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Moonhead
2008-Oct-13, 09:41 AM
Hello,

Reading in this thread (http://www.bautforum.com/astronomy/77962-alien-civilizations-orbiting-black-holes.html#post1342007), the following question came up:

Is there is a simple* formula to calculate X if Y is given, and vice versa,

* (simple being defined as doable on the standard Windows calculator :) )

where

X is the mass of a central stellar object, expressed in solar masses
and

Y is the 'rate of time'°, expressed in a floating point number; the 'rate of time' on earth-as-we-know-it, is 1.0
° (This might be expressed a bit poorly but in this matter I am not willing to go into semantics or a philosophical discussion about the nature of time - unless, of course, this will be necessary for getting an answer.)

It was Eburacum's remark:

You could put things into close orbit around the black hole for storage- because of time dilation, an object orbiting just above the event horizon (or at the lowest possible orbit ) would experience slower time than something orbiting further out.

that got me wondering about this. I've been thinking in this line too, and I thought of civilizations existing near a supermassive black hole, for whom time would go much slower than for the more exterior civilizations. There would be several pros and cons of living e.g. 10 times as slow and 10 times as long as your 'neighbours'...

_____________

EDIT: I first posted this question in the thread I refer to, because I couldn't open a new thread. Thanks to To-Seek, the new thread is here now.

alainprice
2008-Oct-13, 06:44 PM
The wiki page for gravitational time dilation shows solutions for both non-rotating and orbiting situations:
http://en.wikipedia.org/wiki/Gravitational_time_dilation

To=Tf(sqrt(1 - Ro/R))

Ro is the Schwarzchild radius
R is your radial coordinate (in the Schwarzchild metric)
To is the BH observer
Tf is a far away observer

Moonhead
2008-Oct-13, 08:44 PM
The wiki page for gravitational time dilation shows solutions for both non-rotating and orbiting situations:
http://en.wikipedia.org/wiki/Gravitational_time_dilation

To=Tf(sqrt(1 - Ro/R))

Ro is the Schwarzchild radius
R is your radial coordinate (in the Schwarzchild metric)
To is the BH observer
Tf is a far away observer

Thanks. Right now my time is limited (it's about 22:30 here and my girlfriend... well you understand :) No time dilation for me, not now). So I can't go thru the article and your formula right now. Yet, a short question that comes to mind: if the heavy stellar object is not a black hole? The sun does cause a time dilation effect on mercury (as compared to here-and-now-on-earth), right? (As solving why and how Mercury deviated from Newtonian predictions, was one of the early triumphs of Einstein's new theory). Then, with what should the Schwarzschild-radius in the formula be substituted?

thorkil2
2008-Oct-14, 04:25 AM
I'm interested in Moonhead's last question also. There is a formula for determining the relationship between mass, distance, and time dilation; difference between clock time on the surface of the earth and in orbit at some distance, for example. I've been looking for it for some time, but have been unable to locate the formula (lae).

mugaliens
2008-Oct-14, 03:26 PM
Yet, a short question that comes to mind: if the heavy stellar object is not a black hole? The sun does cause a time dilation effect on mercury (as compared to here-and-now-on-earth), right? (As solving why and how Mercury deviated from Newtonian predictions, was one of the early triumphs of Einstein's new theory). Then, with what should the Schwarzschild-radius in the formula be substituted?

The Schwarzschild radius is independant of whether or not a mass is a black hole. It's 2*G*m/c2, where G is the gravitational constant, and m is the mass of the gravitating object. Thus, for the Sun, the radius is 2.96 km. Since it's proportional to the mass, the radius for any other body can be expressed in terms of solar masses: a BH of 10 solar masses would have a radius of 29.6 km.

Moonhead
2008-Oct-15, 07:48 PM
The Schwarzschild radius is independant of whether or not a mass is a black hole. It's 2*G*m/c2, where G is the gravitational constant, and m is the mass of the gravitating object. Thus, for the Sun, the radius is 2.96 km. Since it's proportional to the mass, the radius for any other body can be expressed in terms of solar masses: a BH of 10 solar masses would have a radius of 29.6 km.

Thanks! Didn't know that, about every gravitating object having a schwarzschild radius. (Indeed the earth has one too, according to the Wiki article, about the size of a peanut. So, if we squeeze the earth to the size of a peanut, it would be[come] a black hole, right?)

So, the next step is, to transform the formula Alainprice gave, into one that, instead of comparing two observers at different distances of the same heavy mass object, into one that compares two observers at the same distance to different heavy mass objects. :think:

Ken G
2008-Oct-15, 08:53 PM
Thanks! Didn't know that, about every gravitating object having a schwarzschild radius. (Indeed the earth has one too, according to the Wiki article, about the size of a peanut. So, if we squeeze the earth to the size of a peanut, it would be[come] a black hole, right?)Sort of, but in fact it would depend on what we did with all the gravitational energy released by the squeezing. Usually, a significant fraction of that energy would have to be extracted from the system to get it to squeeze in the first place-- and that would reduce the mass of the object you end up with to below the Earth's current mass. Thus you could get it to a peanut and it would still not quite be a black hole-- but pretty close!

So, the next step is, to transform the formula Alainprice gave, into one that, instead of comparing two observers at different distances of the same heavy mass object, into one that compares two observers at the same distance to different heavy mass objects.The formula you need is that the time dilation factor is the square root of 1 - r_o/r, where r_o is the Schwarzschild radius. To get the factor between any two r's, ratio the factors for each r.

Moonhead
2008-Oct-16, 07:58 PM
Sort of, but in fact it would depend on what we did with all the gravitational energy released by the squeezing. Usually, a significant fraction of that energy would have to be extracted from the system to get it to squeeze in the first place-- and that would reduce the mass of the object you end up with to below the Earth's current mass. Thus you could get it to a peanut and it would still not quite be a black hole-- but pretty close!

A hypothetical question (this is probably obvious as all such questions are): Would two of these not-quite-but-close-to-a-black-hole grains meeting each other, attract each other and form a blachhole? Or would that really depend on how they meet (how close, at what speed, colliding or not etc.)

The formula you need is that the time dilation factor is the square root of 1 - r_o/r, where r_o is the Schwarzschild radius. To get the factor between any two r's, ratio the factors for each r.

Thanks!! That one meets my proposed definition of 'simple'! :)

Ken G
2008-Oct-16, 10:26 PM
A hypothetical question (this is probably obvious as all such questions are): Would two of these not-quite-but-close-to-a-black-hole grains meeting each other, attract each other and form a blachhole? Or would that really depend on how they meet (how close, at what speed, colliding or not etc.)I'd have to say yes and yes-- meaning that one might expect under normal conditions for them to eventually spiral in toward each other and coalesce into a black hole, but by the same token, you could probably come up with some type of circumstance where they would not. It's really a question for more detailed models.

mugaliens
2008-Oct-19, 11:44 AM
I'd have to say yes and yes-- meaning that one might expect under normal conditions for them to eventually spiral in toward each other and coalesce into a black hole, but by the same token, you could probably come up with some type of circumstance where they would not. It's really a question for more detailed models.

If you agree that all black holes with a Schwarzschild radius of r have a distance d, such that d>r, whereby any object placed at d, with an escape velocity v where v<c, there is a range between v and c whereby said object would escape the black hole.

If two black holes are involved, v would change, but it's conceivable that there are a range of relative velocities whereby the two would merely pass like ships in the night.

If it's one BH and a passing object, the orbit would be a hyperbolic one.

Would it still be a hyperbolic orbit when two black holes (or any two massive objects) are involved?