Contents
INTRODUCTION
The light deflection by a Kerr black hole, that is in an axially symmetric gravitational field generated by a non-electrically charged spinning black hole, is a phenomenon predicted by general relativity (Kerr metric).
This metric generalizes the spherical symmetry of a stationary black hole (Schwarzschild’s metric). Refer to appendix for detailed calculations and additional information.
Note: to avoid any confusion, the writing simplification \(c=G=1\) is not used in the present document and all equations are written explicitly.
GENERAL RELATIVITY – KERR METRIC
An elementary displacement of the photon is a light-like vector and its scalar product is zero,1 (hence the name « null geodesics » for the photon trajectories).
Assuming that the gravitational field is axially symmetrical and applying the Kerr metric (see its limits in the conclusion), the scalar product of the elementary displacement \(\overrightarrow{ds}\) \((cdt, dr, d\theta, d\varphi)\) in Boyer-Lindquist’s coordinates can be written:
\(ds^2=-\left(1-\frac{2GM}{c^2\Sigma}\right)c^2dt^2-\frac{4GMar\sin^2{\theta}}{c^2\Sigma}cdtd\varphi+\frac{\Sigma}{\Delta}dr^2+\Sigma d\theta^2\)
\(+\left(r^2+a^2+\frac{2GMa^2r\sin^2{\theta}}{c^2\Sigma}\right)\sin^2{\theta}d\varphi^2=0\)2,
with \(G\) gravitational constant, \(c\) speed of light in a vacuum, \(M\) mass of the black hole, \(a=\frac{J}{cM}\) with \(J\) spin angular momentum of the black hole, \(\Delta=r^2-\frac{2GM}{c^2}r+a^2\) and \(Σ=r^2+a^2\cos^2{\theta}\).
The coefficients of the metric are independent of \(t\) and \(\varphi\): the geometry of Kerr spacetime is therefore stationary and axially symmetrical.
Note: in the asymptotic region \(r \gg \frac{2GM}{c^2}\), the coordinate \(r\) is interpreted as the physical distance between the photon and the center of the black hole.
Parametric equations of motion
The invariance of the energy \(\varepsilon\), the angular momentum component \(l_z\) on the spinning axis of the black hole and the Carter constant \(Q\) enables to get the four parametric equations of motion of the photon and to calculate the light deflection by Kerr black holes:
\(\left(\frac{dr}{d\lambda}\right)^2=\left(\left(r^2+a^2-ac\frac{l_z}{\varepsilon}\right)^2-\Delta\left(\left(a-c\frac{l_z}{\varepsilon}\right)^2+c^2\frac{Q}{\varepsilon^2}\right)\right)\frac{\varepsilon^2}{ c^2\Sigma^2}\),
\(\left(\frac{d\theta}{d\lambda}\right)^2=\left(c^2\frac{Q}{\varepsilon^2}+\cos^2\theta\left(a^2-c^2\frac{l_z^2}{\varepsilon^2\sin^2\theta}\right)\right)\frac{\varepsilon^2}{ c^2\Sigma^2}\),
\(\frac{d\varphi}{d\lambda}=\left(2mar+(\Sigma-2mr)c\frac{l_z}{\varepsilon\sin^2\theta}\right)\frac{\varepsilon}{c\Delta\Sigma}\), and
\(\frac{dct}{d\lambda}=\left((r^2+a^2)^2-\Delta a^2\sin^2\theta-2mar\ c\frac{l_z}{\varepsilon}\right)\frac{\varepsilon}{c\Delta\Sigma}\),
with \(r\) radial coordinate, \(\theta\) colatitude, \(\varphi\) longitude, \(t\) time measured by a static observer, \(\lambda\) an affine parameter and \(m=\frac{GM}{c^2}\) homogeneous to the metre.
Note that the coordinate system is undefined at the poles \(\theta=0\) and \(\theta=\pi\).
In the following, the value \(R_s=2m\) and the dimensionless values \(\bar{r}=\frac{r}{m}\) and Kerr parameter \(\bar{a}=\frac{a}{m}\) are used.
\(\)
Convention used for \(\bar{a}\ne 0\): the \(z\)-axis is the spinning axis of the black hole. When \(\bar{a}>0\), the spin of the black hole is trigonometric and when \(\bar{a}<0\), its spin is clockwise.
It is assumed that \(|\bar{a}|\) lies between \(0\) and \(1\), limits included, except for the over extreme Kerr spacetime described briefly before the conclusion.
Photon trajectories
In the general case, photon trajectories near a spinning black hole can be found by integration of each of the 4 parametric equations, according to the affine parameter \(\lambda\).
The initial values to be taken into account are \(r_0\), \(\theta_0\), \(\varphi_0\), \(t_0\), and the signs of \(\frac{dr}{d\lambda}_0\) and\(\frac{d\theta}{d\lambda}_0\).
The trajectory of the photon is fully determined by the constants \(M\), \(a\), \(\frac{l_z}{\varepsilon}\) and \(\frac{Q}{\varepsilon^2}\).
Fig. A – Photon trajectory coming from \(\infty\)
and deflected by an extreme Kerr black hole
\(\bar{a}=1\) \(\frac{cl_z}{m\varepsilon}=-6\) \(\frac{c^2Q}{m^2\varepsilon^2}\simeq 9.863\)©
For a given value of \(\frac{l_z}{\varepsilon}\), there is a critical value \(\frac{Q_{crit}}{\varepsilon^2}\):
– if \(\frac{Q}{\varepsilon^2}>\frac{Q_{crit}}{\varepsilon^2}\) the photon coming from \(\infty\) will be deflected by the black hole and continue towards \(\infty\),
– if \(\frac{Q}{\varepsilon^2}<\frac{Q_{crit}}{\varepsilon^2}\) the photon coming from \(\infty\) will be absorbed by the black hole,
– if \(\frac{Q}{\varepsilon^2}=\frac{Q_{crit}}{\varepsilon^2}\) the photon coming from \(\infty\) will be captured by the black hole on an orbit.
Fig. B – Photon trajectory coming from \(\infty\)
and virtually captured by an extreme Kerr black hole
\(\bar{a}=1\) \(\frac{cl_z}{m\varepsilon}=-6\) \(\frac{c^2Q}{m^2\varepsilon^2}\simeq\frac{c^2Q_{crit}}{m^2\varepsilon^2}\simeq 9.627\)©
Photon orbits
A constant radial coordinate \(r\) is given by cancelling out the potential of the first parametric equation and its derivative with respect to \(r\), which leads after calculation to \(\bar{r}\) being a root of the polynomial
\(q(\bar{r})=\bar{r}^5-3\bar{r}^4+2\bar{a}^2\bar{r}^3\sin^2i-2\bar{a}^2\bar{r}^2+\bar{a}^4\bar{r}\sin^2i+\bar{a}^4\sin^2i\)
\(+2\bar{a}\bar{r}\cos i\sqrt{3\bar{r}^4+(1-3\sin^2i)\bar{a}^2\bar{r}^2-\bar{a}^4\sin^2i}\),
\(i\) being the constant angle of inclination of the angular momentum \(\overrightarrow{l}\) with respect to the spinning axis of the black hole.
For given \(m\), \(a\) and \(i\), there are at least one real root \(\bar{r}\) between 0 and 4 giving an orbit for the photon.
If \(i\in[0,\pi/2[\) the orbit is prograde (same direction of spin as the black hole), and if \(i\in ]\pi/2,\pi]\) the orbit is retrograde (opposite direction of spin to the black hole).
The roots of the polynomial \(q(\bar{r})\) are difficult to calculate analytically except in the following cases:
– polar orbit (\(\sin^2i=1\)) \(\Rightarrow\bar{r}_{polar}=1+2\sqrt{1-\frac{\bar{a}^2}{3}}\cos\left(\frac{1}{3}\arccos\left(\frac{1-\bar{a}^2}{\left(1-\frac{\bar{a}^2}{3}\right)^\frac{3}{2}}\right)\right)\),
around an extreme Kerr black hole
\(\bar{a}=1\) \(\frac{c^2Q}{m^2\varepsilon^2}=11+8\sqrt{2}\)©
More details
– equatorial prograde orbit (\(\cos i=1\)) \(\Rightarrow\bar{r}_{prograde}=2\left (1+\cos\left (\frac{2}{3}\arccos\left(-\bar{a}\right)\right)\right)\),
– equatorial retrograde orbit (\(\cos i=-1\)) \(\Rightarrow\bar{r}_{retrograde}=2\left (1+\cos\left (\frac{2}{3}\arccos\left(\bar{a}\right)\right)\right)\).
Note: when \(\bar{a}=0\), the above formulas lead to the special case of the Schwarzschild’s metric \(\Rightarrow\bar{r}_{polar}=\bar{r}_{prograde}=\bar{r}_{retrograde}=3\) that is, \(r=3m=\frac{3}{2}R_s\) with \(R_s=\frac{2GM}{c^2}\).
around an extreme Kerr black hole
\(\bar{a}=1\) \(\frac{c^2Q}{m^2\varepsilon^2}=11+8\sqrt{2}\)
(top view)©
General
Each orbit is defined by its constant dimensionless radial coordinate value \(\bar{r}_c\) and by the constant inclination \(i\) of the angular momentum of the photon \(\overrightarrow{l}\), associated with this value.
There are therefore an infinite number of photon « spheres » with constant dimensionnless radial coordinates \(\bar{r}\in[0,4]\), the bound 4 being reached for \(|\bar{a}|=1\).
an extreme Kerr black hole
\(\bar{a}=1\) \(\frac{cl_z}{m\varepsilon}=-1\) \(\frac{c^2Q}{m^2\varepsilon^2}\simeq 25.856\)©
Furthermore, the geometric shape of each orbit is not really a sphere, but an ellipsoid of radius \(\sqrt{r^2+a^2}\sin\theta\) (in Cartesian Boyer-Lindquist’s coordinates) and colatitude \(\theta\) between a value \(\theta_{lim}\) and a value \(\pi-\theta_{lim}\), a function of \(\bar{a},\bar{r}_c\) and \(\sin^2i\).
Parametric equations
With \(l_z=l\cos i\), \(Q=l^2\sin^2i\) and defining the impact parameter \(b_{crit}=c\frac{l}{\varepsilon}\), the equations of the trajectory are:
\(r=r_c\),
with the 3 parametric equations which become:
\(\left(\frac{d\theta}{d\lambda}\right)^2=\left(b_{crit}^2\left(1-\frac{\cos^2i}{\sin^2\theta}\right)+a^2\cos^2\theta\right)\frac{\varepsilon^2}{\Sigma^2 c^2}\),
\(\frac{d\varphi}{d\lambda}=\left(2mar_c+\left(\Sigma-2mr_c\right)b_{crit}\frac{\cos i}{\sin^2\theta}\right)\frac{\varepsilon}{\Delta\Sigma c}\), and
\(\frac{cdt}{d\lambda}=\left(\left(r_c^2+a^2\right)^2-\Delta a^2 \sin^2\theta-2mar_cb_{crit}\cos i\right)\frac{\varepsilon}{\Delta\Sigma c}\).
The value of the critical impact parameter can be calculated using the formula:
\(b_{crit}=m\sqrt{\frac{3\bar{r}_c^4+\bar{a}^2\bar{r}_c^2}{\bar{r}_c^2-\bar{a}^2\sin^2i}}\).
Animated trajectories
Examples of photon trajectories with near-capture by an extreme Kerr black hole
Examples of photon orbits with different \(\bar{a}\) and \(\frac{cl_z}{m\varepsilon}\)
\(\bar{a}=1\) \(\frac{cl_z}{m\varepsilon}=0\) \(b_{crit}\simeq 4.724\)©
\(\bar{a}=1\) \(\frac{cl_z}{m\varepsilon}=-1\) \(b_{crit}\simeq 5.182\)©
\(\bar{a}=0.5\) \(\frac{cl_z}{m\varepsilon}=2\) \(b_{crit}\simeq 4.658\)©
DESCRIPTION OF BLACK HOLE REGIONS, SINGULARITIES AND SHADOW
A Kerr black hole mathematically has four centered regions, each included in the other and defined by mathematical hypersurfaces. From largest to smallest:
– outer ergosphere \(r_{ergoext}=m (1+\sqrt{1-\bar{a}^2\cos^2\theta})\),
– event horizon \(r_h=m (1+\sqrt{1-\bar{a}^2})\),
– Cauchy’s horizon \(r_{Cauchy}=m (1-\sqrt{1-\bar{a}^2})\), and
– inner ergosphere \(r_{ergoint}=m (1-\sqrt{1-\bar{a}^2\cos^2\theta})\).
\(r_{ergoext}\) and \(r_{ergoint}\) are the roots of the equation \(\Sigma-2mr=0\) and \(r_h\) and \(r_{Cauchy}\) are the roots of the equation \(\Delta=0\).
For \(|\bar{a}|\in ]0,1[\), the four regions are distinct, and for \(|\bar{a}|=1\), the event horizon and Cauchy’s horizon are merged.
\(\bar{a}=0\) corresponds to the Schwarzschild’s black hole, where the outer ergosphere and the event horizon are merged (\(r_{ergoext}=r_h=R_s=2m\)), and there is no Cauchy’s horizon or inner ergosphere.
The hypersurface that delimits the outer ergosphere is a stationarity limit, which means that any particle – material or photon – that crosses it cannot be motionless.
Note: once it has crossed the event horizon, a particle can return to it, but can never cross it in the other direction.
Presence of regions
If the central body is a Kerr black hole, the two regions defined by the outer ergosphere and the event horizon (merged with the Cauchy’s horizon for an extreme Kerr black hole) physically exist.
The other regions (defined by the Cauchy’s horizon for a non-extreme Kerr black hole and by the inner ergosphere) can only exist if the physical body of the black hole is « inside » them.
(side view)
Outer ergosphere, event horizon, Cauchy horizon and inner ergosphere©
(top view)©
(exploded view)
with the singularity circle bordering the inner ergosphere©
Singularities
The parametric equations seen above show that a zero value of either \(\Delta\) or \(\Sigma\) does not give a definition of the motion of the photon.
\(\Delta=r^2-\frac{2GM}{c^2}r+a^2= 0\) occurs when the photon crosses the event horizon or the Cauchy’s horizon: it is a simple singularity of the Boyer-Lindquist’s coordinates, which generalizes the singularity of the Schwarzschild’s coordinates in \(r=R_s\)3.
The singularity in \(r\) such as \(Σ=r^2+a^2\cos^2{\theta}=0\) is a true singularity, just as the singularity in \(r=0\) of the Schwarzschild’s metric4.
This is the circle of Cartesian radius \(|a|\) whose center is that of the black hole, located in its equatorial plane. This circle borders the inner ergosphere.
Apparent image or shadow
No photon with an impact parameter below \(b_{crit}\) can reach an outside observer, which results in a « shadow » without any star image.
If this observer is located at a great distance from the black hole and in its equatorial plane, the apparent outline of a Kerr black hole can be determined by the 2 coordinates5:
\(\alpha=-c\frac{l_z}{\varepsilon}\) and \(\beta=\pm c\frac{\sqrt{Q}}{\varepsilon}\), that is:
\(\frac{\alpha}{m}=\frac{\bar{r}_c^3-3\bar{r}_c^2+\bar{a}^2\bar{r}_c+\bar{a}^2}{\bar{a}(\bar{r}_c-1)}\) and \(\frac{\beta}{m}=\pm\sqrt{\frac{-\bar{r}_c^3(\bar{r}_c^3-6\bar{r}_c^2+9\bar{r}_c-4\bar{a}^2)}{\bar{a}^2(\bar{r}_c-1)^2}}\),
with \(\bar{r}_c\) dimensionless radial coordinates of photon orbits varying between the value \(\bar{r}_{c_{min}}\) and the value \(\bar{r}_{c_{max}}\) corresponding to the value of \(\bar{a}\).
(side view)
with its regions and its shadow©
Over extreme Kerr spacetime
When \(|\bar{a}|>1\), the Kerr spacetime is said to be over extreme and \(\Delta\) has no root, so there is no event horizon or Cauchy’s horizon, implying that the massive object is not a black hole. It has a naked singularity (circle of Cartesian radius \(|a|\)) with adjacent outer and inner ergospheres with colatitude \(\in [\arccos{\frac{1}{|\bar{a}|}},\pi-\arccos{\frac{1}{|\bar{a}|}}]\) that form a kind of open torus.
It is a mathematical object whose physical existence is currently unlikely.
(exploded view)
with its outer (medium grey) and inner (dark grey) ergospheres©
CONCLUSION
As most celestial objects rotate on themselves, the axially symmetric Kerr metric provides an absolutely accurate representation of the countless black holes that populate the universe, the Schwarzschild’s metric being a special case, obtained with a zero Kerr parameter.
The structure of a spinning black hole is extremely simple: just two real numbers, m and a, are needed to describe it fully.
The light deflection by Kerr black holes and the trajectories or orbits of photons can be precisely calculated using the Kerr metric.
Note that it does not apply to a spinning star whose metric cannot be described by just a few scalar parameters, even outside the star. It depends on the distribution of mass and momentum inside the star.
- https://luth.obspm.fr/~luthier/gourgoulhon/fr/master/relatM2.pdf ↩︎
- https://luth.obspm.fr/~luthier/gourgoulhon/fr/master/relatM2.pdf ↩︎
- https://luth.obspm.fr/~luthier/gourgoulhon/fr/master/relatM2.pdf ↩︎
- https://luth.obspm.fr/~luthier/gourgoulhon/fr/master/relatM2.pdf ↩︎
- https://arxiv.org/pdf/2105.07101 ↩︎