# How much of a gravitational time dilation did the three planets in Interstellar experience?

In Interstellar, there are multiple planets that are accessible via a worm hole/gravitational anomaly near Saturn. Of twelve, only 3 were considered the most promising, and the ones that returned a signal from the survey missions. All 3 orbited a massive blackhole Gargantua.

### How much of a gravitational time dilation, relative to Earth, did each planet experience? How much of this dilation is due to Gargantua, compared to each planet's own gravity?

Even on Earth, GPS satellites in orbit around the planet experience relative time dilation, due to speed and distance from the surface. On average, 38 µS per day.

We know that Miller's Planet was the worst, with 7 years Earth to 1 hour Miller's Surface relative dilation. But it also experiences 130% gravity at the surface compared to Earth. How much of the 7 years is from the surface gravity compared to Gargantua's gravity?

What about Edmund's and Mann's? Or were they Earth typical?

• Am fighting the temptation to flag this question to export to Physics StackExchange :D :D
– user30432
Commented Feb 8, 2016 at 22:19
• A 30% increase in weight is a relatively minor difference. The planet would have a slightly larger mass than ours, but nothing significant as far as time dilation... they'd only lose a fraction of a second per year or so. I'm no physicist, so I can't do the math for you for that. However, the black hole probably had 10 or 20 solar masses (or more, can't remember if they said in the movie), and they were in close to it... it was responsible for the vast majority of any relativistic effects. Commented Feb 8, 2016 at 23:11

Now first of all, I'm not a huge expert in general relativity, so I'll go from a layman's understanding of it, combined with what the film actually tells us and a few bits and "known" facts about that galaxy from executive producer and scientific advisor Kip Thorne's book The Science of Interstellar.

# Time dilation from the planets alone

Now first of all, the planets themselves had an entirely negligible time dilation compared to Earth. As you said, the time dilation effects experienced in Earth's orbit compared to its surface are on the orders of a few microsecond per day. Now keep in mind, that the planets had largely similar gravity compared to earth (in particular 130% and 80% of Earth's gravity for Miller's and Mann's planet respectively, and judging from Brand's bevahiour at the end we can assume Edmunds' planet to be largely similar). So those planets are largely similar to earth in their density, radii and masses.1 This means the time dilation should be largely similar among all of them and thus also similar to Earth, since that factor only depends on radius and mass, too (while not being exactly linearly proportional to gravity). Such gravitational effects only come into play at a significant degree for very large masses and speeds, not if a puny planet's gravity is just a bit larger than Earth's.

But for Miller's planet we can actually compute the time dilation based on its perfectly specified properties1 and the formulas for gravitational time dilation. Using the Schwarzschild metric we obtain that time on Miller's surface moves about 2.33 µs/h (microseconds per hour) slower compared to a non-dilated observer at an arbitrarily far away point. If we compare this to Earth's time dilation of 2.506 µs/h we see that in fact Miller's planet's own time dilation (disregarding Gargantua) is even less than Earth's (i.e. time moves faster there).2 In other words, without Gargantua's influence 1 hour on Miller's planet would only be ~99.999999995% of an hour on Earth rather than 7 years (but this leaves out any other possible time dilating factors that could be involved here. As said, I'm not a physicist and those values are based on the planets' gravitional potential only). So the entirety (plus a tiny little more) of the time dilation experienced on Miller's planet comes from Gargantua alone.

But whatever the exact time dilation of the planets is, we have to realize that this is absolutely irrelevant at the scales of the other time dilational effects the film explores, especially the effects due to Gargantua. After all that black hole has a mass of about 100 million suns (and thus tens of trillion times that of Earth or the other planets for that matter) and even more than that, the majority of its time dilation is not only caused by its mass alone, but by its very high spin.3 If you compare a few microseconds to 7 years it is clear that we can entirely ignore those planets' own time dilation in every aspect. In fact their time dilational effects are so comparatively small, that they probably lie below any of the rather crude and simple approximations made with all the other measurements.

# Time dilation from Gargantua

But what about the time dilation effects that Gargantua has on the other two planets? Unfortunately, this isn't really elaborated much in the film and all we can really say is, that they're neglibible, too, compared to the scale of the time dilation experienced throughout the movie. What we can base this on is the fact that neither Mann's nor Edmunds' planet are actually that close to Gargantua at all, as they repeatedly say in the film that it would take them months to reach the other planets.4

We then also have to take into account that Gargantua's time dilation drops rather rapidly the farther we get from it. I don't have my general relativity formulas ready (and the fact that it's spinning really fast makes the simpler Schwarzschild metric not applicable), but this can be seen directly in the movie, when Romilly actually stays in a larger orbit around Gargantua while the crew explores Miller's planet. Now it's clear Romilly isn't that far from the planet, since they just get from and to the Endurance with the Ranger in a matter of minutes (hours at worst). But he still does not experience any time dilational effects (or again, not any compared to the heavy effects the crew experiences).

# Conclusion

So on the bottom line, the individual planets' time dilation is largely similar to Earth's and entirely negligible compared to Gargantua's and Miller's planet was the only one that was sufficiently near to the black hole to experience a significant time dilation from it. Everything else and any specific numbers lie entirely beyond the scope of what the film or even its scientific advisor's actual book about the science behind it can offer us.

1) In particular, Miller's planet's density is about 181% of Earth's (10,000 kg/m^3, as given by Kip Thorne), from which we get with a little computation the planet's radius as ~4,562 km, which is about 72% of Earth's radius, and it's mass as ~3.977*10^24 kg, which is about 67% of Earth's mass. Those numbers are just to show you that not only the planets' gravities are largely similar, but also their other properties on which the gravity as well as the time dilation depend. For Mann's and Edmunds' planet Thorne didn't give any more information, we can however assume that they are similar to those of Earth/Miller due to them being structured similarly (though, Mann's planet is a admittedly a little more special).

2) This on first sight unusual discrepancy between the planet having more gravity but less time dilation can be explained by the fact that Miller's planet does actually have a lower mass than Earth and it's higher gravity comes from its larger density and thus smaller size, putting its surface nearer to the center. Yet the planet's radius has a weaker influence on its time dilation that it has on its gravity.

3) Again employing information given by Kip Thorne himself. In fact Thorne went out of his way to justify the humongous time dilation that Nolan wanted for Miller's planet and only achieved it by letting the black hole spin at nearly the maximum rotation speed possible.

4) In fact Kip Thorne does say a few things about the orbit of Mann's planet. From the events in the film he deduces that the planet has a rather skewed and wide orbit around Gargantua that only rarely brings it near to the black hole (and even then by far not as close as Miller's planet, because in this case it would simply get trapped in that stable orbit). And Edmunds' planet is even farther than Mann's planet.