# Can perfectly stable orbits exist in GR?

Defining "stable orbit" between two bodies as one where, in the absence of other bodies or non-gravitational forces, the distance stays between some value pair $$r_{min}>0$$ and $$r_{max}$$. It can follow the same path every cycle, like Newtonian circular orbits, but it's not required to.

Contrasting with:

• Inward spiral orbits - those where the distance either eventually reaches or asymptotically approaches $$0$$
• Outward spiral orbits - those where the distance eventually exceeds any given finite value

I know Newtonian mechanics predicts the existence of stable orbits. The easiest of these is perfectly circular orbits, where the $$r_{min}$$ and $$r_{max}$$ values are the same. There's also elliptical orbits, which are a bit more complex.

Thing is, gravitational waves exist.

Gravitational waves result in energy loss, which reduces the distance between the bodies. Under Newtonian Mechanics, a stable orbit having its distance reduced turns it into an inward spiral orbit.

So, in light of gravitational waves, can stable orbits possibly exist? Or does every orbit eventually break either every possible $$r_{min}>0$$ or every possible $$r_{max}$$

• Mathematically it is obvious that the gravitational waves imply the answer you want. However, the rate is so slow that it might be way longer than the age of the universe. Physically, can we really say that the mathematical answer that you want actually exist when it does not happen for universes on end? Commented Jul 18 at 21:32
• Who knows? may be in quantum gravity small orbits do not emit gravitational radiation and become stable, like an electron in an atom Commented Jul 18 at 23:01
• Does "quantum gravity" count as a personal/unverified theory, or is it considered valid here? Commented Jul 19 at 8:36
• @m4r35n357 I guess you think all people doing research in that field are crackpots? and I used the conditional "who knows", "maybe". Everyone agrees that GR is not the ultimate theory of gravity. Commented Jul 19 at 15:36
• Appreciate that you say "in GR." GR and all other models are just that: models describing what has been observed, to be discarded when inconsistent with new observations. Many of us make the mistake of thinking the model is the reality instead of a method to predict the reality. Commented Jul 20 at 15:57

In the literature, the orbits you are looking for a called “floating orbits”. Floating orbits are not possible in plain GR (See e.g. 1302.1016).

• Is there a reference to why floating orbits are not possible in plain GR? While (Press and Teukolsky 1972) show that they don't work for the non-hairy Kerr hole, this is just an important special case. Commented Jul 18 at 22:16
• @AndersSandberg It has nothing to do with general relativity in the first place. Self-consistent field theories have infinite phase space and there is always a leakage of energy towards infinity. It will happen with gravitation and in quantum field theory. The transition to the ground state solution always dissipates all energy towards infinity. Commented Jul 18 at 22:57
• Barring thermodynamically-perfect systems, of course. But since those don't exist, and orbits always radiate gravitationally, there's always energy loss over time, and thus no orbit is perfectly stable. Commented Jul 19 at 0:19
• @FlatterMann The typical concern with the possible existence comes from superradiant effects which could potentially stall the inspiral. This does not happen in GR, but can happen in various scalar-Tensor theories. Note that this would still not produce a perfectly stable orbit, since you are now drawing from mass and spin of the black hole to sustain the orbit, which will deplete over time. Commented Jul 19 at 7:28

As you say, GR implies that all orbits lose energy (very slowly) over time, due to gravitational waves. Also the vacuum of space is not true vacuum and there is some drag from the intergalactic medium (also extremely small). The implication is that given enough time (in the limit as the age of the universe goes to infinite) everything will end up in black holes except for some rogue stars with high peculiar velocities. The truth is an exact answer cannot be given until we know the exact nature of dark matter or whether there is any validity in the MOND theory of gravity. The consensus is that dark energy or the expansion of the universe does not increase orbit radii, so we can ignore that.

Another issue is known as "the final parsec problem" where black holes in-spiral much faster than expected over short distance, indicating we possibly don't know the full story of how orbits decay.

This excellent website "Orbits in strongly curved spacetime" examines the instability of orbits near a black hole although it does not directly address decay due to gravitational waves.

In the 1979 paper "Time without end", Freeman Dyson calculates a time in the order of 1020 years until the earth would fall into the sun due to gravitational decay alone, based on the following calculation:

If a mass is orbiting around a fixed center with velocity V, period P, and kinetic energy E, it will lose energy by gravitational radiation at a rate of order
$$E_{g}=(V/c)^{5}(E/P)$$
Any gravitationally bound system of objects orbiting around each other will decay by this mechanism of radiation drag with a time scale
$$T_{g}=(c/V)^{5}P$$

After a duration in the order of 1024 years, the majority of planetary orbits in our galaxy would have fallen into their stars because of gravitational decay.

However, he notes that the likelihood of a close encounter between two stars, that would disrupt the orbits of their planets, is such that in the order of 1015 years, most planets will have been flung away from their stars.

Since this is far shorter than the 1024 years for their orbits to decay, a planet eventually falling into their star due to gravitational decay is extremely unlikely.

All this happens long after the sun or even the longest-lived stars have turned into white dwarfs. (longest-lived stars ~ 1014 years, sun ~ 1010 years)

The paper does not take into account any of the newer discoveries, like dark matter, dark energy, etc. But the general timescales of orbital decay compared to star life cycles and disruption of orbits due to chance encounters should still hold.

• According to en.wikipedia.org/wiki/… the Sun becomes a white dwarf in approximately $10$ billion ($10^{10}$) years, not $10$ million ($10^7$) years. Commented Jul 19 at 16:25
• Thanks, I made a conversion error there. Commented Jul 19 at 17:17
• Two thoughts: for Earth I vaguely remember Sun will go through a red giant phase and engulf Earth, but I'm not sure when. For solar system more generally, the chaotic motion will eventually fling planets off into space (but I think the timescale is so large that your other effects may happen first). Commented Jul 19 at 18:11