View Full Version : time slower or fastr near black hole, compared to outsider's frame?

Moonhead

2007-Jul-05, 11:18 PM

Does time go slower near a black hole, compared to anoutsider's point of view?

So, if you would be located near the central massive black hole of the milky way, would people located at earth move relatively faster, live relatively shorter etc.?

Or is it exactly the other way around? (Actually in a way that makes more sense, as planets closer to the sun move fater than those frather away...)

Noclevername

2007-Jul-05, 11:22 PM

Does time go slower near a black hole, compared to anoutsider's point of view?

So, if you would be located near the central massive black hole of the milky way, would people located at earth move relatively faster, live relatively shorter etc.?

Or is it exactly the other way around? (Actually in a way that makes more sense, as planets closer to the sun move fater than those frather away...)

The former. Time goes slower in a dense gravity field. Assuming you got away from the black hole, you'd be welcomed home by your great-great-grandchildren (or more distant descendants).

Moonhead

2007-Jul-05, 11:27 PM

The former. Time goes slower in a dense gravity field. Assuming you got away from the black hole, you'd be welcomed home by your great-great-grandchildren (or more distant descendants).

In that case, we could imagine a cosmic civilization that concentrates it's brightest minds near the galactic center, where they will live longer lifespan. Just a thought, maybe a nice background to a galactic novel that I'll never write :lol:

grant hutchison

2007-Jul-05, 11:34 PM

In that case, we could imagine a cosmic civilization that concentrates it's brightest minds near the galactic center, where they will live longer lifespan. Just a thought, maybe a nice background to a galactic novel that I'll never write :lol:Too late :cry:. Frederik Pohl covered that one in his Heechee series, IIRC.

Grant Hutchison

Noclevername

2007-Jul-06, 12:06 AM

In that case, we could imagine a cosmic civilization that concentrates it's brightest minds near the galactic center, where they will live longer lifespan. Just a thought, maybe a nice background to a galactic novel that I'll never write :lol:

Hmm, that'd just be putting them where no one could benefit from their genius for a few million years. By which time they'd be obsolete. :doh:

Hornblower

2007-Jul-06, 01:36 AM

The former. Time goes slower in a dense gravity field. Assuming you got away from the black hole, you'd be welcomed home by your great-great-grandchildren (or more distant descendants).

You might end up in a zoo.

publius

2007-Jul-06, 01:41 AM

One would have to get very close the event horizon for that much time dilation. In Schwarschild, the time dilation (for stationary observers) goes simply as g_00, which is just:

dT/dt = sqrt(1 - R/r), where R = 2GM/c^2, the Schwarzschild radius, which is the event horizon. Note this doesn't do much until R/r becomes significant. Hovering at r = 2R, that factor is just 1/sqrt(2), or roughly 70%. Not all that much. 7 seconds pass for every 10 seconds of the asymptotic observer.

For a stationary observer, the proper acceleration required to remain stationary is this:

"g(r)" = -GM/r^2 * 1/sqrt(1 - R/r)

That is just inverse square Newton, but with a correction that makes it go to infinity at the horizon. IOW, one cannot remain stationary at r = R, for it requires infinite force. (I put 'g' in quotes, because this is not really what 'g' means -- the coordinate acceleration of something falling in will be different from this, but converge to the same asymptotic, Newtonian value in the weak field, low velocity limit)

So we can note something interesting. The time factor depends only on R/r, M doesn't enter into it. But, the proper g necessary to remain stationary does depend on M, and goes in inverse fashion. For a given R/r, the force will be less for larger M.

For a stellar mass black hole, you will find that proper acceleration for significant R/r will be "millions and billions -- heck trillions -- of earth g" as Carl Sagan would say.

So that is out, unless you do it near a supermassive black hole. But if you look at how much energy (you would think according to what were doing locally) you would have to expend to hold youself there, and then get back out of that very long radial field, it would be a humdinger.

One can imagine orbiting. But one is limited to a radius of 3R there. The time factor for a moving observer is a big more complex and orbital speed will dilate it further, but there you still have to worry about getting out of the well, which takes energy, and the tidal forces. The tidal forces of a stellar mass black hole can themselves reach millions of g over 1m.

So manuevering close enough to a black hole to get significant time dilation but without killing your fool self looks to be a tricky proposition, and will require lots of energy, probably comparable to getting close to light speed in flat space-time anyway.

-Richard

publius

2007-Jul-06, 02:28 AM

It might be fun to run the numbers here, and just how many "millions and billions" we have here. Let's consider a stellar mass black hole, 1 Sol's worth of mass packed into its Schwarzschild radius.

1 Sol mass is ~2 *10^30 kg. The Schwarzschild radius, is therefore

R = 2GM/c^2 ~ 2.97km

According to our asymptotic observer, 1 solar mass is packed into a little sphere of just 3km in radius.

So what is the proper acceleration required to remain stationary at

r = 2R, so that our proper time runs at 70% of that far away observer. Well, Newton would say

GM/r^2 = 3.78 *10^12 m/s^2, 3.78 trillion m/s^2.

From above, there is a correction of sqrt(2), 1.414 times that, which comes out to 5.35 *10^12 m/s^2.

That comes out to about 5.46 x 10^11 g.

Or a tad over 1/2 trillion earth g. "Millions and billions" indeed. A might bit steep. Coal into diamond. Flat as a pancake..........

I leave it as an exercise for the reader to calculate the tidal force across a man length of 2m at that same radius, as well as the same g force calculation for a million stellar masses, and a billion masses, and a trillion. One may simply use Newton, because in the high Schwarzschild symmetry, that tidal force for a stationary observer, as well as a radially moving observer is the same as Newton. It is bit different for observers with tangential velocity, directionally so. You get some corrections involving the "magic factor", (1 - R/r) in different directions.

-Richard

publius

2007-Jul-06, 02:31 AM

I was having with Google's search line calculator. I wanted to see if it recognized "1 solar mass" as a valid entry, and if it knew G, c. It did, as well as letting me spell out units.

-Richard

DaveC426913

2007-Jul-06, 03:45 AM

In that case, we could imagine a cosmic civilization that concentrates it's brightest minds near the galactic center, where they will live longer lifespan. Just a thought, maybe a nice background to a galactic novel that I'll never write :lol:

Other way around. You want the people to be slowed down, not the scientists. Here's why:

Let's quantify the contributions of the great scientific minds. Say they are capable of churning out a major new idea, invention or theory once per year and that they live for one hundred subjective years. That's one hundred inventions.

Hang them near the black hole and their lifetime increases - but their rate of contribution to the population drops. If their lifetime is attentuated over one million years of the planet's existnece, that means the planet only gets one scientific advance every 10,000 years!

But -

Hang the planet's population near the black hole, attenuating its lifetime by one millionfold, and now you have the population able to reap the benefit of 100 advances in a mere 53 minutes.

Of course, you have to replenish the scientists...

Twinsun

2007-Jul-06, 05:04 AM

well for an outsider, the captured object would be moving WAY slower ( in fact the so called frozen frame would be shown ) but for the captured object, time would flow normally for him and the outsiders would move faster than they should :) so each of them would point fingers to each other

Noclevername

2007-Jul-06, 05:10 AM

Hang the planet's population near the black hole, attenuating its lifetime by one millionfold, and now you have the population able to reap the benefit of 100 advances in a mere 53 minutes.

Of course, you have to replenish the scientists...

If you put a planet close enough to the black hole to experience extreme relativistic time dilation, you'd probably have to replenish the entire population as well. :doh:

publius

2007-Jul-06, 07:09 AM

Let's have some more fun with this, and start considering moving objects near the black hole. We simply write the metric in ds^2 form (and I'll leave out the phi coordinate -- we'll just consider motion in a plane -- even with crazy non-Newtonian strong field gravity, orbits confine themselves to a plane):

ds^2 = (1 - R/r) (c dt)^2 - dr^2/(1 - R/r) - (r dO)^2.

Now the proper time along that path is just dT = ds/c, so we have:

dT^2 = (1 - R/r) dt^2 - 1/c^2 [ dr^2/(1 - R/r) + (r dO)^2]

If something is moving, we're coordinatizing that as our r and O coordinates as a function of time. Thus dr = (dr/dt) dt, r dO = (rdO/dt) dt, which we can write as dr = v_r dt and r dO = v_O dt. So,

(dT/dt)^2 = (1 - R/r) - [v_r^2/(1 - R/r) + v_o^2]/c^2

= (1 - R/r) *[1 - (v_r^2/(c^2(1 - R/r)^2) + v_O^2/(c^2(1-R/r)) )

Whoaa what a mess, you may say. But note that v^2/c^2 looking thing. Does that look sort of familiar? All that mess in brackets is the ratio of our coordinate speed to the coordinate speed of light(which varies with direction as the difference in powers of (1 - R/r) attests). That ratio is simply the local speed as measured by a stationary observer using his own local ruler and clock. So we can write that mess simply as:

(dT/dt)^2 = g_00*[1 - (V/c)^2] or

(dT/dt)^2 = g_00* gamma_local^2

How about that? Not that bad at all really. This stuff is just as simple as pie (once you learn how to do it, which ain't so simple.... :whistle: )

And there's another little trick we can do, that will be good enough for govt. work. Newtonian stuff more closely corresponds to that local speed than it does the coordinate speeds of our asymptotic observer. So, staying outside of r = 3R where things will get really crazy, one can solve for Newton's circular orbital speed, and plug that right into the gamma factor and multiply that by g_00 at the radius and have a ballpark estimate of the clock rate of something orbiting down there. That's a ballpark estimate, now, because orbital speed will be different, and start diverging greatly somewhere around r = 3R -- just figure you're in the ballpark, and will overestimate the clock rate a bit. It will tick slower than what we get with this, but it will be in the ol' ballpark.

-Richard

Moonhead

2007-Jul-06, 02:32 PM

Hmm, that'd just be putting them where no one could benefit from their genius for a few million years. By which time they'd be obsolete.

Other way around. You want the people to be slowed down, not the scientists. Here's why:

Let's quantify the contributions of the great scientific minds. Say they are capable of churning out a major new idea, invention or theory once per year and that they live for one hundred subjective years. That's one hundred inventions.

Hang them near the black hole and their lifetime increases - but their rate of contribution to the population drops. If their lifetime is attentuated over one million years of the planet's existnece, that means the planet only gets one scientific advance every 10,000 years!

I was thinking from a rather "elitist" point of view. Indeed, the less than brightest minds would in their lifetime not have much direct benefit of the achievements of those brightest minds. But these brightest minds -let's call them the Inner Galactic Elite- would benefit from the observations of the rest of mankind mapping the galaxy. So, indeed common man might not experience a fast-paced progress in terms of scientific development, but the IGE would experience a fast-paced input of observational and experimental data.

One would have to get very close the event horizon for that much time dilation. [...] So that is out, unless you do it near a supermassive black hole. But if you look at how much energy (you would think according to what were doing locally) you would have to expend to hold youself there, and then get back out of that very long radial field, it would be a humdinger.

I was indeed thinking of a supermassive black hole: the central black hole of the milky way. Indeed it would not be easy - but it needn't be.

O well, as Grant points out, it has already been covered by an SF author anyway.

DaveC426913

2007-Jul-06, 04:45 PM

If you put a planet close enough to the black hole to experience extreme relativistic time dilation, you'd probably have to replenish the entire population as well. :doh:

Well... you'd use a BH with a large radius, minimizing tidal effects.

Powered by vBulletin® Version 4.2.3 Copyright © 2019 vBulletin Solutions, Inc. All rights reserved.