Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: Vajira Kashyapa on 05/06/2011 11:30:03
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Vajira Kashyapa asked the Naked Scientists:
Dear Naked Astronomy,
Your postcast is brilliant (http://www.thenakedscientists.com/HTML/podcasts/)! I have been listening since 1st episode.Â
I have a follow up question regarding an answer given to a listener questions your last episode (May 25th).
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Original Question: A listener asked the questions, If you place two particles of mass in an empty universes, would they actually attract each other and move closer together and collide?
Original Answer: Yes, if u take an example of 1gram particles, set 10000KM apart from each other they would move towards each other as long as they were completely stationary.
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My questions is:Â
Would the particles not stay completely stationary, if the same mass particles started out completely stationary to each other, and at an equal distance away from each other?
My thinking regarding this is: If you place two stationary objects of equal mass, at equal distance from each other, would they not stay the same unless one or both particles are forced to either not stay stationary or are increased in mass or increased in distance?Â
If you were to have two skaters of equal mass spinning around each other at a constant speed, and they are not acted upon by any other force, they will continue to spin until a force acts upon them, They will not collide into each other.
Vajira Kashyapa - From Sri Lanka :)
What do you think?
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No, you have to remember that space is in 3-D. So somehow in has a 'depth' to it. The idea of gravity is that wherever you get a 'hole' in 'Space' it will 'attract' matter even if the slope(s) created are incredibly small. Why it is so is because there is no 'friction' macroscopically. If we assumed a friction it might be that at some 'distribution' of mass relative space's friction the mass would refuse to move relative each other as the slopes inclinations would be too small.
But without any 'friction' I believe the 'equilibrium' of space will crave them to move together. And we haven't found any friction, as far as I know.
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I think I answered this question in another thread on this board. (It may have been a differen forum.) Briefly, Newton's law of gravity requires that any two non-zero masses a finite distance apart (in an otherwise empty universe)- must accelerate toward one another. If their relative velocity is only radial [edit: and less than escape velocity], they must eventually crash into one another, even if they begin moving apart. If they start with some non-radial velocity, they must orbit one another (it may be a very long thin elliptical orbit).
However, Newton wrote his law of gravity for non-expanding space. Since space is expanding, the two objects begin with a certain radial acceleration away from one another in addition to the acceleration of gravity. As they move farther apart, the acceleration caused by expanding space increases and the acceleration of gravity decreases. If the velocity is zero and the acceleration due to expansion is greater than that due to gravity, they will never get any closer together. For non-zero initial velocity, the formula is a bit more complicated. (To satisfy the purists, when I say "velocity" and "acceleration", I'm talking about the time deriviatives of distance. Cosmologists use comoving coordinates system, in which the time derivatives of distance are considered to be only "apparent" velocity and acceleration.)
The rate of expansion is Hubble's constant. Hₒ ≈ 2.5 x 10^-18/s. The Hubble velocity at r is Hₒr, and the Hubble acceleration at r is ½Hₒ²r.
Comoving coordinates expand with space, so space doesn't expand relative to the coordinates. There is another way to look at the universe, besides comoving coordinates. If the coordinate system does not expand, then the expansion of space is equivalent to a parabolic gravity hill centered on the origin. (At billions of light years, the shape is not parabolic because of relativistic effects.) The acceleration of gravity between two objects is like a depression in the top of that gravity hill. The two objects can roll up the sides of the depression; if they roll over the rim, they will keep on going.
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Phract
If their relative velocity is only radial, they must eventually crash into one another, even if they begin moving apart.
Agree - other than fact that if the radial velocity apart is greater than the escape velocity, which for two particles of a gram 10^7 m apart would not be very high (to the order of 10^-10 m/s).
I must admit I struggled to get straight in my head how to quantify the intrinsic expansion of space and see if this would have an effect. I would be interested to see if you came to a conclusion.
I haven't listened to poscast yet - but I wonder if they gave a time scale; I guess it is a second order nonlinear differential equation (which is no fun at all). from five minutes messing around on excel I estimate the impact time to be in the order of billions of years.
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Phract
If their relative velocity is only radial, they must eventually crash into one another, even if they begin moving apart.
Agree - other than fact that if the radial velocity apart is greater than the escape velocity, which for two particles of a gram 10^7 m apart would not be very high (to the order of 10^-10 m/s).
Thanks for the correction. I have edited it into my post.
I must admit I struggled to get straight in my head how to quantify the intrinsic expansion of space and see if this would have an effect. I would be interested to see if you came to a conclusion.
I am not a mathematician. My head hurts just thinking about writing the equation.
I think total acceleration is a = (½GHₒ²)-(Gm²/r²).
EDIT: I meant to write a = ½(r²Hₒ²) - (Gm²/r²), but I'm not sure if that's right, either. This math is still hurting my head.
Velocity is v = (ds/dt) = rHₒ = 2.5 x 10^-18/s.
Acceleration is a = (dv/dt) = d/dt(r x 2.5 x 10^-18/s).
I think I might have taken the derivative with respect to r, instead of t. Ain't there a math whiz, out there who can set me straight?
Distance (in the one dimensional problem) is s = sₒ + ∫t(vₒ - at) dt. (With positive vectors pointing away from each particle toward the other.)
P.S.: If you're using comoving coordinates (mainstream convention), the acceleration and velocity due to expansion are only "apparent", so they are not subject to relativity. If you use non-expanding coordinates, relativity makes a big difference when you get close to the Hubble limit (about 13.7 Gly).
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You're right Phractality, I forgot to account for 'expansion'
If we assume an expansion it should be possible for that to overcome the slopes, assuming some critical distance, as defined between the objects. But then we have Einsteins definition of a 'space' to consider too. He defined it as 'gravity'. 'Gravity' becoming 'Space's' true metric, and that's the way I think of it too. Space is defined by two things to me, gravity and 'distance'. If you assume that the metric will be what 'gravity' there can exist, and then define two particles in a 'space' defined by those particles coupling to 'gravity'. Then I doubt the expansion can stop it.
If that is wrong though, you will be perfectly correct, and in the 'SpaceTime' we know, defined by a uncountable amount of particles, one easily can see that your definition must become the correct one if we account for a expansion.
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As this is obviously a hypothetical question I hope I will be allowed a little leeway in answering it.
For space-time to have any meaning it must contain energy and mass, gravity and time. The only energy is of the gravitational potential of the two particles. It is likely that space time would be a relatively small 'volume' and not expanding, quite possibly contracting.
If space were expanding, I don't think it would make any difference anyway, as distance expands so time dilates maintaining the balance and the speed of light remains constant.
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Actually Phractality's hypothesis seems supported per our own universe. The galaxies do move from each other due to our expansion. And they shouldn't, if we assumed that gravity always will overcome a distance, at least they should 'slow down' as gravity 'fights back' :)
Which brings me a question I've been wondering about some time. If I define gravity as the metric of space. Then gravity is only 'coupled' to mass, motion and energy. If it would be otherwise, that is if gravity would be some inherent/intrinsic property, I can see several possibilities for that 'new space' created in a expansion from not being able to exist too ? dimensions.
However you argue :) I believe you will have to agree with me that either there becomes a gravity 'instantly', or we will be forced to allow a 'space' without its metric (as I define gravity), for however short a time, before gravity has 'propagated' to it at 'c'.
This is assuming that 'new space' somewhat mysteriously 'pops up' between galaxies filling in 'new points'.
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It was necessary and proper to bring the expansion of space into this discussion in order to properly answer the OP. The question is quantitative, not qualitative. The rate of expansion is relevant, here; the cause and meaning of expanion are not. This is not the New Theories section.
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Phract - agree with yours above - btw I did the sums (after swearing I wouldn't waste my time). By my back of an envelope sums done in a pub waiting for a friend to turn up - the expansion of space contributes much less than an already tiny gravity. By looking at escape velocity and expansion velocity - ie if the velocity of two coordinate parts of space is greater than the escape velocity. Well - the expansion is 10^-5 too small - even for these tiny masses.
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Vajira Kashyapa asked the Naked Scientists:
Dear Naked Astronomy,
Your postcast is brilliant (http://www.thenakedscientists.com/HTML/podcasts/)! I have been listening since 1st episode.
I have a follow up question regarding an answer given to a listener questions your last episode (May 25th).
---------
Original Question: A listener asked the questions, If you place two particles of mass in an empty universes, would they actually attract each other and move closer together and collide?
Original Answer: Yes, if u take an example of 1gram particles, set 10000KM apart from each other they would move towards each other as long as they were completely stationary.
---------
My questions is:
Would the particles not stay completely stationary, if the same mass particles started out completely stationary to each other, and at an equal distance away from each other?
My thinking regarding this is: If you place two stationary objects of equal mass, at equal distance from each other, would they not stay the same unless one or both particles are forced to either not stay stationary or are increased in mass or increased in distance?
If you were to have two skaters of equal mass spinning around each other at a constant speed, and they are not acted upon by any other force, they will continue to spin until a force acts upon them, They will not collide into each other.
Vajira Kashyapa - From Sri Lanka :)
What do you think?
It is probably worth mentioning particles are never at complete ''rest''. This violates the uncertainty principle.
It is said all matter in the universe attracts the rest through curvature, or the presence of a gravitational field. This be the case, two particles in a universe will eventually meet if gravity does act on large distances. I say this hesitently, because there has been some question of whether dark energy and matter are a result of gravity failing on very large distances, meaning the equations of General Relativity fail at large distances as well.