GRAVITATION

DEFINITION

The calculation of light deflection by Kerr black holes can use the Kerr metric tensor matrix expressed in the Boyer-Lindquist coordinate 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, \(a=\frac{J}{cM}\) (\(\gt 0\) for a trigonometric spin, \(\lt 0\) for a clockwise spin) with \(J\) angular momentum of spin of the black hole, \(\Delta=r^2-2mr+a^2\) and \(Σ=r^2+a^2\cos^2{\theta}\).
The coefficients of \((g_{\mu\nu})\) are independent of \(t\) and \(\varphi\): the geometry of Kerr spacetime is therefore stationary and axially symmetrical.
Since the null geodesics are of the light-type, their length is zero² and the scalar product of an elementary motion of a photon in Kerr spacetime is therefore written
\(g_{\mu\nu}dx^\mu dx^\nu=ds^2=(1-\frac{2mr}{\Sigma})c^2dt^2-\frac{4mar\sin^2{\theta}}{\Sigma}cdtd\varphi+\frac{\Sigma}{\Delta}dr^2+\Sigma d\theta^2+(r^2+a^2+\frac{2ma^2r\sin^2{\theta}}{\Sigma})\sin^2{\theta}d\varphi^2=0\)

SETTING THE PARAMETRIC EQUATIONS

Following the Hamilton-Jacobi 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 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 energy \(\varepsilon\) and the norm of angular momentum \(\overrightarrow{l}\) all along the motion of the photon gives \(p_0=-\frac{\varepsilon}{c}\) and \(p_\varphi=l_z\), component of angular momentum \(\overrightarrow{l}\) on the spin axis of the black hole, leading to a function such as:
\(S=-\frac{\varepsilon}{c} ct+S^{(r)}(r)+S^{(\theta)}(\theta)+l_z\varphi\hspace{2cm}\)(2.a) looking for a separable solution in \(r\) and \(\theta\).⁴
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 \((-\frac{\varepsilon}{c},\frac{dS(r)}{dr},\frac{dS(\theta)}{d\theta},l_z)\), or:
\(H=\frac{1}{2}\left [\left (-\frac{(r^2+a^2)^2}{\Sigma\Delta}+\frac{a^2\sin^2\theta}{\Sigma}\right )\frac{\varepsilon^2}{c^2}+\frac{4mar}{\Sigma\Delta}\frac{\varepsilon}{c}l_z+\frac{\Delta}{\Sigma}(\frac{dS^{(r)}}{dr})^2+\frac{1}{\Sigma}(\frac{dS^{(\theta)}}{d\theta})^2+\left (\frac{1}{\Sigma\sin^2\theta}-\frac{a^2}{\Sigma\Delta}\right )l_z^2\right ]=0\hspace{2cm}\)(2.b)
in Kerr spacetime because the null geodesics are light-type.
Furthermore \((\frac{dx}{d\lambda})^\mu=\frac{\delta H}{\delta p_\mu}\hspace{2cm}\)(2.c)

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

After multiplying by \(2\Sigma\), equation (2.b) can be written as:
\(-\Delta (\frac{dS^{(r)}}{dr})^2+\frac{(r^2+a^2)^2}{\Delta}\frac{\varepsilon^2}{c^2}-\frac{4mar}{\Delta}\frac{\varepsilon}{c} l_z+\frac{a^2l_z^2}{\Delta}=(\frac{dS^{(\theta)}}{d\theta})^2+a^2\sin^2\theta\frac{\varepsilon^2}{c^2}+\frac{l_z^2}{\sin^2\theta}\)
and subtracting \(a^2\frac{\varepsilon^2}{c^2}+l_z^2\) from each member:
\(-\Delta (\frac{dS^{(r)}}{dr})^2+\frac{(r^2+a^2)^2}{\Delta}\frac{\varepsilon^2}{c^2}-\frac{4mar}{\Delta}\frac{\varepsilon}{c} l_z+\frac{a^2l_z^2}{\Delta}-a^2\frac{\varepsilon^2}{c^2}-l_z^2=(\frac{dS^{(\theta)}}{d\theta})^2-a^2\cos^2\theta\frac{\varepsilon^2}{c^2}+\frac{\cos^2\theta}{\sin^2\theta}l_z^2\hspace{2cm}\)(2.d)
The left-hand member of equation (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 gives the 2 equations:
\(-\Delta (\frac{dS^{(r)}}{dr})^2+\frac{(r^2+a^2)^2}{\Delta}\frac{\varepsilon^2}{c^2}-\frac{4mar}{\Delta}\frac{\varepsilon}{c} l_z+\frac{a^2l_z^2}{\Delta}-a^2\frac{\varepsilon^2}{c^2}-l_z^2=Q\hspace{2cm}\)(2.e)
\((\frac{dS^{(\theta)}}{d\theta})^2-a^2\cos^2\theta\frac{\varepsilon^2}{c^2}+\frac{\cos^2\theta}{\sin^2\theta}l_z^2=Q\hspace{2cm}\)(2.f)
Noting that \(2a-2a\frac{r^2+a^2}{\Delta}=-\frac{4mar}{\Delta}\), equation (2.e) becomes:
\(-\Delta (\frac{dS^{(r)}}{dr})^2+\frac{\left (\left(r^2+a^2\right )\frac{\varepsilon}{c}-al_z\right )^2}{\Delta}-(a\frac{\varepsilon}{c}-l_z)^2=Q\), which gives the 2 equations:
\(\Delta (\frac{dS^{(r)}}{dr})^2=\frac{\left (\left(r^2+a^2\right )\frac{\varepsilon}{c}-al_z\right )^2}{\Delta}-(a\frac{\varepsilon}{c}-l_z)^2-Q\hspace{2cm}\)(2.g)
\((\frac{dS^{(\theta)}}{d\theta})^2=Q+\cos^2\theta(a^2\frac{\varepsilon^2}{c^2}-\frac{l_z^2}{\sin^2\theta})\hspace{2cm}\)(2.h)
Posing
\(V_r=\left (\left(r^2+a^2\right )\frac{\varepsilon}{c}-al_z\right )^2-\Delta\left (\left (a\frac{\varepsilon}{c}-l_z\right )^2+Q\right )=\Delta^2(\frac{dS^{(r)}}{dr})^2\hspace{2cm}\)(2.i) and
\(V_\theta=Q+\cos^2\theta(a^2\frac{\varepsilon^2}{c^2}-\frac{l_z^2}{\sin^2\theta})=(\frac{dS^{(\theta)}}{d\theta})^2\hspace{2cm}\)(2.j)
equation (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))
\(p_\theta=\frac{\delta S}{\delta\theta}=\pm\sqrt{V_\theta}=\Sigma\frac{d\theta}{d\lambda}\) (by applying (2.c))
With \(i\in [0,\pi]\) the constant angle of inclination of the angular momentum \(\overrightarrow{l}\) with respect to the spin axis of the black hole, posing \(l\) the norm of the angular momentum \(\overrightarrow{l}\), \(l_z=l\cos i\), it can be demonstrated \(Q=l^2\sin^2i\) and equations (2.i) et (2.j) become after development for the null geodesics in Kerr spacetime:
\(V_r=\Sigma^2(\frac{dr}{d\lambda})^2=\frac{\varepsilon^2}{c^2}\left (r^4+\left (a^2-c^2\frac{l^2}{\varepsilon^2}\right )r^2+2m\left (a^2-2ac\frac{l}{\varepsilon}\cos i+c^2\frac{l^2}{\varepsilon^2}\right )r-a^2c^2\frac{l^2}{\varepsilon^2}\sin^2i\right )\hspace{2cm}\)(2.k) and
\(V_\theta=\Sigma^2(\frac{d\theta}{d\lambda})^2=\frac{\varepsilon^2}{c^2}\left (a^2\cos^2\theta+c^2\frac{l^2}{\varepsilon^2}\left (\sin^2i-\frac{\cos^2i}{\tan^2\theta}\right )\right )\hspace{2cm}\)(2.l).
Note: due to the subtraction of \(a^2\frac{\varepsilon^2}{c^2}+l_z^2\) from the above equations, the constant \(Q\) is referred to as the modified Carter’s constant, the link with the Carter’s constant \(K\) being \(Q=K-\frac{a^2\varepsilon^2}{c^2}-l_z^2\).

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

According to (2.c), \(\frac{d\varphi}{d\lambda}=\frac{\delta H}{\delta l_z}\) and \(\frac{dct}{d\lambda}=\frac{\delta H}{\delta (-\frac{\varepsilon}{c})}\)
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.k), (2.l), (2.m) and (2.n) lead respectively to the 4 parametric equations of motion of the photon that enable the calculation of the null geodesics in Kerr spacetime:
\((\frac{dr}{d\lambda})^2=\left (r^4+\left (a^2-c^2\frac{l^2}{\varepsilon^2}\right )r^2+2m\left (a^2-2ac\frac{l}{\varepsilon}\cos i+c^2\frac{l^2}{\varepsilon^2}\right )r-a^2c^2\frac{l^2}{\varepsilon^2}\sin^2i\right )\frac{\varepsilon^2}{\Sigma^2 c^2}\hspace{2cm}\)(2.o)
\((\frac{d\theta}{d\lambda})^2=\left (a^2\cos^2\theta+c^2\frac{l^2}{\varepsilon^2}\left (\sin^2i-\frac{\cos^2i}{\tan^2\theta}\right )\right )\frac{\varepsilon^2}{\Sigma^2 c^2}\hspace{2cm}\)(2.p)
\(\frac{d\varphi}{d\lambda}=\left( 2mar+(\Sigma-2mr)c\frac{l}{\varepsilon}\frac{\cos i}{sin^2\theta}\right )\frac{\varepsilon}{\Delta\Sigma c}\hspace{2cm}\)(2.q)
\(\frac{dct}{d\lambda}=\left ((r^2+a^2)^2-\Delta a^2\sin^2\theta-2mar\ c\frac{l}{\varepsilon}\cos i\right )\frac{\varepsilon}{\Delta\Sigma c}\hspace{2cm}\)(2.r)
Note: in the following, it is assumed that \(|a|\) lies between \(0\) and \(m\), limits included.

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 constant parameters \(i\) angle of inclination of angular momentum \(\overrightarrow{l}\) and \(b\) « impact parameter » \(=c\frac{l}{\varepsilon}\), positive or zero.

Typology of trajectories

For a given angle \(i\) and for a photon coming from infinity, there are 3 types of trajectory depending on the value of the trajectory impact parameter \(b\) with respect to a value \(b_{crit}\) (see definition in paragraph 4.3.):
– \(b>b_{crit}\): the photon is deflected by the black hole and continues towards infinity,
– \(b=b_{crit}\): the photon is captured in an orbit with a constant radial coordinate,
– \(b<b_{crit}\): the photon is absorbed by the black hole.

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 \(z\) its spin axis, using Boyer-Lindquist 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 effect

The expression of \(\frac{d\varphi}{dt}\) is obtained from equations (2.q) and (2.r):
\(\frac{d\varphi}{dt}=\frac{2marc+(\Sigma-2mr)c^2\frac{l}{\varepsilon}\frac{\cos i}{sin^2\theta}}{(r^2+a^2)^2-\Delta a^2\sin^2\theta-2mar\ c\frac{l}{\varepsilon}\cos i}\)
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:
\(\frac{d\varphi}{dt}=\frac{2marc+(r^2+a^2\cos^2\theta-2mr)c^2\frac{l}{\varepsilon}\frac{\cos i}{sin^2\theta}}{r^4+a^2r^2+(a^4+a^2r^2)\cos^2\theta+2mar(a\sin^2\theta-c\frac{l}{\varepsilon}\cos i)}\hspace{2cm}\)(3.a)
For trajectories with an angle of inclination \(i=\frac{\pi}{2}\), the terms in \(\cos i\) cancel out, giving:
\(\frac{d\varphi}{dt}=\frac{2marc}{r^4+a^2r^2+(a^4+a^2r^2)\cos^2\theta+2ma^2r\sin^2\theta}\) which is of the sign of \(a\).
This effect is particularly apparent for polar orbits:

Equation (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) when:
\(\bar{a}\gt 0\) and \(i\in [\frac{\pi}{2},\pi]\) \(\Rightarrow\) \(\frac{d\varphi}{dt}\gt 0\),
or \(\bar{a}\lt 0\) and \(i\in [0,\frac{\pi}{2}]\) \(\Rightarrow\) \(\frac{d\varphi}{dt}\lt 0\).

Example of a photon trajectory in the equatorial plane (\(i=\pi\)) 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).

Maximum values of impact parameter \(b\)

Limit according to \(r\)

\((\frac{dr}{d\lambda})^2\) must be positive or zero which means according to (2.o) and with \(b=\frac{cl}{\varepsilon}\) :
\(r^4+(a^2-b^2)r^2+2m(a^2-2ab\cos i+b^2)r-a^2b^2\sin^2i\ge 0\) soit \((-r^2+2mr- a^2sin^2i)b^2-4mar\cos i\ b+r^4+a^2r^2+2ma^2r\ge 0\).
The left-hand side of this inequality is a 2nd-degree polynomial in \(b\) whose reduced discriminant is:
\(D=(2mar\cos i)^2-(-r^2+2mr- a^2sin^2i)(r^4+a^2r^2+2ma^2r)\)
When \(-r^2+2mr- a^2sin^2i\) is negative that is for
\(r\lt m-\sqrt{m^2-a^2sin^2i}\) or \(r\gt m+\sqrt{m^2-a^2sin^2i}\),
\(D\) is positive and the 2nd degree polynomial in \(b\) is positive or zero for \(b\) between the roots \(\frac{2mar\cos i+\sqrt{D}}{-r^2+2mr- a^2sin^2i}\) and \(\frac{2mar\cos i-\sqrt{D}}{-r^2+2mr- a^2sin^2i}\) which means, since \(b\) is positive or zero:
\(0\le b\le b_{maxr}=\frac{2mar\cos i-\sqrt{D}}{-r^2+2mr- a^2sin^2i}\hspace{2cm}\)(3.a).
There is therefore a maximum value of \(b\) depending on the parameters \(m\) and \(a\) of the black hole and on the values of the radial coordinate \(r\) of the photon and the angle of inclination \(i\) of the angular momentum \(\overrightarrow{l}\) of the photon.
When \(-r^2+2mr- a^2sin^2i\) is positive that is, for \(0\lt m-\sqrt{m^2-a^2sin^2i}\lt r \lt\ m+\sqrt{m^2-a^2sin^2i}\lt 2m\), \(D\) is negative and the 2nd degree polynomial in \(b\) remains positive whatever the value of \(b\).

Limit according to \(\theta\)

\((\frac{d\theta}{d\lambda})^2\) must be positive or zero which according to (2.p) and with \(b=\frac{cl}{\varepsilon}\) means:
\(a^2\cos^2\theta+b^2(\sin^2i-\frac{\cos^2i}{\tan^2\theta})\ge 0\)
When \(\sin^2i-\frac{\cos^2i}{\tan^2\theta}\) is negative, the above inequality gives:
\(0\le b\le b_{max\theta}=\frac{|a|\cos\theta}{\sqrt{\frac{\cos^2i}{\tan^2\theta}-sin^2i}}\)
There is therefore a maximum value of \(b\) depending on the parameter \(a\) of the black hole and on the values of the radial coordinate \(r\) of the photon and the angle of inclination \(i\) of the angular momentum \(\overrightarrow{l}\) of the photon.
When \(\sin^2i-\frac{\cos^2i}{\tan^2\theta}\) is positive, \((\frac{d\theta}{d\lambda})^2\) remains positive whatever the value of \(b\).

PHOTON ORBITS

Null geodesics in Kerr 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.k) and its derivative \(\frac{dV_r}{dr}\).
After calculation, these conditions mean with \(\bar{r_c}=\frac{r_c}{m}\) and \(\bar{a}=\frac{a}{m}\) that:
– on the one hand \(\frac{cl_z}{m\varepsilon}=-\frac{(\bar{r_c}^3-3\bar{r_c}^2+\bar{a}^2\bar{r_c}+\bar{a}^2)}{\bar{a}(\bar{r_c}-1)}\), \(\frac{c^2Q}{m^2\varepsilon^2}=-\frac{\bar{r_c}^3(\bar{r_c}^3-6\bar{r_c}^2+9\bar{r_c}-4\bar{a}^2)}{\bar{a}^2(\bar{r_c}-1)^2}\) and \(\frac{c^2l^2}{m^2\varepsilon^2}=\frac{2\bar{r_c}^4+(\bar{a}^2-6)\bar{r_c}^2+2\bar{a}^2\bar{r_c}+\bar{a}^2}{(\bar{r_c}-1)^2}\),
– on the other hand \(\bar{r_c}\) is a root of the polynomial in \(\bar{r}^6\):
\(p(\bar{r})=\bar{r}^6-6\bar{r}^5+(9+2\nu u)\bar{r}^4-4u\bar{r}^3-\nu u(6-u)\bar{r}^2+2\nu u^2 \bar{r}+\nu u^2\)⁶
with \(u=\bar{a}^2\) and \(\nu=\sin^2⁡i\).
\(p(\bar{r_c})=0\hspace{2cm}\)(4.a) 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 is another polynomial in \(\bar{r}^5\) whose root for an angle \(i\) gives the reduced radial coordinate \(\bar{r_c}\) of the photon orbit :
\(q(\bar{r})=\bar{r}^5-3\bar{r}^4+2\bar{a}^2\bar{r}^3\sin^2i-2\bar{a}^2\bar{r}^2+\bar{a}^4\bar{r}\sin^2i+\bar{a}^4\sin^2i+2|\bar{a}|\bar{r}\cos i\sqrt{3\bar{r}^4+(1-3\sin^2i)\bar{a}^2\bar{r}^2-\bar{a}^4\sin^2i}\)⁷
\(q(\bar{r_c})=0\hspace{2cm}\)(4.b)
If \(i\in [0,\frac{\pi}{2}]\), the orbit is prograde and if \(i\in [\frac{\pi}{2},\pi]\), the orbit is retrograde.
Note that for small values of \(i\), (4.a) and (4.b) provide two solutions \(0\le\bar{r_c}\le 1\).
There are no simple analytical solutions to equations (4.a) or (4.b), except for the equatorial orbits (\(i=0\) or \(i=\pi\)) or the polar orbits (\(i=\frac{\pi}{2}\)).
Finally, all orbits are unstable, since the calculation gives \(\frac{d^2V_r}{dr^2}\gt 0\).

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.a), we get:
\(p(\bar{r})=\bar{r}^3(\bar{r}^3-6\bar{r}^2+9\bar{r}-4u)=0\) or
\(\bar{r}^3-6\bar{r}^2+9\bar{r}-4u=0\) and the trivial solution \(\bar{r}=0\) (central or annular singularity).
With the change of variable \(\bar{r}=x+2\), this 3rd degree equation reduces to:
\(x^3-3x+2-4u=0\hspace{2cm}\)(4.c)
Using Cardan’s method, the discriminant of (4.c) \(D=-(4A^3+27B^2)\) with \(A=-3\) and \(B=2-4u\) is \(432u(1-u)\) and given that \(u\in[0,1]\), 2 cases can be identified:
– \(u=0\) or \(u=1\Rightarrow D=0\) and (4.c) 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 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:
\(\bar{r_0}=2+2\cos(\frac{2}{3}\arccos(\bar{a}))\)^{8 9}
\(\bar{r_1}=4\sin^2(\frac{1}{3}\arcsin(\bar{a}))\)¹⁰
\(\bar{r_2}=2+2\cos(\frac{2}{3}\arccos(-\bar{a}))\)^{11 12}

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.a), 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.d)
Using Cardan’s method, the discriminant of (4.d) \(D=-(4A^3+27B^2)\) with \(A=u-3\) and \(B=2u-2\) is \(4u(-u^2-18u^2+27)\) and given 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 solution) and \(r_1=r_2=0\) trivial solution (central singularity).
– \(u\in ]0,1]\) : the sign of \(D\) is that of the polynomial \(-u^2-18u^2+27\) whose determinant is \(108\), which means that the polynomial has 2 real roots: \(-9-\sqrt{108}\) and \(-9+\sqrt{108}\).
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 \(\bar{r_2}\):
\(\bar{r_2}=1-2\sqrt{1-\frac{u}{3}}\sin\left ({1\over 3}\arcsin\left (\frac{1-u}{\left (1-\frac{u}{3}\right )^\frac{3}{2}}\right )\right )\)¹³

Critical impact parameter

As seen above, the value of \(i\) sets the reduced 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:
\(b_{crit}=m\sqrt{\frac{3\bar{r_c}^4+\bar{a}^2\bar{r_c}^2}{|\bar{r_c}^2-\bar{a}^2\sin^2i|}}\)¹⁴
or for \(\bar{r_c}\ne 1\) :
\(b_{crit}=m\sqrt{\frac{2\bar{r_c}^4+(\bar{a}^2-6)\bar{r_c}^2+2\bar{a}^2\bar{r_c}+\bar{a}^2}{(\bar{r_c}-1)^2}}\)

Colatitude limit

The parametric equation (2.p) is written with \(c\frac{l}{\varepsilon}=b_{crit}\):
\(\left(\frac{d\theta}{d\lambda}\right )^2=\left (a^2\cos^2\theta+b_{crit}^2\left (\sin^2i-\frac{\cos^2i}{\tan^2\theta}\right )\right )\frac{\varepsilon^2}{\Sigma^2 c^2}\)
\(\frac{d\theta}{d\lambda}\) therefore cancels for \(a^2\cos^2\theta+b_{crit}^2\left (\sin^2i-\frac{\cos^2i}{\tan^2\theta}\right )=0\) which is written after development and in reduced 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.e)
The left-hand side of (4.e) is a 2nd-degree polynomial in \(\cos^2\theta\) which has a positive root \(\cos^2\theta_{lim}=\frac{\bar{a}^2-(\frac{b_{crit}}{m})^2+\sqrt{(\bar{a}^2-(\frac{b_{crit}}{m})^2)^2+4\bar{a}^2(\frac{b_{crit}}{m})^2\sin^2i}}{2\bar{a}^2}\) 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 \(\cos\theta_{lim}=\sqrt{\frac{\bar{a}^2-(\frac{b_{crit}}{m})^2+\sqrt{(\bar{a}^2-(\frac{b_{crit}}{m})^2)^2+4\bar{a}^2(\frac{b_{crit}}{m})^2\sin^2i}}{2\bar{a}^2}}\hspace{2cm}\)(4.f)
In the case of polar orbits, \(i=\frac{\pi}{2}\) and (4.f) becomes:
\(\cos\theta_{lim}=\sqrt{\frac{\bar{a}^2-(\frac{b_{crit}}{m})^2+\sqrt{(\bar{a}^2+(\frac{b_{crit}}{m})^2)^2}}{2\bar{a}^2}}=1\) which confirms that the colatitude of a polar orbit is defined on \(]0,\pi[\).

Limit inclinations

Extreme Kerr black hole

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

Other Kerr black holes

When the value of the angle of inclination \(i\) is below a value \(i_{lim}\) which can be determined numerically as a function of \(\bar{a}\), 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 horizon).
The larger of the latter two orbits has the particularity of an angle \(\theta_{lim}\) increasing with \(r_c\), unlike the other prograde photon orbits (\(i\lt\frac{\pi}{2}\)).
The value \(i_{lim}\) is less than \(i_{lim}\) for \(|\bar{a}|=1\) and decreasing with \(|\bar{a}|\).
For \(i\gt i_{lim}\), 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.

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 the 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 paragraphe 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 entering the event horizon, notably because of the factor \(\Delta\) in the numerators of \(\frac{dr}{dt}\) and \(\frac{d\theta}{dt}\) which means that the radial coordinate \(r\) and the colatitude \(\theta\) no longer vary temporally on entering the event horizon (\(\Delta=0\), singularity of Boyer-Lindquist coordinates).
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 \(r=r_c=\) constant 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

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

Photons arriving from infinity – examples of quasi-capture

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

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

Schwarzschild black hole \(\bar{a}=0\) (trajectories in a plane)

Photon « spheres »

The plots in this paragraph are arbitrarily made with initial conditions \(\theta_0=\frac{\pi}{2}\) and \((\frac{d\theta}{d\lambda})_0\ge 0\).

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

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

Some other Kerr black holes

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

Two first oscillations (\(\theta_0=90^\circ\) and \((\frac{d\theta}{d\lambda}_0)\gt 0\))

Outlines and boundaries of photon orbits around Kerr 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 reduced radial coordinate \(\bar{r_c}\) varying continuously from \(0\) to \(4\)).
The regions (outer ergosphere, merged event and Cauchy 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:

APPARENT IMAGE OF A KERR BLACK HOLE (SHADOW)

and \(\frac{\beta}{m}=\bar{\beta}=\pm\sqrt{\frac{-\bar{r_c}^3(\bar{r_c}^3-6\bar{r_c}^2+9\bar{r_c}-4\bar{a}^2)}{\bar{a}^2(\bar{r_c}-1)^2}+\bar{a}^2\cos\theta_0^2-(\frac{-(\bar{r_c}^3-3\bar{r_c}^2+\bar{a}^2\bar{r_c}+\bar{a}^2)}{\bar{a}(\bar{r_c}-1)})^2\cot\theta_0^2}\)
with \(\bar{r_c}\) reduced 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 \(\sin\varphi_{obs}=-1\)¹⁶.
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:
\(\sin\theta_{obs}=\frac{2\bar{r_c}\sqrt{\bar{r_c}^2-2\bar{r_c}+\bar{a}^2}\sqrt{\bar{r}_0^2-2\bar{r}_0+\bar{a}^2}}{\bar{r}_0^2\bar{r_c}-\bar{r}_0^2+\bar{r_c}^3-3\bar{r_c}^2+2\bar{r_c}\bar{a}^2}\)¹⁷ 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 \(y(\bar{r_c})=\mp2\tan(\frac{\theta_{obs}}{2})\cos\varphi_{obs}\)¹⁸.
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 \(\bar{a}\).

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