This page extends the study of null geodesics and provides a comparative analysis of timelike and spacelike geodesics in the vicinity of a Kerr black hole. The trajectories in Kerr spacetime depend on three fundamental constants of motion — \(\varepsilon\), \(l_z\), \(Q\) — whose values determine the possible dynamical regimes: quasi-capture, bound orbits, or unbound trajectories.
The relativistic speeds reached by physical objects on eccentric orbits close to the rotating black hole are examined, and the stability of circular orbits is analyzed using the associated radial effective potential.
Finally, the value of the total relativistic energy defines the physical constraints required to place probes or spacecraft on stable orbits around a rotating black hole.
Contents
- 1 INTRODUCTION
- 2 GRAPHICAL COMPARISON OF GEODESICS
- 3 SPEEDS OF A PHYSICAL OBJECT
- 4 Speed for an observer at \(\infty\)
- 5 Actual physical speed
- 6 POTENTIALS
- 7 INNERMOST STABLE CIRCULAR ORBITS
- 8 STABLE ORBITS OF PHYSICAL OBJECTS
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.
\(\bar{a}=0.95\) \(\varepsilon/c^2=1.000001\) \(\frac{cl_z}{m\varepsilon}=-1.9\)©
SPEEDS OF A PHYSICAL OBJECT
Speed for an observer at \(\infty\)
As a comparison with classical mechanics, the observable speed of a physical object by an observer in the asymptotic region 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 timeline expansion caused by the gravitational field of the black hole, this speed does not represent the actual speed of the physical object.
Actual physical speed
The actual physical speed of a physical object according to its proper time \(\tau\) can be measured locally by a Zero Angular Momentum Observer (ZAMO), 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 \(\alpha^2=\frac{\Delta\Sigma}{(r^2+a^2)^2-\Delta a^2\sin^2\theta}\) (\(\alpha\) is the lapse function or the local gravitational expansion factor, that is, \(\frac{d\tau_{ZAMO}}{dt}\)).
\(v_{phys}\) is calculated using the values \(r\), \(\theta\), \(\varphi\) and \(t\) obtained by numerical integration of the parametric equations.
It should be noted that these speeds are independent of the mass of the black hole.
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 \(\simeq 194\ 000\ km/s\).
as seen by a static observer at \(\infty\)
and by a ZAMO
\(\bar{a}=0.95\) \(\frac{\varepsilon}{c^2}=1.000001\) \(\frac{cl_z}{m\varepsilon}=-1.9\)©
Note: in the case of a stellar black hole, the tidal force is so strong due to the low value of \(r_{pericentre}\) that the asteroid or star will be destroyed long before reaching the pericentre. However, these orbits are applicable to a supermassive black hole.
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 (“spherical” 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 spherical orbits can be theoretically reached from \(\infty\) 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).
INNERMOST STABLE CIRCULAR ORBITS
Specific case of a Schwarzschild black hole
The Innermost Stable Circular Orbit (ISCO) around a Schwarzschild black hole is the stability limit orbit and corresponds to the triple root of the effective radial potential \(V_r\) with \(\bar{a}=0\) and \(\frac{c^2l_z^2}{m^2\varepsilon^2}+\frac{c^2Q}{m^2\varepsilon^2}=\frac{c^2l^2}{m\varepsilon^2}\).
Calculation of \(\frac{\varepsilon}{c^2}\) and \(\frac{cl}{m\varepsilon}\)
1) The cancellation of \(V_r\) can be written as:
\(\left(1+\kappa \frac{c^4}{\varepsilon^2}\right)\bar{r}^4=2\kappa \frac{c^4}{\varepsilon^2}\bar{r}^3+\frac{c^2l^2}{m^2\varepsilon^2}\bar{r}^2-2\frac{c^2l^2}{m^2\varepsilon^2}\bar{r}\) or, with the condition \(\bar{r}\ne 0\)
\(1= \left(1-\frac{2}{\bar{r}}\right)\left(-\kappa\frac{c^4}{\varepsilon^2}+\frac{c^2l^2}{m^2\varepsilon^2}\frac{1}{\bar{r}^2}\right)\).
Based on this equation, the calculation can be performed using \(\varepsilon^2\) or \(\frac{c^2l^2}{m^2}\).
Calculation of \(\varepsilon^2\) as a function of \(r\) and \(\frac{c^2l^2}{m^2}\)
\(\varepsilon^2=\left(1-\frac{2}{\bar{r}}\right)\left(-\kappa c^4+\frac{c^2l^2}{m^2}\frac{1}{\bar{r}^2}\right)\).
Calculation of \(\frac{c^2l^2}{m^2}\) as a function of \(r\) and \(\varepsilon^2\)
\(\frac{c^2l^2}{m^2}=\bar{r}^2\left(\kappa c^4+\varepsilon^2\frac{\bar{r}}{\bar{r}-2}\right)\), with the condition \(\bar{r}\ne 2\).
2) The cancellation of \(\frac{dV_r}{dr}\) can be written as \(2(\varepsilon^2+\kappa c^4)\bar{r}^3-3\kappa c^4\bar{r}^2+\frac{c^2l^2}{m^2}(1-\bar{r})=0\).
Calculation of \(\varepsilon^2\) as a function of \(r\)
Replacing \(\frac{c^2l^2}{m^2}\) with its value and after expansion and grouping, we obtain:
\(\varepsilon^2=-\kappa c^4\frac{(\bar{r}-2)^2}{\bar{r}(\bar{r}-3)}\),
with the condition \(\bar{r}\ne 3\).
Calculation of \(\frac{c^2l^2}{m^2}\) as a function of \(r\)
Replacing \(\varepsilon^2\) with its value and after expansion and grouping, we obtain:
\(\frac{c^2l^2}{m^2}=-\kappa c^4\frac{\bar{r}^2}{\bar{r}-3}\),
with the condition \(\bar{r}\ne 3\).
3) Finally, the cancellation of \(\frac{d^2V_r}{dr^2}\) can be written as \(12(\varepsilon^2+\kappa c^4)\bar{r}^2-12\kappa c^4\bar{r}-2\frac{c^2l^2}{m^2}=0\).
Calculation of \(r_{ISCO}\) – Innermost Stable Circular Orbit
The ISCO can be calculated by replacing \(\varepsilon^2\) and \(\frac{c^2l^2}{m^2}\) in the above equation with their values as functions of \(r\), which gives, after grouping:
\(\frac{d^2V_r}{dr^2}\) for circular orbits \(=2\kappa c^4\frac{\bar{r}(\bar{r}-6)}{\bar{r}-3}\), which is \(0\) for
\(\bar{r}=6\Rightarrow\bar{r}_{ISCO}=6\) or \(r_{ISCO}=6m=3R_s\).
For a physical object (\(\kappa=-1\)), the above equation shows that \(\frac{d^2V_r}{dr^2}\) is \(<0\) for \(\bar{r}>6\) which means that circular orbits with constant radial coordinate \(\bar{r}_c>6\) are stable.
Finally, replacing \(\bar{r}\) with the value \(6\) in the expressions of \(\varepsilon^2\) and \(\frac{c^2l^2}{m^2}\) seen above, we have
\(\frac{\varepsilon}{c^2}=\frac{2\sqrt{2}}{3}\simeq 0.943\) and \(\frac{cl}{m}=2\sqrt{3}c^2\), or \(\frac{cl}{m\varepsilon}=\sqrt{\frac{27}{2}}\simeq 3.674\).
Case of space-type geodesics
The equations seen above become, with \(\kappa=1\):
\(\varepsilon^2=c^4\frac{(\bar{r}-2)^2}{\bar{r}(3-\bar{r})}\) and \(\frac{c^2l^2}{m^2}=c^4\frac{\bar{r}^2}{3-\bar{r}}\)
which implies \(0<\bar{r}<3\) for circular orbits (\(V_r=0\) and \(\frac{dV_r}{dr}=0\)).
Furthermore, the cancellation of \(\frac{d^2V_r}{dr^2}\) does not depend on the sign of \(\kappa\) as shown by the equation \(2\kappa c^4\frac{\bar{r}(\bar{r}-6)}{\bar{r}-3}=0\), and gives the same solution as for a time-like geodesic: \(\bar{r}=6\).
This value is not compatible with the condition \(\bar{r}<3\), which shows that there is no stable orbit for space-like geodesics around a Schwarzschild black hole.
General case \(\bar{a}\ne 0\)
Calculation of \(r_{ISCOs}\)
For a physical object around a Kerr blackhole, there is a prograde ISCO and a retrograde ISCO in the equatorial plane which are given by the formulas:
\(\bar{r}_{ISCO\ prograde}=3+z2-\sqrt{(3-z1)(3+z1+2z2)}\) and
\(\bar{r}_{ISCO\ retrograde}=3+z2+\sqrt{(3-z1)(3+z1+2z2)}\)
with \(z1=1+(1-\bar{a}^2)^{1/3}((1+\bar{a})^{1/3}+(1-\bar{a})^{1/3})\) and \(z2=\sqrt{3\bar{a}^2+z1^2}\).
For an extreme Kerr black hole, these formulas give \(\bar{r}_{ISCO\ prograde}=1\) or \(r_{ISCO\ prograde}=m\) and \(\bar{r}_{ISCO\ retrograde}=9\) or \(r_{ISCO\ retrograde}=9m\).
Calculation of \(\frac{\varepsilon}{c^2}\) and \(\frac{cl_z}{m\varepsilon}\)
The associated values of \(\frac{\varepsilon}{c^2}\) and \(\frac{cl_z}{m\varepsilon}\) can be calculated directly using the values of \(\bar{r}_{ISCO}\), following the standard formulas for circular equatorial orbits in Kerr spacetime:
– prograde orbit:
-> if \(|\bar{a}|\ne 1\Rightarrow\frac{\varepsilon}{c^2}=\frac{{\bar{r}_{ISCO\ prograde}}^{3/2}-2{\bar{r}_{ISCO\ prograde}}^{1/2}+|\bar{a}|}{{\bar{r}_{ISCO\ prograde}}^{3/4}\ \sqrt{{\bar{r}_{ISCO\ prograde}}^{3/2}-3{\bar{r}_{ISCO\ prograde}}^{1/2}+2|\bar{a}|}}\)
and \(\frac{cl_z}{m\varepsilon}=\frac{\bar{a}}{|\bar{a}|}\frac{{\bar{r}_{ISCO\ prograde}}^{2}-2|\bar{a}|{\bar{r}_{ISCO\ prograde}}^{1/2}+\bar{a}^2}{{\bar{r}_{ISCO\ prograde}}^{3/2}-2{\bar{r}_{ISCO\ prograde}}^{1/2}+|\bar{a}|}\),
-> if \(|\bar{a}|=1\Rightarrow\frac{\varepsilon}{c^2}=\frac{1}{\sqrt{3}}\simeq 0.577\) and \(\frac{cl_z}{m\varepsilon}=2\frac{\bar{a}}{|\bar{a}|}\).
– retrograde orbit:
\(\frac{\varepsilon}{c^2}=\frac{{\bar{r}_{ISCO\ retrograde }}^{3/2}-2{\bar{r}_{ISCO\ retrograde }}^{1/2}-|\bar{a}|}{{\bar{r}_{ISCO\ retrograde }}^{3/4}\ \sqrt{{\bar{r}_{ISCO\ retrograde }}^{3/2}-3{\bar{r}_{ISCO\ retrograde }}^{1/2}-2|\bar{a}|}}\)
and \(\frac{cl_z}{m\varepsilon}=-\frac{\bar{a}}{|\bar{a}|}\frac{{\bar{r}_{ISCO\ retrograde }}^{2}+2|\bar{a}|{\bar{r}_{ISCO\ retrograde }}^{1/2}+\bar{a}^2}{{\bar{r}_{ISCO\ retrograde }}^{3/2}-2{\bar{r}_{ISCO\ retrograde }}^{1/2}-|\bar{a}|}\),
which gives if \(|\bar{a}|=1\Rightarrow\frac{\varepsilon}{c^2}=\frac{5}{3\sqrt{3}}\simeq 0.962\) and \(\frac{cl_z}{m\varepsilon}=-4.4\frac{\bar{a}}{|\bar{a}|}\).
There is no ISCO for null or space-like geodesics.
STABLE ORBITS OF PHYSICAL OBJECTS
Eccentric orbits
General case
When the effective radial potential \(V_r\) is positive between 2 real roots \(r\) with \(\frac{\varepsilon}{c^2}<1\), these 2 roots correspond to the pericentre and apocentre of a stable orbit of the physical object.
An example is given below with the rosette of an equatorial orbit between \(\bar{r}_{pericentre}\simeq 6\) and \(\bar{r}_{apocentre}\simeq 193\) which are 2 simple roots of the potential \(V_r\) which is positive between these 2 values.
Note: the spinning direction of the Kerr black hole is marked on the orbit figures below.
\(\frac{\varepsilon}{c^2}=0.995\)©
The example below shows a polar orbit between \(\bar{r}_{pericentre}=6\) et \(\bar{r}_{apocentre}\simeq 192\).
In the 3D view, the trajectory appears “disordered,” but the verticality (inclination \(i=90^\circ\)) of the trajectories for large values of \(r\) is particularly visible in the top view.
\(\frac{\varepsilon}{c^2}=0.995\) (3d view)©
\(\frac{\varepsilon}{c^2}=0.995\) (top view)©
The following animated figures highlight the prograde precessions of the prograde equatorial orbit and the polar orbit.
prograde equatorial orbit
\(\frac{\varepsilon}{c^2}=0.995\) (top view)©
polar orbit
\(\frac{\varepsilon}{c^2}=0.995\) (top view)©
Zoom-whirl orbits
If \(r_{pericentre}\) is close to \(r_{ISCO}\), the physical object travels along an eccentric orbit with one or more revolutions around the black hole near the pericentre (zoom-whirl orbit).
The animated figures here are those of retrograde equatorial orbits with values \(\bar{r}_{pericentre}\) close to \(\bar{r}_{ISCO\ retrograde}\).
The first figure below shows an orbit with prograde precession and the second an orbit with retrograde precession, enabled by the large value of its apocentre.
retrograde equatorial orbit
\(\frac{\varepsilon}{c^2}=0.995\) (top view)©
retrograde equatorial orbit
\(\frac{\varepsilon}{c^2}=0.999\) (top view)©
Periodic closed orbits
The orbits can be space-closed and periodic (“3D resonant”): the physical object retraces exactly the trajectory it has already traveled.
In this case there is a relationship between the rotational speeds along each axis such that
\(\frac{\Omega_r}{n_r}=\frac{\Omega_\theta}{n_\theta}=\frac{\Omega_\varphi}{n_\varphi}\) with \(n_r\), \(n_\theta\) and \(n_\varphi\) \(\in\mathbb{Z}\).
The first animated figure below shows a prograde equatorial orbit with a resonance factor of 4 and prograde precession, while the second animated figure shows a retrograde equatorial zoom-whirl orbit with a resonance factor of 2 and retrograde precession.
prograde equatorial orbit
\(\frac{\varepsilon}{c^2}=0.991\) (top view)©
retrograde equatorial orbit
\(\frac{\varepsilon}{c^2}=0.995994\) (top view)©
Limit speeds, periods and distances covered
The limit speeds are obtained from the pericentre and apocentre and can be calculated using the formulas shown above.
The periods corresponding to the proper time of physical objects and the arc lengths (curvilinear abscissa) covered on their orbits, between 2 successive pericentres or 2 successives apocentres, can be obtained numerically from the values \(\tau,r,x,y,z\) calculated by integrating the 4 parametric equations.
For information purposes, the table below gives the rounded extreme values of the speeds among the eccentric orbits seen in this paragraph.
The periods and covered distances have been calculated for an hypothetical black hole with the mass of the sun, and the magnitude of the tidal force at the pericentre implies that only physical objects smaller than a few centimeters can follow these orbits.
| Orbit | vobs max (km/s) | vphys min (km/s) | vobs max (km/s) | vphys max (km/s) | Period (s) | Covered distance (km) |
| \(\frac{\varepsilon}{c^2}=0.999\ \ \) zoom-whirl | \(1\ 481\ \ \ \ \) | \(1\ 483\ \) | \(0.343\) | \(3\ 190\) | ||
| \(\frac{\varepsilon}{c^2}=0.995\ \ \) prograde | \(145\ 000\) | \(177\ 000\) | ||||
| \(\frac{\varepsilon}{c^2}=0.991\ \ \) resonant | \(0.0129\) | \(414\) |
Orbits with constant radial coordinate \(r\)
Another illustration can be made by now considering stable “circular” orbits in radial perturbation (constant \(r_c\)) 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 (\(\bar{r}_c\simeq 5.833\)) :
\(\bar{a}=0.95\) \(\varepsilon/c^2\simeq 0.938651\)
\(\frac{cl_z}{m\varepsilon}=0\) \(\frac{c^2Q}{m^2\varepsilon}\simeq 12.924\) ©
\(\frac{cl_z}{m\varepsilon}=0\) \(\frac{c^2Q}{m^2\varepsilon}\simeq 12.924\)©
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).
Thus, since the energy of a physical object at rest per unit mass is \(c^2\), reaching the ISCO around an extreme Kerr black hole releases an 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.