l μ f ( B L ) μ = 0 , (91) l^{\mu}f_{(\mathrm{BL})\mu}=0, \tag*{(91)} l μ f ( BL ) μ = 0 , (91) where l μ l^{\mu} l μ (89) and (91) provide consistency checks on the construction of the initial polarization state.
To describe the parallel transport of the polarization vector along the null geodesic, we introduce the null tetrad ( l μ , e 2 μ , e 3 μ , k μ ) (l^{\mu},e_{2}^{\mu},e_{3}^{\mu},k^{\mu}) ( l μ , e 2 μ , e 3 μ , k μ ) l μ l^{\mu} l μ e 2 μ e_{2}^{\mu} e 2 μ e 3 μ e_{3}^{\mu} e 3 μ k μ k^{\mu} k μ
Following [69 ] , the parallel transport of the polarization vector can be described through its projections onto the tetrad:
( l μ E ~ μ ν ) f f ( b h , f ) ν = ( l μ E ~ μ ν ) s f ( b h , s ) ν = 0 , (92) \left(l^{\mu}\widetilde{E}^{\nu}_{\ \mu}\right)_{f}f_{(\mathrm{bh,f})\nu}
=\left(l^{\mu}\widetilde{E}^{\nu}_{\ \mu}\right)_{s}f_{(\mathrm{bh,s})\nu}=0, \tag*{(92)} ( l μ E μ ν ) f f ( bh , f ) ν = ( l μ E μ ν ) s f ( bh , s ) ν = 0 , (92) ( e 2 μ E ~ μ ν ) f f ( b h , f ) ν = ( e 2 μ E ~ μ ν ) s f ( b h , s ) ν = C 1 , (93) \left(e_{2}^{\mu}\widetilde{E}^{\nu}_{\ \mu}\right)_{f}f_{(\mathrm{bh,f})\nu}
=\left(e_{2}^{\mu}\widetilde{E}^{\nu}_{\ \mu}\right)_{s}f_{(\mathrm{bh,s})\nu}=C_{1}, \tag*{(93)} ( e 2 μ E μ ν ) f f ( bh , f ) ν = ( e 2 μ E μ ν ) s f ( bh , s ) ν = C 1 , (93) ( e 3 μ E ~ μ ν ) f f ( b h , f ) ν = ( e 3 μ E ~ μ ν ) s f ( b h , s ) ν = C 2 . (94) \left(e_{3}^{\mu}\widetilde{E}^{\nu}_{\ \mu}\right)_{f}f_{(\mathrm{bh,f})\nu}
=\left(e_{3}^{\mu}\widetilde{E}^{\nu}_{\ \mu}\right)_{s}f_{(\mathrm{bh,s})\nu}=C_{2}. \tag*{(94)} ( e 3 μ E μ ν ) f f ( bh , f ) ν = ( e 3 μ E μ ν ) s f ( bh , s ) ν = C 2 . (94) Here, C 1 C_{1} C 1 C 2 C_{2} C 2
At asymptotic infinity, the spacetime is locally flat and the polarization vector is transverse to the final propagation direction. We therefore express the final polarization vector in terms of its spherical components, f θ , f f_{\theta,f} f θ , f f ϕ , f f_{\phi,f} f ϕ , f
f ^ b h , f = f θ , f e θ , f + f ϕ , f e ϕ , f ∣ f θ , f e θ , f + f ϕ , f e ϕ , f ∣ , (95) \hat{f}_{\mathrm{bh,f}}=
\frac{f_{\theta,f}\boldsymbol{e}_{\theta,f}+f_{\phi,f}\boldsymbol{e}_{\phi,f}}
{\left|f_{\theta,f}\boldsymbol{e}_{\theta,f}+f_{\phi,f}\boldsymbol{e}_{\phi,f}\right|}, \tag*{(95)} f ^ bh , f = ∣ f θ , f e θ , f + f ϕ , f e ϕ , f ∣ f θ , f e θ , f + f ϕ , f e ϕ , f , (95) where
e θ , f = [ cos θ b h , f cos ϕ b h , f , cos θ b h , f sin ϕ b h , f , − sin θ b h , f ] , (96) \boldsymbol{e}_{\theta,f}=
\left[\cos\theta_{\mathrm{bh,f}}\cos\phi_{\mathrm{bh,f}},
\cos\theta_{\mathrm{bh,f}}\sin\phi_{\mathrm{bh,f}},
-\sin\theta_{\mathrm{bh,f}}\right], \tag*{(96)} e θ , f = [ cos θ bh , f cos ϕ bh , f , cos θ bh , f sin ϕ bh , f , − sin θ bh , f ] , (96) and
e ϕ , f = [ − sin ϕ b h , f , cos ϕ b h , f , 0 ] . (97) \boldsymbol{e}_{\phi,f}=
\left[-\sin\phi_{\mathrm{bh,f}},\cos\phi_{\mathrm{bh,f}},0\right]. \tag*{(97)} e ϕ , f = [ − sin ϕ bh , f , cos ϕ bh , f , 0 ] . (97) The final polarization direction is transformed into the observer sky-plane coordinate system according to
f ^ s k y , f = R z ( η b h ) R y ( λ b h ) f ^ b h , f , (98) \hat{f}_{\mathrm{sky,f}}=R_{z}(\eta_{\mathrm{bh}})R_{y}(\lambda_{\mathrm{bh}})\hat{f}_{\mathrm{bh,f}}, \tag*{(98)} f ^ sky , f = R z ( η bh ) R y ( λ bh ) f ^ bh , f , (98) and into the pulsar inertial frame according to
f ^ I , f = R y ( − λ p ) R z ( − η p ) R z ( η b h ) R y ( λ b h ) f ^ b h , f . (99) \hat{f}_{\mathrm{I,f}}=R_{y}(-\lambda_{\mathrm{p}})R_{z}(-\eta_{\mathrm{p}})R_{z}(\eta_{\mathrm{bh}})R_{y}(\lambda_{\mathrm{bh}})\hat{f}_{\mathrm{bh,f}}. \tag*{(99)} f ^ I , f = R y ( − λ p ) R z ( − η p ) R z ( η bh ) R y ( λ bh ) f ^ bh , f . (99) As a further consistency check, the final polarization vector obtained from the parallel-transport calculation remains transverse to the final propagation direction:
f ^ I , f ⋅ n ^ I , f = 0. (100) \hat{f}_{\mathrm{I,f}}\cdot\hat{n}_{\mathrm{I,f}}=0. \tag*{(100)} f ^ I , f ⋅ n ^ I , f = 0. (100) Thus, the initial and final polarization states satisfy the required transversality conditions, while the parallel-transport calculation determines the change in the polarization direction accumulated along the curved null geodesic.
The initial polarization vector of the bent ray and the corresponding final polarization vectors from the flat-spacetime reference construction and the curved-spacetime parallel-transport calculation are then used to determine the geometric polarization shift Δ ψ g e o \Delta\psi_{\mathrm{geo}} Δ ψ geo Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt
Numerical Study
We implement the general relativistic framework developed in the previous sections to quantitatively investigate the effects of gravitational light bending and polarization transport on the polarization direction of a pulsar in a binary system. In particular, we study the variations of the polarization rotation angles defined in Section IV as a function of the orbital phase and investigate the corresponding changes in the classical polarization “S-curve”, which is governed by the intrinsic emission geometry [3 ] .
Since no pulsar–BH binary has yet been confirmed observationally, we consider a hypothetical binary configuration for our numerical calculations. Following our previous works [21 , 22 ] , we adopt the same values for most of the system parameters. These include the pulsar mass M p M_p M p P s P_s P s P ˙ s \dot{P}_s P ˙ s P b P_b P b e e e λ p \lambda_p λ p η p \eta_p η p λ b h \lambda_{\rm bh} λ bh η b h \eta_{\rm bh} η bh
We consider a companion intermediate-mass black hole (IMBH) with mass M c = 10 5 M ⊙ M_c = 10^5 M_\odot M c = 1 0 5 M ⊙ i = 85 ∘ i = 85^\circ i = 8 5 ∘ η p \eta_p η p λ p \lambda_p λ p λ b h \lambda_{\rm bh} λ bh η b h \eta_{\rm bh} η bh α \alpha α β \beta β λ p \lambda_p λ p η p \eta_p η p
For the binary configuration considered here, the initial values of λ p \lambda_p λ p η p \eta_p η p 130 ∘ 130^\circ 13 0 ∘ 45 ∘ 45^\circ 4 5 ∘ (25) , we find that over one orbital period of 0.5 yr, the values of λ p \lambda_p λ p η p \eta_p η p 0.015 ∘ 0.015^\circ 0.01 5 ∘ 0.01 ∘ 0.01^\circ 0.0 1 ∘ ω \omega ω P b P_b P b i i i ω \omega ω 0.04 ∘ 0.04^\circ 0.0 4 ∘ i i i P b P_b P b 2.28 × 10 − 13 ∘ 2.28 \times10^{-13}\ ^\circ 2.28 × 1 0 − 13 ∘ 4.95 × 10 − 7 s 4.95 \times10^{-7}\ \mathrm{s} 4.95 × 1 0 − 7 s α = 49 ∘ \alpha= 49^\circ α = 4 9 ∘ β = 1 ∘ \beta= 1^\circ β = 1 ∘ λ p \lambda_p λ p ζ N \zeta_N ζ N λ p = 180 ∘ − ζ N \lambda_p = 180^\circ- \zeta_N λ p = 18 0 ∘ − ζ N [21 ] .
To determine the initial direction of the polarization vector, we calculate the emission height by fixing the observing frequency at ν = 1.4 G H z \nu= 1.4\ \mathrm{GHz} ν = 1.4 GHz (1) , we obtain h e m 1.4 = 437.19 k m h_{\mathrm{em}}^{1.4} = 437.19\ \mathrm{km} h em 1.4 = 437.19 km (2) , is W 1.4 = 4.51 ∘ W^{1.4} = 4.51^\circ W 1.4 = 4.5 1 ∘
We first consider the case in which the companion is non-spinning, i.e., a ~ = 0 \tilde{a} = 0 a ~ = 0 Δ ψ p t = 0 \Delta\psi_{\mathrm{pt}} = 0 Δ ψ pt = 0 Figure 4 shows the variation of the geometric polarization shift Δ ψ g e o \Delta\psi_{\mathrm{geo}} Δ ψ geo A T + ω A_T + \omega A T + ω Δ ψ g e o \Delta\psi_{\mathrm{geo}} Δ ψ geo Δ ψ g e o \Delta\psi_{\mathrm{geo}} Δ ψ geo 7 ∘ 7^\circ 7 ∘ A T + ω = 90 ∘ A_T + \omega= 90^\circ A T + ω = 9 0 ∘
Figure 4. Variation of the geometric polarization shift Δ ψ g e o \Delta\psi_{\rm geo} Δ ψ geo A T + ω A_T + \omega A T + ω a ~ = 0 \tilde{a}=0 a ~ = 0 t = 0 t=0 t = 0 A T + ω = 73.804 ∘ A_T+\omega=73.804^\circ A T + ω = 73.80 4 ∘
Table 1. Parameters adopted for the hypothetical pulsar–IMBH binary considered in the numerical calculations.
junction.
In Figure 4 , the initial longitude of periastron is chosen to be ω = 73.804 ∘ \omega= 73.804^\circ ω = 73.80 4 ∘ t = 0 t = 0 t = 0 A T + ω = 73.804 ∘ A_T+\omega= 73.804^\circ A T + ω = 73.80 4 ∘ A T + ω A_T+\omega A T + ω
The dashed magenta curve represents the calculation without spin–orbit precession, whereas the solid green curve includes the evolution of the pulsar spin and orbital geometry due to spin–orbit coupling. Without spin–orbit precession, the geometrical configuration repeats after each orbital period, and the polarization shift returns to its initial values in successive orbits. When spin–orbit precession is included, the changing orientation of the pulsar spin modifies the emission geometry. Consequently, the polarization shift does not necessarily return to the same value after one orbital period. Since the companion is non-spinning in this case, Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt
We next consider a spinning companion with a ~ = 0.5 \tilde{a}=0.5 a ~ = 0.5 Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt Figure 5 show these quantities, respectively, as functions of orbital phase A T + ω A_T+\omega A T + ω Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt
(a) Frame-dragging contribution to the geometric polarization shift Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo
(b) Frame-dragging contribution to the polarization transport rotation Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt
Figure 5. Spin-induced changes in the polarization rotation for a hypothetical pulsar–IMBH binary with a ~ = 0.5 \tilde{a}=0.5 a ~ = 0.5 Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt
A sharp discontinuity is visible at A T + ω = 90 ∘ A_T+\omega=90^\circ A T + ω = 9 0 ∘ Figure 5 . This feature arises from the change in the orientation of the photon trajectory relative to the spin axis of the companion. During one half of the orbit, the deflected light rays follow prograde trajectories, whereas during the other half, they follow retrograde trajectories. The transition between these configurations produces the discontinuity in the frame-dragging contribution. As derived in Debnath and Bagchi [22 ] , the boundary between these configurations satisfies
N ^ s ⋅ ( r ^ s × S ^ b h ) = 0. (101) \hat{\mathbf{N}}_s\cdot\left(\hat{\mathbf{r}}_s\times\hat{\mathbf{S}}_{\mathrm{bh}}\right)=0. \tag*{(101)} N ^ s ⋅ ( r ^ s × S ^ bh ) = 0. (101) The vertical reference lines in Figure 4 and Figure 5 indicate the initial orbital phase, A T + ω = 73.804 ∘ A_T+\omega=73.804^\circ A T + ω = 73.80 4 ∘ Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt Figure 5 .
We now investigate the dependence of the polarization rotations on the binary parameters M c M_c M c P b P_b P b e e e i i i η p \eta_p η p Figure 6 , Figure 7 , and Figure 8 show the variations of Δ ψ g e o \Delta\psi_{\mathrm{geo}} Δ ψ geo Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt
(a) Δ ψ g e o \Delta\psi_{\mathrm{geo}} Δ ψ geo M c M_{\mathrm{c}} M c
(b) Δ ψ g e o \Delta\psi_{\mathrm{geo}} Δ ψ geo i i i
(c) Δ ψ g e o \Delta\psi_{\mathrm{geo}} Δ ψ geo e e e
(d) Δ ψ g e o \Delta\psi_{\mathrm{geo}} Δ ψ geo P b P_{\mathrm{b}} P b
(e) Δ ψ g e o \Delta\psi_{\mathrm{geo}} Δ ψ geo η p \eta_{\mathrm{p}} η p
(f) Δ ψ g e o \Delta\psi_{\mathrm{geo}} Δ ψ geo η p \eta_{\mathrm{p}} η p
Figure 6. Dependence of the geometric polarization shift Δ ψ g e o \Delta\psi_{\mathrm{geo}} Δ ψ geo M c M_{\mathrm{c}} M c i i i e e e P b P_{\mathrm{b}} P b η p \eta_{\mathrm{p}} η p
(a) Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo M c M_c M c
(b) Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo i i i
(c) Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo e e e
(d) Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo P b P_b P b
(e) Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo η p \eta_p η p
(f) Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo η p \eta_p η p
Figure 7. Dependence of the frame-dragging contribution to the geometric polarization shift, Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo M c M_c M c i i i e e e P b P_b P b η p \eta_p η p
Figure 8. (a) Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt M c M_{\mathrm{c}} M c Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt i i i Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt e e e Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt P b P_{\mathrm{b}} P b Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt η p \eta_{\mathrm{p}} η p Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt η p \eta_{\mathrm{p}} η p Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt M c M_{\mathrm{c}} M c i i i e e e P b P_{\mathrm{b}} P b η p \eta_{\mathrm{p}} η p
Figure 6a , Figure 7a , and Figure 8 a show the variations of Δ ψ g e o \Delta\psi_{\mathrm{geo}} Δ ψ geo Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt M c = 2 × 10 5 M_c=2\times10^5 M c = 2 × 1 0 5 4 × 10 5 4\times10^5 4 × 1 0 5 8 × 10 5 8\times10^5 8 × 1 0 5 10 6 M ⊙ 10^6\,M_\odot 1 0 6 M ⊙
Figure 6b , Figure 7b , and Figure 8 b show the corresponding variations for orbital inclinations i = 70 ∘ i=70^\circ i = 7 0 ∘ 75 ∘ 75^\circ 7 5 ∘ 80 ∘ 80^\circ 8 0 ∘ 85 ∘ 85^\circ 8 5 ∘
The effect of orbital eccentricity is shown in Figure 6c , Figure 7c , and Figure 8 c for e = 0.1 e=0.1 e = 0.1 0.3 0.3 0.3 0.5 0.5 0.5 0.7 0.7 0.7 Δ ψ g e o \Delta\psi_{\mathrm{geo}} Δ ψ geo Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt [21 ] .
Figure 6d , Figure 7d , and Figure 8 d show the variations for orbital periods P b = 0.5 P_b=0.5 P b = 0.5 0.8 0.8 0.8 1.0 1.0 1.0 1.2 y r 1.2\ \mathrm{yr} 1.2 yr
We finally examine the dependence on the pulsar spin orientation parameter η p \eta_p η p Figure 6e , Figure 6f , Figure 7e , Figure 7f , Figure 8 e, and Figure 8 f show the results for positive and negative values of η p \eta_p η p Δ ψ g e o \Delta\psi_{\mathrm{geo}} Δ ψ geo Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo η p \eta_p η p 90 ∘ 90^\circ 9 0 ∘ [21 ] . Consequently, the geometric polarization shift and its spin-induced change also depend on η p \eta_p η p Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt η p \eta_p η p
We also find that the orbital phase at which the geometric polarization shift reaches its maximum or minimum depends on η p \eta_p η p i i i A T + ω = 90 ∘ A_T+\omega=90^\circ A T + ω = 9 0 ∘ λ p \lambda_p λ p λ p \lambda_p λ p [21 ] .
We next investigate the dependence of the polarization rotations Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt a ~ \tilde{a} a ~ η b h \eta_{\mathrm{bh}} η bh λ b h \lambda_{\mathrm{bh}} λ bh Figure 9 and Figure 10 show the corresponding variations for different values of these parameters.
Figure 9. (a) Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo a ~ \tilde{a} a ~ Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo η b h \eta_{\mathrm{bh}} η bh Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo λ b h \lambda_{\mathrm{bh}} λ bh Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo a ~ \tilde{a} a ~ η b h \eta_{\mathrm{bh}} η bh λ b h \lambda_{\mathrm{bh}} λ bh
(a) (a) Δ ψ p t \Delta\psi_{\rm pt} Δ ψ pt a ~ \tilde{a} a ~
(b) (b) Δ ψ p t \Delta\psi_{\rm pt} Δ ψ pt η b h \eta_{\rm bh} η bh
(c) (c) Δ ψ p t \Delta\psi_{\rm pt} Δ ψ pt λ b h \lambda_{\rm bh} λ bh
Figure 10. Dependence of the polarization transport rotation Δ ψ p t \Delta\psi_{\rm pt} Δ ψ pt a ~ \tilde{a} a ~ η b h \eta_{\rm bh} η bh λ b h \lambda_{\rm bh} λ bh
Figure 9 a and Figure 10a show the dependence on the companion spin parameter a ~ \tilde{a} a ~ Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt
stronger spin-induced contribution to the polarization rotation.
The dependence on the spin orientation angle η b h \eta_{\mathrm{bh}} η bh Figure 9 b and Figure 10b . Changing η b h \eta_{\mathrm{bh}} η bh (101) .
Finally, Figure 9 c and Figure 10c show the dependence on the spin orientation angle λ b h \lambda_{\mathrm{bh}} λ bh η b h \eta_{\mathrm{bh}} η bh λ b h \lambda_{\mathrm{bh}} λ bh
One of the most important predictions of the rotating vector model (RVM) is the characteristic polarization position-angle curve, commonly referred to as the polarization S-curve. It describes the variation of the polarization position angle ψ \psi ψ Φ m \Phi_m Φ m Φ m \Phi_m Φ m [37 ] . If F ^ s k y ( Φ m ) \hat{\boldsymbol{F}}_{\mathrm{sky}}(\Phi_m) F ^ sky ( Φ m ) Φ m \Phi_m Φ m
ψ = atan2 ( F ^ s k y , y ( Φ m ) , F ^ s k y , x ( Φ m ) ) (102) \psi= \operatorname{atan2}\left(\hat{F}_{\mathrm{sky},y}(\Phi_m), \hat{F}_{\mathrm{sky},x}(\Phi_m)\right) \tag*{(102)} ψ = atan2 ( F ^ sky , y ( Φ m ) , F ^ sky , x ( Φ m ) ) (102) where F ^ s k y , x \hat{F}_{\mathrm{sky},x} F ^ sky , x F ^ s k y , y \hat{F}_{\mathrm{sky},y} F ^ sky , y x x x y y y
In Figure 11 , we show the polarization position angle as a function of the pulsar rotational phase at the orbital phases A T + ω = 89 ∘ A_T+\omega=89^\circ A T + ω = 8 9 ∘ 90 ∘ 90^\circ 9 0 ∘ 91 ∘ 91^\circ 9 1 ∘ 200 ∘ 200^\circ 20 0 ∘ A T + ω = 89 ∘ A_T+\omega=89^\circ A T + ω = 8 9 ∘ 90 ∘ 90^\circ 9 0 ∘ 91 ∘ 91^\circ 9 1 ∘ A T + ω = 90 ∘ A_T+\omega=90^\circ A T + ω = 9 0 ∘
(a) (a) Orbital phase A T + ω = 89 ∘ A_T+\omega=89^\circ A T + ω = 8 9 ∘
(b) (b) Orbital phase A T + ω = 90 ∘ A_T+\omega=90^\circ A T + ω = 9 0 ∘
(c) (c) Orbital phase A T + ω = 91 ∘ A_T+\omega=91^\circ A T + ω = 9 1 ∘
(d) (d) Orbital phase A T + ω = 200 ∘ A_T+\omega=200^\circ A T + ω = 20 0 ∘
Figure 11. Variations of the polarization position angle ψ \psi ψ Φ m \Phi_m Φ m A T + ω = 89 ∘ A_T+\omega=89^\circ A T + ω = 8 9 ∘ 90 ∘ 90^\circ 9 0 ∘ 91 ∘ 91^\circ 9 1 ∘ A T + ω = 200 ∘ A_T+\omega=200^\circ A T + ω = 20 0 ∘
In contrast, at the orbital phase A T + ω = 200 ∘ A_T+\omega=200^\circ A T + ω = 20 0 ∘
A Test Case of a Pulsar–Supermassive Black Hole Close Binary
We next consider a hypothetical pulsar–supermassive black hole (SMBH) close binary. In such a system, strong-field general relativistic (GR) effects can significantly modify the viewing geometry of the pulsar. The orbital and spin evolution can also be sufficiently strong that spin-orbit evolution cannot be ignored, even over a single orbital period. Since no such binary has been discovered so far, we adopt the baseline parameters in
Table I, except that we set the companion mass to M c = 10 6 M ⊙ M_{\mathrm{c}} = 10^{6}\,M_{\odot} M c = 1 0 6 M ⊙ P b = 5.5 P_{\mathrm{b}} = 5.5 P b = 5.5 i = 70 ∘ i = 70^{\circ} i = 7 0 ∘ 833.83 G M c / c 2 833.83\,GM_{\mathrm{c}}/c^{2} 833.83 G M c / c 2 401.5 G M c / c 2 401.5\,GM_{\mathrm{c}}/c^{2} 401.5 G M c / c 2
We choose a relatively low inclination angle because the Einstein radius of the massive companion can be much larger than the pulsar beam, leading to substantial beam and pulse-profile distortion [22 ] . In [22 ] , we showed that this distortion remains significant for inclination angles up to 80 ∘ 80^{\circ} 8 0 ∘ i = 70 ∘ i = 70^{\circ} i = 7 0 ∘
We first evaluate the changes in the system parameters over one complete orbit. We adopt the baseline values at t = 0 t = 0 t = 0 M c = 10 6 M ⊙ M_{\mathrm{c}} = 10^{6}\,M_{\odot} M c = 1 0 6 M ⊙ P b = 5.5 P_{\mathrm{b}} = 5.5 P b = 5.5 i = λ b h = 70 ∘ i = \lambda_{\mathrm{bh}} = 70^{\circ} i = λ bh = 7 0 ∘ ω \omega ω 2.01 ∘ 2.01^{\circ} 2.0 1 ∘ 1.95 ∘ 1.95^{\circ} 1.9 5 ∘ a ~ = 0 \tilde{a} = 0 a ~ = 0 0.5 0.5 0.5 η p \eta_{\mathrm{p}} η p 0.207 ∘ 0.207^{\circ} 0.20 7 ∘ 0.204 ∘ 0.204^{\circ} 0.20 4 ∘ λ p \lambda_{\mathrm{p}} λ p 0.66 ∘ 0.66^{\circ} 0.6 6 ∘ 0.65 ∘ 0.65^{\circ} 0.6 5 ∘ Δ P b = − 2.37 × 10 − 5 \Delta P_{\mathrm{b}} = -2.37 \times10^{-5} Δ P b = − 2.37 × 1 0 − 5 Δ e = − 1.23 × 10 − 11 \Delta e = -1.23 \times10^{-11} Δ e = − 1.23 × 1 0 − 11 Δ i = 7.95 × 10 − 12 ∘ \Delta i = 7.95 \times10^{-12\,\circ} Δ i = 7.95 × 1 0 − 12 ∘
More rigorous formalisms have been developed for tight and highly relativistic binary systems. In particular, [70 ] , [71 ] , and [72 ] investigated pulsars orbiting massive and supermassive black holes by including spin-spin and spin-curvature interactions through the Mathisson–Papapetrou–Dixon (MPD) equations in Kerr spacetime. These studies considered fast millisecond pulsars with P s = 1 P_{\mathrm{s}} = 1 P s = 1 10 10 10 100 G M c / c 2 100\,GM_{\mathrm{c}}/c^{2} 100 G M c / c 2 [70 ] found that out-of-plane orbital motion remains negligible when the mean orbital separation is larger than approximately 40 G M c / c 2 40\,GM_{\mathrm{c}}/c^{2} 40 G M c / c 2
We first consider a Schwarzschild companion and then examine the effect of companion spin. Figure 12 shows the geometric polarization shift, Δ ψ g e o \Delta\psi_{\mathrm{geo}} Δ ψ geo A T + ω A_{\mathrm{T}}+\omega A T + ω ω = 73.804 ∘ \omega= 73.804^{\circ} ω = 73.80 4 ∘ t = 0 t = 0 t = 0 A T + ω = 73.804 ∘ A_{\mathrm{T}}+\omega= 73.804^{\circ} A T + ω = 73.80 4 ∘ A T + ω = 71.12 ∘ A_{\mathrm{T}}+\omega= 71.12^{\circ} A T + ω = 71.1 2 ∘
Figure 12. Geometric polarization shift, Δ ψ g e o \Delta\psi_{\mathrm{geo}} Δ ψ geo A T + ω A_{\mathrm{T}}+\omega A T + ω
To understand this behavior of Δ ψ g e o \Delta\psi_{\mathrm{geo}} Δ ψ geo
Figs. 13a and 13b the Cartesian coordinates ( x m , y m ) (x_{\mathrm{m}},y_{\mathrm{m}}) ( x m , y m ) m m m Figure 13 a shows the case without spin-orbit evolution, while Figure 13 b shows the case with spin-orbit evolution.
Figure 13. (a) Emission points on the X m Y m X_mY_m X m Y m L L L X m X_m X m X m Y m X_mY_m X m Y m M M M L L L
In these figures, we show the X m Y m X_{\mathrm{m}}Y_{\mathrm{m}} X m Y m h e m h_{\mathrm{em}} h em Z m Z_{\mathrm{m}} Z m m m m M M M Z m Z_{\mathrm{m}} Z m h e m h_{\mathrm{em}} h em X m Y m X_{\mathrm{m}}Y_{\mathrm{m}} X m Y m L L L
We compare the polarization shift at different orbital phases when the pulsar is at rotational phase zero, so that the LoS vector, the magnetic axis, and the pulsar spin axis lie in the same plane. The LoS therefore lies on the X m X_{\mathrm{m}} X m X m X_{\mathrm{m}} X m Figure 13 b. The emission points from which the observer receives gravitationally bent radiation are shown in green when the spin-orbit evolution is neglected and in blue when it is included. Representative orbital phases are also marked.
In Figure 13 a, when spin-orbit evolution is neglected, the emission point A A A M M M L L L M M M L L L M M M A A A A A A
tan − 1 ( tan η p cos i ) . (103) \tan^{-1}\left(\frac{\tan\eta_p}{\cos i}\right). \tag*{(103)} tan − 1 ( cos i tan η p ) . (103) We find that the orbital phase corresponding to point A A A A T + ω = 71.12 ∘ A_T+\omega=71.12^\circ A T + ω = 71.1 2 ∘ 180 ∘ 180^\circ 18 0 ∘
When spin-orbit evolution is included, point A A A M M M L L L A T + ω = 71.12 ∘ A_T+\omega=71.12^\circ A T + ω = 71.1 2 ∘ M M M L L L M M M A A A Δ ψ g e o \Delta\psi_{\rm geo} Δ ψ geo
The apparent discontinuity in Figure 12 when spin-orbit evolution is neglected therefore occurs because the absolute value of the geometric shift reaches 180 ∘ 180^\circ 18 0 ∘ Δ ψ g e o \Delta\psi_{\rm geo} Δ ψ geo
When spin-orbit evolution is included, Δ ψ g e o \Delta\psi_{\rm geo} Δ ψ geo 150 ∘ 150^\circ 15 0 ∘ t = 0 t=0 t = 0 A T + ω = 73.804 ∘ A_T+\omega=73.804^\circ A T + ω = 73.80 4 ∘ Figure 12 ). Initially, its variation closely follows the case without spin-orbit evolution, but the two curves subsequently diverge. The curve that includes spin-orbit evolution does not return to its initial value after one complete orbit because the orbital configuration evolves during the motion. In particular, the final orbital phase is approximately 75 ∘ 75^\circ 7 5 ∘ 73.804 ∘ 73.804^\circ 73.80 4 ∘
Next, we examine the effect of the SMBH spin on the polarization direction. We set the dimensionless companion spin parameter to a ~ = 0.5 \tilde{a}=0.5 a ~ = 0.5 Δ ψ F D , g e o \Delta\psi_{\rm FD,geo} Δ ψ FD , geo Δ ψ p t \Delta\psi_{\rm pt} Δ ψ pt Figure 14 shows Δ ψ F D , g e o \Delta\psi_{\rm FD,geo} Δ ψ FD , geo Δ ψ p t \Delta\psi_{\rm pt} Δ ψ pt Δ ψ p t \Delta\psi_{\rm pt} Δ ψ pt
Figure 14. (a) Absolute frame-dragging contribution to the geometric polarization shift, Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt a ~ = 0.5 \tilde{a}=0.5 a ~ = 0.5 A T + ω A_T+\omega A T + ω Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt
Next, we examine how the polarization S-curve changes with orbital phase. The S-curve shows the polarization position angle ψ \psi ψ (102) , as a function of pulsar rotational phase. Figure 15 shows the S-curves at orbital phases A T + ω = 85 ∘ A_T+\omega=85^\circ A T + ω = 8 5 ∘ 90 ∘ 90^\circ 9 0 ∘ 92 ∘ 92^\circ 9 2 ∘ 95 ∘ 95^\circ 9 5 ∘ 100 ∘ 100^\circ 10 0 ∘ 200 ∘ 200^\circ 20 0 ∘
Figure 15. Polarization S-curves at different orbital phases for the hypothetical pulsar–SMBH binary. The polarization position angle ψ \psi ψ A T + ω = 85 ∘ A_T+\omega= 85^\circ A T + ω = 8 5 ∘ 90 ∘ 90^\circ 9 0 ∘
At A T + ω = 85 ∘ A_T+\omega=85^\circ A T + ω = 8 5 ∘ 90 ∘ 90^\circ 9 0 ∘ A T + ω = 92 ∘ A_T+\omega=92^\circ A T + ω = 9 2 ∘ Figure 15 d and Figure 15 e. At A T + ω = 200 ∘ A_T+\omega=200^\circ A T + ω = 20 0 ∘
This behavior can be understood from the motion of the LoS on the pulsar beam plane. In the co-rotating m m m X m Y m X_mY_m X m Y m [37 ] . Gravitational bending displaces the apparent LoS trajectory on the beam plane. We refer to the displacement along the X m X_m X m Y m Y_m Y m Y m Y_m Y m Figure 1 ).
The reversal in the overall S-shaped variation can be understood from the latitudinal displacement of the LoS trajectory. At orbital phases A T + ω = 85 ∘ A_T+\omega=85^\circ A T + ω = 8 5 ∘ 90 ∘ 90^\circ 9 0 ∘
Near A T + ω = 92 ∘ A_T+\omega=92^\circ A T + ω = 9 2 ∘
A further feature is the shift of the distorted S-curve along the pulsar rotational-phase axis. This shift is visible when bending is strong, at A T + ω = 85 ∘ A_T+\omega=85^\circ A T + ω = 8 5 ∘ 90 ∘ 90^\circ 9 0 ∘ 92 ∘ 92^\circ 9 2 ∘ 95 ∘ 95^\circ 9 5 ∘ 100 ∘ 100^\circ 10 0 ∘ X m Y m X_mY_m X m Y m A T + ω = 200 ∘ A_T+\omega=200^\circ A T + ω = 20 0 ∘
shift of the polarization S-curve are both very small.
Conclusion
In this work, we investigated the effect of gravitational light bending on the polarization of radiation from a pulsar in a binary system with a black hole companion. We developed a formalism to calculate the null geodesic connecting the emission point to a distant observer and to track the polarization vector along the photon trajectory. The null geodesics are calculated using the formalism of [66 , 68 ] , while the parallel transport of the polarization vector is treated using the null tetrad given in [69 ] . Since the formalism assumes that the motion of the companion can be neglected during photon propagation, we applied it to hypothetical pulsar–intermediate-mass black hole (IMBH) and pulsar–supermassive black hole (SMBH) binary systems.
For the IMBH system, we adopted orbital parameters close to those used in our previous work [21 ] . Although that study considered a stellar-mass black hole companion, the adopted parameters are comparable to those considered in other studies of pulsar–IMBH systems [73 ] . We considered companion masses of 10 5 M ⊙ 10^5\,M_\odot 1 0 5 M ⊙ 10 6 M ⊙ 10^6\,M_\odot 1 0 6 M ⊙ [74 , 75 ] .
Our results show that gravitational light bending produces an apparent geometric polarization shift, Δ ψ g e o \Delta\psi_{\mathrm{geo}} Δ ψ geo
reach 180 ∘ 180^\circ 18 0 ∘ Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt [58 ] . Such an analysis is beyond the scope of this work. We therefore interpret Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt
For the systems considered here, Δ ψ F D , g e o \Delta\psi_{\mathrm{FD,geo}} Δ ψ FD , geo Δ ψ p t \Delta\psi_{\mathrm{pt}} Δ ψ pt
Gravitational light bending also modifies the polarization position angle (PPA) profile. For the IMBH system, the characteristic S-shaped profile undergoes a relatively small but noticeable distortion. For the SMBH system, strong bending can reverse the overall sense of the S-shaped variation near the superior conjunction. The profile is also shifted in the pulsar rotational phase due to the longitudinal displacement of the light ray. These results demonstrate that relativistic photon propagation can modify both the shape and the phase dependence of the observed polarization profile.
From an observational perspective, the orbital-phase-dependent distortion of the PPA profile may provide a potential signature of relativistic photon propagation in pulsar–BH binaries. Modeling the polarization profile over the binary orbit could complement pulsar timing measurements and help constrain the geometry of the system [76 ] . However, the frame-dragging contribution to the geometric polarization shift and the polarization rotation associated with parallel transport are much smaller than the main geometric contribution for the systems considered here. This makes them more challenging to isolate observationally. Their detectability will depend on the properties of the binary system, the precision of the polarization measurements, and the ability to distinguish relativistic propagation effects from intrinsic and interstellar contributions to the observed signal.
Future studies incorporating realistic pulsar magnetosphere models together with relativistic photon propagation may provide a more complete interpretation of pulsar polarization profiles [77 , 78 , 79 , 80 , 81 ] . The capabilities of the Square Kilometre Array Observatory (SKAO) may also expand the scope of high-precision polarization studies. Such studies could help establish whether the predicted relativistic signatures can be distinguished from other effects that influence the observed polarization.
Acknowledgments
JD, DP and PT acknowledge financial support from the Istituto Nazionale di Astrofisica (INAF) 2023 Large Grant: “Gravitational Wave Detection using Pulsar Timing Arrays” (PI: D. Perrodin). AC and DP also acknowledge financial support under the National Recovery and Resilience Plan (NRRP), Mission 4, Component 2, Investment 1.1, Call for tender No. 104 published on 2.2.2022 by the Italian Ministry of University and Research (MUR), funded by the European Union - NextGenerationEU - Project Title GUVIP-Gravity tests in the UltraViolet and InfraRed with Pulsar timing (PI: E. Barausse & D. Perrodin).
Asymptotic Tetrad and Polarization Vector at Infinity
In Section V B, we showed that the initial direction of the polarization vector and its final direction after parallel transport to infinity are related through Eq. 80. The transformation matrix E ~ ν μ \widetilde{E}^{\mu}_{\nu} E ν μ [67 ]
E ~ ν μ = ( e ν 0 0 − Ω e ψ 0 e μ 1 0 0 0 0 e μ 2 0 0 0 0 e ψ ) , (104) \widetilde{E}^{\mu}_{\nu} =
\begin{pmatrix}
e^{\nu} & 0 & 0 & -\Omega e^{\psi} \\
0 & e^{\mu_1} & 0 & 0 \\
0 & 0 & e^{\mu_2} & 0 \\
0 & 0 & 0 & e^{\psi}
\end{pmatrix}, \tag*{(104)} E ν μ = e ν 0 0 0 0 e μ 1 0 0 0 0 e μ 2 0 − Ω e ψ 0 0 e ψ , (104) where
e ν = ρ 2 Δ Σ , (105) e^{\nu} = \sqrt{\frac{\rho^{2}\Delta}{\Sigma}}, \tag*{(105)} e ν = Σ ρ 2 Δ , (105) e μ 1 = ρ 2 Δ , (106) e^{\mu_1} = \sqrt{\frac{\rho^{2}}{\Delta}}, \tag*{(106)} e μ 1 = Δ ρ 2 , (106) e μ 2 = ρ 2 , (107) e^{\mu_2} = \sqrt{\rho^{2}}, \tag*{(107)} e μ 2 = ρ 2 , (107) e ψ = sin θ Σ ρ 2 . (108) e^{\psi} = \sin\theta\sqrt{\frac{\Sigma}{\rho^{2}}}. \tag*{(108)} e ψ = sin θ ρ 2 Σ . (108) In the Boyer–Lindquist (BL) coordinate system, the null tetrad ( l μ , e 2 μ , e ~ 3 μ , n ~ μ ) (l^\mu,e_2^\mu,\tilde e_3^\mu,\tilde n^\mu) ( l μ , e 2 μ , e ~ 3 μ , n ~ μ ) [69 ]
l μ = 1 ρ 2 ( − ( a ~ G M c c 2 ) [ ( a ~ G M c c 2 ) sin 2 θ − D λ ] + r 2 + ( a ~ G M c c 2 ) 2 Δ [ r 2 + ( a ~ G M c c 2 ) 2 − ( a ~ G M c c 2 ) D λ ] ) , (109) l^\mu= \frac{1}{\rho^2}\left(-\left(\tilde a\frac{GM_c}{c^2}\right)\left[\left(\tilde a\frac{GM_c}{c^2}\right)\sin^2\theta-D_\lambda\right] + \frac{r^2+\left(\tilde a\frac{GM_c}{c^2}\right)^2}{\Delta}\left[r^2+\left(\tilde a\frac{GM_c}{c^2}\right)^2-\left(\tilde a\frac{GM_c}{c^2}\right)D_\lambda\right]\right), \tag*{(109)} l μ = ρ 2 1 ( − ( a ~ c 2 G M c ) [ ( a ~ c 2 G M c ) sin 2 θ − D λ ] + Δ r 2 + ( a ~ c 2 G M c ) 2 [ r 2 + ( a ~ c 2 G M c ) 2 − ( a ~ c 2 G M c ) D λ ] ) , (109) e 2 μ = 1 ρ 2 Δ C ( ( a ~ G M c c 2 ) cos θ [ r 2 + ( a ~ G M c c 2 ) 2 ] sign ( p r ) R − ( a ~ G M c c 2 ) r Δ sin θ sign ( p θ ) Θ , e_2^\mu= \frac{1}{\rho^2\Delta\sqrt{C}}((\tilde a\frac{GM_c}{c^2})\cos\theta[r^2+(\tilde a\frac{GM_c}{c^2})^2]\operatorname{sign}(p_r)\sqrt{\mathcal R}
-(\tilde a\frac{GM_c}{c^2})r\Delta\sin\theta\operatorname{sign}(p_\theta)\sqrt{\Theta}, e 2 μ = ρ 2 Δ C 1 (( a ~ c 2 G M c ) cos θ [ r 2 + ( a ~ c 2 G M c ) 2 ] sign ( p r ) R − ( a ~ c 2 G M c ) r Δ sin θ sign ( p θ ) Θ , ( a ~ G M c c 2 ) Δ cos θ [ r 2 + ( a ~ G M c c 2 ) 2 − ( a ~ G M c c 2 ) D λ ] , \left(\tilde a\frac{GM_c}{c^2}\right)\Delta\cos\theta\left[r^2+\left(\tilde a\frac{GM_c}{c^2}\right)^2-\left(\tilde a\frac{GM_c}{c^2}\right)D_\lambda\right], ( a ~ c 2 G M c ) Δ cos θ [ r 2 + ( a ~ c 2 G M c ) 2 − ( a ~ c 2 G M c ) D λ ] , − r Δ sin θ [ ( a ~ G M c c 2 ) − D λ sin 2 θ ] , -r\Delta\sin\theta\left[\left(\tilde a\frac{GM_c}{c^2}\right)-\frac{D_\lambda}{\sin^2\theta}\right], − r Δ sin θ [ ( a ~ c 2 G M c ) − sin 2 θ D λ ] , ( a ~ G M c c 2 ) 2 cos θ sign ( p r ) R − r Δ csc θ sign ( p θ ) Θ ) , (110) (\tilde a\frac{GM_c}{c^2})^2\cos\theta\operatorname{sign}(p_r)\sqrt{\mathcal R}-r\Delta\csc\theta\operatorname{sign}(p_\theta)\sqrt{\Theta}), \tag*{(110)} ( a ~ c 2 G M c ) 2 cos θ sign ( p r ) R − r Δ csc θ sign ( p θ ) Θ ) , (110) e ~ 3 μ = − 1 ρ 2 Δ C ( [ r 2 + ( a ~ G M c c 2 ) 2 ] sign ( p r ) R + ( a ~ G M c c 2 ) 2 sin θ cos θ Δ sign ( p θ ) Θ , \tilde e_3^\mu=-\frac{1}{\rho^2\Delta\sqrt{C}}([r^2+(\tilde a\frac{GM_c}{c^2})^2]\operatorname{sign}(p_r)\sqrt{\mathcal R}
+(\tilde a\frac{GM_c}{c^2})^2\sin\theta\cos\theta\Delta\operatorname{sign}(p_\theta)\sqrt{\Theta}, e ~ 3 μ = − ρ 2 Δ C 1 ([ r 2 + ( a ~ c 2 G M c ) 2 ] sign ( p r ) R + ( a ~ c 2 G M c ) 2 sin θ cos θ Δ sign ( p θ ) Θ , r Δ [ r 2 + ( a ~ G M c c 2 ) 2 − ( a ~ G M c c 2 ) D λ ] , r\Delta\left[r^2+\left(\tilde a\frac{GM_c}{c^2}\right)^2-\left(\tilde a\frac{GM_c}{c^2}\right)D_\lambda\right], r Δ [ r 2 + ( a ~ c 2 G M c ) 2 − ( a ~ c 2 G M c ) D λ ] , ( a ~ G M c c 2 ) Δ sin θ cos θ [ ( a ~ G M c c 2 ) − D λ sin 2 θ ] , \left(\tilde a\frac{GM_c}{c^2}\right)\Delta\sin\theta\cos\theta\left[\left(\tilde a\frac{GM_c}{c^2}\right)-\frac{D_\lambda}{\sin^2\theta}\right], ( a ~ c 2 G M c ) Δ sin θ cos θ [ ( a ~ c 2 G M c ) − sin 2 θ D λ ] , ( a ~ G M c c 2 ) r sign ( p r ) R + ( a ~ G M c c 2 ) Δ cot θ sign ( p θ ) Θ ) , (111) (\tilde a\frac{GM_c}{c^2})r\operatorname{sign}(p_r)\sqrt{\mathcal R}
+(\tilde a\frac{GM_c}{c^2})\Delta\cot\theta\operatorname{sign}(p_\theta)\sqrt{\Theta}), \tag*{(111)} ( a ~ c 2 G M c ) r sign ( p r ) R + ( a ~ c 2 G M c ) Δ cot θ sign ( p θ ) Θ ) , (111) n ~ μ = 1 2 C ( ( a ~ G M c c 2 ) [ ( a ~ G M c c 2 ) sin 2 θ − D λ ] + r 2 + ( a ~ G M c c 2 ) 2 Δ [ r 2 + ( a ~ G M c c 2 ) 2 \tilde n^\mu=\frac{1}{2C}((\tilde a\frac{GM_c}{c^2})[(\tilde a\frac{GM_c}{c^2})\sin^2\theta-D_\lambda]
+\frac{r^2+(\tilde a\frac{GM_c}{c^2})^2}{\Delta}[r^2+(\tilde a\frac{GM_c}{c^2})^2 n ~ μ = 2 C 1 (( a ~ c 2 G M c ) [( a ~ c 2 G M c ) sin 2 θ − D λ ] + Δ r 2 + ( a ~ c 2 G M c ) 2 [ r 2 + ( a ~ c 2 G M c ) 2 − ( a ~ G M c c 2 ) D λ ] ) , (112) \left.\left.-\left(\tilde a\frac{GM_c}{c^2}\right)D_\lambda\right]\right), \tag*{(112)} − ( a ~ c 2 G M c ) D λ ] ) , (112) where C C C
C = D q + ( D λ − a ~ G M c c 2 ) 2 . (113) C=D_q+\left(D_\lambda-\tilde a\frac{GM_c}{c^2}\right)^2. \tag*{(113)} C = D q + ( D λ − a ~ c 2 G M c ) 2 . (113) However, e ~ 3 μ \tilde e_3^\mu e ~ 3 μ n ~ μ \tilde n^\mu n ~ μ [69 ]
e 3 μ = e ~ 3 μ + Φ ~ l μ , (114) e_3^\mu=\tilde e_3^\mu+\tilde\Phi l^\mu, \tag*{(114)} e 3 μ = e ~ 3 μ + Φ ~ l μ , (114) k μ = n ~ μ + Φ ~ e ~ 3 μ + Φ ~ 2 2 l μ , (115) k^\mu=\tilde n^\mu+\tilde\Phi\tilde e_3^\mu+\frac{\tilde\Phi^2}{2}l^\mu, \tag*{(115)} k μ = n ~ μ + Φ ~ e ~ 3 μ + 2 Φ ~ 2 l μ , (115) Φ ~ = 1 C J ( r , θ ) , (116) \tilde\Phi=\frac{1}{\sqrt{C}}J(r,\theta), \tag*{(116)} Φ ~ = C 1 J ( r , θ ) , (116) The function J ( r , θ ) J(r,\theta) J ( r , θ ) [69 ]
J ( r , θ ) = sign ( p r ) ∫ r r 2 sign ( p r ) R ( r ) d r + a ~ G M c c 2 ∫ θ cos 2 θ sign ( p θ ) Θ ( θ ) d θ . (117) J(r,\theta)=\operatorname{sign}(p_r)\int_r\frac{r^2}{\operatorname{sign}(p_r)\sqrt{\mathcal R(r)}}\,dr
+\tilde a\frac{GM_c}{c^2}\int_\theta\frac{\cos^2\theta}{\operatorname{sign}(p_\theta)\sqrt{\Theta(\theta)}}\,d\theta. \tag*{(117)} J ( r , θ ) = sign ( p r ) ∫ r sign ( p r ) R ( r ) r 2 d r + a ~ c 2 G M c ∫ θ sign ( p θ ) Θ ( θ ) cos 2 θ d θ . (117) The tetrad ( l μ , e 2 μ , e 3 μ , k μ ) (l^\mu,e_2^\mu,e_3^\mu,k^\mu) ( l μ , e 2 μ , e 3 μ , k μ )
In Eq. 80, the quantities at the source, ( l μ E ~ ν μ ) s (l^\mu\tilde E^\nu{}_\mu)_s ( l μ E ~ ν μ ) s ( e 2 μ E ~ ν μ ) s (e_2^\mu\tilde E^\nu{}_\mu)_s ( e 2 μ E ~ ν μ ) s ( e 3 μ E ~ ν μ ) s (e_3^\mu\tilde E^\nu{}_\mu)_s ( e 3 μ E ~ ν μ ) s r = r b h , s r=r_{\mathrm{bh,s}} r = r bh , s θ = θ b h , s \theta=\theta_{\mathrm{bh,s}} θ = θ bh , s J ( r , θ ) J(r,\theta) J ( r , θ )
J ( r b h , s , θ b h , s ) = 0 (118) J(r_{\mathrm{bh,s}},\theta_{\mathrm{bh,s}})=0 \tag*{(118)} J ( r bh , s , θ bh , s ) = 0 (118) at the position of the source. Consequently, Φ ~ = 0 \tilde\Phi=0 Φ ~ = 0
e 3 μ = e 3 μ and k μ = n μ . (119) e_3^\mu=e_3^\mu\quad\text{and}\quad k^\mu=n^\mu. \tag*{(119)} e 3 μ = e 3 μ and k μ = n μ . (119) To evaluate the corresponding quantities at infinity, ( μ μ E ~ μ ν ) f \left(\mu^\mu\tilde{E}^\nu_{\ \mu}\right)_f ( μ μ E ~ μ ν ) f ( e 2 μ E ~ μ ν ) f \left(e^\mu_2\tilde{E}^\nu_{\ \mu}\right)_f ( e 2 μ E ~ μ ν ) f ( e 3 μ E ~ μ ν ) f \left(e^\mu_3\tilde{E}^\nu_{\ \mu}\right)_f ( e 3 μ E ~ μ ν ) f J ( r , θ ) J(r,\theta) J ( r , θ ) [66 , 68 ] . Following their work, we decompose J ( r , θ ) J(r,\theta) J ( r , θ )
J ( r , θ ) = sign ( p r ) R r − r 3 + J ( r , θ ) , (120) J(r,\theta)=\operatorname{sign}(p_r)\frac{\sqrt{\mathcal{R}}}{r-r_3}+\mathcal{J}(r,\theta), \tag*{(120)} J ( r , θ ) = sign ( p r ) r − r 3 R + J ( r , θ ) , (120) where r 3 r_3 r 3 R \mathcal{R} R
r 1 < r 2 < r 3 < r 4 . (121) r_1<r_2<r_3<r_4. \tag*{(121)} r 1 < r 2 < r 3 < r 4 . (121) The quantity J ( r , θ ) \mathcal{J}(r,\theta) J ( r , θ )
J ( r , θ ) = − sign ( p r ) f ~ [ r 1 r 4 + r 2 r 3 ( r 3 − r 1 ) ( r 4 − r 2 ) F ( sin − 1 X ( r ) , K ) + ( r 3 − r 1 ) ( r 4 − r 2 ) E ( sin − 1 X ( r ) , K ) ] + q sign ( p r ) f ~ [ R ( r i n i t i a l ) r i n i t i a l − r 3 − r 1 r 4 + r 2 r 3 ( r 3 − r 1 ) ( r 4 − r 2 ) F ( sin − 1 X ( r i n i t i a l ) , K ) − ( r 3 − r 1 ) ( r 4 − r 2 ) E ( sin − 1 X ( r i n i t i a l ) , K ) ] + a ~ G M c c 2 ∫ θ cos 2 θ sign ( p θ ) Θ ( θ ) d θ . (122) \begin{aligned}
\mathcal{J}(r,\theta)={}&-\operatorname{sign}(p_r)\tilde{f}\left[\frac{r_1r_4+r_2r_3}{\sqrt{(r_3-r_1)(r_4-r_2)}}F\left(\sin^{-1}X(r),K\right)\right.\\
&\left.+\sqrt{(r_3-r_1)(r_4-r_2)}E\left(\sin^{-1}X(r),K\right)\right]\\
&+q\operatorname{sign}(p_r)\tilde{f}\left[\frac{\sqrt{\mathcal{R}(r_{\mathrm{initial}})}}{r_{\mathrm{initial}}-r_3}
-\frac{r_1r_4+r_2r_3}{\sqrt{(r_3-r_1)(r_4-r_2)}}F\left(\sin^{-1}X(r_{\mathrm{initial}}),K\right)\right.\\
&\left.-\sqrt{(r_3-r_1)(r_4-r_2)}E\left(\sin^{-1}X(r_{\mathrm{initial}}),K\right)\right]\\
&+\tilde{a}\frac{GM_c}{c^2}\int_\theta\frac{\cos^2\theta}{\operatorname{sign}(p_\theta)\sqrt{\Theta(\theta)}}\,d\theta. \tag*{(122)}
\end{aligned} J ( r , θ ) = − sign ( p r ) f ~ [ ( r 3 − r 1 ) ( r 4 − r 2 ) r 1 r 4 + r 2 r 3 F ( sin − 1 X ( r ) , K ) + ( r 3 − r 1 ) ( r 4 − r 2 ) E ( sin − 1 X ( r ) , K ) ] + q sign ( p r ) f ~ [ r initial − r 3 R ( r initial ) − ( r 3 − r 1 ) ( r 4 − r 2 ) r 1 r 4 + r 2 r 3 F ( sin − 1 X ( r initial ) , K ) − ( r 3 − r 1 ) ( r 4 − r 2 ) E ( sin − 1 X ( r initial ) , K ) ] + a ~ c 2 G M c ∫ θ sign ( p θ ) Θ ( θ ) cos 2 θ d θ . (122) Here, F F F E E E r i n i t i a l r_{\mathrm{initial}} r initial r i n i t i a l = r b h , s r_{\mathrm{initial}}=r_{\mathrm{bh,s}} r initial = r bh , s K K K X ( r ) X(r) X ( r )
K = ( r 3 − r 2 ) ( r 4 − r 1 ) ( r 3 − r 1 ) ( r 4 − r 2 ) , X ( r ) = ( r − r 4 ) ( r 3 − r 1 ) ( r − r 3 ) ( r 4 − r 1 ) . (123) K=\frac{(r_3-r_2)(r_4-r_1)}{(r_3-r_1)(r_4-r_2)},\qquad X(r)=\sqrt{\frac{(r-r_4)(r_3-r_1)}{(r-r_3)(r_4-r_1)}}. \tag*{(123)} K = ( r 3 − r 1 ) ( r 4 − r 2 ) ( r 3 − r 2 ) ( r 4 − r 1 ) , X ( r ) = ( r − r 3 ) ( r 4 − r 1 ) ( r − r 4 ) ( r 3 − r 1 ) . (123) The parameter q q q + 1 +1 + 1 − 1 -1 − 1 J ( r , θ ) \mathcal{J}(r,\theta) J ( r , θ ) r 4 ≤ r ≤ ∞ r_4\leq r\leq\infty r 4 ≤ r ≤ ∞ r = ∞ r=\infty r = ∞ θ = θ b h , f \theta=\theta_{\mathrm{bh,f}} θ = θ bh , f J ( r , θ ) \mathcal{J}(r,\theta) J ( r , θ ) r → ∞ r\to\infty r → ∞ θ = θ b h , f \theta=\theta_{\mathrm{bh,f}} θ = θ bh , f [66 , 68 ] .
Using Eqs. (104) , (109) , (110) and taking the limit r → ∞ r\to\infty r → ∞
( μ μ E ~ μ ν ) f = ( 1 , sign ( p r ) f , 0 , 0 ) . (124) \left(\mu^\mu\tilde{E}^\nu_{\ \mu}\right)_f=\left(1,\operatorname{sign}(p_r)_f,0,0\right). \tag*{(124)} ( μ μ E ~ μ ν ) f = ( 1 , sign ( p r ) f , 0 , 0 ) . (124) and,
( e 2 μ E ~ μ ν ) f = 1 C ( sign ( p r ) f ( a ~ G M c c 2 ) cos θ b h , f , ( a ~ G M c c 2 ) cos θ b h , f , − sin θ b h , f [ ( a ~ G M c c 2 ) − D λ sin 2 θ b h , f ] , − sign ( p θ ) f Θ ( θ b h , f ) ) . (125) \begin{aligned}
\left(e^\mu_2\tilde{E}^\nu_{\ \mu}\right)_f={}&\frac{1}{\sqrt{C}}\left(\operatorname{sign}(p_r)_f\left(\tilde{a}\frac{GM_c}{c^2}\right)\cos\theta_{\mathrm{bh,f}},\right.\\
&\left.\left(\tilde{a}\frac{GM_c}{c^2}\right)\cos\theta_{\mathrm{bh,f}},\right.\\
&\left.-\sin\theta_{\mathrm{bh,f}}\left[\left(\tilde{a}\frac{GM_c}{c^2}\right)-\frac{D_\lambda}{\sin^2\theta_{\mathrm{bh,f}}}\right],\right.\\
&\left.-\operatorname{sign}(p_\theta)_f\sqrt{\Theta(\theta_{\mathrm{bh,f}})}\right). \tag*{(125)}
\end{aligned} ( e 2 μ E ~ μ ν ) f = C 1 ( sign ( p r ) f ( a ~ c 2 G M c ) cos θ bh , f , ( a ~ c 2 G M c ) cos θ bh , f , − sin θ bh , f [ ( a ~ c 2 G M c ) − sin 2 θ bh , f D λ ] , − sign ( p θ ) f Θ ( θ bh , f ) ) . (125) Similarly, using Eqs. (104) , (111) , A8 and (120) , we obtain
( e 3 μ E ~ μ ν ) f = 1 C ( J ( ∞ , θ b h , f ) + sign ( p r ) f r 3 , sign ( p r ) f [ J ( ∞ , θ b h , f ) + sign ( p r ) f r 3 ] , sign ( p r ) f sign ( p θ ) f Θ ( θ b h , f ) , sign ( p r ) f sin θ b h , f [ D λ sin 2 θ b h , f − a ~ G M c c 2 ] ) . (126) \begin{aligned}
\left(e^\mu_3\tilde{E}^\nu_{\ \mu}\right)_f=\frac{1}{\sqrt{C}}\bigg(&\mathcal{J}(\infty,\theta_{\mathrm{bh,f}})+\operatorname{sign}(p_r)_f r_3,\\
&\operatorname{sign}(p_r)_f\left[\mathcal{J}(\infty,\theta_{\mathrm{bh,f}})+\operatorname{sign}(p_r)_f r_3\right],\\
&\operatorname{sign}(p_r)_f\operatorname{sign}(p_\theta)_f\sqrt{\Theta(\theta_{\mathrm{bh,f}})},\\
&\operatorname{sign}(p_r)_f\sin\theta_{\mathrm{bh,f}}\left[\frac{D_\lambda}{\sin^2\theta_{\mathrm{bh,f}}}-\tilde{a}\frac{GM_c}{c^2}\right]\bigg). \tag*{(126)}
\end{aligned} ( e 3 μ E ~ μ ν ) f = C 1 ( J ( ∞ , θ bh , f ) + sign ( p r ) f r 3 , sign ( p r ) f [ J ( ∞ , θ bh , f ) + sign ( p r ) f r 3 ] , sign ( p r ) f sign ( p θ ) f Θ ( θ bh , f ) , sign ( p r ) f sin θ bh , f [ sin 2 θ bh , f D λ − a ~ c 2 G M c ] ) . (126) Here, sign ( p r ) f \operatorname{sign}(p_r)_f sign ( p r ) f
sign ( p r ) f = 1. (127) \operatorname{sign}(p_r)_f=1. \tag*{(127)} sign ( p r ) f = 1. (127) The sign of the polar momentum at infinity is given by
sign ( p θ ) f = ( − 1 ) m sign ( p θ ) s . (128) \operatorname{sign}(p_\theta)_f=(-1)^m\operatorname{sign}(p_\theta)_s. \tag*{(128)} sign ( p θ ) f = ( − 1 ) m sign ( p θ ) s . (128) Here, m m m p θ p_\theta p θ (128) . For the pulsar–BH binary configurations considered in this work, we consider trajectories with either m = 0 m=0 m = 0 m = 1 m=1 m = 1 m = 0 m=0 m = 0 m = 1 m=1 m = 1 m = 0 m=0 m = 0 m = 1 m=1 m = 1 m > 1 m>1 m > 1 (124) , (125) , and (126) , we solve Eq. 80 to determine the final direction of the polarization vector, f ^ ( b h , f ) μ \hat{f}_{(\mathrm{bh,f})\mu} f ^ ( bh , f ) μ
At infinity, the radiation propagates in an asymptotically flat spacetime, and the polarization vector must remain transverse to the final propagation direction. Therefore,
f ^ ( b h , f ) ν ( l μ E ~ ν μ ) f = 0. (129) \hat{f}_{(\mathrm{bh},\mathrm{f})\,\nu}\left(l^\mu\widetilde{E}^\nu{}_\mu\right)_{\mathrm{f}}=0. \tag*{(129)} f ^ ( bh , f ) ν ( l μ E ν μ ) f = 0. (129) From Eq. (124) , the final propagation direction is given by
( l μ E ~ ν μ ) f = ( 1 , 1 , 0 , 0 ) , (130) \left(l^\mu\widetilde{E}^\nu{}_\mu\right)_{\mathrm{f}}=(1,1,0,0), \tag*{(130)} ( l μ E ν μ ) f = ( 1 , 1 , 0 , 0 ) , (130) where we have used sign ( p r ) f = 1 \operatorname{sign}(p_r)_{\mathrm{f}}=1 sign ( p r ) f = 1 f ^ ( b h , f ) μ \hat{f}_{(\mathrm{bh},\mathrm{f})\,\mu} f ^ ( bh , f ) μ f ^ ( b h , f ) t = − f ^ ( b h , f ) r \hat{f}_{(\mathrm{bh},\mathrm{f})\,t}=-\hat{f}_{(\mathrm{bh},\mathrm{f})\,r} f ^ ( bh , f ) t = − f ^ ( bh , f ) r
f ^ ( b h , f ) μ → f ^ ( b h , f ) μ + κ ( E ν μ l ν ) f , (131) \hat{f}_{(\mathrm{bh},\mathrm{f})\,\mu}\to\hat{f}_{(\mathrm{bh},\mathrm{f})\,\mu}+\kappa\left(E^\nu{}_\mu l_\nu\right)_{\mathrm{f}}, \tag*{(131)} f ^ ( bh , f ) μ → f ^ ( bh , f ) μ + κ ( E ν μ l ν ) f , (131) without changing the physical polarization state, where κ \kappa κ l μ l_\mu l μ l μ l^\mu l μ [69 ] . Here, E ν μ E^\nu{}_\mu E ν μ E ~ ν μ \widetilde{E}^\nu{}_\mu E ν μ
( E ν μ l ν ) f = ( − 1 , 1 , 0 , 0 ) . (132) \left(E^\nu{}_\mu l_\nu\right)_{\mathrm{f}}=(-1,1,0,0). \tag*{(132)} ( E ν μ l ν ) f = ( − 1 , 1 , 0 , 0 ) . (132) Since the polar and azimuthal components of ( E ν μ l ν ) f \left(E^\nu{}_\mu l_\nu\right)_{\mathrm{f}} ( E ν μ l ν ) f κ = f ^ ( b h , f ) t \kappa=\hat{f}_{(\mathrm{bh},\mathrm{f})\,t} κ = f ^ ( bh , f ) t
In other words, the gauge freedom allows us to set the temporal and radial components of f ^ ( b h , f ) μ \hat{f}_{(\mathrm{bh},\mathrm{f})\,\mu} f ^ ( bh , f ) μ
For completeness, we also give the expression for ( k μ E ~ ν μ ) f \left(k^\mu\widetilde{E}^\nu{}_\mu\right)_{\mathrm{f}} ( k μ E ν μ ) f
\begin{aligned}
\left(k^\mu\widetilde{E}^\nu{}_\mu\right)_{\mathrm{f}}={}&\left(\frac{1}{2C}\left[\left(\mathcal{J}(\infty,\theta_{\mathrm{bh},\mathrm{f}})+\operatorname{sign}(p_r)_{\mathrm{f}}r_3\right)^2+\left(\tilde{a}\frac{GM_c}{c^2}\right)^2\cos^2\theta_{\mathrm{bh},\mathrm{f}}+C\right],\\
&\frac{\operatorname{sign}(p_r)_{\mathrm{f}}}{2C}\left[\left(\mathcal{J}(\infty,\theta_{\mathrm{bh},\mathrm{f}})+\operatorname{sign}(p_r)_{\mathrm{f}}r_3\right)^2+\left(\tilde{a}\frac{GM_c}{c^2}\right)^2\cos^2\theta_{\mathrm{bh},\mathrm{f}}-C\right],\\
&-\operatorname{sign}(p_r)_{\mathrm{f}}\frac{\tilde{a}GM_c}{c^2}\frac{\sin\theta_{\mathrm{bh},\mathrm{f}}\cos\theta_{\mathrm{bh},\mathrm{f}}}{C}\left(\tilde{a}\frac{GM_c}{c^2}-\frac{D_\lambda}{\sin^2\theta_{\mathrm{bh},\mathrm{f}}}\right)\\
&+\operatorname{sign}(p_r)_{\mathrm{f}}\mathcal{J}(\infty,\theta_{\mathrm{bh},\mathrm{f}})\operatorname{sign}(p_\theta)_{\mathrm{f}}\frac{\sqrt{\Theta(\theta_{\mathrm{bh},\mathrm{f}})}}{C}
+\operatorname{sign}(p_\theta)_{\mathrm{f}}\frac{r_3\sqrt{\Theta(\theta_{\mathrm{bh},\mathrm{f}})}}{C},\\
&-\frac{\sin\theta_{\mathrm{bh},\mathrm{f}}}{C}\tilde{a}\frac{GM_c}{c^2}\cot\theta_{\mathrm{bh},\mathrm{f}}\operatorname{sign}(p_r)_{\mathrm{f}}\operatorname{sign}(p_\theta)_{\mathrm{f}}\sqrt{\Theta(\theta_{\mathrm{bh},\mathrm{f}})}\\
&-\frac{\sin\theta_{\mathrm{bh},\mathrm{f}}}{C}\left(\tilde{a}\frac{GM_c}{c^2}-\frac{D_\lambda}{\sin^2\theta_{\mathrm{bh},\mathrm{f}}}\right)\operatorname{sign}(p_r)_{\mathrm{f}}\mathcal{J}(\infty,\theta_{\mathrm{bh},\mathrm{f}})\\
&-r_3\frac{\sin\theta_{\mathrm{bh},\mathrm{f}}}{C}\left(\tilde{a}\frac{GM_c}{c^2}-\frac{D_\lambda}{\sin^2\theta_{\mathrm{bh},\mathrm{f}}}\right)+\tilde{a}\left(\frac{GM_c}{c^2}\right)^2\frac{\sin\theta_{\mathrm{bh},\mathrm{f}}}{C}\right). \tag*{(133)}
\end{aligned}
References 