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In the movie Interstellar, Miller's planet is so close to the black hole that it experiences severe time dilation - one hour on the planet is seven years for every one else.

The planet is (presumably) in a stable orbit around the black hole and in no danger of falling in any time soon. Otherwise, it wouldn't be a very good candidate for new Earth. Thus, it must be speeding around the black hole very quickly to have enough angular momentum to avoid falling in.

Given that the gravity is causing such a severe time dilation, how fast are they travelling?

I suspect that it is very fast since even at 90% of the speed of light (something we are not even close to achieving for macroscopic objects), there is little time dilation.

EDIT: I was incorrectly equating time dilation due to speed with gravitational time dilation, so ignore the above sentence. They are related, but not in the way I suggest.

So, how fast did they need to accelerate the ship to match Miller's orbit and land? And again, when they take off, they would need to decelerate to match that of the mother ship. Wouldn't that delta-v have used up most of their fuel? That alone should have precluded them from attempting to visit this planet until they exhausted Mann's and Edmund's planets.

Highly related question: Since there is such a severe time dilation, if you measured your speed while on Miller's planet, you'd see that it was substantially higher than when measured from the mother-ship.

Would the planet appear to be orbiting the black hole very slowly from the perspective of the mother-ship, even though it is orbiting very fast?

Does that mean that as objects fall into black holes, outside observers perceive it to take longer for the object to fall in due to the time dilation? I believe this is the case.

So as the Endeavour approaches the planet, they would perceive that the planet is accelerating and would need to speed up to catch it. Bromily from the mother ship might see them as travelling a constant speed, or (most likely) even slowing down as the approach the planet, when in reality they are accelerating towards it.

Likewise, when they leave the planet, Bromily would see them very slowly coming towards him, but accelerating as they got nearer. In reality, however, they would be decelerating to match the speed of the mother ship.

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    The speed of the planet isn't what causes the time dilation- see scifi.stackexchange.com/questions/72173/…. – PointlessSpike Jan 27 '15 at 16:24
  • @PointlessSpike You are right - the acceleration to that speed is what causes it. – Trenin Jan 27 '15 at 16:26
  • @PointlessSpike But the question remains - in order to achieve that time dilation, one must accelerate to that speed. – Trenin Jan 27 '15 at 16:27
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    @Trenin Gravity itself causes time dilation, not only high velocity/acceleration. Even if you're at rest in such strong gravity, you would experience time dilation. – Martin Ender Jan 27 '15 at 17:06
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    @Trenin By, "at rest" I mean literally at rest, i.e. that you'd somehow balance the gravitational force (e.g. through a very powerful jet blast) - ignoring the fact that this is impossible for humans to achieve technically, if you were actually at rest near a black hole, you would experience time dilation (compared to someone further away from the black hole) without being in motion. Therefore, you can't attribute the time dilation purely to the motion, and I think the idea was even that the dilation is predominantly due to the gravity, and not the motion. – Martin Ender Jan 27 '15 at 17:21
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Chapter 17 of The Science of Interstellar by physicist Kip Thorne (who was a consultant on the movie, and co-wrote the original script treatment) discusses Miller's planet and its orbit around Gargantua (the supermassive black hole seen in the movie, said in the book to have a mass about 100 million times greater than the Sun), and says:

Einstein's laws dictate that, as seen from afar, for example, from Mann's planet, Miller's planet travels around Gargantua's billion-kilometer circumference orbit once each 1.7 hours. This is roughly half the speed of light! Because of time's slowing, the Ranger's crew measures an orbital period sixty thousand times smaller than this: a tenth of a second. Ten trips around Gargantua per second. That's really fast! Isn't it far faster than light? No, because of the space whirl induced by Gargantua's fast spin. Relative to the whirling space at the planet's location, and using time as measured there, the planet is moving slower than light, and that's what counts. That's the sense in which the speed limit is enforced.

The "space whirl" he mentions refers to an effect called frame dragging, which can be thought of as space being whirled around the rotating black hole, which he discussed earlier in chapter 5. So from the point of view of distant observers, the planet completes an orbit once every 1.7 hours, so if they are using a coordinate system where the circumference is a billion kilometers this would be 588.24 million kilometers per hour, or about 163,400 kilometers per second, or about 55% the speed of light.

The quote above also answers your question "Would the planet appear to be orbiting the black hole very slowly from the perspective of the mother-ship, even though it is orbiting very fast?" In a way it depends on what you mean by "very slowly", but the answer is that the period of an orbit observed on the planet is much faster than the period observed from afar, by the same time dilation factor of around 61,000 that relates aging on the planet to the aging of faraway observers (since an orbit every tenth of a second is 61,200 times more than an orbit every 1.7 hours).

As for how they manage to navigate from one orbit to another, Thorne discusses this in chapter 7. Basically his answer is that although their rockets alone wouldn't be sufficient, they use gravitational slingshots past other massive objects in orbit around Gargantua, including smaller black holes and neutron stars. Quoting Thorne again:

In my science interpretation of Interstellar, the Endurance, parked at ten Gargantua radii while the crew visit Miller's planet, moves at one-third the speed of light: c/3, where c represents the speed of light. Miller's plaet moves at 55 percent the speed of light, 0.55c.

To reach Miller's planet from the parking orbit in my interpretation (Figure 7.1), the Ranger must slow its forward motion from c/3 to far less than that, so Gargantua's gravity can pull it downwards. And when it reaches the vicinity of the planet, the Ranger must turn from downward to forward. And, having picked up far too much speed while falling, it must slow by about c/4 to reach the planet's 0.55c speed and rendezvous with it.

...

Fortunately, Nature provides a way to achieve the huge speed changes, c/3, required in Interstellar: gravitational slingshots around black holes far smaller than Gargantua.

Stars and small black holes congregate around gigantic black holes like Gargantua (more on this in the next section). In my science interpretation of the movie, I imagine that Cooper and his team make a survey of all the small black holes orbiting Gargantua. They identify one that is well positioned to gravitationally deflect the Ranger from its nearly circular orbit and send it plunging downward towards Miller's planet (Figure 7.2). This gravity-assisted maneuver is called a "gravitational slingshot," and has often been used by NASA in the solar system—though with the gravity coming from planets rather than a black hole (see the end of the chapter).

This slingshot maneuver is not seen or discussed in Interstellar, but the next one is mentioned, by Cooper: "Look, I can swing around that neutron star to decelerate," he says.

...

To change velocities by as much as c/3 or c/4, the Ranger must come close enough to the small black hole and neutron star to feel their intense gravity. At those close distances, if the deflector is a neutron star or is a black hole with a radius less than 10,000 kilometers, the humans and rangers will be torn apart by tidal forces (Chapter 4). For the Ranger and humans to survive, the deflector must be a black hole at least 10,000 kilometers in size (about the size of the Earth).

Now, black holes that size do occur in Nature. They are called intermediate-mass black holes, or IMBHs, and despite their big size, they are tiny compared to Gargantua: ten thousand times smaller.

So Christopher Nolan should have used an Earth-sized IMBH to slow down the Ranger, not a neutron star. I discussed this with Chris early in his rewrites of Jonah's screenplay. After our discussion, Chris chose the neutron star. Why? Because he didn't want to confuse his mass audience by having more than one black hole in the movie. One black hole, one wormhole, and also a neutron star, along with Interstellar's other rich science, all to be absorbed in a fast-paced two-hour film; that was all Chris thought he could get away with. Recognizing that strong gravitational slingshots are needed to navigate around Gargantua, Chris included one slingshot in Cooper's dialog, at the price of using a scientifically implausible deflector: the neutron star instead of a black hole.

Finally, this may be more than you need to know, but one tricky aspect of this stuff is that in relativity there is no absolute notion of "speed", and likewise no absolute notion of the "circumference" of a black hole, these notions depend on the spacetime coordinate system you use to label physical events with position and time coordinates (and speed is then the rate of change in coordinate position with respect to coordinate time). In special relativity, which deals with the effects of high velocity in regions far from gravity, the notion that light always travels at the same speed, denoted by the constant c, is only true of a particular class of coordinate systems known as inertial frames; if you choose a "non-inertial" coordinate system such as Rindler coordinates in which a group of accelerating observers are treated as being at rest, then light's speed can be non-constant. In general relativity, which analyzes gravity in terms of the idea that mass curves the fabric of spacetime, all large-scale coordinate systems on curved spacetime are non-inertial, but if you zoom in a very small region of spacetime, in the limit as its size approaches zero you can define "local inertial frames" in that region, in which objects in free fall (not being acted on by any non-gravitational forces) are treated as moving at constant velocity, and such free fall observers using local inertial frames will measure the basic laws of physics in this small region to work the same way as in an inertial frame far from gravity (the equivalence principle), including the fact that light rays move at c. So in general relativity it's still true that light always has a velocity of c as measured locally by free-falling observers, and I believe that's what Thorne means in the first quote by "Relative to the whirling space at the planet's location, and using time as measured there, the planet is moving slower than light", despite the fact that the planet is completing an orbit ten times every second from the perspective of observers standing on it.

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    Wow! That is crazy... I don't think I understand it all, but it makes more sense than before. Thanks for the info!! – Trenin Jan 29 '15 at 17:35
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Looks like there is already an answer here.

The planet is travelling about 50% the speed of light in orbit around the black hole.

As at how fast they would need to go to leave the planet and return to the mother ship, it is 82% of the speed of light, shown here.

  • That second linked answer isn't correct for Interstellar because it's based on a non-rotating black hole, whereas the black hole in Interstellar is said to be rotating extremely rapidly, which changes the properties of the spacetime quite a bit. – Hypnosifl Jan 27 '15 at 18:24
  • @Hypnosifl Do you have the correct or closer answer? – Trenin Jan 27 '15 at 18:37
  • I have a little more info on the speed of the planet and some thoughts from Kip Thorne about how they do relativistic orbital maneuvers, I'll post later today or tomorrow. – Hypnosifl Jan 27 '15 at 19:22
  • @Hypnosifl Also, do you know what this would look like from the observer? See my second question in the original question. I assume that it would look like the craft slows down as it approaches the planet, but speeds up as it comes back, even though it is doing the opposite. – Trenin Jan 27 '15 at 19:42

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