GRAVITATION

DEFINITION

The calculation of light deflection by Kerr black holes can use the Kerr metric tensor matrix expressed in the Boyer-Lindquist’s coordinates system (\(ct, r, \theta, \varphi\)):
\((g_{\mu\nu})=\pmatrix{-1+\frac{2mr}{\Sigma}&0&0&-\frac{2mar\sin^2\theta}{\Sigma}\\0&\frac{\Sigma}{\Delta}&0&0\\0&0&\Sigma&0\\-\frac{2mar\sin^2\theta}{\Sigma}&0&0&\left(r^2+a^2+\frac{2ma^2r\sin^2\theta}{\Sigma}\right)\sin^2\theta}\)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 spacetime is therefore stationary and axially symmetrical.
The coordinate system is undefined at the poles, since \(g_{\varphi\varphi}\) is null for \(\theta=0\) or \(\theta=\pi\). In addition, coordinates are invalid when \(\Delta=0\) where \(g_{rr}\) diverges (coordinate singularity) or when \(\Sigma=0\) where \(g_{00}\), \(g_{0\varphi}\), \(g_{\varphi 0}\) and \(g_{\varphi\varphi}\) diverge (ring or central singularity if \(a=0\)).
Since the null geodesics are light-like their length is zero² and the scalar product of an elementary motion \(\overrightarrow{ds}\) of a photon in Kerr 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, with \(\frac{\delta S}{\delta\lambda}=0\) because photon geodesics are light-like.
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 spinning 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 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 spacetime because photon geodesics are light-like.
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 \(Q\), and this 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).
Note: the constant \(Q\) is referred to as Carter constant in the following, as it is related to Carter original constant \(K\) by the relation \(Q=K+(l_z-a\varepsilon)^2\).

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 differential equations of 1st order of motion of the photon that enable the calculation of the null geodesics in Kerr 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), and
\(\frac{dct}{d\lambda}=\left((r^2+a^2)^2-\Delta a^2\sin^2\theta-2mar\ c\frac{l_z}{\varepsilon}\right)\frac{\varepsilon}{c\Delta\Sigma}\hspace{2cm}\)(2.r).
Note: axial symmetry is reflected in the 4 equations above which 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 \(-a\), \(-l_z\) are symmetrical with respect to the spinning \(z\)-axis of the black hole (opposite \(\varphi-\varphi_{initial}\) values).
On the other hand, (2.q) shows that in Kerr spacetime, a given photon trajectory cannot be followed in the other direction except in the special case \(a=0\) (Schwarzschild’s spacetime) which is the only solution for \(\frac{d\varphi_2}{d\lambda}=-\frac{d\varphi_1}{d\lambda}\) with \(l_{z_2}=-l_{z_1}\).
Finally, (2.r) has as denominator the term \(\Delta\), which cancels out for the event and Cauchy horizons, showing that these hypersurfaces are light-like, and that, an observer arriving to the event horizon would perceive outside radiations with an infinite red shift (time infinite dilatation), and arriving to the Cauchy horizon, would perceive them with an infinite blue shift (time infinite compression).
In the following, \(r_s=R_s=2m\), \(a>0\) corresponds to a trigonometric spin of the black hole and \(a<0\) to a clockwise spin, the dimensionless variables \(\bar{a}=\frac{a}{m}\) and \(\bar{r}=\frac{r}{m}\) are used, and \(|a|\) lies between \(0\) and \(m\), limits included, excluding the over extreme Kerr spacetime described before the conclusion.

PHOTON TRAJECTORIES

The calculation of light deflection by Kerr 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 spinning axis of the black hole and \(Q\) Carter constant.

Cartesian expression of the trajectory

The null geodesics in Kerr 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 \(O_z\) its spinning 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):
\(\frac{d\varphi}{dt}=\frac{2marc+(\Sigma-2mr)c^2\frac{l_z}{\varepsilon\sin^2\theta}}{(r^2+a^2)^2-\Delta a^2\sin^2\theta-2marc\frac{l_z}{\varepsilon}}\),
which causes the photon to be « dragged along » by the spinning black hole.
Replacing \(\Delta\) and \(\Sigma\) by their respective value and after developing the denominator, we get:
\(\frac{d\varphi}{dt}=\frac{2marc+(r^2+a^2\cos^2\theta-2mr)c^2\frac{l_z}{\varepsilon\sin^2\theta}}{(r^2+a^2)(r^2+a^2\cos^2\theta)+2mar\left(a\sin^2\theta-c\frac{l_z}{\varepsilon}\right)}\hspace{2cm}\)(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): \(a\gt 0\Rightarrow\) \(\frac{d\varphi}{dt}\gt 0\) or \(a\lt 0\Rightarrow\) \(\frac{d\varphi}{dt}\lt 0\). Thus, the outer ergosphere hypersurface is a stationarity limit.

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 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 \(\bar{\Delta}=\bar{r}^2-2\bar{r}+\bar{a}^2\) that:
– if \(\bar{\Delta}>0\) (photon outside the event horizon or inside the Cauchy’s horizon of the black hole): \(\frac{c^2Q}{m^2\varepsilon^2}\le\frac{\left(\bar{r}^2+\bar{a}^2-\bar{a}\frac{cl_z}{m\varepsilon}\right)^2}{\bar{\Delta}}-(\bar{a}-\frac{cl_z}{m\varepsilon})^2\),
– if \(\bar{\Delta}<0\) (photon between the event horizon and the Cauchy’s horizon of the black hole): \(\frac{c^2Q}{m^2\varepsilon^2}\ge\frac{\left(\bar{r}^2+\bar{a}^2-\bar{a}\frac{cl_z}{m\varepsilon}\right)^2}{\bar{\Delta}}-(\bar{a}-\frac{cl_z}{m\varepsilon})^2\).
There are therefore limit values for \(\frac{c^2Q}{m^2\varepsilon^2}\) which depend on the characteristics \(m\) and \(a\), 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 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 characteristics \(m\) and \(a\), the colatitude \(\theta\) of the photon and \(\frac{cl_z}{m\varepsilon}\).
Note that \(\frac{c^2l_z^2}{m^2\varepsilon^2}\frac{1}{\tan^2\theta}\) 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 constant \(Q\) in the case of Kerr black holes.

Extreme values of \(l_z\) component of angular momentum on the spinning 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 \(\bar{\Delta}=\bar{r}^2-2\bar{r}+\bar{a}^2\):
\(\bar{r}^4+\left(\bar{a}^2-\frac{c^2l_z^2}{m^2\varepsilon^2}\right)\bar{r}^2+2\left(\bar{a}-\frac{cl_z}{m\varepsilon}\right)^2-\bar{\Delta}\frac{c^2Q}{m^2\varepsilon^2}\ge 0\), 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 regrouping:
\(D’=\left(\bar{r}^4+\bar{r}(2-\bar{r})\frac{c^2Q}{m^2\varepsilon^2}\right)\bar{\Delta}\).
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 characteristics \(m\) 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 \(\bar{a}>0\), \(\frac{cl_z}{m\varepsilon}\le\frac{\bar{r}^4+\bar{a}^2\bar{r}^2+2\bar{a}^2\bar{r}-\bar{\Delta}\frac{c^2Q}{m^2\varepsilon^2}}{4\bar{a}\bar{r}}=\frac{2}{\bar{a}}+\bar{a}\left(1-\frac{c^2Q}{8m^2\varepsilon^2}\right)\),
– if \(\bar{a}<0\), \(\frac{cl_z}{m\varepsilon}\ge\frac{\bar{r}^4+\bar{a}^2\bar{r}^2+2\bar{a}^2\bar{r}-\bar{\Delta}\frac{c^2Q}{m^2\varepsilon^2}}{4\bar{a}\bar{r}}=\frac{2}{\bar{a}}+\bar{a}\left(1-\frac{c^2Q}{8m^2\varepsilon^2}\right)\).

Limit according to \(\theta\)

The expression (2.p) must remain positive or zero, which results in:
\(\frac{c^2l_z^2}{m^2\varepsilon^2}\le\sin^2\theta\left(\bar{a}^2+\frac{c^2Q}{m^2\varepsilon^2}\frac{1}{\cos^2\theta}\right)\).
There is therefore a maximum value for \(\frac{c^2l_z^2}{m^2\varepsilon^2}\) which depend on the characteristics \(m\) and \(a\), of the colatitude \(\theta\) of the photon and \(\frac{c^2Q}{m^2\varepsilon^2}\).

PHOTON ORBITS

Null geodesics in Kerr spacetime may have a constant radial coordinate \(r\) generating a photon orbit.

Constant radial coordinate

The constant radial coordinate value \(r_c\) is obtained by cancelling out the potential \(V_r\) (2.i) and its derivative \(\frac{dV_r}{dr}\).
These conditions lead, after calculation (see paragraph 4.6) 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 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 \(l=\sqrt{l_z^2+Q}\): \(\frac{cl}{m\varepsilon}=\sqrt{\frac{2\bar{r}_c^4+(\bar{a}^2-6)\bar{r}_c^2+2\bar{a}^2\bar{r}_c+\bar{a}^2}{(\bar{r}_c-1)^2}}\),
– 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, with one or three real solutions depending on the values of \(\bar{a}\) and \(i\) (see paragraph 4.8).
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.
– finally, 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:
\(p(\bar{r})=\bar{r}^6-6\bar{r}^5+(9+2\bar{a}^2\sin^2⁡i)\bar{r}^4-4\bar{a}^2\bar{r}^3-\bar{a}^2(6-\bar{a}^2)\bar{r}^2\sin^2⁡i+2\bar{a}^4\bar{r}\sin^2⁡i+\bar{a}^4\sin^2⁡i\).
\(p(\bar{r}_c)=0\hspace{2cm}\)(4.b) is a characteristic equation which gives for the same value of \(\sin^2i\) 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 \(i=\pi\)), the polar orbits (\(i=\frac{\pi}{2}\)) or the orbits around an extreme Kerr black hole (see paragraph 4.6).
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 symmetry according to \(\varphi\) 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 value \(l^2\sin^2i\) 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 spinning 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, \(\sin^2i=0\) in (4.b), we get with \(u=\bar{a}^2\):
\(p(\bar{r})=\bar{r}^3(\bar{r}^3-6\bar{r}^2+9\bar{r}-4u)=0\) or
the trivial solution \(\bar{r}=0\) (central or ring singularity) and \(\bar{r}^3-6\bar{r}^2+9\bar{r}-4u=0\)
(or \(Q=l^2\sin^2i=0\)).
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 has 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 \(\bar{r}_2=\bar{r}_0=3\) (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 has 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 value 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 \(0\lt\bar{r}_1\le 1\le\bar{r}_2\le 3\le\bar{r}_0\le 4\), and
with \(\bar{r}_1\) and \(\bar{r}_2\) prograde orbits (\(i=0\)), and \(\bar{r}_0\) retrograde orbit (\(i=\pi\)).
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 spinning axis of the black hole (\(i=\frac{\pi}{2}\)), which results in a trajectory that passes through the 2 poles.
Note that they must satisfy the condition \(V_\theta\ge 0\) whatever the value of \(\theta\in]0,\pi[\), that is, \(Q\ge 0\).
Applying \(i=\frac{\pi}{2}\) that is, \(\sin^2i=1\) in (4.b), we get with \(u=\bar{a}^2\):
\(p(\bar{r})=(\bar{r}^3-3\bar{r}^2+u\bar{r}+u)^2=0\) (or\(l_z=l\cos i=0\)).
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 has 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 \(u\) is between these 2 roots, which indicates that the polynomial is positive for \(u\in ]0,1]\Rightarrow D>0\) and (4.d) therefore has 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 value gives:
\(\bar{r}_0=1+2\sqrt{1-\frac{u}{3}}\cos\left({1\over 3}\arccos\left(\frac{1-u}{\left(1-\frac{u}{3}\right)^\frac{3}{2}}\right)\right)\), which satisfies the condition \(Q\ge 0\),
\(\bar{r}_1=1+2\sqrt{1-\frac{u}{3}}\cos\left({1\over 3}\arccos\left(\frac{1-u}{\left(1-\frac{u}{3}\right)^\frac{3}{2}}\right)+\frac{2\pi}{3}\right)<0\Rightarrow\) unacceptable solution since the radial coordinate of the photon is positive or zero, and
\(\bar{r}_2=1+2\sqrt{1-\frac{u}{3}}\cos\left({1\over 3}\arccos\left(\frac{1-u}{\left(1-\frac{u}{3}\right)^\frac{3}{2}}\right)+\frac{4\pi}{3}\right)\), which cannot be kept because it leads to a negative value of \(Q\),
and 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.

Orbits with vertical motion at the equator

When \(\theta\) is \(\frac{\pi}{2}\), the numerator of \(\frac{d\varphi}{d\lambda}\) cancels out if \(\frac{b_{crit}}{m}\cos i\), that is, \(\frac{cl_z}{m\varepsilon}=\frac{2a}{2m-r_c}=\frac{2\bar{a}}{2-\bar{r}_c}\) which means that the trajectory of the photon crosses the equatorial plane vertically.
By replacing \(\frac{cl_z}{m\varepsilon}\) with the value given above \(\frac{cl_z}{m\varepsilon}=-\frac{(\bar{r}_c^3-3\bar{r}_c^2+\bar{a}^2\bar{r}_c+\bar{a}^2)}{\bar{a}(\bar{r}_c-1)}\) and noting the factorisation by \(\bar{r}_c-3\), we have after regrouping and simplification \(\frac{d\varphi}{d\lambda}_{\theta=\frac{\pi}{2}}=\frac{3-\bar{r}_c}{\bar{r}_c-1}\frac{1}{a}\frac{\varepsilon}{c}\).
\(\frac{d\varphi}{d\lambda}_{\theta=\frac{\pi}{2}}\) therefore cancels out for \(\bar{r}_c=3\) which gives \(\frac{cl_z}{m\varepsilon}=-2\bar{a}\) and as seen above \(\frac{c^2Q}{m^2\varepsilon^2}=-\frac{\bar{r}_c^3(\bar{r}_c^3-6\bar{r}_c^2+9\bar{r}_c-4\bar{a}^2)}{\bar{a}^2(\bar{r}_c-1)^2}=27\frac{4\bar{a}^2}{4\bar{a}^2}=27\), and this is whatever the value of the Kerr parameter \(\bar{a}\).
The Lense-Thirring effect described in paragraph 3.2 precisely cancels out at the equator the retrograde motion of the photon.
The specific case of the Schwarzschild’s black hole is found with \(\bar{a}=0\) that is, \(\frac{b_{crit}}{m}=\frac{cl}{m\varepsilon}=\sqrt{\frac{c^2l_z^2}{m^2\varepsilon^2}+\frac{c^2Q}{m^2\varepsilon^2}}=\sqrt{27}\) and \(r_c=3m=\frac{3}{2}r_s\) with \(r_s=\frac{2GM}{c^2}\).

Critical impact parameter

As seen in paragraph 4.1, 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 spacetime can be calculated by the following formulae:
\(\frac{b_{crit}}{m}=\bar{b}_{crit}=\sqrt{\frac{3\bar{r}_c^4+\bar{a}^2\bar{r}_c^2}{\bar{r}_c^2-\bar{a}^2\sin^2i}}\), with \(\bar{r}_c^2>\bar{a}^2\sin^2i\), see calculation in paragraph 4.6
or
\(\frac{b_{crit}}{m}=\bar{b}_{crit}m\sqrt{\frac{2\bar{r}_c^4+(\bar{a}^2-6)\bar{r}_c^2+2\bar{a}^2\bar{r}_c+\bar{a}^2}{(\bar{r}_c-1)^2}}\) for \(\bar{r}_c\ne 1\) and using the expression for \(\frac{cl}{m\varepsilon}\) given in paragraph 4.1.

Colatitude limit

According to (4.c), \(\frac{d\theta}{d\lambda}\) is null 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:
\(-\bar{a}^2\cos^4\theta-(\bar{b}_{crit}^2-\bar{a}^2)\cos^2\theta+\bar{b}_{crit}^2\sin^2i=0\hspace{2cm}\)(4.f).
The left-hand side of (4.f) is a 2nd-degree polynomial in \(\cos^2\theta\) which has a positive root \(\cos^2\theta_{lim}=\frac{\bar{a}^2-\bar{b}_{crit}^2+\sqrt{\left(\bar{a}^2-\bar{b}_{crit}^2\right)^2+4\bar{a}^2\bar{b}_{crit}^2\sin^2i}}{2\bar{a}^2}\) 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 \(\cos\theta_{lim}=\sqrt{\frac{\bar{a}^2-\bar{b}_{crit}^2+\sqrt{\left(\bar{a}^2-\bar{b}_{crit}^2\right)^2+4\bar{a}^2\bar{b}_{crit}^2\sin^2i}}{2\bar{a}^2}}\hspace{2cm}\)(4.g).
In the case of polar orbits, \(i=\frac{\pi}{2}\) and (4.g) becomes:
\(\cos\theta_{lim}=\sqrt{\frac{\bar{a}^2-\bar{b}_{crit}^2+\sqrt{(\bar{a}^2+\bar{b}_{crit}^2)^2}}{2\bar{a}^2}}=1\) which confirms that the colatitude of a polar orbit is defined on \(]0,\pi[\).
After replacing \(\bar{b}_{crit}^2\) and \(\bar{b}_{crit}^2\sin^2i\) by their respective value \(\frac{c^2l^2}{m^2\varepsilon^2}\) and \(\frac{c^2Q}{m^2\varepsilon^2}\) seen above, (4.g) can be expressed as a function of \(\bar{r}_c\) and \(\bar{a}\):
\(\cos\theta_{lim}=\sqrt{\frac{\bar{r}_c(-\bar{r}_c^3+3\bar{r}_c-2\bar{a}^2+2\sqrt{2\bar{r}_c^5-7\bar{r}_c^4+6\bar{r}_c^3+2\bar{a}^2\bar{r}_c(\bar{r}_c^2-\bar{r}_c- 1)+\bar{a}^4})}{\bar{a}^2(\bar{r}_c-1)^2}}\).
For an extreme Kerr black hole, the function simplifies to
\(\cos\theta_{lim}=\sqrt{-\bar{r}_c^2+2\bar{r}_c(\sqrt{2\bar{r}_c+1}-1)}\).

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
\((r^2+a^2)\frac{\varepsilon}{c}-al_z-\frac{2r}{r-m}\frac{\varepsilon}{c}(r^2-2mr+a^2)=0\) 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).
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 \(c^2\frac{Q}{\varepsilon}^2=-\frac{r^4}{a^2}\) cannot be considered.
\(l=\sqrt{l_z^2+Q}\) and (4.j) and (4.k) then give after regrouping:
\(c\frac{l}{\varepsilon}=\sqrt{\frac{2r^4+(a^2-6m^2)r^2+2ma^2r+m^2a^2}{(r-m)^2}}\) or \(\frac{cl}{m\varepsilon}=\sqrt{\frac{2\bar{r}_c^4+(\bar{a}^2-6)\bar{r}_c^2+2\bar{a}^2\bar{r}_c+\bar{a}^2}{(\bar{r}_c-1)^2}}\hspace{2cm}\)(4.l).
For an extreme Kerr black hole, (4.j), (4.k) and (4.l) are simplified and can be written as follows:
\(\frac{cl_z}{m\varepsilon}=\bar{a}\left(2-(\bar{r}_c-1)^2\right)\), \(\frac{c^2Q}{m^2\varepsilon^2}=\bar{r}_c^3(4-\bar{r}_c)\) and \(\frac{cl}{m\varepsilon}=\sqrt{2(\bar{r}_c+1)^2-1}\).
For \(\bar{r}_c=1\) which only exists for an extreme Kerr black hole, the conditions \(V_r=0\) and \(\frac{dV_r}{dr}=0\) lead to \(\lim_{\bar{r}_c\to 1}\frac{cl_z}{m\varepsilon}=2\bar{a}\) whatever the value of \(Q\) which can be negative, zero or positive.

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 criteria for 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 value (4.j) and (4.k), it comes after developing for an orbit of radial coordinate \(\bar{r}_c\):
\(\frac{c^2}{\varepsilon^2m^4}\frac{d^2V_r}{dr^2}=\frac{8\bar{r}_c}{(\bar{r}_c-1)^2}(\bar{r}_c^3-3\bar{r}_c^2+3\bar{r}_c-\bar{a}^2)=\frac{8\bar{r}_c}{(\bar{r}_c-1)^2}\left(\left(\bar{r}_c-1\right)^3+1-\bar{a}^2\right)\) which has only one real root \(\bar{r}_{c_{stab}}=1-(1-\bar{a}^2)^{1/3}\).
This value \(\bar{r}_{c_{stab}}\) is less or equal to \(\bar{r}_{Cauchy}\) 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 unstable in radial perturbation,
– for \(\bar{r}_c<\bar{r}_{c_{stab}}\), \(\frac{d^2V_r}{dr^2}\) is \(<0\) and the orbit is stable in radial perturbation.
In the latter case, any variation of the radial coordinate \(r\) around \(r_c\) results in a negative value of \(V_r\), which shows that a photon cannot join this orbit. To follow it, it must be emitted at coordinates \(r_{em}=r_c\) with the corresponding \(l_z\) and \(Q\) values, and \(\theta_{em}\in [\theta_{lim},\pi-\theta_{lim}]\).

Inclinations and stabilities

The value \(\bar{r}_{c_{stab}}\) seen above corresponds to an 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 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 very close to the event and Cauchy’s horizons \(\bar{r}_c\simeq 1\), the respective values of \(\frac{cl_z}{m\varepsilon}\) and \(\frac{cl}{m\varepsilon}\) are \(2\bar{a}\) and \(\sqrt{7}\), which leads to an inclination angle \(i_{stab}=\arccos\frac{2\bar{a}}{\sqrt{7}}\) or \(\simeq 40.9^\circ\) if \(\bar{a}=1\) and \(\simeq 139.1^\circ\) if \(\bar{a}=-1\).
For \(i<i_{stab}\) if \(\bar{a}=1\) or \(>i_{stab}\) if \(\bar{a}=-1\), photon orbits are stable and inside the Cauchy’s horizon, that is, with a value \(\bar{r}_c\in [0,1[\). Otherwise, they are unstable and outside the event horizon, with \(\bar{r}_c\in ]1,4]\).

Other Kerr 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 is greater than \(r_h\) that is, a photon orbit outside the event horizon and unstable,
– one is between \(r_{stab}\) and \(r_{Cauchy}\), that is, a photon orbit inside the Cauchy’s horizon and unstable (unlike the other orbits, this orbit has an angle \(\theta_{lim}\) increasing with \(r_c\) if \(\bar{a}>0\), or decreasing if \(\bar{a}<0\)),
– one is less than \(r_{stab}\), that is, a stable photon orbit inside the Cauchy’s horizon.
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 orbit is outside the event horizon and is unstable.

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 \((r_0, \theta_0, \varphi_0)\) 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 on the radial coordinate doesn’t help since \(r\) remains 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 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 near-capture

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

Extreme Kerr 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 with an arbitrary number of oscillations unless otherwise stated.

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

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

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

Two first oscillations

Outlines and boundaries of photon orbits around Kerr black holes

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

Using the formulae for dimensionless radial coordinates of equatorial orbits discussed in paragraph 4.3.1, two cases can be differentiated for the position of photon orbits with respect to the regions of a Kerr 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, and
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, and
2) orbit entirely between the Cauchy’s horizon and the inner ergosphere.
Note: whatever the value of \(|\bar{a}|\), there is no photon orbit crossing the event horizon or the Cauchy’s horizon, or located between the 2 horizons.

APPARENT IMAGE OF A KERR BLACK HOLE (SHADOW)

The calculation shows that the apparent image of a Kerr black hole is significantly larger than its event horizon, and it is to an outside observer a “shadow” without any star image.

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 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 \(y(\bar{r}_c)=-2\tan(\frac{\theta_{obs}}{2})\cos\varphi_{obs}\)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 SPACETIME

The Kerr spacetime is said to be « over extreme » when \(|\bar{a}|\), the absolute value of the Kerr parameter, is greater than \(1\). Since the physical existence of an over extreme Kerr spacetime is currently considered unlikely, this paragraph is a mathematical description.

Over extreme Kerr object

As an over extreme Kerr object has no event horizon, it does not belong to the black hole category and is directly observable (no shadow).
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 parameter is obtained for a spinning that drives a point of the object at the speed of light in vacuum.
Formally, therefore, there is no mathematical upper limit on \(|a|\) if the over extreme Kerr 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 near-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 object.

Photon orbits

Equatorial orbit

(4.d) has the discriminant \(D=432u(1-u)\) 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 \(x\), \(B\) and \(D\) by their respective value:
\(\bar{r}_0=2+\sqrt[3]{2u-1+2\sqrt{u(u-1)}}+\sqrt[3]{2u-1-2\sqrt{u(u-1)}}\).
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 \(|\bar{a}|=1\) and corresponds to a retrograde orbit (\(i=\pi\)).

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 and which with \(|\bar{a}|>1\) satisfy the condition \(Q\ge 0\) or \(V_\theta\ge 0\) whatever the value of \(\theta\in]0,\pi[\):
\(\bar{r}_0=1+2\sqrt{1-\frac{u}{3}}\cos\left({1\over 3}\arccos\left(\frac{1-u}{\left(1-\frac{u}{3}\right)^\frac{3}{2}}\right)\right)\), and
\(\bar{r}_2=1+2\sqrt{1-\frac{u}{3}}\cos\left({1\over 3}\arccos\left(\frac{1-u}{\left(1-\frac{u}{3}\right)^\frac{3}{2}}\right)+\frac{4\pi}{3}\right)\),
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 double: \(x_1=\frac{3B}{A}\) and \(x_2=x_0=-\frac{3B}{2A}\) which, replacing \(x\), \(A\) and \(B\) by their respective value leads to: \(\bar{r}_1=3-2\sqrt{3}\) which cannot be considered because negative, and \(\bar{r}_2=\bar{r}_0=\sqrt{3}\).
– \(u>-9+6\sqrt{3}\ \Rightarrow D<0\) and (4.e) has only one real solution \(x_1=\sqrt[3]{\frac{-B+\sqrt{\frac{-D}{27}}}{2}}+\sqrt[3]{\frac{-B-\sqrt{\frac{-D}{27}}}{2}}\), which, replacing \(x\) and \(B\) by their respective value leads to:
\(\bar{r}_1=1+\sqrt[3]{1-u+\frac{1}{6}\sqrt{\frac{-D}{3}}}+\sqrt[3]{1-u-\frac{1}{6}\sqrt{\frac{-D}{3}}}\) which cannot be considered because negative 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 \(\bar{r}_{c_{stab}}=1-(1-\bar{a}^2)^{1/3}\) greater than 1, there are 2 other angles \(i_{lim1}\) and \(i_{lim2}\) defined by \(\sin^2i_{lim}=\frac{1}{\bar{a}^2}\).
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.

\(\bar{a}>1\) (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 \(\bar{r}_0\).

\(\bar{a}<-1\) (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 increases from \(1\) to \(\bar{r}_{c_{stab}}\),
a group with \(\bar{r}_c\) which decreases from \(\bar{r}_{c_{lim1}}\) 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 with an arbitrary number of oscillations unless otherwise stated.

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

« Last » polar orbit

Example of 2 orbits with the same inclination angle and the same Kerr 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 ↩︎