General Relativistic Aberration Equation and Measurable Angle of Light Ray in Kerr–de Sitter Spacetime

Abstract

As an extension of our previous paper, instead of the total deflection angle $\alpha$, we will mainly focus on the discussion of measurable angle of the light ray ${\psi }_P$ at the position of observer P in Kerr–de Sitter spacetime, which includes the cosmological constant $\mathsf{\Lambda }$. We will investigate the contribution of the radial and transverse motion of the observer which are connected with radial velocity $v^r$ and transverse velocity $b{v}^{\varphi }$ (b is the impact parameter) as well as the spin parameter a of the central object which induces the gravito-magnetic field or frame dragging and the cosmological constant $\mathsf{\Lambda }$. The general relativistic aberration equation is employed to take into account the influence of motion of the observer on the measurable angle ${\psi }_P$. The measurable angle ${\psi }_P$ derived in this paper can be applicable to the observer placed within the curved and finite-distance region in the spacetime. The equation of light trajectory will be obtained in such a sense that the background is de Sitter spacetime instead of Minkowski one. As an example, supposing the cosmological gravitational lensing effect, we assume that the lens object is the typical galaxy and the observer is in motion with respect to the lensing object at a recession velocity $v^r=b{v}^{\varphi }=v_H=H_{0}D$ (where $H_0$ is a Hubble constant and D means the distance between the observer and the lens object). The static terms $\mathcal{O}(\mathsf{\Lambda }bm,\mathsf{\Lambda }ba)$ are basically comparable with the second order deflection term $\mathcal{O}(m^2)$, and they are almost one order smaller that the Kerr deflection $-4ma/b^2$. The velocity-dependent terms $\mathcal{O}(\mathsf{\Lambda }bm{v}^r,\mathsf{\Lambda }ba{v}^r)$ for radial motion and $\mathcal{O}(\mathsf{\Lambda }{b}^{2}m{v}^{\varphi },\mathsf{\Lambda }{b}^{2}a{v}^{\varphi })$ for transverse motion are at most two orders of magnitude smaller than the second order deflection $\mathcal{O}(m^2)$. We also find that even when the radial and transverse velocity have the same sign, asymptotic behavior as $\varphi$ approaches 0 is different from each other, and each diverges to opposite infinity.

1. Introduction

The cosmological constant problem is an old but unsolved issue in astrophysics and cosmology that is closely related to the general theory of relativity; see reviews by, e.g., [1,2]. After the establishment of the general theory of relativity in 1915–1916, Einstein incorporated the cosmological term $\mathsf{\Lambda }{g}_{\mu \nu }$ into the field equation to represent the static Universe. Although the discovery of cosmic expansion by Hubble caused Einstein to withdraw the cosmological term from the field equation, it is currently widely considered that the cosmological constant $\mathsf{\Lambda }$, or the dark energy in a more general sense, is the most promising candidate for explaining the observed accelerating expansion of the Universe [3,4,5] despite the fact that its details are not at all clear.

One straightforward way to tackle the cosmological constant problem from another viewpoint is to investigate the effect of the cosmological constant $\mathsf{\Lambda }$ on the light deflection. The deflection of a light ray is the basis of gravitational lensing which is a powerful tool in astrophysics and cosmology; see, e.g., [6,7] and the references therein. Particularly, in cosmological gravitational lensing, it is important to take into account the cosmological constant to the formula of the total deflection angle $\alpha$; then it is expected to provide a new verification of the cosmological constant.

The influence of the cosmological constant $\mathsf{\Lambda }$ on the total deflection angle $\alpha$ had been the subject of a long debate and was investigated mainly under the static and spherically symmetric vacuum solution, namely the Schwarzschild–de Sitter/Kottler solution. See [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24] and the references therein. However, since celestial objects such as stars and galaxies are generally rotating, then it is necessary to formulate the effect of rotation as well as the effect of the cosmological constant $\mathsf{\Lambda }$ to consider light deflection. To date, several authors have discussed the total light deflection $\alpha$ in Kerr–de Sitter spacetime, which is the stationary and axially symmetric vacuum solution and includes the spin parameter a of the central object and the cosmological constant $\mathsf{\Lambda }$; see, e.g., [25,26,27,28] and the references therein. The same consideration is further extended to the more general Kerr-type solutions, see, e.g., [29,30,31,32,33,34,35,36].

Moreover, it is also significant to incorporate the effect of the relative motion between the lens object and the observer. In many previous papers, the light deflection is mainly discussed under the assumption that the observer is stationary in space. However, in general, the observer is in motion with respect to the lens object, thus it is important to take velocity effect into account for more accurate formulation. In addition, not only to solve the problem of the cosmological constant but also to verify the theory of gravity using gravitational lensing, it is crucial to clarify how the rotation of celestial body, the cosmological constant, and the velocity of observer as well as the mass of celestial body are connected with each other and contribute to the bending of light ray.

In [37], accounting for the effect of the observer’s velocity, we examined the measurable angle of the light ray ${\psi }_P$ at the position of the observer P in the Kerr spacetime instead of the discussing on the total deflection angle $\alpha$. In order to deal with velocity effect, we adopted the general relativistic aberration equation,

$$\begin{array}{c}\hfill cos{\psi }_P=\frac{g_{\mu \nu }{k}^{\mu }{w}^{\nu }}{(g_{\mu \nu }{u}^{\mu }{k}^{\nu })(g_{\mu \nu }{u}^{\mu }{w}^{\nu })}+1.\end{array}$$

(1)

Equation (1) gives the measurable angle ${\psi }_P$ at the position P between the tangent vector $k^{\mu }$ of the light ray ${\mathsf{\Gamma }}_k$ that we investigate and the tangent vector $w^{\mu }$ of the radial null geodesic ${\mathsf{\Gamma }}_w$ connecting the center O and the position of observer P, and $u^{\mu }$ is the 4-velocity of the observer P. See Figure 1 and for the derivation of Equation (1), refer to [20] and see also [37,38,39]. Equation (1) enables us to compute the effect of the motion of the observer easily and straightforwardly because the equations of the null geodesic of ${\mathsf{\Gamma }}_k$ and ${\mathsf{\Gamma }}_w$ do not depend on the motion of the observer; the velocity effect is incorporated in the formula as the form of the 4-velocity of the observer $u^{\mu }$.

In this paper, we will extend our paper [37] to Kerr–de Sitter spacetime containing the cosmological constant $\mathsf{\Lambda }$ as well as the spin parameter of the central object a. Our purpose is to examine not only the contribution of the cosmological constant $\mathsf{\Lambda }$ and the spin parameter a of the central object but also the effect of the motion of the observer on the measurable angle ${\psi }_P$. The 4-velocity of the observer $u^{\mu }$ appearing in Equation (1) is converted to the coordinate radial velocity $v^r=dr/dt$ and coordinate transverse velocity $b{v}^{\varphi }=bd\varphi /dt$ (b is the impact parameter and $v^{\varphi }=d\varphi /dt$ denotes the coordinate angular velocity), respectively. Previously, the radial motion of the observer in Kerr–Newman and Kerr–de Sitter spacetimes have been discussed by [40,41], respectively.

Before closing this section, it is noteworthy to explain why we deal with measurable angle ${\psi }_P$ instead of the total deflection angle $\alpha$ of light ray in this paper. This is because the concept and definition of the total deflection angle of the light ray $\alpha$ is a counterintuitive and difficult problem. One of the reasons for this is that because the Schwarzschild–de Sitter and Kerr–de Sitter solutions are not asymptotically flat unlike the Schwarzschild and Kerr solutions, it is ambiguous and unclear how the total deflection angle $\alpha$ should be defined in curved spacetime. Although to overcome this difficulty, a method for calculating the total deflection angle is independently investigated and proposed on the basis of the Gauss–Bonnet theorem by e.g., [23,24,42,43], it seems that further consideration is needed to settle the argument. However it is always possible to determine the measurable (local) angle ${\psi }_P$ at the position of the observer P.

This paper is organized as follows: in Section 2, the trajectory of a light ray in Kerr–de Sitter spacetime is derived from the first-order differential equation of the null geodesic. In Section 3, the measurable angle ${\psi }_P$ in Kerr–de Sitter spacetime is calculated for the cases of the static observer, the observer in radial motion and the observer in transverse motion. Finally, Section 4 is devoted to presenting the conclusions.

2. Light Trajectory in Kerr–de Sitter Spacetime

The Kerr–de Sitter solution—see Equations of (5.65) and (5.66) in [44], and also, e.g., [25,26,27,28]—in Boyer–Lindquist type coordinates $(t,r,\theta ,\varphi )$ can be rearranged as

$$\begin{array}{cc}\hfill d{s}^2& =g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }\hfill \\ \hfill & =-\frac{{\mathsf{\Delta }}_r-{\mathsf{\Delta }}_{\theta }{a}^2{sin}^2\theta }{{\rho }^2{\mathsf{\Xi }}^2}d{t}^2+\frac{{\rho }^2}{{\mathsf{\Delta }}_r}d{r}^2+\frac{{\rho }^2}{{\mathsf{\Delta }}_{\theta }}d{\theta }^2-\frac{2a{sin}^2\theta }{{\rho }^2{\mathsf{\Xi }}^2}\left[{\mathsf{\Delta }}_{\theta }(r^2+a^2)-{\mathsf{\Delta }}_r\right]dtd\varphi \hfill \\ \hfill & +\frac{{sin}^2\theta }{{\rho }^2{\mathsf{\Xi }}^2}\left[{\mathsf{\Delta }}_{\theta }{(r^2+a^2)}^2-{\mathsf{\Delta }}_r{a}^2{sin}^2\theta \right]d{\varphi }^2,\hfill \\ \hfill {\mathsf{\Delta }}_r& =r^2+a^2-2mr-\frac{\mathsf{\Lambda }}{3}{r}^2(r^2+a^2),\hfill \\ \hfill {\mathsf{\Delta }}_{\theta }& =1+\frac{\mathsf{\Lambda }}{3}{a}^2{cos}^2\theta ,{\rho }^2=r^2+a^2{cos}^2\theta ,\mathsf{\Xi }=1+\frac{\mathsf{\Lambda }}{3}{a}^2.\hfill \end{array}$$

(2)

$g_{\mu \nu }$ is the metric tensor; Greek indices, e.g., $\mu ,\nu$, run from 0 to 3; $\mathsf{\Lambda }$ is the cosmological constant; m is the mass of the central object; $a\equiv J/m$ is a spin parameter of the central object (J is the angular momentum of the central object) and we use the geometrical unit $c=G=1$ throughout this paper. For the sake of brevity, we restrict the trajectory of the light ray to the equatorial plane $\theta =\pi /2,d\theta =0$, and rewrite Equation (2) in symbolic form as

$$\begin{array}{cc}\hfill d{s}^2& =-A(r)d{t}^2+B(r)d{r}^2+2C(r)dtd\varphi +D(r)d{\varphi }^2,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill A(r)& ={\left(1+\frac{\mathsf{\Lambda }}{3}{a}^2\right)}^{-2}\left[1-\frac{2m}{r}-\frac{\mathsf{\Lambda }}{3}(r^2+a^2)\right],\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill B(r)& ={\left[\left(1-\frac{\mathsf{\Lambda }}{3}{r}^2\right)\left(1+\frac{a^2}{r^2}\right)-\frac{2m}{r}\right]}^{-1},\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill C(r)& =-{\left(1+\frac{\mathsf{\Lambda }}{3}{a}^2\right)}^{-2}a\left[\frac{2m}{r}+\frac{\mathsf{\Lambda }}{3}(r^2+a^2)\right],\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill D(r)& ={\left(1+\frac{\mathsf{\Lambda }}{3}{a}^2\right)}^{-2}\left[(r^2+a^2)\left(1+\frac{\mathsf{\Lambda }}{3}{a}^2\right)+\frac{2m{a}^2}{r}\right].\hfill \end{array}$$

(7)

Two constants of motion of the light ray, the energy E and the angular momentum L, are given by

$$\begin{array}{c}\hfill E=A(r)\frac{dt}{d\lambda }-C(r)\frac{d\varphi }{d\lambda },L=C(r)\frac{dt}{d\lambda }+D(r)\frac{d\varphi }{d\lambda },\end{array}$$

(8)

where $\lambda$ is the affine parameter. Solving for $dt/d\lambda$ and $d\varphi /d\lambda$, we obtain two relations:

$$\begin{array}{c}\hfill \frac{dt}{d\lambda }=\frac{ED(r)+LC(r)}{A(r)D(r)+C^2(r)},\frac{d\varphi }{d\lambda }=\frac{LA(r)-EC(r)}{A(r)D(r)+C^2(r)}.\end{array}$$

(9)

From the null condition $d{s}^2=0$, the geodesic equation of the light ray is expressed as

$$\begin{array}{c}\hfill {\left(\frac{dr}{d\varphi }\right)}^2=\frac{A(r)D(r)+C^2(r)}{B(r){[bA(r)-C(r)]}^2}[-b^{2}A(r)+2bC(r)+D(r)],\end{array}$$

(10)

where b is the impact parameter defined as

$$\begin{array}{c}\hfill b\equiv \frac{L}{E}.\end{array}$$

(11)

Replacing r by $u=1/r$ and using Equations (4), (5), (6), (7) and (10), the first-order differential equation of the light ray becomes 1

$$\begin{array}{cc}\hfill {\left(\frac{du}{d\varphi }\right)}^2& =\frac{{\left[\left(1-\frac{\mathsf{\Lambda }}{3u^2}\right)(1+a^2{u}^2)-2mu\right]}^2}{{\left(1+\frac{\mathsf{\Lambda }}{3}{a}^2\right)}^2{\left[b-2mu(b-a)-\frac{\mathsf{\Lambda }}{3u^2}(b-a)(1+a^2{u}^2)\right]}^2}\hfill \\ \hfill & \times \left[1+(a^2-b^2)u^2+2m{u}^3{(b-a)}^2+\frac{\mathsf{\Lambda }}{3}{(b-a)}^2(1+a^2{u}^2)\right].\hfill \end{array}$$

(12)

Expanding Equation (12) up to the second order with respect to $m/b$, $a/b$, and $\mathsf{\Lambda }{b}^2$, we have

$$\begin{array}{c}\hfill {\left(\frac{du}{d\varphi }\right)}^2=\frac{1}{b^2}+\frac{\mathsf{\Lambda }}{3}-u^2+2m{u}^3\begin{array}{c}\hfill -2a^2{u}^4\end{array}+\frac{3a^2{u}^2}{b^2}-\frac{4amu}{b^3}-\frac{2\mathsf{\Lambda }a}{3b^3{u}^2}+\mathcal{O}({\epsilon }^3).\end{array}$$

(13)

Note that for the sake of simplicity, we introduced the notation for the small expansion parameters $m/b$, $a/b$ and $\mathsf{\Lambda }{b}^2$ as

$$\begin{array}{c}\hfill \epsilon \sim m/b\sim a/b\sim \mathsf{\Lambda }{b}^2,\epsilon \ll 1,\end{array}$$

(14)

then $\mathcal{O}({\epsilon }^3)$ in Equation (13) denotes combinations of these three parameters, $m/b$, $a/b$, and $\mathsf{\Lambda }{b}^2$. Henceforth, we use the same notation to represent the order of the approximation and residual terms.

It is instructive to discuss how to choose a zeroth-order solution $u_0$ of the light trajectory u. If $m=0$ and $a=0$, then Equation (13) reduces to the null geodesic equation in de Sitter spacetime

$$\begin{array}{c}\hfill {\left(\frac{du}{d\varphi }\right)}^2=\frac{1}{b^2}+\frac{\mathsf{\Lambda }}{3}-u^2,\end{array}$$

(15)

which can be also derived immediately from Equation (12). Because we assume a nonzero cosmological constant $\mathsf{\Lambda }$ a priori, we cannot take $\mathsf{\Lambda }$ to be zero, and Equation (15) cannot be reduced to the null geodesic equation in Minkowski spacetime; in fact the action

$$\begin{array}{c}\hfill \mathcal{S}=\int \left[\frac{c^4}{16\pi G}(R-2\mathsf{\Lambda })+{\mathcal{L}}_M\right]\sqrt{-g}{d}^{4}x,\end{array}$$

(16)

and the field equation

$$\begin{array}{c}\hfill R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R+\mathsf{\Lambda }{g}_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \nu },\end{array}$$

(17)

include the cosmological constant $\mathsf{\Lambda }$ explicitly; here $g=det(g_{\mu \nu })$, ${\mathcal{L}}_M$ denotes the Lagrangian for the matter field; $R_{\mu \nu }$ and R are the Ricci tensor and Ricci scalar, respectively; $T_{\mu \nu }$ is the energy-momentum tensor. Because m and a are the integration or arbitrary constants in the Kerr–de Sitter solution, it is possible to put $m=0$ and $a=0$. Therefore, zeroth-order solution of the u of the light ray should be taken as the form

$$\begin{array}{c}\hfill u_0=\frac{sin\varphi }{B},\frac{1}{B^2}\equiv \frac{1}{b^2}+\frac{\mathsf{\Lambda }}{3},\end{array}$$

(18)

in which we introduced another constant B. Note that Equation (18) should be evaluated in de Sitter spacetime.

Let us obtain the equation of the light trajectory in accordance with the standard perturbation scheme. We take the solution $u=u(\varphi )$ of the light trajectory as

$$\begin{array}{c}\hfill u=\frac{sin\varphi }{B}+\epsilon \delta u_1+{\epsilon }^2\delta u_2,\end{array}$$

(19)

where $\epsilon \delta u_1$ and ${\epsilon }^2\delta u_2$ are the first order $\mathcal{O}(\epsilon )$ and second order $\mathcal{O}({\epsilon }^2)$ corrections to the zeroth-order solution $u_0=sin\varphi /B$, respectively.

According to the standard perturbation method, we insert Equation (19) into Equation (13), and obtain the differential equation for $d\delta u_1/d\varphi$ and $d\delta u_2/d\varphi$ by expanding with respect to $\epsilon$ and combining the same order terms of $\epsilon$. Then, integrating $d\delta u_1/d\varphi$ and $d\delta u_2/d\varphi$ iteratively, $\delta u_1$ and $\delta u_2$ are obtained, and inserting the $\delta u_1$ and $\delta u_2$ into Equation (19), the equation of the light trajectory in Kerr–de Sitter spacetime is given up to second-order $\mathcal{O}({\epsilon }^2)$ by

$$\begin{array}{cc}\hfill \frac{1}{r}& =\frac{sin\varphi }{B}+\frac{m}{2B^2}(3+cos2\varphi )\hfill \\ \hfill & +\frac{1}{16B^3{b}^3}\left\{b^3\left(3m^2+2a^2\right)(3sin\varphi -sin3\varphi )+4b\left[7b^2{m}^2+6a^2(B^2-b^2)\right]sin\varphi \right.\hfill \\ \hfill & +\left.\left(30b^3{m}^2-12a^2{b}^3+12B^2{a}^{2}b\right)\left(\pi -2\varphi \right)cos\varphi -32B^{3}am\right\}\hfill \\ \hfill & -\frac{\mathsf{\Lambda }{B}^{3}a\left(2{sin}^2\varphi -1\right)}{3b^{3}sin\varphi }+\mathcal{O}({\epsilon }^3),\hfill \end{array}$$

(20)

where the integration constants of $\delta u_1$ and $\delta u_2$ are chosen so as to maximize u (or to minimize r) at $\varphi =\pi /2$.

We note that because the differential equations, $d\delta u_1/d\varphi$ and $d\delta u_2/d\varphi$, are contain two constants, B and b (see Equations (13) and (19)), and the expansion parameter $\epsilon$ is also related to the constant b instead of B (see Equation (14)). As a consequence, the expression of $u=1/r$, Equation (20), includes two constants B and b, and leads to a complicated expression for the measurable angle ${\psi }_P$. To avoid the complex expression for ${\psi }_P$, we expand B in $\mathsf{\Lambda }$ and express B by b and $\mathsf{\Lambda }$, obtaining

$$\begin{array}{cc}\hfill \frac{1}{r}& =\left(\frac{1}{b}+\frac{b\mathsf{\Lambda }}{6}-\frac{b^3{\mathsf{\Lambda }}^2}{72}\right)sin\varphi +\frac{m}{2}(3+cos2\varphi )\left(\frac{1}{b^2}+\frac{\mathsf{\Lambda }}{3}\right)\hfill \\ \hfill & +\frac{1}{16b^3}\left\{m^2\left[37sin\varphi +30(\pi -2\varphi )cos\varphi -3sin3\varphi \right]+8a^2{sin}^3\varphi -32am\right\}\hfill \\ \hfill & -\frac{\mathsf{\Lambda }a\left(2{sin}^2\varphi -1\right)}{3sin\varphi }+\mathcal{O}({\epsilon }^3).\hfill \end{array}$$

(21)

Note that if we use the approximate solution of Equation (13) used by, e.g., [27], as,

$$\begin{array}{c}\hfill u=\frac{sin\varphi }{b}+\delta u_1+\delta u_2,\end{array}$$

(22)

the following terms in Equation (21) disappear:

$$\begin{array}{c}\hfill \left(\frac{b\mathsf{\Lambda }}{6}-\frac{b^3{\mathsf{\Lambda }}^2}{72}\right)sin\varphi ,\frac{m\mathsf{\Lambda }}{6}(3+cos2\varphi ).\end{array}$$

(23)

The existence of the above terms in Equation (21) reflects the fact that the background spacetime is de Sitter spacetime instead of Minkowski spacetime.

Before concluding this section, it is noteworthy that, unlike Schwarzschild–de Sitter spacetime, the trajectory equation of the light ray in Kerr–de Sitter spacetime depends on the cosmological constant $\mathsf{\Lambda }$; from the condition

$$\begin{array}{c}\hfill {\left.\frac{du}{d\varphi }\right|}_{u=u_0}=0,u_0=\frac{1}{r_0},\end{array}$$

(24)

and Equation (13), we have following relation:

$$\begin{array}{c}\hfill \frac{1}{B^2}=\frac{1}{b^2}+\frac{\mathsf{\Lambda }}{3}=\frac{1}{r_0^2}-\frac{2m}{r_0^3}+\frac{2a^2}{r_0^2}-\frac{3a^2}{b^2{r}_0^2}+\frac{4am}{b^3{r}_0}-\frac{2\mathsf{\Lambda }a{r}_0^2}{3b^3}+\mathcal{O}({\epsilon }^3),\end{array}$$

(25)

in which $r_0$ is the radial coordinate value of the light ray at the point of closest approach (in our case $\varphi =\pi /2$), and $r_0$ can be obtained by the observation in principle as the circumference radius ${\ell }_0=2\pi r_0$. It is also possible to obtain a similar relation from Equations (12) and (13); but the expression becomes more complicated. Equation (25) means that unlike the Schwarzschild–de Sitter case, B cannot be expressed only by $r_0$, m, and a; $\mathsf{\Lambda }$ and b too are required. As a result, the trajectory equation of the light ray in Kerr–de Sitter spacetime depends on the cosmological constant $\mathsf{\Lambda }$ and b; whereas the equation of the light trajectory in Schwarzschild–de Sitter spacetime is independent of $\mathsf{\Lambda }$ and b; in fact setting $a=0$ in Equation (25) yields

$$\begin{array}{c}\hfill \frac{1}{B_{\mathrm{SdS}}^2}=\frac{1}{b^2}+\frac{\mathsf{\Lambda }}{3}=\frac{1}{r_0^2}-\frac{2m}{r_0^3}.\end{array}$$

(26)

Equation (26) shows that the constant $B_{\mathrm{SdS}}$ in Schwarzschild–de Sitter spacetime can be determined without knowing $\mathsf{\Lambda }$ and b.

3. Measurable Angle in Kerr–de Sitter Spacetime

Henceforth, in accordance with the procedure and formulas described in [37], we calculate the measurable angle ${\psi }_P$ at the position of observer P and in this paper, we only gives the final results of the measurable angle ${\psi }_P$. See [37] for details of the calculation.

3.1. Measurable Angle by Static Observer

In the case of the static observer, the component of the 4-velocity of the observer $u^{\mu }$ becomes

$$\begin{array}{c}\hfill u^{\mu }=(u^t,0,0,0),\end{array}$$

(27)

and the condition for the time-like observer $g_{\mu \nu }{u}^{\mu }{u}^{\nu }=-1$ gives $u^t$ as

$$\begin{array}{c}\hfill u^t=\frac{1}{\sqrt{A(r)}},\end{array}$$

(28)

where we take $u^t$ to be positive.

Expanding up to the order $\mathcal{O}({\epsilon }^2)$, the measurable angle ${\psi }_{\mathrm{stat}}$ is obtained as, for the range $0\le \psi \le \pi /2$,

$$\begin{array}{cc}\hfill {\psi }_{\mathrm{stat}}& =\varphi +\frac{2m}{b}cos\varphi +\frac{1}{8b^2}\left\{m^2\left[15(\pi -2\varphi )-sin2\varphi \right]-16macos\varphi \right\}\hfill \\ \hfill & -\frac{\mathsf{\Lambda }{b}^2}{6}cot\varphi +\frac{\mathsf{\Lambda }b}{3}cos\varphi \left[m(1+{csc}^2\varphi )+2acsc\varphi \right]-\frac{{\mathsf{\Lambda }}^2{b}^4}{288}{csc}^4\varphi sin4\varphi \hfill \\ \hfill & +\mathcal{O}({\epsilon }^3).\hfill \end{array}$$

(29)

and for the range $\pi /2\le \varphi \le \pi$

$$\begin{array}{cc}\hfill {\psi }_{\mathrm{stat}}& =\pi -\varphi -\frac{2m}{b}cos\varphi -\frac{1}{8b^2}\left\{m^2\left[15(\pi -2\varphi )-sin2\varphi \right]-16macos\varphi \right\}\hfill \\ \hfill & +\frac{\mathsf{\Lambda }{b}^2}{6}cot\varphi -\frac{\mathsf{\Lambda }b}{3}cos\varphi \left[m(1+{csc}^2\varphi )+2acsc\varphi \right]+\frac{{\mathsf{\Lambda }}^2{b}^4}{288}{csc}^4\varphi sin4\varphi \hfill \\ \hfill & +\mathcal{O}({\epsilon }^3).\hfill \end{array}$$

(30)

As in [37], we divided the expression for ${\psi }_{\mathrm{stat}}$ into two cases, Equations (29) and (30). The purpose of this was to utilize trigonometric identities such as $\sqrt{1-{sin}^2\varphi }=cos\varphi$ for $0\le \varphi \le \pi /2$ and $\sqrt{1-{sin}^2\varphi }=-cos\varphi$ for $\pi /2\le \varphi \le \pi$. Henceforth we adopt a similar procedure when calculating angle measured by the observer in radial motion, ${\psi }_{\mathrm{rad}}$ and in transverse motion ${\psi }_{\mathrm{trans}}$ below. Although this procedure may not be necessary for computing ${\psi }_{\mathrm{stat}}$ and ${\psi }_{\mathrm{rad}}$, it is required when computing ${\psi }_{\mathrm{trans}}$; see Equation (38) and observe the case for $\varphi \to \pi$.

The first line in Equations (29) and (30) are in agreement with the measurable angle of the static observer in Kerr spacetime derived in [37], and the second line in Equations (29) and (30) are due to the influence of the cosmological constant $\mathsf{\Lambda }$.

Before concluding this section, we note that if Equation (22) (see also Equation (23)) is adopted, the measurable angle ${\tilde{\psi }}_{\mathrm{stat}}$ for the range $0\le \varphi \le \pi /2$ becomes

$$\begin{array}{cc}\hfill {\tilde{\psi }}_{\mathrm{stat}}& =\varphi +\frac{2m}{b}cos\varphi +\frac{1}{8b^2}\left\{m^2\left[15(\pi -2\varphi )-sin2\varphi \right]-16macos\varphi \right\}\hfill \\ \hfill & -\frac{\mathsf{\Lambda }{b}^2}{6}csc\varphi sec\varphi +\frac{\mathsf{\Lambda }b}{3}\left[m(cot\varphi csc\varphi -sec\varphi )+2acot\varphi \right]\hfill \\ \hfill & -\frac{{\mathsf{\Lambda }}^2{b}^4}{9}cot2\varphi {csc}^{2}2\varphi +\mathcal{O}({\epsilon }^3),\hfill \end{array}$$

(31)

where the first line is also in agreement with the measurable angle of the static observer in the Kerr spacetime as derived in [37].

Comparing Equations (29) and (31), we find the following: first, in spite of the different correction terms due to the cosmological constant $\mathsf{\Lambda }$, the measurable angles ${\psi }_{\mathrm{stat}}$ and ${\tilde{\psi }}_{\mathrm{stat}}$ take a large value rapidly and diverge to negative infinity when $\varphi$ approaches 0. This property is related to the existence of the de Sitter horizon. Second, when $\varphi \to \pi /2$, Equation (29) leads to the result ${\psi }_{\mathrm{stat}}\to \pi /2$, which is consistent with the initial condition, ${du/d\varphi |}_{\varphi =\pi /2}=0$. However, Equation (31) diverges, and ${\tilde{\psi }}_{\mathrm{stat}}\to \infty$, which contradicts the initial condition, ${du/d\varphi |}_{\varphi =\pi /2}=0$. Therefore, Equation (18) should be employed as the zeroth-order solution of u; as a consequence, Equation (20) or at least Equation (21) should be used as the trajectory equation of the light ray when investigating light bending in Kerr–de Sitter spacetime. The same holds for Schwarzschild–de Sitter spacetime.

3.2. Measurable Angle by Observer in Radial Motion

The component of the 4-velocity $u^{\mu }$ of the radially moving observer is

$$\begin{array}{c}\hfill u^{\mu }=(u^t,u^r,0,0),\end{array}$$

(32)

and from the condition $g_{\mu \nu }{u}^{\mu }{u}^{\nu }=-1$, $u^t$ can be expressed in terms of $u^r$ as

$$\begin{array}{c}\hfill u^t=\sqrt{\frac{B(r){(u^r)}^2+1}{A(r)}},\end{array}$$

(33)

where $u^t$ is taken to be positive.

Here, we use the radial velocity $v^r$ converted from the component of the 4-velocity $u^r$, and impose the slow motion approximation for the radial velocity of the observer, $v^r\ll 1$. Then, the measurable angle ${\psi }_{\mathrm{rad}}$ for the range $0\le \varphi \le \pi /2$ is given by up to the order $\mathcal{O}({\epsilon }^2,{\epsilon }^2{v}^r)$ as,

$$\begin{array}{cc}\hfill {\psi }_{\mathrm{rad}}& =\varphi +v^{r}sin\varphi +\frac{2m}{b}(cos\varphi +v^r)+\frac{1}{8b^2}\left(m^2[15(\pi -2\varphi )-sin2\varphi ]-16amcos\varphi \right)\hfill \\ \hfill & +\frac{v^r}{16b^2}\left\{m^2\left[30(\pi -2\varphi )cos\varphi +95sin\varphi -sin3\varphi \right]\right.\hfill \\ \hfill & \left.-16am(1+cos2\varphi )-2a^2(3sin\varphi -sin3\varphi )\right\}\hfill \\ \hfill & -\frac{\mathsf{\Lambda }{b}^2}{6}cot\varphi -\frac{\mathsf{\Lambda }{b}^2{v}^r}{12}(cos2\varphi -3)+\frac{\mathsf{\Lambda }b}{3}cos\varphi \left[m(1+{csc}^2\varphi )+2acsc\varphi \right]\hfill \\ \hfill & +\frac{\mathsf{\Lambda }b{v}^r}{3}csc\varphi \left[a+mcsc\varphi +(a-2mcsc\varphi )cos2\varphi \right]-\frac{{\mathsf{\Lambda }}^2{b}^4}{288}{csc}^4\varphi sin4\varphi \hfill \\ \hfill & -\frac{{\mathsf{\Lambda }}^2{b}^4{v}^r}{144}(cos2\varphi -7){cot}^2\varphi csc\varphi +\mathcal{O}({\epsilon }^3,{(v^r)}^2),\hfill \end{array}$$

(34)

and for the range $\pi /2\le \varphi \le \pi$

$$\begin{array}{cc}\hfill {\psi }_{\mathrm{rad}}& =\pi -\varphi +v^{r}sin\varphi +\frac{2m}{b}\left(-cos\varphi +v^r\right)-\frac{1}{8b^2}\left(m^2[15(\pi -2\varphi )-sin2\varphi ]-16amcos\varphi \right)\hfill \\ \hfill & +\frac{v^r}{16b^2}\left\{m^2\left[30(\pi -2\varphi )cos\varphi +95sin\varphi -sin3\varphi \right]\right.\hfill \\ \hfill & \left.-16am(1+cos2\varphi )-2a^2(3sin\varphi -sin3\varphi )\right\}\hfill \\ \hfill & +\frac{\mathsf{\Lambda }{b}^2}{6}cot\varphi -\frac{\mathsf{\Lambda }{b}^2{v}^r}{12}(cos2\varphi -3)-\frac{\mathsf{\Lambda }b}{3}cos\varphi \left[m(1+{csc}^2\varphi )+2acsc\varphi \right]\hfill \\ \hfill & +\frac{\mathsf{\Lambda }b{v}^r}{3}csc\varphi \left[a+mcsc\varphi +(a-2mcsc\varphi )cos2\varphi \right]+\frac{{\mathsf{\Lambda }}^2{b}^4}{288}{csc}^4\varphi sin4\varphi \hfill \\ \hfill & -\frac{{\mathsf{\Lambda }}^2{b}^4{v}^r}{144}(cos2\varphi -7){cot}^2\varphi csc\varphi +\mathcal{O}({\epsilon }^3,{(v^r)}^2).\hfill \end{array}$$

(35)

The first three lines in Equations (34) and (35) coincide with the measurable angle in Kerr spacetime which has already been investigated in [37], and the remaining terms, lines 4 to 6, are the correction due to the cosmological constant $\mathsf{\Lambda }$.

3.3. Measurable Angle by Observer in Transverse Motion

As the third case, let us investigate the observer in transverse motion which is the motion in a direction perpendicular to the radial direction in the orbital plane. The component of the 4-velocity of the observer $u^{\mu }$ is

$$\begin{array}{c}\hfill u^{\mu }=(u^t,0,0,u^{\varphi }),\end{array}$$

(36)

and the condition $g_{\mu \nu }{u}^{\mu }{u}^{\nu }=-1$ gives

$$\begin{array}{c}\hfill u^t=\frac{C(r)u^{\varphi }+\sqrt{{\left[C(r)u^{\varphi }\right]}^2+A(r)[D(r){(u^{\varphi })}^2+1]}}{A(r)},\end{array}$$

(37)

in which we chose the sign of $u^t$ to be positive.

As in the radial case, we convert from the component of 4-velocity $u^{\varphi }$ to the angular velocity $v^{\varphi }$ and impose a slow motion approximation $v^{\varphi }\ll 1$. Further, because $v^{\varphi }=d\varphi /dt$ is the coordinate angular velocity, we regard $b{v}^{\varphi }$ as the coordinate transverse velocity. $b{v}^{\varphi }\ll 1$. Expanding up to the order $\mathcal{O}({\epsilon }^2,{\epsilon }^{2}b{v}^{\varphi })$, the measurable angle ${\psi }_{\mathrm{trans}}$ becomes for $0\le \varphi \le \pi /2$,

$$\begin{array}{cc}\hfill {\psi }_{\mathrm{trans}}& =\varphi +b{v}^{\varphi }tan\frac{\varphi }{2}+\frac{2m}{b}cos\varphi \left(1+\frac{b{v}^{\varphi }}{1+cos\varphi }\right)\hfill \\ \hfill & +\frac{1}{8b^2}\left\{m^2[15(\pi -2\varphi )-sin2\varphi ]-16amcos\varphi \right\}\hfill \\ \hfill & +\frac{b{v}^{\varphi }}{8b^2(1+cos\varphi )}\{m^2[15(\pi -2\varphi )-16sin\varphi +7sin2\varphi +16tan\frac{\varphi }{2}]\hfill \\ \hfill & +8ma(1-2cos\varphi -cos2\varphi )\}\hfill \\ \hfill & -\frac{\mathsf{\Lambda }{b}^2}{6}cot\varphi -\frac{\mathsf{\Lambda }{b}^3{v}^{\varphi }cot\varphi }{6(1+cos\varphi )}+\frac{\mathsf{\Lambda }b}{3}cos\varphi \left[m(1+{csc}^2\varphi )+2acsc\varphi \right]\hfill \\ \hfill & +\frac{\mathsf{\Lambda }{b}^3{v}^{\varphi }}{6b(cos\varphi -1){(1+cos\varphi )}^2}\left[m(1-4cos\varphi +cos2\varphi )-2a(sin\varphi +sin2\varphi )\right]\hfill \\ \hfill & -\frac{{\mathsf{\Lambda }}^2{b}^4}{288}{csc}^4\varphi sin4\varphi -\frac{{\mathsf{\Lambda }}^2{b}^5{v}^{\varphi }}{2304}cos\varphi (1-2cos\varphi +3cos2\varphi ){csc}^3\frac{\varphi }{2}{sec}^5\frac{\varphi }{2}\hfill \\ \hfill & +\mathcal{O}({\epsilon }^3,{(b{v}^{\varphi })}^2),\hfill \end{array}$$

(38)

and for $\pi /2\le \varphi \le \pi$,

$$\begin{array}{cc}\hfill {\psi }_{\mathrm{trans}}& =\pi -\varphi +b{v}^{\varphi }cot\frac{\varphi }{2}-\frac{2m}{b}cos\varphi \left(1+\frac{b{v}^{\varphi }}{1-cos\varphi }\right)\hfill \\ \hfill & -\frac{1}{8b^2}\{m^2[15(\pi -2\varphi )-sin2\varphi ]-16amcos\varphi \}\hfill \\ \hfill & +\frac{b{v}^{\varphi }}{8b^2(1-cos\varphi )}\left\{m^2\left[-15(\pi -2\varphi )-16sin\varphi -7sin2\varphi +16cot\frac{\varphi }{2}\right]\right.\hfill \\ \hfill & +\left.8am(1+2cos\varphi -cos2\varphi )\right\}\hfill \\ \hfill & +\frac{\mathsf{\Lambda }{b}^2}{6}cot\varphi +\frac{\mathsf{\Lambda }{b}^3{v}^{\varphi }cot\varphi }{6(1-cos\varphi )}-\frac{\mathsf{\Lambda }b}{3}cos\varphi \left[m(1+{csc}^2\varphi )+2acsc\varphi \right]\hfill \\ \hfill & -\frac{\mathsf{\Lambda }{b}^3{v}^{\varphi }}{6b{(cos\varphi -1)}^2(1+cos\varphi )}\left[m(1+4cos\varphi +cos2\varphi )-2a(sin\varphi -sin2\varphi )\right]\hfill \\ \hfill & +\frac{{\mathsf{\Lambda }}^2{b}^4}{288}{csc}^4\varphi sin4\varphi +\frac{{\mathsf{\Lambda }}^2{b}^5{v}^{\varphi }}{2304}cos\varphi (1+2cos\varphi +3cos2\varphi ){csc}^5\frac{\varphi }{2}{sec}^3\frac{\varphi }{2}\hfill \\ \hfill & +\mathcal{O}({\epsilon }^3,{(b{v}^{\varphi })}^2).\hfill \end{array}$$

(39)

As is the case of Equations (34) and (35), the first four lines in Equations (38) and (39) are equivalent to the measurable angle in Kerr spacetime obtained in [37], and the remaining terms, lines 5 to 7, are the additional terms due to the cosmological constant $\mathsf{\Lambda }$.

3.4. Comparison of Static, Radial and Transverse Cases and Their Properties

Here, let us examine how the cosmological constant $\mathsf{\Lambda }$, the spin parameter a as well as the radial and transverse velocities, $v^r$ and $b{v}^{\varphi }$, contribute to the measurable angle ${\psi }_P$ of the light ray. We assume that the observer is located within the range $0\le \varphi \le \pi /2$, and we adopt typical galaxy with mass $M_{\mathrm{gal}}$, radius $R_{\mathrm{gal}}$, and angular momentum $J_{\mathrm{gal}}$ as the lens object; the impact parameter b is comparable with $R_{\mathrm{gal}}$. See

Table 1. Numerical values. We use the following numerical values in this paper. As the value of the total angular momentum $J_{\mathrm{gal}}$, we adopt that of our Galaxy $J_{\mathrm{gal}}\simeq 1.0\times {10}^{67}$ kg m${}^2$/s from [45].

Name	Symbol	Value
Mass of the Galaxy	$M_{\mathrm{gal}}={10}^{12}{M}_{\odot }$	$2.0\times {10}^{42}\mathrm{kg}$
	$m=G{M}_{\mathrm{gal}}/c^2$	$1.5\times {10}^{15}$ m
Impact Parameter	$b=R_{\mathrm{gal}}=26\mathrm{kly}$	$2.5\times {10}^{20}$ m
Angular Momentum of the Galaxy [45]	$J=J_{\mathrm{gal}}$	$1.0\times {10}^{67}$ kg m${}^2$/s
Spin Parameter	$a=J/(M_{\mathrm{gal}}c)$	$1.7\times {10}^{16}$ m
Cosmological Constant	$\mathsf{\Lambda }$	${10}^{-52}$ m${}^{-2}$
Hubble Constant	$H_0=c\sqrt{\mathsf{\Lambda }/3}$	$1.73\times {10}^{-18}{\mathrm{s}}^{-1}$
Distance from Lens Object	$D=1.0\mathrm{Gly}$	$9.4\times {10}^{24}\mathrm{m}$
Recession Velocity	$v_H=H_{0}D$	$1.6\times {10}^7\mathrm{m}/\mathrm{s}$
Radial Velocity	$v^r=v_H/c$	$0.05$
Transverse Velocity	$b{v}^{\varphi }=v_H/c$	$0.05$

.

In order to clarify the contribution of the cosmological constant $\mathsf{\Lambda }$ to the total deflection angle of light ray in Kerr spacetime, we compute the following terms using the values summarized in Table 1:

$$\begin{array}{cc}\hfill \frac{4m}{b}& \approx 2.4\times {10}^{-5}\mathrm{rad},\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill \frac{15\pi m^2}{4b^2}& \approx 4.2\times {10}^{-10}\mathrm{rad},\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill -\frac{4ma}{b^2}& \approx \mp 1.6\times {10}^{-9}\mathrm{rad}\mathrm{for}\pm a.\hfill \end{array}$$

(42)

First, we deal with the case of the static observer. Because the Kerr contributions appearing in Equations (29) and (30) are examined in [37], we extract the terms concerning the cosmological constant $\mathsf{\Lambda }$ from Equation (29) and put,

$$\begin{array}{cc}\hfill {\psi }_{\mathrm{stat}}^1& ={\psi }_{\mathrm{stat}}(\varphi ;\mathsf{\Lambda },b)=-\frac{\mathsf{\Lambda }{b}^2}{6}cot\varphi -\frac{{\mathsf{\Lambda }}^2{b}^4}{288}{csc}^4\varphi sin4\varphi ,\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill {\psi }_{\mathrm{stat}}^2& ={\psi }_{\mathrm{stat}}(\varphi :m,a,\mathsf{\Lambda },b)=\frac{\mathsf{\Lambda }b}{3}cos\varphi \left[m(1+{csc}^2\varphi )+2acsc\varphi \right],\hfill \end{array}$$

(44)

Figure 2 and Figure 3 show the $\varphi$ dependences of Equations (43) and (44), respectively. From Figure 2, Equation (43) is a monotonic function of $\varphi$ and increases rapidly, diverging to negative infinity as $\varphi$ approaches 0. This property is due to the existence of the de Sitter horizon at $r_{\mathrm{dS}}\approx \sqrt{3/\mathsf{\Lambda }}$. The order $\mathcal{O}(\mathsf{\Lambda }b,{\mathsf{\Lambda }}^2{b}^2)$ terms in Equation (43) take a negative value which diminishes the measurable angle $\psi$. Equation (44) contains the order $\mathcal{O}(\mathsf{\Lambda }bm,\mathsf{\Lambda }ba)$ terms and its magnitude is $\mathcal{O}({10}^{-10})$ which is almost comparable to the second order deflection angle, Equation (41). In accordance with the sign of the spin parameter a, Equation (44) takes a different sign; for $a>0$, Equation (44) is positive and vice versa. However, regardless of the sign of the spin parameter a, Equation (44) diverges to positive infinity as $\varphi$ approaches 0.

Next, to observe the case of radially moving observer, we extract the coupling terms for cosmological constant $\mathsf{\Lambda }$ and radial velocity $v^r$ from Equation (34) except the part of Equations (43) and (44):

$$\begin{array}{cc}\hfill {\psi }_{\mathrm{rad}}^1& ={\psi }_{\mathrm{rad}}(\varphi ;\mathsf{\Lambda },b,v^r)=-\frac{\mathsf{\Lambda }{b}^2{v}^r}{12}(cos2\varphi -3)-\frac{{\mathsf{\Lambda }}^2{b}^4{v}^r}{144}(cos2\varphi -7){cot}^2\varphi csc\varphi ,\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill {\psi }_{\mathrm{rad}}^2& ={\psi }_{\mathrm{rad}}(\varphi ;m,a,\mathsf{\Lambda },b,v^r)=\frac{\mathsf{\Lambda }b{v}^r}{3}csc\varphi \left[a+mcsc\varphi \right.\left.+(a-2mcsc\varphi )cos2\varphi \right].\hfill \end{array}$$

(46)

Because the background spacetime of Kerr–de Sitter is de Sitter spacetime, we assume that radial velocity obeys Hubble’s law:

$$\begin{array}{c}\hfill v^r\approx v_H=H_{0}D,H_0=c\sqrt{\frac{\mathsf{\Lambda }}{3}},\end{array}$$

(47)

where D is the distance between the lens (central) object O and observer P, and we take $D\approx 1\mathrm{Gly}\simeq 9.4\times {10}^{24}\mathrm{m}$ which is the typical distance scale of the galaxy lensing. We note that because we are now concerned with cosmological gravitational lensing effect, it seems appropriate to assume that the expansion of the universe is dominant than the effect of local gravity due to the central (lens) object and the observer is receding from the lens object (and the source) with velocity $v_H$.2

We mention that we are not only dealing with a receding observer with velocity $v_H>0$, but also, as an example we also discuss the case where the observer approaches the lens object with $v_H<0$ though the observer does not actually approach the lens object at the recession velocity $v_H$.

Figure 4 and Figure 5 illustrate the $\varphi$ dependence of Equations (45) and (46), respectively. Equation (45) diverges to positive or negative infinity when $\varphi$ approaches 0 depending on the sign of the velocity $v^r$. On the other hand, when $\varphi \to \pi /2$, Equation (45) approaches $\mathsf{\Lambda }{b}^2{v}^r/3$. For the positive velocity $v^r>0$, Equation (46) becomes negative infinity when $\varphi$ approaches 0, while for the negative velocity $v^r<0$, Equation (46) diverges to positive infinity. These properties are independent of the sign of the spin parameter a. Next, when $\varphi \to \pi /2$, Equation (46) converges to $\mathsf{\Lambda }bm{v}^r$ which depends on the radial velocity $v^r$ but is independent of the spin parameter a. The magnitude of the velocity-dependent part, Equation (46), is at most $\mathcal{O}({10}^{-12})$ which is two orders of magnitude smaller then the second order contribution $\mathcal{O}(m^2)$ in Equation (41).

At last, in the case of an observer in transverse motion, The parts depending on the cosmological constant $\mathsf{\Lambda }$ and the transverse velocity $b{v}^{\varphi }$ can be extracted from Equations (38) and (39) as,

$$\begin{array}{cc}\hfill {\psi }_{\mathrm{trans}}^1& ={\psi }_{\mathrm{trans}}(\varphi ;\mathsf{\Lambda },b,b{v}^{\varphi })\hfill \\ \hfill & =-\frac{\mathsf{\Lambda }{b}^3{v}^{\varphi }cot\varphi }{6(1+cos\varphi )}-\frac{{\mathsf{\Lambda }}^2{b}^5{v}^{\varphi }}{2304}cos\varphi (1-2cos\varphi +3cos2\varphi ){csc}^3\frac{\varphi }{2}{sec}^5\frac{\varphi }{2},\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill {\psi }_{\mathrm{trans}}^2& ={\psi }_{\mathrm{trans}}(\varphi ;m,a,\mathsf{\Lambda },b,b{v}^{\varphi })\hfill \\ \hfill & =\frac{\mathsf{\Lambda }{b}^3{v}^{\varphi }}{6b(cos\varphi -1){(1+cos\varphi )}^2}\left[m(1-4cos\varphi +cos2\varphi )-2a(sin\varphi +sin2\varphi )\right],\hfill \end{array}$$

(49)

and we also assume here that the transverse velocity is comparable with the recessional velocity $v_H$

$$\begin{array}{c}\hfill b{v}^{\varphi }\approx v_H,\end{array}$$

(50)

see also Table 1.

Figure 6 and Figure 7 show the $\varphi$ dependences of Equations (48) and (49), respectively. Equation (48) consists of $\mathcal{O}(\mathsf{\Lambda }{b}^3{v}^{\varphi },{\mathsf{\Lambda }}^2{b}^5{v}^{\varphi })$ terms; however, unlike Equation (45), for the positive transverse velocity $b{v}^{\varphi }>0$, Equation (48) diverges to negative infinity when $\varphi$ approaches 0 and vice versa. When $\varphi \to \pi /2$, Equation (48) converges to 0 regardless of the sign of the transverse velocity $b{v}^{\varphi }$. From Figure 7, we find that regardless of the sign of the spin parameter a, Equation (49) becomes positive infinity for the positive transverse velocity $b{v}^{\varphi }>0$ and negative infinity for the negative transverse velocity $b{v}^{\varphi }<0$ when $\varphi$ approaches 0. When $\varphi \to \pi /2$, Equation (49) converges to $\mathsf{\Lambda }{b}^{2}a{v}^{\varphi }/3$ which depends on both the spin parameter a and the transverse velocity $b{v}^{\varphi }$ unlike Equation (46). As in Equation (46), the magnitude of Equation (49) is at most $\mathcal{O}({10}^{-12})$ and it is two orders of magnitude of smaller than the second order contribution $\mathcal{O}(m^2)$ in Equation (41).

Before closing this section, we summarize the behavior of Equations (43), 44, 45, 46, (48) and (49) within the range $0<\varphi <\pi /2$ in

Table 2. Behavior of Equations (43), (44), (45), (46), (48) and (49).

Motion of Observer	Equation Number	$\mathit{\varphi }\to 0$	$0<\mathit{\varphi }<\mathit{\pi }/2$	$\mathit{\varphi }\to \mathit{\pi }/2$
Static	Equation (43)	$-\infty$	Negative	0
	Equation (44)	∞	$\mathrm{Positive}\mathrm{for}a>0$	0
			Mostly Negative for $a<0$
Radial	Equation (45)	∞ for $v^r>0$	Positive for $v^r>0$	$\mathsf{\Lambda }{b}^2{v}^r/3$
		$-\infty$ for $v^r<0$	Negative for $v^r<0$
	Equation (46)	$-\infty$ for $v^r>0$	Mostly Positive for $a>0$	$\mathsf{\Lambda }bm{v}^r$
			Mostly Negative for $a<0$
		∞ for $v^r<0$	Mostly Negative for $a>0$
			Mostly Positive for $a<0$
Transverse	Equation (48)	$-\infty$ for $v^{\varphi }>0$	Negative for $v^{\varphi }<0$	0
		∞ for $v^{\varphi }<0$	Positive for $v^{\varphi }<0$
	Equation (49)	∞ for $v^{\varphi }>0$	Positive for $a>0$	$\mathsf{\Lambda }{b}^{2}a{v}^{\varphi }/3$
			Mostly Negative for $a<0$
		$-\infty$ for $v^{\varphi }<0$	Negative for $a>0$
			Mostly Positive for $a<0$

.

4. Conclusions

In this paper, instead of the total deflection angle $\alpha$, we mainly focused on discussing the measurable angle of the light ray $\psi$ at the position of the observer in Kerr–de Sitter spacetime which includes the cosmological constant $\mathsf{\Lambda }$. We investigated the contributions of the radial and transverse motions of the observer which are related to the radial velocity $v^r$ and the transverse velocity $b{v}^{\varphi }$ as well as the influence of the gravitomagnetic field or frame dragging described by the spin parameter a of the central object and the cosmological constant $\mathsf{\Lambda }$.

The general relativistic aberration equation was employed to incorporate the effect of the motion of the observer on the measurable angle $\psi$. To obtain the measurable angle $\psi$, the equation of the light trajectory was obtained in such a way that the background is de Sitter spacetime instead of Minkowski spacetime. At the end of Section 3.1, we mentioned that the choice of the zeroth-order solution $u_0$ is important; a zeroth-order solution $u_0$ in Kerr–de Sitter and Schwarzschild–de Sitter spacetimes should be chosen in such a way that the background is de Sitter spacetime, Equation (18). Further, Equation (20) or at least Equation (21) should be used as the trajectory equation of the light ray.

If we assume that the lens object is the typical galaxy, the static terms $\mathcal{O}(\mathsf{\Lambda }bm,\mathsf{\Lambda }ba)$ in Equation (44) are basically comparable with the second order deflection term $\mathcal{O}(m^2)$ but almost one order smaller than the Kerr deflection $-4ma/b^2$. The velocity-dependent terms $\mathcal{O}(\mathsf{\Lambda }bm{v}^r,\mathsf{\Lambda }ba{v}^r)$ in Equation (46) for radial motion and $\mathcal{O}(\mathsf{\Lambda }{b}^{2}m{v}^{\varphi },\mathsf{\Lambda }{b}^{2}a{v}^{\varphi })$ in Equation (49) for transverse motion are at most two orders of magnitude smaller than the second order deflection $\mathcal{O}(m^2)$. Therefore, if the second order deflection term $\mathcal{O}(m^2)$ becomes detectable by gravitational lensing, it may be possible to detect the cosmological constant $\mathsf{\Lambda }$ from the static terms in Equation (44). We also find that even when the radial and transverse velocities have the same sign, their asymptotic behavior when $\varphi$ approaches 0 is differs, and each diverges to the opposite infinity.

Our future work is to formulate the total deflection angle of light ray in Kerr–de Sitter spacetime, taking into account the velocity effect of the observer, and apply it to the cosmological lens equation to discuss the observability of the contribution to the cosmological constant by gravitational lensing. The formulation of the total deflection angle of the light ray in Kerr–de Sitter spacetime may be realized by using the Gauss–Bonnet theorem, and we expect that the results of this study will be useful in the future work.