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Forum » SpaceEngine » Science and Astronomy Discussions » Science and Astronomy Questions
Science and Astronomy Questions
midtskogenDate: Thursday, 13.11.2014, 20:10 | Message # 286
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The key is your speed around the black hole. You need to go extremely fast to make one hour equivalent to 7 years. I don't know the equations, though.

Some info on the black hole in Interstellar here. For it to work the black hole needs to be extremely large and be spinning extremely fast. Something found in the centre of galaxies.





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HarbingerDawnDate: Thursday, 13.11.2014, 21:13 | Message # 287
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Quote midtskogen ()
The key is your speed around the black hole. You need to go extremely fast to make one hour equivalent to 7 years.

Gravity can cause time dilation by itself. Despite their significantly higher speed, clocks on GPS satellites actually tick faster than those on Earth, because the decreased gravitational influence more than offsets the increased velocity.

Merely being close to a black hole would slow time for you, even if you were (somehow) not moving. Of course, in practice, velocity time dilation must also be factored in as you must orbit the black hole.





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WatsisnameDate: Thursday, 13.11.2014, 23:28 | Message # 288
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Time dilation is not simply a function of your speed (derivative of your position in 3-space with respect to coordinate time). This is only true in flat space-time, which falls into the realm of special relativity. You can also derive the equation for this fairly easily, using the 4D version of Pythagorus' Theorem, and some algebra. (Maybe I'll show this later, because it's a fun example which illustrates foundations of special relativity.)

In curved space-time, figuring out time dilation is a lot harder. Instead of velocity through space, you need velocity through space-time (derivative with respect to proper time, tau), and the metric describing the space-time geometry.

I'll go over how to do this in detail later. (Celebrating my Dad's birthday today). smile Also, I'm not going to do it properly, because Gargantua (the black hole in Interstellar) rotates. A rotating black hole is described by the Kerr metric, which is significantly more complicated than the Schwarzschild metric (describing an uncharged, non-rotating black hole).

For now, I'll say that time dilation effects from a non-rotating black hole would not be significant enough to match the film's description. You'd have to be extremely close to the event horizon, where there are no stable orbits. But for a rapidly rotating black hole like Gargantua, the spin is very important. It literally drags space-time around the hole, an effect called "frame dragging". Kip Thorne wanted the science of Interstellar to be as accurate as possible, and he actually did use correct equations describing this for the movie. How closely the film's use of time dilaton follows it, I don't know, but it's probably not far off! From what I've seen of discussion by other astrophysicists, you actually can have stable orbits with such large time dilation factors.





 
DoctorOfSpaceDate: Friday, 14.11.2014, 03:02 | Message # 289
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Quote midtskogen ()
Some info on the black hole in Interstellar here.


Image from that page


Interesting stuff to think about.





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WatsisnameDate: Friday, 14.11.2014, 05:30 | Message # 290
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That is really cool. A couple descriptive errors (e.g. describing how to make a black hole), but otherwise great info. The graphic of Gargantua really helps show how gargantuan it truly is. That is a monster black hole! For comparison, the black hole at the center of our galaxy fits easily inside the orbit of Mercury.

Also, I think I'll post the derivation for the special relativistic time dilation formula here tonight, the one that assumes you're in flat space-time (or at least flat enough that contribution from gravitational field is negligible for your desired precision). It's a little bit of a grind, but the math itself is surprisingly easy considering we're talking about something most people view as arcane -- the relativity of time and space. But if you know even the bare bones of geometry and algebra, you can do it yourself and impress your friends. smile

Start with a thought experiment.

Consider the following apparatus. Let’s build a box. Inside is a laser emitter, aimed at a detector on the other side, some known distance away. The laser activates, the beam travels across the box, and hits the detector. The span of time it takes for this to happen makes for one tick of what we will call a clock. When the detector receives the laser pulse, the laser may fire again. Actually, heck, let's idealize this even further -- instead of a laser and detector, have a single photon bouncing between two perfect mirrors. Each bounce corresponds to a tick of the clock.

How long do these ticks take? First let’s look at the frame of reference moving with the clock – the rest frame. Easy: the time it takes for the pulse to cover the distance is simply the distance divided by the speed of the pulse: t=d/c. Or, d=ctr, where r indicates that this is the rest frame we're dealing with.

What if the clock is moving relative to the observer (let’s say perpendicular to the line connecting the mirrors)? The observer follows the path of the photon as it bounces back and forth, and notices that, in the time between bounces, the mirrors have moved a distance vtm (m for moving frame), owing to the motion of the clock itself. For the photon to move from one mirror to another, it must follow an angled path. The faster the clock is moving, the more severe that angle. The distance covered is thus greater as well.

So, in the moving frame, the clock moves a distance vtm, and the distance travelled by the photon is ctm.

The Newtonian thinker might find this absurd, that instead it is the speed of light that must change. But we know from experiment that this is not true! The speed of light is a constant for everyone, regardless of the state of motion. If c is constant, then d and t must vary. So in both frames, we use the same c, and we’ll calculate the change in t.

Here are our observations:

-In the rest frame, the clock moves a distance zero, by definition.
-Again in the rest frame, the photon covers a distance ctr.
-In the moving frame, the clock moves a distance vtm
-Again in the moving frame, the photon covers a distance ctm

We must find a formula relating all of these. As usual in solving problems of a geometric nature, it is helpful to draw a picture:



This is a right triangle! You know what to do:

By the Pythagorean Theorem, equate the square of the hypotenuse to the sum of squares of the other two sides. We obtain



The rest is just algebraic manipulation. Admittedly, this is not so obvious at first, and it takes a few steps. Ultimately, we want to isolate the terms involving time.




and finally, taking the square root and defining tm over tr as the time dilation factor, we find:


Plotting it (and setting c=1), we get the classic asymptotic curve. Special relativistic time dilation is minuscule for every-day speeds, but grows enormously as you approach the speed of light.

One may very well ask if this result is valid, since we started with a pretty contrived definition of a clock which itself uses the speed of light. I assure you, it doesn't matter. Any clock you could possibly conceive of will exhibit this time dilation effect in exactly the same way -- digital clocks, mechanical, atomic, biological (your heart; brain), whatever you want. The effect is demonstrable in numerous experiments, and as HarbingerDawn noted earlier, it is important to the proper functioning of GPS! If we did not account for relativity (special and general), GPS would incur a rapidly accumulating error and be rendered useless.

More to follow on general relativity, including time dilation for a circular orbit around a non-rotating black hole, next time!





 
midtskogenDate: Friday, 14.11.2014, 07:48 | Message # 291
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If not moving fast enough to be in orbit, I think the main concern would be the experienced G forces, not the experienced time dilation. smile

I've one issue with Miller's planet: I suppose it must have been formed elsewhere and then found an orbit close to the black hole, otherwise it would be less than 200,000 years old.

How would it be to observe the sky from Miller's planet?





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SpaceEngineerDate: Friday, 14.11.2014, 11:14 | Message # 292
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The first planet is simply impossible. Lets calculate:

M = 108 Msol
Rg = 2gM / c2 = 2 AU (not 1 AU like on those infographics! They forgot the 2 multiplier, lol)

gamma = 7 years / 1 hour = 61360 (time dilation)
gamma = 1 / sqrt(1 - Rg/R) (gravitational time dilation)

where R is a distance to the black hole's center. Lets calculate it:

R = Rg / (1 - 1 / gamma2) = 1.0000000002656 Rg
R - Rg = Rg (1 / (1 - 1 / gamma2) - 1) = 0.0000000002656 Rg = 80 meters

So this planet should orbit just in 80 meters above the event horizon of the black hole!
Taking into account dime dilation due to speed, which must be close to speed of light, might increase this distance. But anyway this planet could not exist - it simply will be rip apart by tidal forces.





 
InariusDate: Friday, 14.11.2014, 12:38 | Message # 293
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And before that, the people on it would be instantly dead, and at least the sea would looks like ice with very powerful volcanos due to tidal forces. An increase of "just" 30% of gravity (like in the movie) just doesn't fit with a 7 years-1 hour time dilatation.

And same thing for the other planets, a monster like gargantua with such a gravity would crush anything around it.
 
midtskogenDate: Friday, 14.11.2014, 13:40 | Message # 294
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Quote SpaceEngineer ()
So this planet should orbit just in 80 meters above the event horizon of the black hole!
Taking into account dime dilation due to speed, which must be close to speed of light, might increase this distance. But anyway this planet could not exist - it simply will be rip apart by tidal forces.

Have you taken into account that the black hole is spinning at 0.998c?





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SalvoDate: Friday, 14.11.2014, 16:59 | Message # 295
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Quote SpaceEngineer ()
it simply will be rip apart by tidal forces.

But actually fidal forces depends on the mass of the black hole.

An 8 solar masses black hole would kill you way before you could reach event horizon, but a black hole with 10^8 solar masses could let you cross the event horizon with "no problems".
Actually, a planet behaves differently from an human body, but being such a massive object means that tidal forces becomes disruptive at a very low distance to the center (compared to event horizon radius), so... I don't know, maybe the planet could resist and behave like Io does related with Jupiter... wacko





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Edited by Salvo - Friday, 14.11.2014, 17:02
 
WatsisnameDate: Friday, 14.11.2014, 17:20 | Message # 296
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@SpaceEngineer: The BadAstronomer made the same criticism himself, but then realized that Gargantua is a rotating black hole. This increases the severity of time dilation effects. Apparently -- though I've not seen it shown explicitly -- you can have stable orbits with arbitrarily large gamma in Kerr geometry. Tidal effects are probably still lethal to the planet, though. If not only from the radial derivative of gravitational field, then probably by frame dragging.

@Midtskogen: A black hole "spinning at some fraction of c" makes no physical sense. They should say what the angular momentum is, or what percentage of maximally rotating. We could assume they mean 99.8% of a maximally rotating black hole.

Also, wow, yeah, whoever made that graphic did forget a factor of 2 in there when figuring its size. Or if they wish to mean it has a radius of 1AU, then the mass needs to be half of that -- 50 million solar masses.





 
WatsisnameDate: Saturday, 22.11.2014, 08:02 | Message # 297
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I hope that a double post can be forgiven in this case. Also, this post is monstrous, probably the longest post I have ever written anywhere. I hope you can forgive me for that, too. smile

Earlier I showed a derivation for special relativistic time dilation – the slowing of clocks due to relative motion of observers.

Now I will bring gravity into the picture, and show the time dilation for
(a) A body maintaining constant position in a gravitational field.
(b) A body in a circular orbit.

For the above, I will assume the gravitational field outside a spherically symmetric, non-rotating distribution of mass, or Schwarzschild geometry. This is a good first approximation for the field of the Earth, a star, or a non-rotating black hole. But I will also discuss (without derivation) the time dilation for a circular orbit of a rotating (Kerr) black hole, and the consequences for the planet in the film Interstellar (re: Salvo’s question from the last page).

For most, these derivations are going to be superfluous. They are purely for any who are curious about where the formulas come from. I tried very hard to make that process as clear as possible, but I must warn it may still be pretty intense, with a fair amount of mathematics and notation introduced along the way. General Relativity is no walk in the park. If it is too much, just skip through it for the final formulas and discussion.

Finally, this is NOT going to be a thorough or rigorous review. It’ll be lengthy though, so I’ll spoilerize it in sections so that it is not too enormous upon loading.

Section 1: Metrics, Tensors, and the Line Element
Wherein the reader remembers an old friend – Pythagorus’ Theorem – and, after equipping him with new tools, sends him forth to measure the heavens.



Section 2: The Schwarzschild Metric
Wherein the reader, now armed with magic potions and incantations of geometrodynamics, confronts the darkness of the black hole.



Section 3: Time Dilation of Circular Orbits
Wherein the reader, triumphant, lassos the black hole.



Section 4: Further discussion, Kerr geometry, and Miller’s Planet
In which the reader encounters whirlpools of space-time.



Added: I just realized a couple of the formulas have both capital R and lowercase r. That is a typo -- they should be the same (lowercase) r, representing radius from the singularity.





 
DoctorOfSpaceDate: Saturday, 22.11.2014, 08:28 | Message # 298
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Watsisname, I want to know you IRL so we can be best friends. With the effort you put in these posts I suspect that discussions with you would be very interesting. These last few posts are great.




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midtskogenDate: Saturday, 22.11.2014, 08:53 | Message # 299
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Great post, great compendium, Watsisname.




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WatsisnameDate: Sunday, 23.11.2014, 05:00 | Message # 300
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Thanks you guys, this is a pleasure. smile

By the way, I saw Kip's The Science of Interstellar on the shelf at the bookstore today. I was skeptical of it at first, thinking it would probably be pretty hokey, but picked it up and started skimming to see. It is actually extremely good! Thorough, excellently written, beautifully illustrated, and easily understandable to popular audiences. Lots of cool science involved that I hadn't even realized (he really did do his homework), plus interesting discussion of his experience working with the rest of the team in Hollywood. Recommended!





 
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