# Planets close to a black hole

Like in the movie Interstellar, when they go near "gargantua" the big black hole that have a planet orbiting the black hole, they go on the planet and every minute equals X years in earth time. If from this planet we would look out at the stars. Would we see stars in fast forward moving?

Does it mean that if we are inside a black hole, time is halted?

The movie had Kip Thorne as the main scientific adviser. Most of the physics was pretty good, with a couple of oddities, and the physics at the end was somewhat speculative. The theme of the movie was disappointing though. It involved humans leaving Earth for new planets, which is a nonsensical idea.

We can look at this from the perspective of the Schwarzschild metric initially. The metric is $$ds^2~=~Adt^2~-~A^{-1}dr^2~-~r^2d\Omega^2,$$ for $A~=~1~-~2GM/rc^2$ $=~1~-~2m/r$. We consider a circular orbit so that $d\Omega~=~d\phi$, $\theta~=~\pi/2$ and $dr~=~0$. This gives us $$ds^2~=~Adt^2~-~r^2d\phi^2~=~\left[A~-~r^2\left(\frac{d\phi}{dt}\right)^2\right]dt^2.$$ We now divide through by $dt^2$ to define the Lorentz gamma factor $\Gamma$ $$\frac{ds}{dt}~=~\Gamma^{-1}~=~\sqrt{1~-~\frac{2GM}{rc^2}~-~r^2\left(\frac{d\phi}{dt}\right)^2}.$$ for the tangential velocity $v~=~r\frac{d\phi}{dt}$ and for zero mass this reduces to the typical Lorentz gamma factor.

This does illustrate a time dilation. However the closest a body can orbit a Schwarzshild black hole is $r~=~3GM/c^2$. In order to get the wildly large time dilation factor in the movie something else is needed.

That something else is angular momentum of the black hole. I can illustrate this simply for an approximate Kerr solution, or one where the angular momentum is comparatively small. In this case we have the metric $$ds^2~=~Adt^2~-~A^{-1}dr^2~-~\frac{a}{r}dtd\phi~-~r^2d\Omega^2,$$ for $a~=~mJ/c$ the angular momentum parameter. This then gives the modified Lorentz gamma factor $$\frac{ds}{dt}~=~\Gamma^{-1}~=~\sqrt{1~-~\frac{2GM}{rc^2}~-~\frac{a}{r}~-~r^2\left(\frac{d\phi}{dt}\right)^2}.$$ The orbit of a mass can in this metric be closer to the black hole than $r~=~3m$ and the Lorentz gamma factor can be larger for a stable orbit. This does require a much more complete analysis with the Kerr metric, which is not entirely easy to do.

Of course while this made for interesting cinematic play, the last planet I would want to colonize is one that close to a black hole. The large wave in the film could be due to the large tidal forces on the planet. For a supermassive black hole it would have to be really super massive to not have such tidal effects across a planet.

There are a few other funnies in the film, such as the whole wormhole part. Kip Thorne is pretty big on traversable wormholes. The other is why did the space mission start with a dramatic launch with the up and coming SLS launch vehicle, but later the small craft seemed very deft at getting on and off planets. There are then of course deeper questions with respect to the black hole interior, which if you do go up a dimension does lead to the sort of alternative histories scenario.

• "the last planet I would want to colonize is one that close to a black hole"- that's the only thing I found a bit daft. They're about to get wiped out by that wave and I was thinking, "You're surprised by this?". I'm not sure the theme - which I too agree is ridiculous - is inappropriate for the movie, though. The storyline seemed to be advancing an Aeternalist viewpoint, which I'm pretty sure Thorne has at least courted in the past if not now and is perhaps about as much an approaching plausible explanation for the Earth Version 2 idea as it gets. In general I loved it. Commented Mar 11, 2017 at 9:29
• As science fiction movies go it was better than most. I would put $\it Star~ Wars$ in the category of "space opera" and not quite the same. A comparison between $\it Interstellar$ and other turkeys like $\it The~Core$ and $\it Event~Horizon$ is clear; two big thumbs down on those. The idea of Earth 2.0 is a bit wrong headed I think. Of course this was a minor effort for Kip Thorne, and he stands a good chance of getting the Nobel Prize in this year for LIGO detection of gravitational radiation. Watch for the early October announcements! Commented Mar 11, 2017 at 15:02

I think for Interstella they actually employed real physicists (I always wondered why anyone would try to do Si-Fi without one, myself). Anyway it has a reputation for accuracy in this kind of area.

The planetary explorers would experience time dilation, as the film showed, and everything in the empty space frame of reference would be affected. Time would speed up for distant objects like stars.

That said - be carefull. The distant stars do not move across the night sky once a night, it is the planet that rotates. For an observer on the planet, the planet is rotating at a normal speed because they are in the same frame of reference, and stars will appear to "move" normally across the sky.

The starlight will be blue shifted, as the frequency of light from them will be slightly higher.

A distant observer (who didn't go with the landing party for example) will see the planet rotate more slowly due to time dilation.

• They didn't just "employ physicists"; Kip Thorne himself was behind the whole thing. Commented Mar 11, 2017 at 2:03
• (The Feynman Professor of Theoretical Physics at California Institute of Technology.) Commented Mar 11, 2017 at 2:28
• @Javier Now, THAT's a physicist (to paraphrase Paul Hogan). Commented Mar 11, 2017 at 9:26
• +1 The normal star motion is a crucial point. Theoretically you'd see the stars evolve more quickly than we would on Earth, but the effect would need to be much more pronounced than even in the film for this to make for a practical observation. Commented Mar 11, 2017 at 9:32
• Ah, Kip bagged (the theoretical part of) this year's Nobel Prize. Commented Oct 3, 2017 at 12:13