This page extends the study of null geodesics and presents a comparative analysis with time-like and space-like geodesics in the vicinity of a Kerr black hole.
The trajectories are governed by three constants of motion — \(\varepsilon\), \(l_z\), \(Q\) — whose values can lead to near-captures or orbits.
The associated radial effective potential is used to analyze the stability of orbits of physical objects, and the total relativistic energy value sets the physical constraints for probes or spacecrafts to enter on stable orbits around a rotating black hole.
Contents
INTRODUCTION
The general equations of geodesics in Kerr spacetime can be written in Boyer-Lindquist coordinates:
\(\left(\frac{dr}{d\lambda}\right)^2=\frac{V_r}{\Sigma^2}\)
\(\left(\frac{d\theta}{d\lambda}\right)^2=\frac{V_\theta}{\Sigma^2}\)
\(\frac{d\varphi}{d\lambda}=\left(2mar+(\Sigma-2mr)c\frac{l_z}{\varepsilon\sin^2\theta}\right)\frac{\varepsilon}{c\Delta\Sigma}\), 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
\(V_r=\left(\left(r^2+a^2\right)\frac{\varepsilon}{c}-al_z\right)^2-\Delta\left(-\kappa c^2r^2+\left(a\frac{\varepsilon}{c}-l_z\right)^2+Q\right)\) and
\(V_\theta=Q+\cos^2\theta\left(a^2\left(\kappa c^2+\frac{\varepsilon^2}{c^2}\right)-\frac{l_z^2}{\sin^2\theta}\right)\).
The parameter \(\kappa\) represents the geodesic type:
– \(\kappa=0\) null,
– \(\kappa=-1\) time-like (physical object),
– \(\kappa=1\) space-like,
and the affine parameter \(\lambda\) corresponds for a physical object to its proper time \(\tau\) and for a space-like to its proper length \(l\) divided by \(c\).
Note: time-like geodesics locally maximize proper time, and space-like geodesics locally minimize proper length.
Refer to the axial symmetry study for definitions of terms.
GRAPHICAL COMPARISON OF GEODESICS
The figures are plotted in normalized Cartesian coordinates \(x/r_s\), \(y/r_s\) and \(z/r_s\) with \(r_s=2m=\frac{2GM}{c^2}\), which makes them independent of the black hole mass \(M\).
A graphical comparison of the three types of Kerr spacetime geodesics is provided in this figure, which shows the trajectories of a physical object, a photon, and a space-like arriving from “\(\infty\)” (that is, from the asymptotic region) on the \(x\) axis, and are nearly captured by a Kerr black hole with parameter \(\bar{a}= 0.95\).
\(\varepsilon\) is one of the constant parameters of space-like or time-like geodesics, and in the latter case it corresponds to the total relativistic energy per unit mass, that is, \(\frac{c^2}{\sqrt{1-v_{\infty}^2/c^2}}\) for an object coming from \(\infty\).
The null geodesic appears as the limit between the time-like and space-like geodesics for \(\varepsilon\rightarrow 0\).
This figure shows the unstable “spherical” orbits around a Kerr black hole with parameter \(\bar{a}= 0.95\) for the three trajectories shown above.
\(\bar{a}=0.95\) \(\frac{cl_z}{m\varepsilon}=-1.9\) \(\varepsilon/c^2=1.000001\)©
SPEEDS OF A PHYSICAL OBJECT
As a comparison with classical mechanics, the observable speed of a physical object by an observer at infinity according to his proper time \(t\) can be written:
\(v_{obs}=\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2+\left(\frac{dz}{dt}\right)^2}\)
and is calculated using the Cartesian coordinates \(x\), \(y\), and \(z\) obtained by numerical integration of the parametric equations.
Due to the temporal “dilatation” caused by the gravitational field of the black hole, this speed does not represent the actual speed of the physical object near the black hole.
This actual physical speed can be measured locally by a Zero Angular Momentum Observer (ZAMO) according to his proper time \(\tau\), and is written as:
\(v_{phys}=\sqrt{\frac{g_{rr}\left(\frac{dr}{d\tau}\right)^2+g_{\theta\theta}\left(\frac{d\theta}{d\tau}\right)^2+g_{\varphi\varphi}\left(\frac{d\varphi}{d\tau}\right)^2}{\alpha^2\left(\frac{dt}{d\tau}\right)^2}}\)
with \(\alpha^2=\frac{\Delta\Sigma}{(r^2+a^2)^2-\Delta a^2\sin^2\theta}\).
It is calculated using the values \(r\), \(\theta\), \(\varphi\) and \(t\) obtained by numerical integration of the parametric equations.
The value \(\varepsilon/c^2=1.000001\) is, for example, that of an asteroid or star with a speed \(v_{\infty}\simeq 400\ km/s\). The maximum physical speed of this object (see trajectory figure above), for a ZAMO is \(\simeq 194\ 000\ km/s\) at its closest point to the black hole, compared to the maximum speed observed “from infinity,” which is \(\simeq 146\ 000\ km/s\).
On the unstable orbit seen above, the physical speed of the asteroid or star oscillates between \(\simeq 190\ 000\ km/s\) and \(\simeq 194\ 000\ km/s\).
POTENTIALS
\(V_r\) and \(V_\theta\) can be considered respectively as the effective radial and colatitude potentials for null or time-like trajectories.
For a space-like geodesic, these are not potentials because the affine parameter \(\lambda\) can vary in either direction and the signs of \(V_r\) and \(V_\theta\) have no causal meaning.
Using the dimensionless variables \(\bar{a}=\frac{a}{m}\) and \(\bar{r}=\frac{r}{m}\), and expanding the previous expression:
\(\frac{c^2}{m^4\varepsilon^2}V_r=\left(1+\kappa \frac{c^4}{\varepsilon^2}\right)\bar{r}^4-2\kappa \frac{c^4}{\varepsilon^2}\bar{r}^3+\left(\bar{a}^2\left(1+\kappa \frac{c^4}{\varepsilon^2}\right)-\frac{c^2l_z^2}{m^2\varepsilon^2}-\frac{c^2Q}{m^2\varepsilon^2}\right)\bar{r}^2+2\left(\left(\bar{a}-\frac{cl_z}{m\varepsilon}\right)^2+\frac{c^2Q}{m^2\varepsilon^2}\right)\bar{r}-\bar{a}^2\frac{c^2Q}{m^2\varepsilon^2}\) is a 4th-degree polynomial in \(\bar{r}\) with constant coefficients, since \(\varepsilon\), \(l_z\) and \(Q\) are constants along the geodesic.
The plot of the radial effective potentials of the previous orbits highlights the double roots (canceling out the potentials and their derivative with respect to \(\bar{r}\)), that is, the values of the constant radial coordinates of the orbits.
Here, \(\bar{r}_c\) are respectively \(\simeq 2.7\), \(=3\) and \(\simeq 4.4\) for the space-like, null and time-like geodesics.
The plot shows that the orbits can be theoretically reached from l’\(\infty\) with \(r\) decreasing \(\left(\frac{dr}{d\lambda}<0\right)\) and that they are unstable (the second derivative of \(V_r\) with respect to \(\bar{r}\) is positive).
Here, the value \(l_z\) is identical for all 3 geodesics, and the potentials therefore have the same value for \(\bar{r}=\bar{r}_{Cauchy}\) (Cauchy horizon) ou \(\bar{r}=\bar{r}_h\) (event horizon).
STABLE ORBITS OF PHYSICAL OBJECTS
Another illustration can be made by now considering stable orbits in radial perturbation for a physical object.
The plot of the potentials \(V_r\) shows that these orbits cannot be reached in free fall from \(\infty\), as the potential values are negative for large values of \(r\). An example is given below with a Kerr parameter \(\bar{a}=0.95\) and the smallest stable polar orbit (\(\bar{r}_c\simeq 5.833\)) :
\(\bar{a}=0.95\) \(\frac{c^2Q}{m^2\varepsilon} ≈ 12.924\) \(\varepsilon/c^2=0.938651\)©
Furthermore, the initial value of \(\varepsilon/c^2>1\) must decrease below \(1\) through braking in order for a physical object coming from \(\infty\) to enter a stable orbit around the black hole. In the case of a probe or even a spacecraft, the intended orbit must have a very large radial coordinate \(r_c\) so that the braking energy required per unit mass \(\Delta\varepsilon=\varepsilon_{initial} (>1)-\varepsilon_{final} (<1)\) remains technically achievable.
Stable orbit insertion of a physical object
For example, reaching a distant orbit \(\bar{r}_c\simeq 50\ 000\) of a Kerr black hole with parameter \(\bar{a}=0.95\), would require an object coming with \(v_{\infty}\simeq 100\ km/s\) or \(\frac{\varepsilon}{c^2}=1.000000056\) to decrease to \(\frac{\varepsilon}{c^2}=0.99999\) that is, a braking energy of \(1.0056\ 10^{-5}c^2/kg\) or approximately \(9\ 10^{11}\ J/kg\) far beyond what an onboard braking system could provide with current technology.
Thus, for a probe or spacecraft coming from \(\infty\) to enter a distant orbit around a black hole, using the best chemical propellant to decelerate, either \(\Delta E_{chim}\simeq 4\ 10^{7}\ J/kg\), \(\frac{\varepsilon}{c^2}\) (its initial total relativistic energy per unit mass divided by \(c^2\)) be less than \(1+\frac{\Delta E_{chim}}{c^2}\simeq 1+4.45\ 10^{-10}\ J/kg\) which corresponds to \(v_{\infty}\simeq 9\ km/s\) only.
Radius of circular orbits of a physical object – Newtonian mechanical energy and total relativistic energy
The radius \(r_c\) of a distant circular orbit as a function of \(\frac{\varepsilon}{c^2}<1\) can be calculated using Newtonian mechanical energy per unit mass of the physical object \(\varepsilon_{meca}=\frac{1}{2}v^2-\frac{mc^2}{r_c}\) and \(v^2=\frac{mc^2}{r_c}\) or \(\varepsilon_{meca}=-\frac{mc^2}{2r_c}\), which allows us to write \(\frac{\varepsilon}{c^2}\simeq 1+\frac{\varepsilon_{méca}}{c^2}=1-\frac{m}{2r_c}=1-\frac{1}{2\bar{r}_c}\), or \(\bar{r}_c\simeq\frac{1}{2\left(1-\frac{\varepsilon}{c^2}\right)}\).
Note that this formula cannot be applied for values of \(\frac{\varepsilon}{c^2}\) significantly less than \(1\) which will result in “close” orbits: the values of \(r_c\) can then be obtained by numerically searching for the double roots of the potential \(V_r\).
\(\kappa=-1\) \(\bar{a}=1\)©
The graph shows with a logarithmic ordinate the constant radial coordinate \(\bar{r}_c\) of the stable prograde equatorial orbits of a physical object around an extreme Kerr black hole, as a function of \(\frac{\varepsilon}{c^2}\) (total relativistic energy of the physical object per unit mass, divided by \(c^2\)).
The last value on the right is \(1/\sqrt{3}\) and corresponds to \(\bar{r}_c=1\), which is the Innermost Stable Circular Orbit (ISCO).
Since the energy of a physical object at rest per unit mass is \(c^2\), reaching the ISCO of an extreme Kerr black hole seen above causes a release of energy equal to \(c^2-c^2/\sqrt{3}\simeq 42\%\) of the total energy at rest, or \(\simeq 3.8\ 10^{16}\ J/kg\), which represents the most efficient energy production system known in the universe, far better than the \(0,7\%\) of thermonuclear reactions in stars (pp chain or CNO cycle) and far beyond any known human technology.