# Are stable orbits within the event horizon of a black hole possible?

This paper https://arxiv.org/abs/1103.6140 theorizes that orbits inside the event horizon of a rotating or charged black hole are not only possible but actually stable enough to potentially support life. How can this possibly be true?

Using the formula for the Schwarzchild radius of a black hole, it seems to me like any radius smaller than that is another event horizon. This would mean that any orbit inside a black hole can only be an inwards spiral, even if you negate electromagnetic or gravitational wave radiation. The paper seems to say this is not true if both the black hole and the planet were charged, but why is that? Even if they had an enormously strong charge, wouldn't the force of gravity still dominate?

Let's consider the charged non-rotating black hole since this is the simplest case. The geometry is described by the Reissner-Nordström metric, though we won't need to go into the gory details to get a basic idea of what is happening.

If you start with zero charge this is just the Schwarzschild metric. All the mass/energy is at the singularity so no matter how close you get to the centre of the black hole all the mass is still ahead of you pulling you inwards. The result is that once you pass through the horizon the $r$ coordinate becomes timelike and remains timelike as you approach and eventually crash into the singularity. No stable orbits are possible inside the horizon.

But once you charge the black hole the electric field is present both inside and outside the event horizon, and this field has an energy so it produces a gravitational force. When you're at some distance $r$ from the centre there is a part of the field behind you pulling you outwards. And the closer you get to the singularity the more of the electric field is behind you pulling you outwards.

The result is that a charged black hole has two horizons. As you pass through the outer horizon the radial coordinate becomes timelike and in this region you are doomed to fall inwards. However there is a second horizon marking the point at which the energy of the field outside balances out the mass at the singularity. As you pass through this horizon the radial coordinate becomes spacelike again and inside the second horizon you are not doomed to fall inwards. Indeed it's possible to find worldlines that travel in through the two horizons then turn round and travel back out again. For more on this see Entering a black hole, jumping into another universe---with questions.

The paper contends that inside the second horizon there are stable orbits where planets could orbit and life exist. I haven't been through the paper so I can't comment. It isn't immediately obvious that stable orbits exist inside the second horizon, after all a Schwarzschild black hole has no stable orbits for $r_s \lt r \lt 3r_s$ even though this is outside the horizon. So the fact the $r$ coordinate is spacelike doesn't guarantee stable orbits exist. However since the paper was peer reviewed I assume they have done the sums correctly.

However we should note that it is extremely unlikely a black hole would ever accumulate enough charge to move the inner horizon any great distance from the singularity, so realistically this is never going to happen for a charged black hole. With a rotating black hole it is more feasible, though I'm unsure if the supermassive black holes at galaxy cores are rotating fast enough to make it possible.

• Very helpful answer. Another thing to consider is how a planet would come to be in that inner island of stability, even if it existed. If a planet somehow made it inside without being shredded into plasma by the accretion disk, or raw material that could become a planet fell into the hole, I assume it would be moving at a very high speed inward. So considering the overall trajectory starting from outside, I imagine the "residence time" between the inner and outer horizons would be far too brief to achieve a stable orbit, or accumulate into a body Jan 3, 2022 at 3:16

They answer is yes and no (depending on what you mean by stable).

John Rennie has given a good intuitive picture of why orbits inside the horizon could be possible is the black hole is rotating or spinning. Let me give a slightly more technical answer. For this lets focus on the astrophysically relevant case of a rotating black hole. In general relativity the metric of a (perfect) rotating black hole is given by the Kerr metric.

The radial motion of a particle in the Kerr metric is controlled by the radial potential:

$$R = (E(r^2+a^2)-a L)^2 - (r^2-2r+a^2)(r^2+(aE-L)^2+Q),$$

where $$E$$ is the (specific) energy, $$L$$ the (specific) axial angular mometum, and $$Q$$ is the so-called Carter constant. (The mass of the black hole has been normalized to $$M=1$$) Geodesics oscillate between the zeros of this polynomial (in regions where R is negative). In general, since R is a fourth order polynomial in $$r$$ it has four roots, and potentially two oscillating solutions for each choice of $$a$$, $$E$$, $$L$$, and $$Q$$ (for some choices there may be only 2 or even no zeroes with real values of $$r$$, in which case we may have 1 or no oscillating solutions).

It is fairly trivial to confirm that choices of $$a$$, $$E$$, $$L$$, and $$Q$$ exist such that this polynomial has zeroes inside the inner horizon. For example, for $$a=0.9$$, $$E=0.1$$, $$L=0.1$$, and $$Q=0$$, $$R$$ has roots \begin{align} r_1 &= 1.4577\\ r_2 &= 0.562259\\ r_3 &= 0.000246485\\ r_4 &= 1.35525 \times 10^{-16}.\\ \end{align} and R is negative between $$r_2$$, and $$r_3$$. This means that there exists a timelike geodesic that oscillates between $$r_2$$, and $$r_3$$, both of which lie inside the inner horizon $$r_{-} = 0.56411$$.

Such orbits are stable in the sense that any small perturbation of the initial conditions (and thus $$E$$, $$L$$, and $$Q$$) would again lead to an oscillating solution. (This follows from the fact that the positions of the roots are smooth functions of $$E$$, $$L$$, and $$Q$$).

However, stability of the geodesic solution is only part of the stability question. While the Kerr metric outside the event horizon is thought to be stable to gravitational perturbations, this is known to be false for the part of the Kerr metric inside the (outer) event horizon. In particular, the area around the inner horizon is highly unstable with linear perturbations tending to blow up and becoming singular at the inner horizon. Consequently, the part of the Kerr metric inside the inner horizon should not be taken seriously from an astrophysical point of view, as any perturbation (such as that caused by the presence of planet in the interior) could lead to a wildly different metric.

So while these orbits maybe stable as geodesics in a fixed metric, the metric itself is unstable to perturbations. Consequently, we cannot consider these solutions as realistic "stable orbits", and any speculation of civilizations existing on a planet on such an orbit is the stuff of fairy tales.

• v. helpful answer with info both on orbits and on metric instability which is precisely what we need to hear about; thanks. Oct 26, 2021 at 9:31

This reference discusses orbits "inside" the Cauchy horizon. This region is unphysical (does not exist in real black holes) since the blueshift effect will convert the Cauchy horizon to a "final singularity" or otherwise catastrophically modify the interior structure. The details of this process are a subject of active research, but the basics are well-established. See, for example, https://arxiv.org/pdf/1912.06047.pdf

What we know is that even if they are in an orbit... They are not coming out of that blackhole's event horizon (maybe they might, as Hawking radiation).

However, it is just a hypothesis. Because we can't really observe anything like that yet and moreover, we haven't really developed our physics to the point where we can theorize about what happens in black hole past the event horizon. There are lots of hypothesis and possibilities that we can imagine and where the math works out but we can't say where they are right or wrong since we don't have tools to observe such predictions.