GRAVITATION

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, without backreaction of the particles or objects on the Kerr spacetime.
The trajectories depend on three fundamental constants of motion — \(\varepsilon\), \(l_z\), \(Q\) — whose values determine the possible dynamical regimes: absorption, 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 relativistic specific energy defines the physical constraints required to place probes or spacecraft on stable orbits around a rotating black hole.

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 \(M\).

POTENTIALS

General case

\(V_r\) and \(V_\theta\) can be considered respectively as the radial and polar effective potentials for null or time-like trajectories.
For a space-like geodesic, it is not really a matter of 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.

Apart from the orbits discussed later on this page, null, time-like or space-like geodesics may, depending on the values of the constants:

• be unbounded (the repulsive point is a simple positive root of the potential \(V_r\) greater than \(r_h\) (event horizon)),
• ”end up in the black hole”, with 3 possible cases:
  • collide with the physical body of the black hole (null or time-like geodesic),
  • exiting our universe (the repulsive point is a simple positive root of the potential \(V_r\) less than \(r_{Cauchy}\) (Cauchy horizon)),
  • reaching the central disk bounded by the ring singularity and entering negative space (no positive root of the potential \(V_r\)).

An example of a time-like geodesic is shown below, with the trajectory as seen by a stationary observer at infinity, and the trajectory actually followed by the physical object, which here reaches the ring singularity.

Example of \(V_r\) potential curves

Special case inside Cauchy horizon

An interesting special case is that of a physical object that follows a “confined” trajectory between the inner ergosphere and the Cauchy horizon: it remains “trapped” in this region, with a radial coordinate varying between two values \(r_{min}\) and \(r_{max}\).
This is an oscillating geodesic, which is in fact the same type as an eccentric orbit (see paragraph below).
The constants of the motion \(\varepsilon\), \(l_z\), and \(Q\) are such that the radial effective potential \(V_r\) has 4 real roots, the 2 smallest of which are less than \(r_{Cauchy}\). The limit case reduces to a circular orbit when these two roots are double.
In the example below, \(\bar{r}_{min}=0.2\) and \(\bar{r}_{max}\simeq 0.286\) with \(\bar{r}_{Cauchy}\simeq 0.688\) for a Kerr parameter \(\bar{a}=0.95\).
In the 3d view, the Cauchy horizon is shown in light gray and the inner ergosphere in dark gray. The outer ergosphere and event horizon are not shown. The trajectory is prograde (same spin as the Kerr black hole spin marked on the z-axis).

Note : for a physical object coming from \(\infty\) to follow such a trajectory, it is necessary first of all that this trajectory does not encounter the physical body constituting the black hole and, above all, that the energy of the physical body per unit mass \(\varepsilon\) divided by \(c^2\) decreases to the value of this example, that is, \(0.999\), which requires a very strong braking, for example by collision with another object.

SPEEDS OF A PHYSICAL OBJECT

Speed for an observer at \(\infty\)

As a way of comparison with classical mechanics, the radial, polar, and azimuthal components of the observable speed of a physical object, as seen by an observer located in the asymptotic region and as a function of his proper time \(t\) are written as:
\(v_{obs\ r}=\sqrt{g_{rr}}\left(\frac{dr}{dt}\right)\), \(v_{obs\ \theta}=\sqrt{g_{\theta\theta}}\left(\frac{d\theta}{dt}\right)\) and \(v_{obs\ \varphi}=\sqrt{g_{\phi\phi}}\left(\frac{d\phi}{dt}\right)\)
which gives a speed \(v_{obs}=\sqrt{v_{obs\ r}^2+v_{obs\ \theta}^2+v_{obs\ \varphi}^2}\)
which is calculated using the values \(r\), \(\theta\), \(\phi\), and \(t\) obtained by numerically integrating the parametric equations.
Due to time dilatation 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 as a function of its proper time \(\tau\) can be measured locally by an observer with zero angular momentum (Zero Angular Momentum Observer – ZAMO).
Its radial, polar, and azimuthal components can be written for \(\Delta>0\) as:
\(v_{phys\ r}=\frac{\sqrt{g_{rr}}}{\gamma}\frac{dr}{d\tau}\), \(v_{phys\ \theta}=\frac{\sqrt{g_{\theta\theta}}}{\gamma}\frac{d\theta}{d\tau}\) and \(v_{phys\ \varphi}=\frac{\sqrt{g_{\varphi\varphi}}}{\gamma}\left(\frac{d\varphi}{d\tau}+c\frac{g_{0\varphi}}{g_{\varphi\varphi}}\frac{dt}{d\tau}\right)\),
where \(\gamma\) is the Lorentz factor (see calculation at the end of the paragraph),
and are calculated using the values \(r\), \(\theta\), \(\varphi\), and \(t\) obtained by numerically integrating the parametric equations.
Finally, the norm of the physical speed is simply written as:
\(v_{phys}=c\sqrt{1-\frac{1}{\gamma^2}}\).

Note: in the case of a stellar black hole, the tidal force is so strong due to the low value of the pericentre \(r_{per}\) that the asteroid or star will be destroyed long before reaching the pericentre. However, these orbits are applicable to a supermassive black hole.

Calculation of the Lorentz factor and the gravitational expansion factor

The Lorentz factor is defined as the negative of the scalar product of the four-velocities of 2 objects divided by \(c^2\), that is, \(\gamma=-g_{\mu\nu}\left(\frac{dx}{d\tau}\right)^{\mu}\left(\frac{dx}{d\tau}\right)^{\nu}/c^2\).

Calculation of the four-velocity of a ZAMO
By definition, the ZAMO has zero angular momentum \(g_{0\varphi}c\frac{dt}{d\tau_{ZAMO}}+ g_{\varphi\varphi}\frac{d\varphi}{d\tau_{ZAMO}}=0\)
which gives \(\frac{d\varphi}{dt}=-c\frac{g_{0\varphi}}{g_{\varphi\varphi}}\).
Furthermore, the normalization of its four-velocity gives with \(r\) and \(\theta\) constant: \(g_{00}c^2\left(\frac{dt}{d\tau_{ZAMO}}\right)^2+2g_{0\varphi}c\frac{dt}{d\tau_{ZAMO}}\frac{d\varphi}{d\tau_{ZAMO}}+g_{\varphi\varphi}\left(\frac{d\varphi}{d\tau_{ZAMO}}\right)^2=-c^2\).
Writing \(\frac{d\varphi}{d\tau_{ZAMO}}=\frac{d\varphi}{dt}\frac{dt}{d\tau_{ZAMO}}\) we obtain \(g_{00}c^2\left(\frac{dt}{d\tau_{ZAMO}}\right)^2+2g_{0\varphi}c\frac{dt}{d\tau_{ZAMO}}\frac{d\varphi}{dt}\frac{dt}{d\tau_{ZAMO}}+g_{\varphi\varphi}\left(\frac{d\varphi}{dt}\frac{dt}{d\tau_{ZAMO}}\right)^2=-c^2\)
which, upon substituting \(\frac{d\varphi}{dt}\) with its value, gives \(g_{00}c^2\left(\frac{dt}{d\tau_{ZAMO}}\right)^2-2c^2\frac{g_{0\varphi}^2}{g_{\varphi\varphi}}\left(\frac{dt}{d\tau_{ZAMO}}\right)^2+c^2\frac{g_{0\varphi}^2}{g_{\varphi\varphi}}\left(\frac{dt}{d\tau_{ZAMO}}\right)^2=-c^2\)
that is, \(\left(\frac{dt}{d\tau_{ZAMO}}\right)^2\left(g_{00}-\frac{g_{0\varphi}^2}{g_{\varphi\varphi}}\right)=-1\) or \(\frac{dt}{d\tau_{ZAMO}}=\frac{1}{\alpha}\)
where \(\alpha=\sqrt{\frac{g_{0\varphi}^2}{g_{\varphi\varphi}}-g_{00}}=\frac{d\tau_{ZAMO}}{dt}\), the lapse function or local gravitational expansion factor.
Note: after expanding the coefficients of the metric tensor \(g_{00}\), \(g_{0\varphi}\), and \(g_{\varphi\varphi}\) and regrouping the terms, we also have:
\(\alpha^2=\frac{\Delta\Sigma}{(r^2+a^2)^2-\Delta a^2\sin^2\theta}\).
The four-velocity of the ZAMO is then defined as: \(\left(\frac{1}{\alpha},0,0,-\frac{c}{\alpha}\frac{g_{0\varphi}}{g_{\varphi\varphi}}\right)\).

Scalar product of the four-velocities of a physical object and a ZAMO
Given the four-velocity of the physical object (\(\frac{dt}{d\tau},\frac{dr}{d\tau},\frac{d\theta}{d\tau},\frac{d\varphi}{d\tau}\)), the scalar product of the 2 four-velocities is written as: \(g_{00}\frac{dt}{d\tau}\frac{c^2}{\alpha}-2\frac{dt}{d\tau}\frac{g_{0\varphi}^2}{g_{\varphi\varphi}}\frac{c^2}{\alpha}-g_{\varphi\varphi}\frac{d\varphi}{d\tau}\frac{g_{0\varphi}}{g_{\varphi\varphi}}\frac{c}{\alpha}\).
Setting \(\frac{d\varphi}{d\tau}=\frac{d\varphi}{dt}\frac{dt}{d\tau}=-c\frac{g_{0\varphi}}{g_{\varphi\varphi}}\frac{dt}{d\tau}\), the scalar product is \(\frac{c^2}{\alpha}\left(g_{00}-\frac{g_{0\varphi}^2}{g_{\varphi\varphi}}\right)\frac{dt}{d\tau}=-\frac{c^2}{\alpha}\alpha^2\frac{dt}{d\tau}=-c^2\alpha\frac{dt}{d\tau}\), which finally gives \(\gamma=\alpha\frac{dt}{d\tau}\).
Let us now apply the normalization of the four-velocity of the physical object: \(g_{00}c^2\left(\frac{dt}{d\tau}\right)^2+2g_{0\varphi}c\frac{dt}{d\tau}\frac{d\varphi}{d\tau}+g_{rr}\left(\frac{dr}{d\tau}\right)^2+g_{\theta\theta}\left(\frac{d\theta}{d\tau}\right)^2+g_{\varphi \varphi}\left(\frac{d\varphi}{d\tau}\right)^2=-c^2\).
Rewriting \(2g_{0\varphi}c\frac{dt}{d\tau}\frac{d\varphi}{d\tau}+g_{\varphi \varphi}\left(\frac{d\varphi}{d\tau}\right)^2=g_{\varphi\varphi}\left(\frac{d\varphi}{d\tau}+c\frac{g_{0\varphi}}{g_{\varphi\varphi}}\frac{dt}{d\tau}\right)^2-c^2\frac{g_{0\varphi}^2}{g_{\varphi\varphi}}\left(\frac{dt}{d\tau}\right)^2\), we obtain
\(c^2\left(g_{00}-\frac{g_{0\varphi}^2}{g_{\varphi\varphi}}\right)\left(\frac{dt}{d\tau}\right)^2+g_{rr}\left(\frac{dr}{d\tau}\right)^2+g_{\theta\theta}\left(\frac{d\theta}{d\tau}\right)^2+g_{\varphi\varphi}\left(\frac{d\varphi}{d\tau}+c\frac{g_{0\varphi}}{g_{\varphi\varphi}}\frac{dt}{d\tau}\right)^2=-c^2\)
or \(-c^2\gamma^2+g_{rr}\left(\frac{dr}{d\tau}\right)^2+g_{\theta\theta}\left(\frac{d\theta}{d\tau}\right)^2+g_{\varphi\varphi}\left(\frac{d\varphi}{d\tau}+c\frac{g_{0\varphi}}{g_{\varphi\varphi}}\frac{dt}{d\tau}\right)^2=-c^2\).
Setting \(v^2=\frac{g_{rr}}{\gamma^2}\left(\frac{dt}{d\tau}\right)^2+\frac{g_{\theta\theta}}{\gamma^2}\left(\frac{d\theta}{d\tau}\right)^2+\frac{g_{\varphi\varphi}}{\gamma^2}\left(\frac{d\varphi}{d\tau}+c\frac{g_{0\varphi}}{g_{\varphi\varphi}}\frac{dt}{d\tau}\right)^2\), the normalization is written as:
\(-c^2\gamma^2+v^2\gamma^2=-c^2\) that is, \(v^2=-\frac{c^2}{\gamma^2}+c^2=c^2\left(1-\frac{1}{\gamma^2}\right)\) which gives \(v=c\sqrt{1-\frac{1}{\gamma^2}}\)
or also \(\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\).

Another expression of the Lorentz factor
Given that \(\frac{dt}{d\tau}=\left((r^2+a^2)^2-\Delta a^2\sin^2\theta-2mar\ c\frac{l_z}{\varepsilon}\right)\frac{\varepsilon}{c^2\Delta\Sigma}\)
and given that \((r^2+a^2)^2-\Delta a^2\sin^2\theta=\Sigma\frac{g_{\varphi\varphi}}{\sin^2\theta}\) and that \(-2mar\ c\frac{l_z}{\varepsilon}=\Sigma\frac{g_{0\varphi}}{\sin^2\theta}\) :
\(\gamma^2=\alpha^2\left(\frac{\Sigma}{\sin^2\theta}g_{\varphi\varphi}\left(1+m\frac{g_{0\varphi}}{g_{\varphi\varphi}}\frac{cl_z}{m\varepsilon}\right)\frac{\varepsilon}{c^2\Delta\Sigma}\right)^2\) and with \(\Delta\Sigma=\alpha^2\frac{\Sigma }{\sin^2\theta}g_{\varphi\varphi}\), we obtain, with the condition \(\Delta>0\):
\(\gamma=\frac{\varepsilon}{c^2}\frac{\left(1+m\frac{g_{0\varphi}}{g_{\varphi\varphi}}\frac{cl_z}{m\varepsilon}\right)}{\alpha}\).

STABLE ORBITS OF PHYSICAL OBJECTS

Eccentric orbits

General case

When the effective radial potential \(V_r\) with \(\frac{\varepsilon}{c^2}<1\) is positive between 2 simple real roots located outside the outer ergosphere, the geodesic is a stable orbit of the physical object, with the 2 roots corresponding to the pericentre and the apocentre.
An example is given below with the rosette of an equatorial orbit between \(\bar{r}_{per}\simeq 6\) and \(\bar{r}_{apo}\simeq 193\) which are 2 simple roots of the potential \(V_r\) which is positive between these 2 values.
Note: the spin of the Kerr black hole is marked on the z-axes of the orbit figures below.

The example below shows a polar orbit between \(\bar{r}_{per}=6\) et \(\bar{r}_{apo}\simeq 192\).
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.

The following animated figures highlight the prograde precessions of the prograde equatorial orbit and the polar orbit.
Note: see the definition of the Innermost Stable Circular Orbit (\(ISCO\)) in the following paragraph.

Zoom-whirl orbits

If \(r_{per}\) 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 \(\bar{r}_{per}\) 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.

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.

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 three 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 min (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	\(1\ 481\ \ \ \ \)	\(1\ 483\ \)	\(135\ 800\)	\(185\ 100\)	\(0.343\)	\(3\ 190\)
\(\frac{\varepsilon}{c^2}=0.995\ \ \) prograde	\(5\ 854\)	\(5\ 870\)	\(145\ 000\)	\(158\ 100\)	\(0.03106\)	\(688\)
\(\frac{\varepsilon}{c^2}=0.991\ \ \) résonante	\(10\ 670\)	\(10\ 720\)	\(143\ 700\)	\(156\ 600\)	\(0.0129\)	\(414\)

ORBITS WITH CONSTANT RADIAL COORDINATE \(r\)

Innermost Stable Circular Orbits – 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 \(\frac{c^2l^2}{m^2}\).

2) The cancellation of \(\frac{dV_r}{dr}\) can be written as \(2(\varepsilon^2+\kappa c^4)\bar{r}^3-3\kappa c^4\bar{r}^2+\frac{c^2l^2}{m^2}(1-\bar{r})=0\).

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 \(r_{ISCO}\)

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 assuming that \(\varepsilon^2>0\) 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.

Innermost stable circular orbits – Kerr black hole with \(\bar{a}\ne 0\)

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:
\(\bar{r}_{ISCO\ prograde}=3+z_2-\sqrt{(3-z_1)(3+z_1+2z_2)}\) and
\(\bar{r}_{ISCO\ retrograde}=3+z_2+\sqrt{(3-z_1)(3+z_1+2z_2)}\)
with \(z_1=1+(1-\bar{a}^2)^{1/3}((1+\bar{a})^{1/3}+(1-\bar{a})^{1/3})\) and \(z_2=\sqrt{3\bar{a}^2+z_1^2}\).
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.

Orbits with constant radial coordinate \(r\)

In general, for a physical object, there exist stable “spherical” orbits in radial perturbation (constant \(r_c\)) with any type of inclination \(i\).
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 (\(\bar{r}_c\simeq 5.833\)) :

Furthermore, the initial value of \(\frac{\varepsilon}{c^2}>1\) 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 relativistic specific 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 relativistic specific energy

The radius \(r_c\) of a distant circular orbit as a function of \(\frac{\varepsilon}{c^2}<1\) can be calculated using the 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 \(V_r\).

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.