# Measuring black hole spin through gravitational lensing of pulsars

Amjad Ashoorioon<EMAIL_ADDRESS>School of Physics, Institute for Research in
Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran  Mohammad Bagher
Jahani Poshteh<EMAIL_ADDRESS>School of Physics, Institute for Research in
Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran  Robert B. Mann
<EMAIL_ADDRESS>Department of Physics and Astronomy, University of
Waterloo, Waterloo, Ontario, N2L 3G1, Canada  Perimeter Institute for
Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada

###### Abstract

We put forward a new procedure for measuring the spin of a black hole with
unprecedented accuracy based on gravitational lensing of millisecond pulsars.
The deflection angle of light increases by increasing the rotation parameter.
For primary and secondary images the angular positions are larger for rotating
black holes by an amount of the order of microarcseconds. Also, the
differential time delay for the case of a rotating black hole is larger than
that for the non-rotating case and the difference could be as large as a few
seconds. We show that this quantity could help us achieve an extremely precise
measurement of the black hole spin, much more accurate than the current and
near future achievable estimation of black hole spin through other methods.

The spin of a black hole is an important property that can reveal information
about its origin, evolution, and environment. For supermassive black holes,
knowledge their spin could help us test and constrain different formation
scenarios, such as gas cloud collapse, mergers from smaller black holes, or
accretion of surrounding matter [1]. The spin also affects the energy output,
radiation efficiency, gravitational wave emission, and spacetime distortion of
the black hole, which have important implications for feedback processes,
observability, and tests of general relativity [2, 3, 4]. Measurement of black
hole spin is therefore a crucial task for understanding these fascinating
objects and their role in cosmic history.

Many methods have been developed or applied to estimate the spins of black
holes. In the Fe-K$\alpha$ method one observes the X-ray emission from the
iron atoms in the accretion disk, which is distorted by the strong gravity and
rotation of its accompanying black hole [5, 6, 7]. Another method is X-ray
polarimetry, which is based on observing the polarization properties of the
X-ray radiation emitted by the accretion disk, because the polarization of
X-ray photons is affected by the strong gravity and rotation of the black hole
[8, 9, 10]. Other methods include using the shape and size of the black hole
shadow [11, 12, 13], gravitational wave observations [14], the observed
transition of the jet boundary shape [15], observed twisted light propagating
near a rotating black hole [16], and modelling the jet power and the accretion
rate [17, 18].

In this Letter, working in the context of general relativity, we propose a new
method for measuring the spin of black holes with extraordinary accuracy. Our
method is based on the gravitational lensing of millisecond (or mildly-
recycled [19]) pulsars with periods $P\lesssim 30{\rm ms}$ ($20{\rm
ms}\lesssim P\lesssim 100{\rm ms}$). Millisecond pulsars are very important
for astrophysics and cosmology as they can be used as precise clocks to
measure various phenomena in the universe. Recently, by observing the tiny
changes in the arrival times of the radio pulses from these pulsars, the
NANOGrav collaboration has found compelling evidence that pulsar timing is
affected by the gravitational wave background, opening a new window for
exploring gravitational wave emission [20].

There have been several attempts to study gravitational lensing by spinning
black holes [21, 22, 23, 24, 25, 26, 27, 28, 29]. Most of them work in either
the strong or weak field limits, in which the deflection angle is expanded
near the photon circle or near infinity, respectively. However in a
gravitational lensing event, the light trajectory starts from a source that is
usually taken to be at a distance far from the black hole (lens), reaches a
turning point near the black hole, and then escapes to the observer, who is
also far from the black hole. The light ray therefore experiences both the
weak and strong field regimes along its single trajectory, and neither weak
nor strong field approximations alone can describe the event accurately.
(Another criticism against the strong field approximation has been raised in
[30].)

Here we generalize the numerical methods of [31, 32, 33] for rotating black
holes to study gravitational lensing by Kerr black holes. In this method, we
numerically solve the lens equation to find the distance of the closest
approach (to the black hole) and other lensing quantities. We note that we
only consider light rays on the equatorial plane of a rotating black hole. We
use units in which $G=c=1$ and metric signature +2.

The scenario we study is presented in Fig. 1. The ray of light that is coming
from the source S (which we assume to be a pulsar) is deflected by the black
hole. The observer cannot see the source in its actual place but sees its
lensed images. One image is on the same side of the black hole as the source.
This image is called the primary image. Another image is on the opposite side
of the black hole and is called the secondary image.

Figure 1: The lens diagram. The light ray which is coming from the source $S$
(which we take to be a pulsar) would be deflected near the black hole (lens)
$L$. The observer $O$ would see images of the source, one on the same side as
the source (with respect to the line of sight to $L$), which is called the
primary image $I_{p}$, and another one on the opposite side of the source,
which is called the secondary image $I_{s}$.

The metric of a Kerr spacetime is

$ds^{2}=g_{tt}dt^{2}+g_{t\phi}dtd\phi+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2},$
(1)

where

$\displaystyle g_{tt}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2mr}{\Sigma}\right),$ (2) $\displaystyle
g_{t\phi}$ $\displaystyle=$ $\displaystyle\frac{4mar\sin^{2}\theta}{\Sigma},$
(3) $\displaystyle g_{rr}$ $\displaystyle=$
$\displaystyle\frac{\Sigma}{r^{2}-2mr+a^{2}},$ (4) $\displaystyle
g_{\theta\theta}$ $\displaystyle=$ $\displaystyle\Sigma,$ (5) $\displaystyle
g_{\phi\phi}$ $\displaystyle=$
$\displaystyle\left(r^{2}+a^{2}+\frac{2ma^{2}r\sin^{2}\theta}{\Sigma}\right)\sin^{2}\theta,$
(6)

and

$\Sigma\equiv r^{2}+a^{2}\cos^{2}\theta.$ (7)

In these relations $m$ is the mass of Kerr black hole and $a$ is its rotation
parameter.

The geodesics are found by using the Lagrangian

$\mathcal{L}=\frac{1}{2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu},$ (8)

where the dot shows differentiation with respect to some affine parameter. The
constants of motion are

$\displaystyle E$ $\displaystyle=$
$\displaystyle-\frac{\partial\mathcal{L}}{\partial\dot{t}}=-\frac{1}{2}\left(2g_{tt}\dot{t}+g_{t\phi}\dot{\phi}\right),$
(9) $\displaystyle L_{z}$ $\displaystyle=$
$\displaystyle-\frac{\partial\mathcal{L}}{\partial\dot{\phi}}=-\frac{1}{2}\left(g_{t\phi}\dot{t}+2g_{\phi\phi}\dot{\phi}\right).$
(10)

Consider null geodesics on the equatorial plane of the black hole, for which
$\theta=\frac{\pi}{2}$ and $\dot{\theta}=0$ and $\mathcal{L}=0$. Hence from
Eq. (8) we have

$g_{tt}\dot{t}^{2}+g_{t\phi}\dot{t}\dot{\phi}+g_{rr}\dot{r}^{2}+g_{\phi\phi}\dot{\phi}^{2}=0,$
(11)

which results in

$\frac{g_{tt}\dot{t}^{2}+g_{t\phi}\dot{t}\dot{\phi}}{g_{\phi\phi}\dot{\phi}^{2}}+\frac{g_{rr}}{g_{\phi\phi}}\left(\frac{dr}{d\phi}\right)^{2}=-1.$
(12)

Substituting for $\dot{t}$ and $\dot{\phi}$ from Eqs. (9) and (10) yields

$\displaystyle\frac{\left(2g_{\phi\phi}E-g_{t\phi}L_{z}\right)\left(2g_{\phi\phi}g_{tt}E+g_{t\phi}g_{tt}L_{z}-g_{t\phi}^{2}E\right)}{g_{\phi\phi}\left(g_{t\phi}E-2g_{tt}L_{z}\right)^{2}}$
$\displaystyle+\frac{g_{rr}}{g_{\phi\phi}}\left(\frac{dr}{d\phi}\right)^{2}=-1.$
(13)

At the point of closest approach, $r=b$, we have $dr=0$, so

$\displaystyle\frac{2g_{\phi\phi}(b)E-g_{t\phi}(b)L_{z}}{g_{\phi\phi}(b)\left[g_{t\phi}(b)E-2g_{tt}(b)L_{z}\right]^{2}}\times$
(14)
$\displaystyle\left[2g_{\phi\phi}(b)g_{tt}(b)E+g_{t\phi}(b)g_{tt}(b)L_{z}-g_{t\phi}^{2}(b)E\right]=-1.$

Eq. (14) can be solved for the ratio $\eta=\frac{L_{z}}{E}$ as a function of
$b$, yielding

$\eta=\frac{g_{t\phi}(b)-\sqrt{g_{t\phi}^{2}(b)-4g_{\phi\phi}(b)g_{tt}(b)}}{2g_{tt}(b)}.$
(15)

for the impact parameter $\eta$, with the sign chosen so that the static case
is recovered as $a\rightarrow 0$ [31, 32, 33].

By using the parameter $\eta$ one can rewrite Eq. (Measuring black hole spin
through gravitational lensing of pulsars) as

$\displaystyle\frac{d\phi}{dr}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{g_{rr}}{g_{\phi\phi}}}$ (16) $\displaystyle\times$
$\displaystyle\left[-1-\frac{\left(2g_{\phi\phi}-g_{t\phi}\eta\right)\left(2g_{\phi\phi}g_{tt}+g_{t\phi}g_{tt}\eta-
g_{t\phi}^{2}\right)}{g_{\phi\phi}\left(g_{t\phi}-2g_{tt}\eta\right)^{2}}\right]^{-1/2}$
$\displaystyle\equiv$ $\displaystyle\Gamma.$

The deflection angle is then

$\hat{\alpha}(b)=2\int_{b}^{\infty}\Gamma\,dr-\pi.$ (17)

Eq. (11) can also be written as

$\displaystyle\frac{\left(2g_{tt}L_{z}-g_{t\phi}E\right)\left(2g_{\phi\phi}g_{tt}L_{z}+g_{\phi\phi}g_{t\phi}E-g_{t\phi}^{2}L_{z}\right)}{g_{tt}\left(g_{t\phi}L_{z}-2g_{\phi\phi}E\right)^{2}}$
$\displaystyle+\frac{g_{rr}}{g_{tt}}\left(\frac{dr}{dt}\right)^{2}=-1.$ (18)

At $r=b$ we have $dr=0$. This can be used to find the same $\eta$ as in Eq.
(15). Therefore from Eq. (Measuring black hole spin through gravitational
lensing of pulsars) we obtain

$\displaystyle\frac{dt}{dr}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{g_{rr}}{g_{tt}}}$ (19) $\displaystyle\times$
$\displaystyle\left[-1-\frac{\left(2g_{tt}\eta-
g_{t\phi}\right)\left(2g_{\phi\phi}g_{tt}\eta+g_{\phi\phi}g_{t\phi}-g_{t\phi}^{2}\eta\right)}{g_{tt}\left(g_{t\phi}\eta-2g_{\phi\phi}\right)^{2}}\right]^{-1/2}$
$\displaystyle\equiv$ $\displaystyle\Pi$

from which we compute the time delay

$\tau(b)=\int_{b}^{r_{s}}\Pi\,dr+\int_{b}^{D_{d}}\Pi\,dr-D_{s}\sec\beta,$ (20)

where $D_{s}=D_{d}+D_{ds}$ and
$r_{s}=\sqrt{D_{ds}^{2}+D_{s}^{2}\tan^{2}\beta}$. Defined in this way, the
time delay is the difference between the time it takes light to travel from
source to observer in the physical spacetime (with the black hole), and the
time it would take to travel from source to observer in flat spacetime (with
no black hole).

Using the geometrical arguments we can write

$\eta=D_{d}\sin\theta.$ (21)

and in turn obtain the Virbhadra-Ellis lens equation [31]

$\tan\beta=\pm\left\\{\tan\theta-\mathcal{D}\left[\tan\theta+\tan\left(\hat{\alpha}-\theta\right)\right]\right\\},$
(22)

where $\mathcal{D}=D_{ds}/D_{s}$. The plus/minus signs correspond to
primary/secondary images and

$\mu=\left(\frac{\sin\beta}{\sin\theta}\frac{d\beta}{d\theta}\right)^{-1}$
(23)

is the image magnification. To find $\frac{d\beta}{d\theta}$ we use the lens
equation (22), which implies computing
$\frac{d\hat{\alpha}}{d\theta}=\frac{d\hat{\alpha}}{db}\frac{db}{d\theta}$.
The second factor can be obtained from (15) and (21). We calculate the first
factor numerically.

Now we investigate the primary and secondary images produced in gravitational
lensing by a rotating black hole and compare these to the non-rotating case of
the same mass. Specifically, we consider the black hole at the center of the
M87 galaxy, with mass $M_{{\rm M87^{*}}}=9.6\times 10^{12}\,{\rm m}\equiv
6.5\times 10^{9}M_{\odot}$ and distance $D_{d}=5.2\times 10^{23}\,{\rm
m}\equiv 16.8$ Mpc [34].

Table 1:  Image positions $\theta$ and deflection angles $\hat{\alpha}$ of primary and secondary images due to lensing by rotating (ro) as well as non-rotating (nr) black holes. $\beta$ is the angular source position and the subscripts $p$ and $s$ refer to primary and secondary images, respectively. All angles are in arcseconds. We have used $m=M_{{\rm M87^{*}}}=9.6\times 10^{12}\,{\rm m}$, $D_{d}=5.2\times 10^{23}\,{\rm m}$ and $\mathcal{D}=0.5$. For the rotating black hole we have taken the rotation parameter to be $a^{*}=0.99$.  $\beta$ | Rotating black hole | Non-rotating black hole
---|---|---
| $\theta_{p,{\rm ro}}$ | $\hat{\alpha}_{p,{\rm ro}}$ | $\theta_{s,{\rm ro}}$ | $\hat{\alpha}_{s,{\rm ro}}$ | $\theta_{p,{\rm nr}}$ | $\hat{\alpha}_{p,{\rm nr}}$ | $\theta_{s,{\rm nr}}$ | $\hat{\alpha}_{s,{\rm nr}}$
$0$ | $1.2533650$ | $2.5067300$ | $1.2533650$ | $2.5067300$ | $1.2533631$ | $2.5067262$ | $1.2533631$ | $2.5067262$
$0.1$ | $1.3043613$ | $2.4087225$ | $1.2043614$ | $2.6087229$ | $1.3043594$ | $2.4087188$ | $1.2043595$ | $2.6087190$
$0.5$ | $1.5280523$ | $2.0561047$ | $1.0280571$ | $3.0561143$ | $1.5280508$ | $2.0561017$ | $1.0280549$ | $3.0561098$
$1$ | $1.8494124$ | $1.6988249$ | $0.8494216$ | $3.6988431$ | $1.8494113$ | $1.6988225$ | $0.8494190$ | $3.6988379$
$2$ | $2.6034054$ | $1.2068108$ | $0.6034216$ | $5.2068432$ | $2.6034047$ | $1.2068094$ | $0.6034185$ | $5.2068371$
$3$ | $3.4547138$ | $0.9094276$ | $0.4547318$ | $6.9094636$ | $3.4547134$ | $0.9094267$ | $0.4547285$ | $6.9094569$
$4$ | $4.3602762$ | $0.7205524$ | $0.3603017$ | $8.7206034$ | $4.3602759$ | $0.7205519$ | $0.3602982$ | $8.7205964$
Table 2:  Magnification $\mu$ and (differential) time delay $\tau$ ($t_{d}=\tau_{s}-\tau_{p}$) of primary and secondary images due to lensing by rotating (ro) as well as non-rotating (nr) black holes. $\beta$ and the subscripts $p$ and $s$ are as in Table 1. (Differential) time delays are in seconds. $m$, $D_{d}$, $\mathcal{D}$, and $a^{*}$ are as in Table 1.  $\beta$ | Rotating black hole | Non-rotating black hole
---|---|---
| $\mu_{p,{\rm ro}}$ | $\tau_{p,{\rm ro}}$ | $\mu_{s,{\rm ro}}$ | $t_{d,{\rm ro}}$ | $\mu_{p,{\rm nr}}$ | $\tau_{p,{\rm nr}}$ | $\mu_{s,{\rm nr}}$ | $t_{d,{\rm nr}}$
$0$ | $\times$ | $1627269.6$ | $\times$ | $0$ | $\times$ | $1627269.3$ | $\times$ | $0$
$0.1$ | $6.7850417$ | $1617251.1$ | $-5.7891987$ | $20444.927$ | $6.7850511$ | $1617250.8$ | $-5.7892080$ | $20444.897$
$0.5$ | $1.8260851$ | $1580930.6$ | $-0.8262814$ | $102870.41$ | $1.8260869$ | $1580930.3$ | $-0.8262832$ | $102870.25$
$1$ | $1.2669222$ | $1542808.2$ | $-0.2670980$ | $209692.19$ | $1.2669230$ | $1542807.9$ | $-0.2670988$ | $209691.89$
$2$ | $1.0565710$ | $1484436.7$ | $-0.0566955$ | $448742.30$ | $1.0565712$ | $1484436.5$ | $-0.0566958$ | $448741.68$
$3$ | $1.0176446$ | $1441783.5$ | $-0.0177282$ | $737895.11$ | $1.0176447$ | $1441783.4$ | $-0.0177284$ | $737894.18$
$4$ | $1.0068406$ | $1408827.1$ | $-0.0068980$ | $1089190.1$ | $1.0068407$ | $1408827.0$ | $-0.0068981$ | $1089188.9$

In Tables 1 and 2 we present the image positions, deflection angles,
magnification, and (differential) time delay of primary and secondary images
for different values of angular source positions. For illustrative purposes,
we have taken the black hole to be halfway between the source and the
observer. Although no pulsars have yet been observed at this distance, pulsars
have been observed at distances comparable to that of M87 [35], and it is
conceivable that future observations will detect pulsars even further away.
Note that the image angle is relative to the lens-observer axis in Fig. 1 and
is always positive.

From Table 1 we see that the deflection angle is larger for rotating black
holes. We also see that the angular positions of primary and secondary images
are larger if the black hole is rotating; for the values considered in this
table and for small values of angular source position, the difference can be
as large as a few microarcseconds. The difference between the rotating and
non-rotating cases for the angular position of the primary image decreases by
with increasing angular source position. On the other hand, the corresponding
value for secondary image increases by increasing $\beta$.

From Table 2 we see that for primary images the magnification is larger if the
black hole is non-rotating. The difference between the primary image
magnification in the rotating and non-rotating cases,
$\mu_{p,a^{*}=0.99}-\mu_{p,a^{*}=0}$, is a negative quantity and its absolute
value decreases with increasing angular position of the source. Furthermore,
the absolute value of the magnification of secondary images is larger if the
lens is non-rotating. Since the magnification of secondary images is defined
to be negative, the quantity $\mu_{s,a^{*}=0.99}-\mu_{s,a^{*}=0}$ is positive,
and its value decreases with increasing $\beta$.

In Table 2 we also see that the time delay of primary images decreases by
increasing the source angular position for both rotating and non-rotating
cases. This quantity is larger if the lens is a rotating black hole. We have
presented the time delay of secondary images. Instead we have shown the
differential time delay $t_{d}=\tau_{s}-\tau_{p}$. This quantity is of
considerable observational importance: if the source is pulsating, every phase
in its period appears in the secondary image $t_{d}$ seconds after it has
appeared in the primary image. This differential time delay increases with
increasing angular position of the source. It is larger if the black hole is
rotating; for the values chosen in Table 2, and for large values of $\beta$
the difference can be as large as a few seconds. The difference between the
differential time delay in the case of rotating black hole and the case of
non-rotating black hole, $t_{d,a^{*}=0.99}-t_{d,a^{*}=0}$, gets larger as
$\beta$ increases.

We have also studied the behavior of lensing parameters for a range of values
of the rotation parameter (details will be presented elsewhere.) The angular
positions of both primary and secondary images increases with increasing the
rotation parameter, whereas the magnification of the primary image and the
absolute value of the magnification of secondary image decreases. The
differential time delay increases with increasing rotation parameter. Details
of the behavior of lensing parameters as a function of $\mathcal{D}$ will be
presented in a forthcoming publication.

We do not see the source in its real place and therefore we cannot measure
$\beta$ directly. However, it is possible to find $\beta$ from the angular
positions of primary and secondary images. Suppose that we observe primary and
secondary images; these can be distinguished from their magnifications. From
Table 2 we see that the absolute value of the magnification of the primary
image is larger than that of the secondary image for both the rotating and
non-rotating cases. We can find the distance to the source by, for example,
using the red shift of the images. Suppose also that we know the mass of the
lens and our distance to it. What we do not know are $\beta$ and if the lens
is rotating or not. The primary image can be due to lensing by a rotating
black hole and a source at position $\beta_{p,{\rm ro}}$ or due to lensing by
a non-rotating black hole and a source at position $\beta_{p,{\rm nr}}$. On
the other hand, The secondary image can be due to a rotating black hole and a
source at position $\beta_{s,{\rm ro}}$ or due to lensing by a non-rotating
black hole and a source at position $\beta_{s,{\rm nr}}$. We should note that
for both primary and secondary images the source is the same source and the
lens is the same black hole. Therefore, either $\beta_{p,{\rm
ro}}=\beta_{s,{\rm ro}}$ or $\beta_{p,{\rm nr}}=\beta_{s,{\rm nr}}$ is true.
This way it is possible to find $\beta$ and determine if the black hole is
rotating.

Most significantly, a measurement of differential time delay could help us
find the rotation parameter with unprecedented precision. In fact, for large
values of angular source position (of the order of arcseconds), a measurement
of differential time delay with accuracy of $\sim 10^{-4}$ seconds would
result in a measurement of the spin parameter with less than 1 percent error.
This is better than the most accurate estimates of M87* spin that we can find,
i.e. [16, 18], which are based on twisted light and accretion-jet analysis.

Recently it has been shown that LISA could constrain the spin to a precision
as high as $\Delta a^{*}\sim 10^{-10}$ [36]. Such a tight measurement is only
possible for a near extremal black hole with spin parameter $a^{*}=1-10^{-9}$.
This value is far past the Thorne limit of $a^{*}=0.998$ [37]. For LISA to be
able to constrain the spin, the black hole should constitute a binary system.
Furthermore, the optimum mass to archive such precision is $\sim
10^{7}M_{\odot}$ [36], which is not a good method for M87*, whose mass is
$\sim 10^{9}M_{\odot}$. By using the differential time-delay method we
propose, it is possible to find the spin with precision better than $\Delta
a^{*}\sim 10^{-10}$ if we could measure the differential time delay with
accuracy $10^{-11}$ seconds. Luckily, millisecond pulsars period can be
measured with a very high precision — of the order of femtosecond [19, 38, 39,
40]. Therefore, if we have a pulsar as our source behind the black hole, a
measurement of differential time delay with accuracy of the order $10^{-11}$
seconds indeed seems feasible.

Our method can do better than LISA even for masses $\sim 10^{7}M_{\odot}$.
Consider for instance the supermassive black hole in M60-UCD1 galaxy with mass
$2.1\times 10^{7}M_{\odot}$ and distance $16.5$ Mpc [41, 42]. To find its spin
with precision better than $\Delta a^{*}\sim 10^{-10}$ would require measuring
the differential time delay with an accuracy of $\sim 10^{-13}$ seconds.
Unfortunately we have yet to observe any millisecond pulsars at such large
distances. But we have several of them in our galactic neighborhood up to
distances $\sim 17.9$ kpc [43]. It is noteworthy that the method we have
presented here can be used to measure the spins of lighter black holes that
are present in our neighborhood. For example for the intermediate-mass black
hole CO-0.40-0.22* with mass $\sim 10^{5}M_{\odot}$ and distance $8.3$ kpc a
precision of $\Delta a^{*}\sim 10^{-10}$ would be achieved if we measure the
differential time delay with accuracy of $\sim 10^{-15}$ seconds, within reach
of present technology [19, 38, 39, 40]. For stellar-mass black holes, however,
a precision of $\sim 10^{-19}$ seconds is needed.

_Acknowledgments:_ This work was supported in part by the Natural Sciences and
Engineering Research Council of Canada.

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