GRAVITATION

DEFINITION

The calculation of light deflection by Kerr’s black holes can use the Kerr’s metric tensor matrix expressed in the Boyer-Lindquist’s coordinates system \((ct, r, \varphi, \theta)\):
\((g_{\mu\nu})=\pmatrix{-1+\frac{2mr}{\Sigma}&0&0&-\frac{2mar\sin^2\theta}{\Sigma}\\0&\frac{\Sigma}{\Delta}&0&0\\0&0&\Sigma&0\\-\frac{2mar\sin^2\theta}{\Sigma}&0&0&\left(r^2+a^2+\frac{2ma^2r\sin^2\theta}{\Sigma}\right)\sin^2\theta}\)¹
with \(r\) radial coordinate of the photon, \(\theta\) its colatitude, \(G\) gravitational constant, \(c\) speed of light in vacuum, \(M\) mass of the black hole, \(m=\frac{GM}{c^2}\) reduced mass homogeneous to the meter, \(a=\frac{J}{cM}\) (\(>0\) for a trigonometric spin, \(<0\) for a clockwise spin) with \(J\) angular momentum of spin of the black hole, \(\Delta=r^2-2mr+a^2\) and \(\Sigma=r^2+a^2\cos^2{\theta}\).
The coefficients of \((g_{\mu\nu})\) are independent of \(t\) and \(\varphi\): the geometry of Kerr’s spacetime is therefore stationary and axially symmetrical.
Since the null geodesics are of the light-type, their length is zero² and the scalar product of an elementary motion \(\overrightarrow{ds}\) of a photon in Kerr’s spacetime is therefore written
\(g_{\mu\nu}dx^\mu dx^\nu=ds^2=\left(1-\frac{2mr}{\Sigma}\right)c^2dt^2-\frac{4mar\sin^2{\theta}}{\Sigma}cdtd\varphi+\frac{\Sigma}{\Delta}dr^2+\Sigma d\theta^2\)
\(+\left(r^2+a^2+\frac{2ma^2r\sin^2{\theta}}{\Sigma}\right)\sin^2{\theta}d\varphi^2=0\).

SETTING UP THE PARAMETRIC EQUATIONS

Following the Hamilton-Jacobi’s approach, we need to find \(S(x^\mu, \lambda)\), a function of the photon coordinates (\(x^\mu)=(ct, r, \theta, \varphi)\) and an affine parameter \(\lambda\), and solution of the Hamilton-Jacobi’s equation \(H\left(x^\mu,\frac{\delta S}{\delta x^\mu}\right)+\frac{\delta S}{\delta\lambda}=0\)³.
It can be shown that if \(S\) is a solution then \(\frac{\delta S}{\delta x^\mu}=p_\mu\) with (\(p_\mu\)) conjugate moment of the photon.
The conservation of the energy \(\varepsilon\) and the component \(l_z\) of the angular momentum \(\overrightarrow{l}\) on the spin axis of the black hole, all along the motion of the photon, gives \(p_0=-\frac{\varepsilon}{c}\) and \(p_\varphi=l_z\), leading to a function such as:
\(S=-\frac{\varepsilon}{c} ct+S^{(r)}(r)+S^{(\theta)}(\theta)+l_z\varphi\)⁴ \(\hspace{2cm}\)(2.a) looking for a separable solution in \(r\) and \(\theta\).
The inverse of the Kerr’s metric tensor matrix is:
\((g^{\mu\nu})=\pmatrix{-\frac{(r^2+a^2)^2}{\Sigma\Delta}+\frac{a^2\sin^2\theta}{\Sigma}&0&0&-\frac{2mar}{\Sigma\Delta}\\0&\frac{\Delta}{\Sigma}&0&0\\0&0&\frac{1}{\Sigma}&0\\-\frac{2mar}{\Sigma\Delta}&0&0&\frac{1}{\Sigma\sin^2\theta}-\frac{a^2}{\Sigma\Delta}}\)⁵
and the Hamiltonian is \(H=\frac{1}{2}g^{\mu\nu}p_\mu p_\nu\), with (\(p_\mu\)) conjugate moment \(\left(-\frac{\varepsilon}{c},\frac{dS(r)}{dr},\frac{dS(\theta)}{d\theta},l_z\right)\), or:
\(H=\frac{1}{2}\left[\left(-\frac{(r^2+a^2)^2}{\Sigma\Delta}+\frac{a^2\sin^2\theta}{\Sigma}\right)\frac{\varepsilon^2}{c^2}+\frac{4mar}{\Sigma\Delta}\frac{\varepsilon}{c}l_z+\frac{\Delta}{\Sigma}\left(\frac{dS^{(r)}}{dr}\right)^2+\frac{1}{\Sigma}\left(\frac{dS^{(\theta)}}{d\theta}\right)^2+\left(\frac{1}{\Sigma\sin^2\theta}-\frac{a^2}{\Sigma\Delta}\right)l_z^2\right]=0\hspace{2cm}\)(2.b)
in Kerr’s spacetime because the null geodesics are light-type.
Furthermore \(\frac{dx^\mu}{d\lambda}=\frac{\delta H}{\delta p_\mu}\hspace{2cm}\)(2.c)

Parametric equations of \(r\) and \(\theta\)

After multiplying by \(2\Sigma\), equation (2.b) can be written as:
\(-\Delta\left(\frac{dS^{(r)}}{dr}\right)^2+\frac{(r^2+a^2)^2}{\Delta}\frac{\varepsilon^2}{c^2}-\frac{4mar}{\Delta}\frac{\varepsilon}{c} l_z+\frac{a^2l_z^2}{\Delta}=\left(\frac{dS^{(\theta)}}{d\theta}\right)^2+a^2\sin^2\theta\frac{\varepsilon^2}{c^2}+\frac{l_z^2}{\sin^2\theta}\)
and subtracting \(a^2\frac{\varepsilon^2}{c^2}+l_z^2\) from each member:
\(-\Delta\left(\frac{dS^{(r)}}{dr}\right)^2+\frac{(r^2+a^2)^2}{\Delta}\frac{\varepsilon^2}{c^2}-\frac{4mar}{\Delta}\frac{\varepsilon}{c} l_z+\frac{a^2l_z^2}{\Delta}-a^2\frac{\varepsilon^2}{c^2}-l_z^2=\left(\frac{dS^{(\theta)}}{d\theta}\right)^2-a^2\cos^2\theta\frac{\varepsilon^2}{c^2}+\frac{\cos^2\theta}{\sin^2\theta}l_z^2\hspace{2cm}\)(2.d)
The left-hand member of (2.d) does not depend on \(\theta\) and the right-hand member does not depend on \(r\), which implies that they keep a constant value \(Q\) known as Carter’s constant and gives the 2 equations:
\(-\Delta\left(\frac{dS^{(r)}}{dr}\right)^2+\frac{(r^2+a^2)^2}{\Delta}\frac{\varepsilon^2}{c^2}-\frac{4mar}{\Delta}\frac{\varepsilon}{c} l_z+\frac{a^2l_z^2}{\Delta}-a^2\frac{\varepsilon^2}{c^2}-l_z^2=Q\hspace{2cm}\)(2.e)
\(\left(\frac{dS^{(\theta)}}{d\theta}\right)^2-a^2\cos^2\theta\frac{\varepsilon^2}{c^2}+\frac{\cos^2\theta}{\sin^2\theta}l_z^2=Q\hspace{2cm}\)(2.f)
Noting that \(2a-2a\frac{r^2+a^2}{\Delta}=-\frac{4mar}{\Delta}\), (2.e) becomes:
\(-\Delta\left(\frac{dS^{(r)}}{dr}\right)^2+\frac{\left(\left(r^2+a^2\right)\frac{\varepsilon}{c}-al_z\right)^2}{\Delta}-(a\frac{\varepsilon}{c}-l_z)^2=Q\), which gives the 2 equations:
\(\Delta\left(\frac{dS^{(r)}}{dr}\right)^2=\frac{\left(\left(r^2+a^2\right)\frac{\varepsilon}{c}-al_z\right)^2}{\Delta}-(a\frac{\varepsilon}{c}-l_z)^2-Q\hspace{2cm}\)(2.g)
\(\left(\frac{dS^{(\theta)}}{d\theta}\right)^2=Q+\cos^2\theta\left(a^2\frac{\varepsilon^2}{c^2}-\frac{l_z^2}{\sin^2\theta}\right)\hspace{2cm}\)(2.h)
Note : (2.g)/\(\Delta\) is mathematically positive or zero which means that for a given value \(r\) there are conditions linking \(\frac{cl_z}{m\varepsilon}\) and \(\frac{c^2Q}{m^2\varepsilon^2}\). Similarly, (2.h) being mathematically positive or zero, there exist for a given value \(\theta\) conditions linking \(\frac{cl_z}{m\varepsilon}\) and \(\frac{c^2Q}{m^2\varepsilon^2}\). These conditions are discussed in paragraphs 3.4 and 3.5.
Setting
\(V_r=\left(\left(r^2+a^2\right)\frac{\varepsilon}{c}-al_z\right)^2-\Delta\left(\left(a\frac{\varepsilon}{c}-l_z\right)^2+Q\right)=\Delta^2\left(\frac{dS^{(r)}}{dr}\right)^2\hspace{2cm}\)(2.i) and
\(V_\theta=Q+\cos^2\theta\left(a^2\frac{\varepsilon^2}{c^2}-\frac{l_z^2}{\sin^2\theta}\right)=\left(\frac{dS^{(\theta)}}{d\theta}\right)^2\hspace{2cm}\)(2.j)
(2.a) seen above is written:
\(S=-\frac{\varepsilon}{c} ct+\int\frac{\sqrt{V_r}}{\Delta}dr+\int\sqrt{V_\theta}\ d\theta+l_z\varphi\)
This leads to:
\(p_r=\frac{\delta S}{\delta r}=\pm\frac{\sqrt{V_r}}{\Delta}=\frac{\Sigma}{\Delta}\frac{dr}{d\lambda}\) (by applying (2.c)), and
\(p_\theta=\frac{\delta S}{\delta\theta}=\pm\sqrt{V_\theta}=\Sigma\frac{d\theta}{d\lambda}\) (by applying (2.c))
that is, \(V_r=\Sigma^2(\frac{dr}{d\lambda})^2\hspace{2cm}\)(2.k)
and \(V_\theta=\Sigma^2(\frac{d\theta}{d\lambda})^2.\hspace{2cm}\)(2.l)

Parametric equations of \(\varphi\) and \(ct\)

According to (2.c), \(\frac{d\varphi}{d\lambda}=\frac{\delta H}{\delta l_z}\) and \(\frac{dct}{d\lambda}=\frac{\delta H}{\delta\left(-\frac{\varepsilon}{c}\right)}\)
which leads to
\(\frac{d\varphi}{d\lambda}=\frac{\varepsilon}{c}\left(\frac{2mar}{\Sigma}+(\Sigma-2mr)\frac{1}{\Sigma\sin^2\theta}c\frac{l_z}{\varepsilon}\right)/\Delta\hspace{2cm}\)(2.m) and
\(\frac{dct}{d\lambda}=\frac{\varepsilon}{c}\left(\left (\frac{(r^2+a^2)^2}{\Sigma}-\frac{\Delta a^2\sin^2\theta}{\Sigma}-\frac{2mar}{\Sigma}c\frac{l_z}{\varepsilon}\right)\right)/\Delta.\hspace{2cm}\)(2.n)

Expression of the 4 parametric equations\(\frac{dr}{d\lambda}\), \(\frac{d\theta}{d\lambda}\), \(\frac {d\varphi}{d\lambda}\) and \(\frac {dct}{d\lambda}\)

Finally, equations (2.i), (2.j), (2.k), (2.l), (2.m) and (2.n) lead to the 4 parametric equations of motion of the photon that enable the calculation of the null geodesics in Kerr’s spacetime:
\(\left(\frac{dr}{d\lambda}\right)^2=\left(\left(r^2+a^2-ac\frac{l_z}{\varepsilon}\right)^2-\Delta\left(\left(a-c\frac{l_z}{\varepsilon}\right)^2+c^2\frac{Q}{\varepsilon^2}\right)\right)\frac{\varepsilon^2}{ c^2\Sigma^2}\hspace{2cm}\)(2.o)
\(\left(\frac{d\theta}{d\lambda}\right)^2=\left(c^2\frac{Q}{\varepsilon^2}+\cos^2\theta\left(a^2-c^2\frac{l_z^2}{\varepsilon^2\sin^2\theta}\right)\right)\frac{\varepsilon^2}{ c^2\Sigma^2}\hspace{2cm}\)(2.p)
\(\frac{d\varphi}{d\lambda}=\left(2mar+(\Sigma-2mr)c\frac{l_z}{\varepsilon\sin^2\theta}\right)\frac{\varepsilon}{c\Delta\Sigma}\hspace{2cm}\)(2.q)
\(\frac{dct}{d\lambda}=\left((r^2+a^2)^2-\Delta a^2\sin^2\theta-2mar\ c\frac{l_z}{\varepsilon}\right)\frac{\varepsilon}{c\Delta\Sigma}\hspace{2cm}\)(2.r)
Note: the 4 equations above demonstrate a double symmetry of \(a\) and \(l_z\) with respect to \(0\) which shows that the trajectories of 2 photons with the same initial conditions and respective parameters \(a\), \(l_z\) and \(-a\), \(-l_z\) will be symmetrical with respect to the spin axis \(z\) of the black hole (opposite \(\varphi\) values).
In the following, except for the over extreme Kerr’s spacetime described before the conclusion, it is assumed that \(|a|\) lies between \(0\) and \(m\), limits included, with a black hole trigonometric spin when \(a>0\) and a clockwise spin when \(a<0\).

PHOTON TRAJECTORIES

The calculation of light deflection by Kerr’s black holes and, more precisely, the trajectories of photons deflected by a spinning black hole with characteristics \(m\) and \(a\) can be obtained by integrating equations (2.o), (2.p), (2.q) and (2.r) with respect to an affine parameter \(\lambda\).
Initial conditions are \(ct_0, r_0, \theta_0, \varphi_0\), the signs of \((\frac{dr}{d\lambda})_0 \) and \((\frac{d\theta}{d\lambda})_0\) and the parameters are \(c\frac{l_z}{\varepsilon}\) and \(c^2\frac{Q}{\varepsilon^2}\) with invariants \(\varepsilon\) photon energy, \(l_z\) component of the angular momentum \(\overrightarrow{l}\) of the photon along the spin axis of the black hole and Carter’s constant \(Q\).

Cartesian expression of the trajectory

The null geodesics in Kerr’s spacetime give trajectories that can be displayed in a fixed reference frame (\(O, x, y, z\)), \(O\) being the center of the black hole and \(z\) its spin axis, using Boyer-Lindquist’s Cartesian coordinates:
\(x=\sqrt{r^2+a^2}\cos\varphi\sin\theta\), \(y=\sqrt{r^2+a^2}\sin\varphi\sin\theta\) and \(z=r\cos\theta\).

Lense-Thirring’s effect

The expression of \(\frac{d\varphi}{dt}\) is obtained from equations (2.q) and (2.r):
\(\frac{d\varphi}{dt}=\frac{2marc+(\Sigma-2mr)c\frac{l_z}{\varepsilon\sin^2\theta}}{(r^2+a^2)^2-\Delta a^2\sin^2\theta-2marc\frac{l_z}{\varepsilon}}\)
which causes the photon to be « dragged along » by the spinning black hole.
Replacing \(\Delta\) et \(\Sigma\) by their respective values and after developing the denominator, we get:
\(\frac{d\varphi}{dt}=\frac{2marc+(r^2+a^2\cos^2\theta-2mr)c\frac{l_z}{\varepsilon\sin^2\theta}}{(r^2+a^2)(r^2+a^2\cos^2\theta)+2mar\left(a\sin^2\theta-c\frac{l_z}{\varepsilon}\right)}\hspace{2cm}\)(3.a)
and for \(l_z=0\) :
\(\frac{d\varphi}{dt}=\frac{2marc}{(r^2+a^2)(r^2+a^2\cos^2\theta)+2ma^2r\sin^2\theta}\) which has the sign of \(a\).
This effect is particularly apparent for polar orbits:

(3.a) also implies that a photon entering the outer ergosphere (that is \(\Sigma-2mr\lt 0\)) is necessarily prograde (rotates in the same direction as the black hole): \(\bar{a}\gt 0\Rightarrow\) \(\frac{d\varphi}{dt}\gt 0\) or \(\bar{a}\lt 0\Rightarrow\) \(\frac{d\varphi}{dt}\lt 0\).

Example of a photon trajectory in the equatorial plane (\(Q=0\)) with inversion of the direction of variation of \(\varphi\) before entering the outer ergosphere, and then rejoining the singularity circle of radius \(a\) (ring singularity).

Condition if Carter’s constant \(Q<0\)

Extreme values of \(Q\)

For a given value of \(\frac{l_z}{\varepsilon}\), there are limits to \(\frac{Q}{\varepsilon^2}\).

Limits according to \(r\)

The expression (2.o) must remain positive or zero, which means with \(\bar{a}=\frac{a}{m}\), \(\bar{r}=\frac{r}{m}\) and \(\bar{\Delta}=\bar{r}^2-2\bar{r}+\bar{a}^2\) that:
– if \(\bar{\Delta}>0\) (photon outside the event horizon or inside the Cauchy’s horizon): \(\frac{c^2Q}{m^2\varepsilon^2}\le\frac{\left(\bar{r}^2+\bar{a}^2-\bar{a}\frac{cl_z}{m\varepsilon}\right)^2}{\bar{\Delta}}-(\bar{a}-\frac{cl_z}{m\varepsilon})^2\)
– if \(\bar{\Delta}<0\) (photon between the event horizon and the Cauchy’s horizon): \(\frac{c^2Q}{m^2\varepsilon^2}\ge\frac{\left(\bar{r}^2+\bar{a}^2-\bar{a}\frac{cl_z}{m\varepsilon}\right)^2}{\bar{\Delta}}-(\bar{a}-\frac{cl_z}{m\varepsilon})^2\)
There are therefore limit values for \(\frac{c^2Q}{m^2\varepsilon^2}\) which depend on the parameters \(m\) et \(a\), the radial coordinate \(r\) of the photon and \(\frac{cl_z}{m\varepsilon}\).

Limit according to \(\theta\)

The expression (2.p) must remain positive or zero, which means with \(\bar{a}=\frac{a}{m}\) that:
\(\frac{c^2Q}{m^2\varepsilon^2}\ge\frac{c^2l_z^2}{m^2\varepsilon^2}\frac{1}{\tan^2\theta}-\bar{a}^2\cos^2\theta\)
There is therefore a minimum value for \(\frac{c^2Q}{m^2\varepsilon^2}\) which depends on the parameters \(m\) et \(a\), the colatitude \(\theta\) of the photon and \(\frac{cl_z}{m\varepsilon}\).

Extreme values of \(l_z\) component of angular momentum on the spin axis of the black hole

For a given value of \(\frac{Q}{\varepsilon^2}\), there are limits to \(\frac{l_z}{\varepsilon}\).

Limits according to \(r\)

Expression (2.o) must remain positive or zero, which after developing results in, with \(\bar{a}=\frac{a}{m}\), \(\bar{r}=\frac{r}{m}\) and \(\bar{\Delta}=\bar{r}^2-2\bar{r}+\bar{a}^2\):
\(\bar{r}^4+\left(\bar{a}^2-\frac{c^2l_z^2}{m^2\varepsilon^2}\right)\bar{r}^2+2\left(\bar{a}-\frac{cl_z}{m\varepsilon}\right)^2-\bar{\Delta}\frac{c^2Q}{m^2\varepsilon^2}\ge 0\), which can be written as:
\(\bar{r}(2-\bar{r})(\frac{cl_z}{m\varepsilon})^2-4\bar{a}\bar{r}\frac{cl_z}{m\varepsilon}+\bar{r}^4+\bar{a}^2\bar{r}^2+2\bar{a}^2\bar{r}-\bar{\Delta}\frac{c^2Q}{m^2\varepsilon^2}\ge 0\hspace{2cm}\)(3.c)
The left-hand side of (3.c) is a 2nd degree polynomial in \(\frac{cl_z}{m\varepsilon}\) and the reduced discriminant can be written after developing:
\(D’=\bar{r}^6-2\bar{r}^5+\bar{a}^2\bar{r}^4+\bar{r}(2-\bar{r})\bar{\Delta}\frac{c^2Q}{m^2\varepsilon^2}\)
Assuming that \(\bar{r}\ne 2\) (see below for the special case \(\bar{r}=2\)), there are therefore limit values for \(\frac{cl_z}{m\varepsilon}\) which depend on the parameters \(m\) and \(a\), the radial coordinate \(r\) of the photon and \(\frac{c^2Q}{m^2\varepsilon^2}\):
– if \(D’>0\) there are 2 roots: \(\frac{2\bar{a}\bar{r}\pm\sqrt{D’}}{\bar{r}(2-\bar{r})}\)
For (3.c) to be valid, when \(\bar{r}>2\), \(\frac{cl_z}{m\varepsilon}\) must be between the 2 roots and when \(\bar{r}<2\), \(\frac{cl_z}{m\varepsilon}\) must be outside the 2 roots.
– if \(D’\le 0\), \(\bar{r}\) must be below 2, whatever the value of \(\frac{cl_z}{m\varepsilon}\).
When \(\bar{r}=2\), (3.c) implies:
– if \(\bar{a}>0\), \(\frac{cl_z}{m\varepsilon}\le\frac{\bar{r}^4+\bar{a}^2\bar{r}^2+2\bar{a}^2\bar{r}-\bar{\Delta}\frac{c^2Q}{m^2\varepsilon^2}}{4\bar{a}\bar{r}}\)
– if \(\bar{a}<0\), \(\frac{cl_z}{m\varepsilon}\ge\frac{\bar{r}^4+\bar{a}^2\bar{r}^2+2\bar{a}^2\bar{r}-\bar{\Delta}\frac{c^2Q}{m^2\varepsilon^2}}{4\bar{a}\bar{r}}\).

Limit according to \(\theta\)

The expression (2.p) must remain positive or zero, which results with \(\bar{a}=\frac{a}{m}\) as:
\(\frac{c^2l_z^2}{m^2\varepsilon^2}\le\sin^2\theta\left(\bar{a}^2+\frac{c^2Q}{m^2\varepsilon^2}\frac{1}{\cos^2\theta}\right)\)
There is therefore a maximum value for \(\frac{c^2l_z^2}{m^2\varepsilon^2}\) which depend on the parameters \(m\) and \(a\), of the colatitude \(\theta\) of the photon and \(\frac{c^2Q}{m^2\varepsilon^2}\).
Note that \(\frac{c^2l_z^2}{m^2\varepsilon^2}\) must remain positive or zero, and the condition seen above leads to \(\frac{c^2Q}{m^2\varepsilon^2}\ge -\bar{a}^2\cos^2\theta\) which means that \(\frac{c^2Q}{m^2\varepsilon^2}\) cannot be less than \(-\bar{a}^2\), a very restrictive condition for negative values of the Carter’s constant \(Q\) in the case of Kerr black holes.

PHOTON ORBITS

Null geodesics in Kerr’s spacetime can admit a constant radial coordinate \(r\) generating a photon orbit.

Constant radial coordinate

The constant radial coordinate value \(r_c\) is obtained by canceling the potential \(V_r\) (2.i) and its derivative \(\frac{dV_r}{dr}\).
These conditions lead, after calculation (see paragraph 4.6) with \(\bar{r_c}=\frac{r_c}{m}\) and \(\bar{a}=\frac{a}{m}\), to \(\frac{cl_z}{m\varepsilon}=-\frac{(\bar{r_c}^3-3\bar{r_c}^2+\bar{a}^2\bar{r_c}+\bar{a}^2)}{\bar{a}(\bar{r_c}-1)}\) and \(\frac{c^2Q}{m^2\varepsilon^2}=-\frac{\bar{r_c}^3(\bar{r_c}^3-6\bar{r_c}^2+9\bar{r_c}-4\bar{a}^2)}{\bar{a}^2(\bar{r_c}-1)^2}\)
On an other hand, (2.p) means that for a photon orbit \(\left(\frac{d\theta}{d\lambda}\right)^2\) has its maximum value for \(\theta=\frac{\pi}{2}\), this maximum being \(Q\).
Consequently, for a photon orbit crossing the equatorial plane, the value of the Carter’s constant is necessarily greater or equal to zero, otherwise \(\left(\frac{d\theta}{d\lambda}\right)^2\) would be negative.
Restricting to the case \(Q\ge 0\), with \(i\in [0,\pi]\) the constant inclination angle of the angular momentum \(\overrightarrow{l}\) with respect to the spin axis of the black hole and \(l\) the constant norm of the angular momentum \(\overrightarrow{l}\), \(l_z=l\cos i\) leads to \(Q=l^2\sin^2i\) (see paragraph 4.9) and the following results can be obtained (see demonstration paragraph 4.6):
– on the one hand with \(l^2=Q+l_z^2\): \(\frac{c^2l^2}{m^2\varepsilon^2}=\frac{2\bar{r_c}^4+(\bar{a}^2-6)\bar{r_c}^2+2\bar{a}^2\bar{r_c}+\bar{a}^2}{(\bar{r_c}-1)^2}\),
– on the other hand \(\bar{r_c}\) is a root of the polynomial in \(\bar{r}^5\):
\(q(\bar{r})=\bar{r}^5-3\bar{r}^4+2\bar{a}^2\bar{r}^3\sin^2i-2\bar{a}^2\bar{r}^2+\bar{a}^4\bar{r}\sin^2i+\bar{a}^4\sin^2i\)
\(+2\bar{a}\bar{r}\cos i\sqrt{3\bar{r}^4+(1-3\sin^2i)\bar{a}^2\bar{r}^2-\bar{a}^4\sin^2i}\)
\(q(\bar{r_c})=0\hspace{2cm}\)(4.a) is a characteristic equation which gives for a value of \(i\) between \(0\) and \(\pi\) the dimensionless radial coordinate \(\bar{r_c}\) of the photon orbit.
If \(i\in [0,\frac{\pi}{2}]\) or \(l_z\ge 0\), the orbit is prograde and if \(i\in [\frac{\pi}{2},\pi]\) or \(l_z\le 0\), the orbit is retrograde.
There is another polynomial in \(\bar{r}^6\) whose root for a value of \(\sin^2i\) also gives the dimensionless radial coordinate \(\bar{r_c}\) of the photon orbit:
\(p(\bar{r})=\bar{r}^6-6\bar{r}^5+(9+2\nu u)\bar{r}^4-4u\bar{r}^3-\nu u(6-u)\bar{r}^2+2\nu u^2 \bar{r}+\nu u^2\)
with \(u=\bar{a}^2\) and \(\nu=\sin^2⁡i\).
\(p(\bar{r_c})=0\hspace{2cm}\)(4.b) is a characteristic equation which gives for the same value of \(\nu\) at least 2 solutions \(\bar{r_c}_{prograde}\) (orbit driven in the spin direction of the black hole) and \(\bar{r_c}_{retrograde}\) (orbit driven in the opposite spin direction of the black hole) such as \(0\le\bar{r_c}_{prograde}\le 3\le\bar{r_c}_{retrograde}\le 4\) with \(i_{prograde}\in [0,\frac{\pi}{2}]\) or \(l_z\ge 0\), \(i_{retrograde}\in [\frac{\pi}{2},\pi]\) or \(l_z\le 0\), and \(\sin i_{retrograde}=-\sin i_{prograde}\).
There are no known simple analytical solutions to equations (4.a) or (4.b), except for the equatorial orbits (\(i=0\) or \(i=\pi\)) or the polar orbits (\(i=\frac{\pi}{2}\)).
Note that if for given \(\bar{a}\) and \(i\), \(q(\bar{r_c})=p(\bar{r_c})=0\) then these equalities also apply for \(\bar{a}=-\bar{a}\) and \(i=\pi-i\) which shows a double symmetry: \(\bar{a}\) with respect to \(0\) and \(i\) with respect to \(\frac{\pi}{2}\) which is the one seen previously in paragraph 2.3.
Refer to paragraph 4.7 for the discussion of the stability of the orbits under radial perturbation.

Expression of the 3 parametric equations \(\frac{d\theta}{d\lambda}\), \(\frac {d\varphi}{d\lambda}\) and \(\frac {dct}{d\lambda}\)

By replacing \(Q\) and \(l_z\) by their respective values \(l^2\sin^2i\) and \(l\cos i\), and with \(b_{crit}=c\frac{l}{\varepsilon}\), equations (2.p), (2.q) and (2.u) are written:
\(\left(\frac{d\theta}{d\lambda}\right)^2=\left(a^2\cos^2\theta+b_{crit}^2\left(1-\frac{\cos^2i}{\sin^2\theta}\right)\right)\frac{\varepsilon^2}{ c^2\Sigma^2}\hspace{2cm}\)(4.c)
\(\frac{d\varphi}{d\lambda}=\left( 2mar_c+(\Sigma-2mr_c)b_{crit}\frac{\cos i}{sin^2\theta}\right)\frac{\varepsilon}{c\Delta\Sigma}\)
\(\frac{dct}{d\lambda}=\left((r_c^2+a^2)^2-\Delta a^2\sin^2\theta-2mar_c\ b_{crit}\cos i\right)\frac{\varepsilon}{c\Delta\Sigma}\)
where \(r_c\) is the value of the constant radial coordinate.

Noteworthy orbits

Equatorial orbits

Equatorial orbits are obtained when the angular momentum of the photon \(\overrightarrow{l}\) is parallel to the spin axis of the black hole (\(i=0\) or \(i=\pi\)), which results in a photon trajectory in the equatorial plane of the black hole.
Applying \(i=0\) or \(i=\pi\) that is \(\nu=0\) in (4.b), we get:
\(p(\bar{r})=\bar{r}^3(\bar{r}^3-6\bar{r}^2+9\bar{r}-4u)=0\) or
the trivial solution \(\bar{r}=0\) (central or annular singularity) and \(\bar{r}^3-6\bar{r}^2+9\bar{r}-4u=0\)
With the change of variable \(\bar{r}=x+2\), this 3rd degree equation reduces to:
\(x^3-3x+2-4u=0\hspace{2cm}\)(4.d)
Using Cardan’s method, the discriminant of (4.d) \(D=-(4A^3+27B^2)\) with \(A=-3\) and \(B=2-4u\) is \(432u(1-u)\) and assuming that \(u\in[0,1]\), 2 cases can be identified:
– \(u=0\) or \(u=1\Rightarrow D=0\) and (4.d) therefore admits 3 real solutions \(\frac{3B}{A}\) and \(-\frac{3B}{2A}\) (double root) which gives:
for \(u=0\): \(\bar{r_1}=0\) trivial solution (central singularity) and \(\bar{r_0}=\bar{r_2}=3\) (Schwarzschild’s solution),
for \(u=1\): \(\bar{r_0}=4\) and \(\bar{r_1}=\bar{r_2}=1\).
– \(u\in ]0,1[\Rightarrow D>0\) and (4.c) therefore admits 3 distinct real solutions:
\(x_k=2\sqrt{\frac{-A}{3}}\cos\left ({1\over 3}\arccos\left (\frac{3B}{2A}\sqrt{\frac{3}{-A}}\right)+\frac{2k\pi}{3}\right)\) with \(k\in\) {0,1,2}
and replacing \(A\) and \(B\) by their respective values gives:
\(\bar{r_0}=2+2\cos\left({1\over 3}\arccos\left (2u-1\right)\right)\)
\(\bar{r_1}=2+2\cos\left({1\over 3}\arccos\left (2u-1\right)+\frac{2\pi}{3}\right)\)
\(\bar{r_2}=2+2\cos\left({1\over 3}\arccos\left (2u-1\right)+\frac{4\pi}{3}\right)\)
with \(0\lt\bar{r_1}\le 1\le\bar{r_2}\le 3\le\bar{r_0}\le 4\).
Note: the formulae below give the same results:
\(\bar{r_0}=2+2\cos\left(\frac{2}{3}\arccos(\bar{a})\right)\)^{6 7}
\(\bar{r_1}=4\sin^2\left(\frac{1}{3}\arcsin(\bar{a})\right)\)⁸
\(\bar{r_2}=2+2\cos\left(\frac{2}{3}\arccos(-\bar{a})\right)\).^{9 10}

Polar orbits

Polar orbits are obtained when the angular momentum of the photon \(\overrightarrow{l}\) is orthogonal to the spin axis of the black hole (\(i=\frac{\pi}{2}\)), which results in a trajectory that passes through the 2 poles.
Applying \(i=\frac{\pi}{2}\) soit \(\nu=1\) in (4.b), we get:
\(p(\bar{r})=(\bar{r}^3-3\bar{r}^2+u\bar{r}+u)^2=0\)
With the change of variable \(\bar{r}=x+1\), this equation reduces to:
\(x^3+(u-3)x+2u-2=0\hspace{2cm}\)(4.e)
Using Cardan’s method, the discriminant of (4.e) \(D=-(4A^3+27B^2)\) with \(A=u-3\) and \(B=2u-2\) is \(4u(-u^2-18u+27)\) and assuming that \(u\in[0,1]\), 2 cases can be identified:
– \(u=0\Rightarrow D=0\) and (4.d) therefore admits 3 real solutions, one of which is double: \(x_0=\frac{3B}{A}\) and \(x_1=x_2=-\frac{3B}{2A}\) which gives \(r_0=3\) (Schwarzschild’s solution) and \(r_1=r_2=0\) trivial solution (central singularity).
– \(u\in ]0,1]\) : the sign of \(D\) is that of the polynomial \(-u^2-18u+27\) whose reduced discriminant is \(108\), which means that the polynomial has 2 real roots: \(-9-\sqrt{108}=-9-6\sqrt{3}\) and \(-9+\sqrt{108}=-9+6\sqrt{3}\).
It is easy to check that \(u\) lies between these 2 roots, which indicates that the polynomial is positive for \(u\in ]0,1]\Rightarrow D>0\) and (4.d) therefore admits 3 distinct real solutions:
\(x_k=2\sqrt{\frac{-A}{3}}\cos\left({1\over 3}\arccos\left(\frac{3B}{2A}\sqrt{\frac{3}{-A}}\right)+\frac{2k\pi}{3}\right)\) with \(k\in\) {0,1,2}
and replacing \(A\) and \(B\) by their respective values gives:
\(\bar{r_0}=1+2\sqrt{1-\frac{u}{3}}\cos\left({1\over 3}\arccos\left(\frac{1-u}{\left(1-\frac{u}{3}\right)^\frac{3}{2}}\right)\right)\)
\(\bar{r_1}=1+2\sqrt{1-\frac{u}{3}}\cos\left({1\over 3}\arccos\left(\frac{1-u}{\left(1-\frac{u}{3}\right)^\frac{3}{2}}\right)+\frac{2\pi}{3}\right)<0\Rightarrow\) unacceptable solution since the radial coordinate of the photon is positive or zero.
\(\bar{r_2}=1+2\sqrt{1-\frac{u}{3}}\cos\left({1\over 3}\arccos\left(\frac{1-u}{\left(1-\frac{u}{3}\right)^\frac{3}{2}}\right)+\frac{4\pi}{3}\right)\)
with \(\bar{r_1}\lt 0\lt\bar{r_2}\le1\le\bar{r_0}\le 3\).
Note: the formula below gives the same result for \(\bar{r_2}\):
\(\bar{r_2}=1-2\sqrt{1-\frac{u}{3}}\sin\left({1\over 3}\arcsin\left(\frac{1-u}{\left(1-\frac{u}{3}\right)^\frac{3}{2}}\right)\right)\).¹¹

Critical impact parameter

As seen above, the value of \(i\) sets the dimensionless radial coordinate \(\bar{r_c}\) of the photon orbit and the critical impact parameter related to the null geodesics in Kerr’s spacetime can be calculated by the following formulae:
\(b_{crit}=m\sqrt{\frac{3\bar{r_c}^4+\bar{a}^2\bar{r_c}^2}{\bar{r_c}^2-\bar{a}^2\sin^2i}}\), with \(\bar{r_c}^2>\bar{a}^2\sin^2i\), see calculation in paragraph 4.6
or
\(b_{crit}=m\sqrt{\frac{2\bar{r_c}^4+(\bar{a}^2-6)\bar{r_c}^2+2\bar{a}^2\bar{r_c}+\bar{a}^2}{(\bar{r_c}-1)^2}}\) for \(\bar{r_c}\ne 1\) and using the expression for \(\frac{c^2l^2}{m^2\varepsilon^2}\) given in paragraph 4.1.

Colatitude limit

According to (4.c), \(\frac{d\theta}{d\lambda}\) cancels for \(a^2\cos^2\theta+b_{crit}^2\left(1-\frac{\cos^2i}{\sin^2\theta}\right)=0\) which is written after developing and in dimensionless values:
\(-\bar{a}^2\cos^4\theta-\left(\left(\frac{b_{crit}}{m}\right)^2-\bar{a}^2\right)\cos^2\theta+\left(\frac{b_{crit}}{m}\right)^2\sin^2i=0\hspace{2cm}\)(4.f)
The left-hand side of (4.f) is a 2nd-degree polynomial in \(\cos^2\theta\) which has a positive root \(\cos^2\theta_{lim}=\frac{\bar{a}^2-\left(\frac{b_{crit}}{m}\right)^2+\sqrt{\left(\bar{a}^2-\left(\frac{b_{crit}}{m}\right)^2\right)^2+4\bar{a}^2\left(\frac{b_{crit}}{m}\right)^2\sin^2i}}{2\bar{a}^2}\) and is positive or zero for \(\cos^2\theta\in [0,\cos^2\theta_{lim}]\).
The photon orbits therefore have a colatitude \(\theta\) that remains within the interval \([\theta_{lim},\pi-\theta_{lim}]\) with \(\cos\theta_{lim}=\sqrt{\frac{\bar{a}^2-\left(\frac{b_{crit}}{m}\right)^2+\sqrt{\left(\bar{a}^2-\left(\frac{b_{crit}}{m}\right)^2\right)^2+4\bar{a}^2\left(\frac{b_{crit}}{m}\right)^2\sin^2i}}{2\bar{a}^2}}\hspace{2cm}\)(4.g)
In the case of polar orbits, \(i=\frac{\pi}{2}\) and (4.g) becomes:
\(\cos\theta_{lim}=\sqrt{\frac{\bar{a}^2-\left(\frac{b_{crit}}{m}\right)^2+\sqrt{\left(\bar{a}^2+\left(\frac{b_{crit}}{m}\right)^2\right)^2}}{2\bar{a}^2}}=1\) which confirms that the colatitude of a polar orbit is defined on \(]0,\pi[\).
For an extreme Kerr black hole \(\bar{a}^2=1\) and after replacing \(\left(\frac{b_{crit}}{m}\right)^2\) and \(\left(\frac{b_{crit}}{m}\right)^2\sin^2i\) by their respective values \(\frac{c^2l^2}{m^2\varepsilon^2}\) and \(\frac{c^2Q}{m^2\varepsilon^2}\) seen above, (4.g) can be expressed as a simplified function of \(\bar{r_c}\):
\(\cos\theta_{lim}=\sqrt{-\bar{r_c}^2+2\bar{r_c}(\sqrt{2\bar{r_c}+1}-1)}\).

Setting up the characteristic equations

Calculation of \(c\frac{l_z}{\varepsilon}\), \(c\frac{Q}{\varepsilon^2}\) and \(c^2\frac{l^2}{\varepsilon^2}\)
The derivative with respect to \(r\) in (2.i) gives \(\frac{dV_r}{dr}=4r\frac{\varepsilon}{c}\left((r^2+a^2)\frac{\varepsilon}{c}-al_z\right)-2(r-m)\left((a\frac{\varepsilon}{c}-l_z)^2+Q\right)\)
and the condition \(\frac{dV_r}{dr}=0\) is then written:
\((a\frac{\varepsilon}{c}-l_z)^2+Q=\frac{2r}{r-m}\frac{\varepsilon}{c}\left((r^2+a^2)\frac{\varepsilon}{c}-al_z\right)\hspace{2cm}\)(4.h) or
\((r^2+a^2)\frac{\varepsilon}{c}-al_z=\frac{r-m}{2r}\frac{c}{\varepsilon}\left((a\frac{\varepsilon}{c}-l_z)^2+Q\right)\hspace{2cm}\)(4.i)
The condition \(V_r=0\) gives with (2.i) by replacing \((a\frac{\varepsilon}{c}-l_z)^2+Q\) by its value given by (4.h):
\(\left((r^2+a^2)\frac{\varepsilon}{c}-al_z\right)\left((r^2+a^2)\frac{\varepsilon}{c}-al_z-\frac{2r}{r-m}\frac{\varepsilon}{c}(r^2-2mr+a^2)\right)=0\) that is, 2 solutions:
\((r^2+a^2)\frac{\varepsilon}{c}-al_z=0\) or
\(\left((r^2+a^2)\frac{\varepsilon}{c}-al_z-\frac{2r}{r-m}\frac{\varepsilon}{c}(r^2-2mr+a^2)\right)=0\) which after developing gives:
\(c\frac{l_z}{\varepsilon}=-\frac{(r^3-3mr^2+a^2r+ma^2)}{a(r-m)}\) or \(\frac{cl_z}{m\varepsilon}=-\frac{(\bar{r_c}^3-3\bar{r_c}^2+\bar{a}^2\bar{r_c}+\bar{a}^2)}{\bar{a}(\bar{r_c}-1)}\hspace{2cm}\)(4.j) with \(\bar{r_c}=\frac{r}{m}\) et \(\bar{a}=\frac{a}{m}\).
The same condition \(V_r=0\) gives with (2.i) by replacing \((r^2+a^2)\frac{\varepsilon}{c}-al_z\) by its value given by (4.i):
\(\left(\left(a\frac{\varepsilon}{c}-l_z\right)^2+Q\right)\left(\left(\frac{r-m}{2r}\right)^2\frac{c^2}{\varepsilon^2}\left(\left(a\frac{\varepsilon}{c}-l_z\right)^2+Q\right)-(r^2-2mr+a^2)\right)=0\) that is, 2 solutions:
\(\left(a\frac{\varepsilon}{c}-l_z\right)^2+Q=0\) or
\(\left(\frac{r-m}{2r}\right)^2\frac{c^2}{\varepsilon^2}\left(\left(a\frac{\varepsilon}{c}-l_z\right)^2+Q\right)-(r^2-2mr+a^2)=0\) which gives after developing, by replacing \(c\frac{l_z}{\varepsilon}\) by its value given by (4.j):
\(c^2\frac{Q}{\varepsilon^2}=-\frac{r^3(r^3-6mr^2+9m^2r-4ma^2)}{a^2(r-m)^2}\) or \(\frac{c^2Q}{m^2\varepsilon^2}=-\frac{\bar{r_c}^3(\bar{r_c}^3-6\bar{r_c}^2+9\bar{r_c}-4\bar{a}^2)}{\bar{a}^2(\bar{r_c}-1)^2}\hspace{2cm}\)(4.k)
Note: the 1st solution seen above \((r^2+a^2)\frac{\varepsilon}{c}-al_z=0\) and \(\left(a\frac{\varepsilon}{c}-l_z\right)^2+Q=0\) gives \(c\frac{l_z}{\varepsilon}=\frac{r^2+a^2}{a}\) and \(c^2\frac{Q}{\varepsilon^2}=-\frac{r^4}{a^2}\), and the left-hand side of condition (4.g) applicable if \(Q<0\) is then \(a^2-(\frac{r^2+a^2}{a})^2+\frac{r^4}{a^2}\) that is \(-2r^2\) which shows that the condition is not met: the solution \(c\frac{l_z}{\varepsilon}=\frac{r^2+a^2}{a}\) and \(c^2\frac{Q}{\varepsilon}^2=-\frac{r^4}{a^2}\) can not be considered.
\(l^2=l_z^2+Q\) and (4.j) et (4.k) then give after regrouping:
\(c^2\frac{l^2}{\varepsilon^2}=\frac{2r^4+(a^2-6m^2)r^2+2ma^2r+m^2a^2}{(r-m)^2}\) or \(\frac{c^2l^2}{m^2\varepsilon^2}=\frac{2\bar{r_c}^4+(\bar{a}^2-6)\bar{r_c}^2+2\bar{a}^2\bar{r_c}+\bar{a}^2}{(\bar{r_c}-1)^2}\hspace{2cm}\)(4.l)
For an extreme Kerr black hole, the formulae seen above are simplified and can be written as follows:
\(\frac{cl_z}{m\varepsilon}=\pm\left(2-(\bar{r_c}-1)^2\right)\), \(\frac{c^2Q}{m^2\varepsilon^2}=-\bar{r_c}^3(\bar{r_c}-4)\) and \(\frac{c^2l^2}{m^2\varepsilon^2}=2(\bar{r_c}+1)^2-1\)
and with the special case \(\bar{r_c}=1\) which only exists for an extreme Kerr black hole:
\(\frac{cl_z}{m\varepsilon}=\pm\ 2\), \(\frac{c^2Q}{m^2\varepsilon^2}=3\) and \(\frac{c^2l^2}{m^2\varepsilon^2}=7\).

Calculation of the characteristic equations
Equation (2.i) is written with \(l_z=l\cos i\), \(Q=l^2\sin^2i\) and \(b=c\frac{l}{\varepsilon}\) :
\(V_r=\frac{\varepsilon^2}{c^2}\left(r^4+(a^2-b^2)r^2+2m(a^2-2ab\cos i+b^2)r-a^2b^2\sin^2i\right)\hspace{2cm}\)(4.m)
and the derivative with respect to \(r\) gives:
\(\frac{dV_r}{dr}=\frac{\varepsilon^2}{c^2}\left(4r^3+2(a^2-b^2)r+2m(a^2-2ab\cos i+b^2)\right)\)
The condition \(\frac{dV_r}{dr}=0\) is then written:
\(2m(a^2-2ab\cos i+b^2)=-4r^3-2(a^2-b^2)r\hspace{2cm}\)(4.n)
The condition \(V_r=0\) gives with (2.k) by replacing \(2m(a^2-2ab\cos i+b^2)\) by its value given by (4.n):
\(3r^4+(a^2-b^2)r^2+a^2b^2\sin^2i=0\), that is
\(b^2=\frac{3r^4+a^2r^2}{r^2-a^2\sin^2i}.\hspace{2cm}\)(4.o)

Equation in \(r^5\)
Replacing in (4.n) \(b\) by its value given by (4.o), we get:
\(\left(2r^3+a^2(r+m)\right)(r^2-a^2\sin^2i)-(r-m)(3r^4+a^2r^2)\)
\(-2ma\cos i\sqrt{(3r^4+a^2r^2)(r^2-a^2\sin^2i)}=0\) that is, after developpment
\(r^5-3mr^4+2a^2\sin^2i\ r^3-2ma^2r^2+a^4\sin^2i\ r+ma^4\sin^2i\)
\(+2ma\cos i\ r\sqrt{3r^4+(1-3\sin^2i)a^2r^2-a^4\sin^2i}=0\) or
\(\bar{r}^5-3\bar{r}^4+2\bar{a}^2\sin^2i\ \bar{r}^3-2\bar{a}^2\bar{r}^2+\bar{a}^4\sin^2i\ \bar{r}+\bar{a}^4\sin^2i\)
\(+2\bar{a}\cos i\ \bar{r}\sqrt{3\bar{r}^4+(1-3\sin^2i)\bar{a}^2\bar{r}^2-\bar{a}^4\sin^2i}=0\).

Equation in \(r^6\)
\(b^2=c^2\frac{l^2}{\varepsilon^2}\) gives by replacing \(b^2\) by its value (4.o) and \(c^2\frac{l^2}{\varepsilon^2}\) by its value (4.l):
\(\frac{3r^4+a^2r^2}{r^2-a^2\sin^2i}=\frac{2r^4+(a^2-6m^2)r^2+2ma^2r+m^2a^2}{(r-m)^2}\) which gives after developing:
\(r^6-6mr^5+(9m^2+2a^2\sin^2i)r^4-4ma^2r^3-a^2\sin^2i(6m^2-a^2)r^2\)
\(+2ma^4\sin^2i\ r+m^2a^4\sin^2i=0\) or
\(\bar{r}^6-6\bar{r}^5+(9+2\bar{a}^2\sin^2i)\bar{r}^4-4\bar{a}^2\bar{r}^3-\bar{a}^2\sin^2i(6-\bar{a}^2)\bar{r}^2\)
\(+2\bar{a}^4\sin^2i\ \bar{r}+\bar{a}^4\sin^2i=0\).

Stability of orbits under radial perturbation

After developing (2.i) :
\(\frac{c^2}{\varepsilon^2}V_r=r^4+\left(a^2-c^2\frac{l_z^2}{\varepsilon^2}-c^2\frac{Q}{\varepsilon^2}\right)r^2+2m\left(\left(a-c\frac{l_z}{\varepsilon}\right)^2+c^2\frac{Q}{\varepsilon^2}\right)r-a^2c^2\frac{Q}{\varepsilon^2}\)
hence \(\frac{c^2}{\varepsilon^2}\frac{dV_r}{dr}=4r^3+2\left(a^2-c^2\frac{l_z^2}{\varepsilon^2}-c^2\frac{Q}{\varepsilon^2}\right)r+2m\left(\left(a-c\frac{l_z}{\varepsilon}\right)^2+c^2\frac{Q}{\varepsilon^2}\right)\)
which gives:
\(\frac{c^2}{\varepsilon^2}\frac{d^2V_r}{dr^2}=12r^2+2\left(a^2-c^2\frac{l_z^2}{\varepsilon^2}-c^2\frac{Q}{\varepsilon^2}\right)\)
Using the dimensionless parameters and variables and replacing \(\frac{cl_z}{m\varepsilon}\) and \(\frac{c^2Q}{m^2\varepsilon^2}\) by their respective values (4.j) and (4.k), it comes after developing for an orbit of radial coordinate \(\bar{r_c}\):
\(\frac{c^2\varepsilon^2}{m^3}\frac{d^2V_r}{dr^2}=\frac{8\bar{r_c}}{(\bar{r_c}-1)^2}\left(\left(\bar{r_c}-1\right)^3+1-\bar{a}^2\right)\) which has a single real root \(\bar{r_{c_{stab}}}=1-(1-\bar{a}^2)^{1/3}\)
This value \(\bar{r_{c_{stab}}}\) is less or equal to \(1\) for Kerr black holes and delimits the stability of photon orbits:
– for \(\bar{r_c}<\bar{r_{c_{stab}}}\), \(\frac{d^2V_r}{dr^2}\) is \(<0\) and the orbit is stable in radial perturbation,
– for \(\bar{r_c}>\bar{r_{c_{stab}}}\), \(\frac{d^2V_r}{dr^2}\) is \(>0\) and the orbit is unstable in radial perturbation.

Limit inclinations

The value \(\bar{r_{c_{stab}}}\) seen above corresponds to a limit inclination angle \(i_{stab}(\bar{a})=\arccos\frac{l_z}{\sqrt{l_z^2+Q}}\), the values \(l_z\) and \(Q\) being given by (4. j) and (4.k) with \(\bar{r_c}=\bar{r_{c_{stab}}}\).

Extreme Kerr’s black hole

When \(|\bar{a}|=1\), each value of the inclination angle \(i\) determines one and only one value of \(\bar{r_c}\).
When the photon is on the event horizon \(\bar{r_c}=1\) with \(Q\ne 0\), the respective values of \(\frac{cl_z}{m\varepsilon}\) and \(\frac{c^2l^2}{m^2\varepsilon^2}\) are \(2\) (if \(\bar{a}=1\)) or \(-2\) (if \(\bar{a}=-1\)) and \(7\), which leads to an inclination angle \(i_{stab}=\arccos\frac{2}{\sqrt{7}}\simeq 40.9^\circ\) if \(\bar{a}=1\), or \(i_{stab}=\arccos\frac{-2}{\sqrt{7}}\simeq 139,1^\circ\) if \(\bar{a}=-1\).
For \(i<i_{stab}\) if \(\bar{a}=1\) or for \(>i_{stab}\) if \(\bar{a}=-1\), photon orbits are inside the Cauchy’s horizon, that is, with a value \(\bar{r_c}\in [0,1[\).

Other Kerr’s black holes

When the value of the inclination angle \(i\) is below \(i_{stab}(\bar{a})\) if \(\bar{a}>0\), or above \(i_{stab}(\bar{a})\) if \(\bar{a}<0\), it corresponds to 3 values of \(r_c\), one of which is greater than \(r_h\) (photon orbit outside the event horizon) and the other two are below \(r_{Cauchy}\) (photon orbits inside the Cauchy’s horizon).
The larger of the latter two orbits has the particularity of an angle \(\theta_{lim}\) increasing with \(r_c\) if \(\bar{a}>0\), or decreasing if \(\bar{a}<0\), unlike the other prograde photon orbits.
For \(i\) above \(i_{stab}(\bar{a})\) if \(\bar{a}>0\), or below \(i_{stab}(\bar{a})\) if \(\bar{a}<0\), there is one and only one value of \(r_c\) which is greater than \(r_h\) and the photon orbits are outside the event horizon.

Definition of \(Q\) as a function of \(l\) and \(i\)

Using the value of \(\frac{cl_z}{m\varepsilon}\) given in paragraph 4.1, we get:
\(\frac{c^2l_z^2}{m^2\varepsilon^2}=\left (-\frac{(\bar{r_c}^3-3\bar{r_c}^2+\bar{a}^2\bar{r_c}+\bar{a}^2)}{\bar{a}(\bar{r_c}-1)}\right)^2\) or \(\frac{\bar{r_c}^6-6\bar{r_c}^5+9\bar{r_c}^4-4\bar{a}^2\bar{r_c}^3}{\bar{a}^2(\bar{r_c}-1)^2}+\frac{2\bar{r_c}^4+(\bar{a}^2-6)\bar{r_c}^2+2\bar{a}^2\bar{r_c}+\bar{a}^2}{(\bar{r_c}-1)^2}\) which is written with the value of \(\frac{c^2Q}{m^2\varepsilon^2}\) given in paragraph 4.1:
\(\frac{c^2l_z^2}{m^2\varepsilon^2}=-\frac{c^2Q}{m^2\varepsilon^2}+\frac{2\bar{r_c}^4+(\bar{a}^2-6)\bar{r_c}^2+2\bar{a}^2\bar{r_c}+\bar{a}^2}{(\bar{r_c}-1)^2}\).
Furthermore, \(l_z=l\cos i\) gives \(\sin^2i=\frac{l^2-l_z^2}{l^2}\) that is, by replacing \(l_z\) with its value calculated above:
\(\sin^2i=\frac{\frac{c^2l^2}{m^2\varepsilon^2}+\frac{c^2Q}{m^2\varepsilon^2}-\left (\frac{2\bar{r_c}^4+(\bar{a}^2-6)\bar{r_c}^2+2\bar{a}^2\bar{r_c}+\bar{a}^2}{(\bar{r_c}-1)^2}\right)}{\frac{c^2l^2}{m^2\varepsilon^2}}\)
and by setting \(\frac{2\bar{r_c}^4+(\bar{a}^2-6)\bar{r_c}^2+2\bar{a}^2\bar{r_c}+\bar{a}^2}{(\bar{r_c}-1)^2}=\frac{c^2l^2}{m^2\varepsilon^2}\) x const, we get:
\(\sin^2i=\frac{Q}{l^2}+1\) – const\(\hspace{2cm}\)(4.p)
In the specific case \(i=0\) and \(\theta=\frac{\pi}{2}\), the constant value of \(i\) implies that the null geodesic remains in the equatorial plane that is, \(\frac{d\theta}{d\lambda}=0\).
Since, as seen above, \(\Sigma\frac{d\theta}{d\lambda}=\pm\sqrt{V_\theta}\), we get \(V_{\theta}=0\) that is, with (2.j) \(Q=0\) and from (4.p), const \(=1\).
Replacing const with its value in (4.p) gives then \(Q=l^2\sin^2i\).

NUMERICAL INTEGRATION

Numerical integration which enables the calculation of the null geodesics in Kerr spacetime and the plotting of corresponding trajectories can be done using parametric equations (integration with respect to an affine parameter) or time derivatives (integration with respect to time \(t\) of a static observer).
Cartesian plots are then done using the equations seen above \(x=\sqrt{r^2+a^2}\cos\varphi\sin\theta\), \(y=\sqrt{r^2+a^2}\sin\varphi\sin\theta\) and \(z=r\cos\theta\).

Trajectories

Affine parameters

Equations (2.o), (2.p), (2.q) and (2.r) written in paragraph 2.3 show an affine parameter \(\Lambda=\frac{\lambda\varepsilon}{\Sigma c}\), dependent on \(\Sigma=r^2+a^2\cos^2\theta\), and they can be integrated according to a constant value of the affine step \(\Lambda\).
Another solution for integrating the 4 parametric equations is to consider the affine parameter \(\Lambda=\frac{\lambda\varepsilon}{c}\) by dividing each of the equations by \(\Sigma\) which allows us to use a constant value of the affine step not including the value \(\Sigma\).
Using a simple 4th-order Runge-Kutta integration method, the 2 solutions give good results, with integration over the affine step including \(\Sigma\) giving more accurate results for low values of the radial coordinate \(r\) while integration on the affine step which does not include \(\Sigma\) gives more accurate results for large values of \(r\).
The best results are obtained logically with an adaptive affine step (set by a targeted precision on the calculation of \(r\)), and the use of the adaptive affine step including the value \(\Sigma\) avoids to unnecessarily complexify the calculations of the 4 RK4 coefficients.
The initial conditions to be considered are explained in paragraph 3 above.

Time integration

1st-order time derivatives \(\frac{dr}{dt},\frac{d\theta}{dt}\) and \(\frac{d\varphi}{dt}\) are obtained by dividing the equations (2.o), (2.p) and (2.q) by (2.r) and multiplying by \(c\), and they can be integrated using a constant-step RK4 method.
The initial conditions to be considered are the photon coordinates \((r_0, \varphi_0, \theta_0)\) and the initial signs of \(\frac{dr}{dt}\) and of \(\frac{d\theta}{dt}\).
It should be noted that time integration does not allow us to plot the trajectories of photons located between the event horizon and the Cauchy’s horizon, due to the time \(t\) of the external observer, which is not defined in this region.
However, time integration remains an interesting option to plot animated trajectories, according to a time step and a sampling of results to be chosen to not unnecessarily weigh the files.

Orbits

The integration solutions discussed in paragraph 5.1 above apply to the 3 1st-order parametric equations \(\frac{d\theta}{d\lambda}\), \(\frac{d\varphi}{d\lambda}\) and \(\frac{dt}{d\lambda}\) with \(r=r_c=\) const for the photon orbit.
The solution using the constant affine step including the value \(\Sigma\) is slightly more accurate than that without the value \(\Sigma\) while the adaptive step method on \(r\) cannot be applied since the radial coordinate \(r\) is constant.
Finally, as seen in paragraph 5.1.2 above, time integration is an interesting way of plotting animated orbits, with a time step and a sampling of results to be chosen to not unnecessarily weigh the files.

EXAMPLES OF PHOTON TRAJECTORIES AND ORBITS

The calculation of the null geodesics in Kerr’s spacetime as described above enables us to draw Cartesian trajectories and orbits, some examples of which are given in this paragraph.

Photons arriving from infinity – examples of quasi-capture

Extreme Kerr’s black hole \(\bar{a}=-1\)

Extreme Kerr’s black hole \(\bar{a}=1\)

Schwarzschild’s black hole \(\bar{a}=0\) (trajectories in a plane \(\theta=\) const)

Photon « spheres »

The plots in this paragraph are arbitrarily made with initial conditions \(\theta_0=\frac{\pi}{2}\) and \((\frac{d\theta}{d\lambda})_0\ge 0\) and the number of oscillations is arbitrary unless otherwise stated.

Extreme Kerr’s black hole \(\bar{a}=1\)

Extreme Kerr’s black hole \(\bar{a}=1\) with orbits inside the Cauchy horizon \(\bar{r_c}\lt 1\)

that is, \(\frac{cl_z}{m\varepsilon}<2\) and \(\frac{c^2Q}{m^2\varepsilon^2}<3\)

Some other Kerr’s black holes

Top views of figures for \(\bar{r_c}\gt 1\)

Two first oscillations

Outlines and boundaries of photon orbits around Kerr’s black holes

orange plot which corresponds to the polar orbit \(\bar{r_c}=1+ \sqrt{2}\).
The equatorial orbit \(\bar{r_c}=4\) is represented by its diameter \(4r_s=8m\).
The red plot is the outline of the orbit \(\bar{r_c}=1\) and the blue plots are the outlines of the orbits \(\bar{r_c}\lt 1\), the plot \(\bar{r_c}=0\) corresponding to the ring singularity of diameter \(r_s=2m\).
The green and cyan plots are the boundaries in the xz plane of all possible orbits (that is, with dimensionless radial coordinate \(\bar{r_c}\) varying continuously from \(0\) to \(4\)).
The regions (outer ergosphere, merged event and Cauchy’s horizons and inner ergosphere) are plotted in magenta.
Note that, with the exception of photons with zero angular momentum \(\overrightarrow{l}\), no photon can be found in the region delimited by the cyan line.
Examples of photon orbits boundaries around some Kerr black holes:

POSITION OF PHOTON ORBITS

Two cases can be distinguished for the position of photon orbits with respect to the regions of a Kerr’s black hole:
– for \(|\bar{a}|\gt\frac{\sqrt{2}}{2}\), there are 4 possible orbit positions:
1) orbit entirely outside the outer ergosphere,
2) orbit partially outside the outer ergosphere (for \(\theta\) close to \(\theta_{lim}\) or \(\pi-\theta_{lim}\)) and partially between the outer ergosphere and the event horizon (for \(\theta\) close to \(\frac{\pi}{2}\)), necessarily prograde (\(i<\frac{\pi}{2}\)) since the photons pass through the outer ergosphere (see example below),
3) orbit entirely between the outer ergosphere and the event horizon,
4) orbit entirely between the Cauchy’s horizon and the inner ergosphere.

– for \(|\bar{a}|\lt\frac{\sqrt{2}}{2}\), there are 2 possible orbit positions:
1) orbit entirely outside the outer ergosphere,
2) orbit entirely between the Cauchy’s horizon and the inner ergosphere.
Note: there are no photon orbits crossing the event horizon or the Cauchy horizon, or located between the 2 horizons.

APPARENT IMAGE OF A KERR’S BLACK HOLE (SHADOW)

and \(\frac{\beta}{m}=\bar{\beta}=\pm\sqrt{\frac{-\bar{r_c}^3(\bar{r_c}^3-6\bar{r_c}^2+9\bar{r_c}-4\bar{a}^2)}{\bar{a}^2(\bar{r_c}-1)^2}+\cos\theta_0^2\left(\bar{a}^2-\left(\frac{-(\bar{r_c}^3-3\bar{r_c}^2+\bar{a}^2\bar{r_c}+\bar{a}^2)}{\bar{a}(\bar{r_c}-1)}\right)^2\frac{1}{\sin\theta_0^2}\right)}\)
with \(\bar{r_c}\) dimensionless radial coordinates of photon orbits varying between a value \(\bar{r}_{c_{min}}\) and a value \(\bar{r}_{c_{max}}\).
The celestian coordinate \(\varphi_{obs}\) of the observer at a colatitude \(\theta_0\) of the black hole can be expressed as:
\(\sin\varphi_{obs}=\frac{\bar{r_c}^3-3\bar{r_c}^2+\bar{r_c}\bar{a}^2+\bar{a}^2+\bar{a}^2\sin^2\theta_0(\bar{r_c}-1)}{2\bar{a}\bar{r_c}\sin\theta_0\sqrt{\bar{r_c}^2-2\bar{r_c}+\bar{a}^2}}\) and the values \(\bar{r}_{c_{min}}\) and \(\bar{r}_{c_{max}}\) are respectively solutions to \(\sin\varphi_{obs}=1\) and \(\sin\varphi_{obs}=-1\)¹³.
For a given observation angle \(\theta_0\), the pairs of values \(\bar{\alpha}\) and \(\bar{\beta}\) are obtained by varying \(\bar{r_c}\) from \(\bar{r}_{c_{min}}\) to \(\bar{r}_{c_{max}}\).
Each outline is symmetrical with respect to the horizontal axis, and the value \(\alpha\) changing its sign with \(\bar{a}\), the outlines are symmetrical with respect to the vertical axis for two opposite values of \(\bar{a}\).

Exact calculation

The celestial coordinate \(\theta_{obs}\) of the static observer located at a distance \(r_0\) from a Kerr’s black hole can be expressed as:
\(\sin\theta_{obs}=\frac{2\bar{r_c}\sqrt{\bar{r_c}^2-2\bar{r_c}+\bar{a}^2}\sqrt{\bar{r}_0^2-2\bar{r}_0+\bar{a}^2}}{\bar{r}_0^2\bar{r_c}-\bar{r}_0^2+\bar{r_c}^3-3\bar{r_c}^2+2\bar{r_c}\bar{a}^2}\)¹⁴ with \(\bar{r}_0=\frac{r_0}{m}\).
The stereographic projection in a plane tangent to the celestial sphere of the observer at the pole \(\theta=0\) gives the Cartesian coordinates of the apparent outline of the black hole in this plane:
\(x(\bar{r_c})=-2\tan(\frac{\theta_{obs}}{2})\sin\varphi_{obs}\) and \(y(\bar{r_c})=\mp\ 2\tan(\frac{\theta_{obs}}{2})\cos\varphi_{obs}\)¹⁵.
For given observation angle \(\theta_0\) and distance \(r_0\), the pairs of values \(x\) and \(y\) are obtained by varying \(\bar{r_c}\) from \(\bar{r}_{c_{min}}\) to \(\bar{r}_{c_{max}}\).
Each outline is symmetrical with respect to the horizontal axis, and the value \(\varphi_{obs}\) changing its sign with \(\bar{a}\), the outlines are symmetrical with respect to the vertical axis for two opposite values of \(\bar{a}\).

OVER EXTREME KERR’S SPACETIME

The Kerr’s spacetime is said to be « over extreme » when the absolute value \(|\bar{a}|\) of the Kerr’s parameter is greater than \(1\).

Over extreme Kerr’s object

As an over extreme Kerr’s object has no event horizon, it does not belong to the black hole category.
It has an outer ergosphere and an inner ergosphere that are adjacent and therefore form a single hypersurface, and its singularity (circle of Cartesian radius \(|a|\)) is said to be « naked » due to the non-existence of an event horizon.
The physical limit of the Kerr’s parameter is obtained for a spin that drives a point of the over extreme Kerr’s object at the speed of light in vacuum.
Formally, therefore, there is no mathematical upper limit on \(|a|\) if the over extreme Kerr’s object is reduced to a material point.

Parametric equations and photon trajectories

All the equations in paragraphs 1 to 3 above apply.

Photons arriving from infinity – examples of quasi-capture

Condition on \(Q\)

For negative values of the Carter constant \(Q\), the condition expressed in paragraph 3.5.2 \(\frac{c^2Q}{m^2\varepsilon^2}\ge -\bar{a}^2\) is slightly less restrictive for an over extreme Kerr’s object.

Photon orbits

Equatorial orbits

(4.d) has the discriminant \(D=432u(u-1)\) with \(u=\bar{a}^2\) which leads to \(D<0\) whatever the value \(|\bar{a}|>1\).
(4.d) therefore has a single real solution \(x=\sqrt[3]{\frac{-B+\sqrt{\frac{-D}{27}}}{2}}+\sqrt[3]{\frac{-B-\sqrt{\frac{-D}{27}}}{2}}\), which gives, by replacing \(x\), \(B\) et \(D\) by their respective values:
\(\bar{r_0}=2+\sqrt[3]{2u-1+2\sqrt{u(u-1)}}+\sqrt[3]{2u-1-2\sqrt{u(u-1)}}\).
Note: for a given value of \(|\bar{a}|\), \(\bar{r_0}\) is the maximum dimensionless radial coordinate \(\bar{r_c}\) of the photon orbits. An increasing function of \(u\), it has \(4\) as its minimum when \(u=1\) or \(|\bar{a}|=1\).

Polar orbits

(4.e) has the discriminant \(D=4u(-u^2-18u+27\)) with \(u=\bar{a}^2\), which leads us to differentiate 3 cases:
– \(u\in ]1,-9+6\sqrt{3}[\ \Rightarrow D>0\) and (4.e) has the 2 positive real solutions seen previously in paragraph 4.3.2:
\(\bar{r_0}=1+2\sqrt{1-\frac{u}{3}}\cos\left({1\over 3}\arccos\left(\frac{1-u}{\left(1-\frac{u}{3}\right)^\frac{3}{2}}\right)\right)\)
\(\bar{r_2}=1+2\sqrt{1-\frac{u}{3}}\cos\left({1\over 3}\arccos\left(\frac{1-u}{\left(1-\frac{u}{3}\right)^\frac{3}{2}}\right)+\frac{4\pi}{3}\right)\)
with \(1<\bar{r_2}<\sqrt{3}<\bar{r_0}<1+\sqrt{2}\).
– \(u=-9+6\sqrt{3}\ \Rightarrow D=0\) and (4.e) has 3 real solutions, one of which is a double: \(x_0=\frac{3B}{A}\) and \(x_1=x_2=-\frac{3B}{2A}\) which, replacing \(x\), \(A\) and \(B\) by their respective values leads to: \(\bar{r_0}=3-2\sqrt{3}\) which cannot be considered because \(<0\), and \(\bar{r_1}=\bar{r_2}=\sqrt{3}\).
– \(u>-9+6\sqrt{3}\ \Rightarrow D<0\) and (4.e) has only one real solution \(x=\sqrt[3]{\frac{-B+\sqrt{\frac{-D}{27}}}{2}}+\sqrt[3]{\frac{-B-\sqrt{\frac{-D}{27}}}{2}}\), which, replacing \(x\) and \(B\) by their respective values leads to:
\(\bar{r_0}=1+\sqrt[3]{1-u+\frac{1}{6}\sqrt{\frac{-D}{3}}}+\sqrt[3]{1-u-\frac{1}{6}\sqrt{\frac{-D}{3}}}\) which cannot be considered because \(<0\), whatever the value \(u>-9+6\sqrt{3}\) or \(|\bar{a}|>\sqrt{-9+6\sqrt{3}}\),
which shows that in this case, there is no longer polar photon orbit.

Limit inclinations

In addition to the inclination angle \(i_{stab}\) seen in paragraph 4.8 corresponding to the limit of radial stability \(\bar{r_{c_{stab}}}=1-(1-\bar{a}^2)^{1/3}\), there are 2 other angles \(i_{lim1}\) and \(i_{lim2}\) defined by \(\sin^2i=\frac{1}{\bar{a}^2}\).
These 2 angles correspond to an infinite limit of \(c^2\frac{l^2}{\varepsilon^2}\) for \(\bar{r_c}=1\) (see equation (4.o)) and they border \(i_{stab}\):
\(0\le i_{lim1}<i_{stab}<i_{lim2}=\pi-i_{lim1}\le\pi\).

Description of orbits

In this paragraph, \(\bar{r_0}\) is the constant dimensionless radial coordinate of the equatorial orbit.

\(\bar{a}>0\) (trigonometric spin):
– for an inclination angle \(i\) varying from \(0\) to \(i_{lim1}\), there exists a photon orbit with \(\bar{r_c}\) which increases from \(0\) to \(1\),
– there is no photon orbit with an inclination angle \(i\in[i_{lim1}, i_{stab}]\),
– for the same inclination angle \(i\) varying from \(i_{stab}\) to \(i_{lim2}\), there are 2 groups of photon orbits:
a group with \(\bar{r_c}\) which decreases from \(\bar{r_{c_{stab}}}\) to \(1\),
a group with \(\bar{r_c}\) which increases from \(\bar{r_{c_{stab}}}\) to \(\bar{r_{c_{lim2}}}\),
– finally, for inclination angle \(i\) varying from \(i_{lim2}\) to \(\pi\), there is a photon orbit with \(\bar{r_c}\) which increases from \(\bar{r_{c_{lim2}}}\) to \(\bar{r_0}\).

\(\bar{a}<0\) (clockwise spin):
– for an inclination angle \(i\) varying from \(0\) to \(i_{lim1}\), there exists a photon orbit with \(\bar{r_c}\) which decreases from \(\bar{r_0}\) to \(\bar{r_{c_{lim1}}}\),
– for the same inclination angle \(i\) varying from \(i_{lim1}\) to \(i_{stab}\), there are 2 groups of photon orbits:
a group with \(\bar{r_c}\) which decreases from \(\bar{r_{c_{lim1}}}\) to \(\bar{r_{c_{stab}}}\),
a group with \(\bar{r_c}\) which increases from \(1\) to \(\bar{r_{c_{stab}}}\),
– there is no photon orbit with an inclination angle \(i\in[i_{stab}, i_{lim2}]\),
– finally, for an inclination angle \(i\) varying from \(i_{lim2}\) to \(\pi\), there exists a photon orbit with \(\bar{r_c}\) which decreases from \(1\) to \(0\).

Examples of photon « spheres »

The plots below are arbitrarily made with initial conditions \(\theta_0=\frac{\pi}{2}\) and \((\frac{d\theta}{d\lambda})_0>0\) and the number of oscillations is arbitrary unless otherwise stated.

Examples of outer and inner polar orbits for the same Kerr’s parameter

« Last » polar orbit

Example of 2 orbits with the same inclination angle and the same Kerr’s parameter

1. https://luth.obspm.fr/~luthier/gourgoulhon/fr/master/relatM2.pdf ↩︎
2. https://luth.obspm.fr/~luthier/gourgoulhon/fr/master/relatM2.pdf ↩︎
3. https://www.roma1.infn.it/teongrav/onde19_20/geodetiche_Kerr.pdf ↩︎
4. https://www.roma1.infn.it/teongrav/onde19_20/geodetiche_Kerr.pdf ↩︎
5. https://luth.obspm.fr/~luthier/gourgoulhon/fr/master/relatM2.pdf ↩︎
6. https://arxiv.org/abs/1210.2486 ↩︎
7. https://arxiv.org/pdf/2009.07012.pdf ↩︎
8. https://arxiv.org/pdf/2009.07012.pdf ↩︎
9. https://arxiv.org/abs/1210.2486 ↩︎
10. https://arxiv.org/pdf/2009.07012.pdf ↩︎
11. https://arxiv.org/pdf/2009.07012.pdf ↩︎
12. https://arxiv.org/pdf/2105.07101 ↩︎
13. https://arxiv.org/pdf/2105.07101 ↩︎
14. https://arxiv.org/pdf/2105.07101 ↩︎
15. https://arxiv.org/pdf/2105.07101 ↩︎