# What effect does time dilation have on bodies orbiting close to Black Holes?

As far as my (somewhat basic) knowledge of astrophysics goes in general the closer to a star your orbit gets smaller (because you travel less distance) and faster (because you're deeper in the gravity well and need to go quicker to avoid falling in.)

But what happens on bodies orbiting close to black holes and subject to time dilation as far as an external observer is concerned?

This question isn't about interstellar but it's a good point of reference. Take the water planet they drop onto where gravitational time dilation is enough to make 20 or 30 years pass to the external observers while only a few hours pass for them. (I don't know if this is a realistic rate(?) of time dilation.)

What would the apparent orbit of their planet be to the external orbit and what would his orbit appear to be for them? Even though he is further from the star would he complete more orbits in the elapsed time despite being further from the centre of the system? And would they appear to be orbiting "slower" than him from his point of view.

Out of interest, though perhaps a second question is more appropriate, how would the apparent difference in relative orbital speed affect rendezvous manoeuvres if at all?

For example would the standard decelerate and fall into a lower orbit still apply but the trajectory plotted from a static observer look weird as time dilation set in? Or would the manoeuvre have to be different to account for the apparent velocity difference? What would a non-circular orbit look like in this system when observed from a circular orbit?

Also is the velocity required for orbit so close to a black hole high enough that speed based relativity also comes into play?

• Too many questions in one question. Commented Aug 28, 2021 at 8:39

## 2 Answers

Trajectories close to a black hole can't be approximated by Newtonian mechanics, General Relativity is needed. The black hole doesn't have a surface, but the Schwarzschild radius is point of no return. However it is not possible to orbit just above the Schwartzchild radius.

If you are less than 1.5 radii from the event horizon, there are no orbits. The orbital velocity closer to a black hole than 1.5 radii would be greater than the speed of light. Inside this radius, the faster you try to go, the less the centrifugal force you experience becomes, and so the faster you would fall towards the blackhole.

In fact you won't find a stable orbit closer than 3 radii from the black hole. If the black hole is spinning (as most younger ones are expected to do) the space around the black hole is dragged along by the spinning black hole. It is possible to gain energy by entering this region of dragged space around a black hole. This all means that a space ship can't remain in the region of ultra-intense gravity for many hours by orbiting, It can't experience (from the point of view of a distant observer) extended gravitation time dilation by orbiting a black hole only a few km above its Schwarzschild radius.

It is possible to approach closer than 1.5 radii. An outside observer would see the spaceship approach the black-hole in a spiral at very high speed, but not cross the event horizon. The orbit would not appear to be elliptical and it would appear time dilated. Outside of 1.5 radii the dilation would be less.

Trajectories close to the black hole would have to be calculated using GR. Newtonian mechanics would not be a close approximation. The calculation in GR would be computationally intensive, since the GR equations can't be integrated exactly, even for a two body problem, so a numerical technique would have to be used.

Provided you stay rather further from the black hole, then the usual Newtonian approximation is valid. There's no certain boundary at which Newtonian mechanics is a good approximation. It depends on the accuracy needed.

Remember also that the region close to a black hole is a violent region of space. It would be dangerous for a ship to be so close to a black hole due to the intense radiation produced by infalling matter, and tidal forces.

• Can you elaborate on where 1.5 radii comes from? I'm assuming by radii, you're referring to the Schwarzschild radius. Is the 1.5 radii the ISCO radius? That should be 3 Schwarzchild radii if that's the case and that applies only for a non-rotating black hole. Commented Feb 28, 2017 at 13:57
• The 1.5 is the radius of the photon sphere. There are no stable orbits inside the photon sphere, though there may be stable orbits far enough outside it. In practice you would need to be further away to have a hope of doing orbit manoeuvres, and you're right the ISCO =3r. I've not mentioned the ergosphere of a rotating black hole. The principle remains: if you are "close" you will have trouble finding a stable orbit, and you need GR to calculate your trajectory. If you are at a safe distance, then Newtonian mechanics does the job, but there will be insignificant time dilation Commented Feb 28, 2017 at 21:38
• Wouldn't you also have to account for the black hole's immense gravitational effect on time dilation in addition to velocity? Commented Mar 1, 2017 at 6:23
• @iMerchant At 3 schwatzchild radii the gravitational time dilation is 0.81, ie. if one hour passes for a distant observer, 48minutes pass for the astronaut near the black hole. Being at an orbital distance is not enough to turn a few hours near a black hole into 20 or 30 years for an outside observer. There is also the dilation due to orbital velocity. I haven't done the maths, but it is probably on the same order. You can't remain in the ultra high gravity region by orbiting. Commented Mar 1, 2017 at 17:43
• In my digging, I found this breaking down the extreme time dilation effect in Interstellar: relativitydigest.com/2014/11/07/on-the-science-of-interstellar A fair chunk of it goes over my head, but it suggests the mass of the blackhole and its rotation contributes to the innermost stable orbits being in an area of significant time dilation. Out of curiosity in a non-rotating black hole does the time dilation at any particular schwarzchild radius remain the same? Commented Mar 3, 2017 at 4:29

A body around a non-rotating black hole (side note: non-spinning black holes are almost impossible at this young age of the universe), as said above, can't find a stable orbit any closer than 3 times the Schwarzschild radius. If this black hole was devouring mass at the time, this area would be filled with tiny partials at billions of K. not a great place to hang around.

If the black hole is spinning (again as said already), then the fabric of space-time will be pulled around in a spiral. (like a whirlpool but with more dimensions). in general relativity, the 'Kerr metric' will be used to calculate the movement of body's around such a black hole.

The effect of the spiral of spacetime is that the minimum orbit will be reduced. How much it's reduced depends on how fast the black hole is spinning and how massive it is. This area of spacetime is not a nice place to be if the blackhole is not very large and is spinning fast, because the tidal forces will reduce you to atoms after stretching you into spaghetti. However, if the black hole is large enough, the tidal forces will be weak enough to allow large objects to continue existing.

As a side note, for a planet with liquid water (as in inside the habitable zone) (like from interstellar) to orbit a black hole, the black hole would have to be greater than 163 million solar masses, spin at a 100-millionth of a percent off of the speed of light, and not have consumed mass for a long time (so it has a small accretion disk). (such a planet would have its atmosphere ripped off and freeze quite quickly)

A critical point in general relativity is that everything is traveling at the speed of light through spacetime, and all we can change is its distribution throughout the dimensions of space and time. Light travels at the speed of light (duh), which means that it sees time go by with a speed of 0. A close orbiting planet would travel very quickly, and to the external viewer, it would probably appear squished sideways (length contraction). To something on the planet, the planet would travel normally, while the rest of the universe flew by around them. The black hole would fill over half the sky as light was pulled around the black hole. In some places on the planet, it might be able too see your own planet when looking up , if the light was pulled all the way around the black hole. In what's left, looking up into the sky would be surreal. Supermassive stars could be born and then die again in a human lifetime, as time dilation slows everything down. To a distant observer, the planet would move slowly (because it's time is slower than the observers), be contracted, and possibly have a slight red tint as light coming from it is pulled around.

A ship approaching the planet would be accelerated to very fast speeds, so it's time would slow down. as it shifts into the reference of the planet, the rest of the universe would start to fly by, while the planet would appear more like a ball. There would be a beautiful moment of watching the planet and the black hole, and then the ship would probably crash.

• If the observer on the planet can see starts forming and dying, the time dilation is huge, so it would not be slight red tint, the planet would be just invisible in visible light. Commented Aug 28, 2021 at 15:37