# The Naked Scientists Forum

### Author Topic: Would two grains of sand find each other in an otherwise empty Universe?  (Read 2612 times)

#### Felix Nielsen

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##### Would two grains of sand find each other in an otherwise empty Universe?
« on: 20/04/2011 10:25:40 »
Felix Nielsen  asked the Naked Scientists:

I actually have more questions that you'd like me to ask, but one of them I tried to figure out on my own, and failed.

Imagine, for a moment, an empty universe, to avoid interference.
We then place two grains of sand, or something else if you wish, a great distance apart, say a light year. (a billion trillion quadrillion light years and rather heavy objects would also be fun ;) )

It is my understanding that these two objects, as long as they've got mass, will attract each other so, my question is:

Given similar parameters as previously explained, how long would it take for these two object to collide and how fast would they be travelling at the time of impact?

I would also be very interested in understanding the maths. as previously explained, I've tried, and failed. But I can understand if it isn't practical.

In any case, thank you for your time, I do hope you'll answer my question.

Best regards.
Felix Nielsen
Copenhagen, Denmark

What do you think?
« Last Edit: 20/04/2011 10:25:40 by _system »

#### yor_on

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##### Would two grains of sand find each other in an otherwise empty Universe?
« Reply #1 on: 20/04/2011 10:42:20 »
Maybe this one can be of assistance?
Gravity calculator.

#### Phractality

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##### Would two grains of sand find each other in an otherwise empty Universe?
« Reply #2 on: 20/04/2011 20:24:42 »
Assuming infinite time and finite starting distance:

If not for the expansion of space, the answer would be simple. If there is no non-radial velocity, they will collide. If there is non-radial velocity, they will orbit.

In an expanding universe, the limiting distance is the distance at which the Hubble expansion moves them apart faster than gravity brings them together. The space between them at radius, r, increases at the rate of H₀r, and accelerates at the rate of H₀˛r. If R is the distance of greatest separation between them, and H₀˛R < G(m₁m₂/R˛), the distance will decrease, and they will eventually collide or orbit. Otherwise, they will never move any closer together——even if the force gravity is faster than light (which has not been proven to everyone's satisfaction).

EDIT: Math ain't my strength. I may have omitted a factor of 2. Acceleration is dv/dt; v = H₀r; H₀ ≈ 2.5 x 10^-18/s. Is it H₀˛r or ˝H₀˛r?

Also, the formula depends on whether you are using comoving coordinates or non-expanding Euclidean coordinates. But the question of collision or no collision must be the same, regardless.
« Last Edit: 21/04/2011 18:57:27 by Phractality »

#### imatfaal

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##### Would two grains of sand find each other in an otherwise empty Universe?
« Reply #3 on: 21/04/2011 12:43:35 »
Further to Phrac - and assuming they are not at a large distance and travelling away from each other rapidly at which point i would say that they will decay to energy and heat bfore they meet / orbit.

The force between two grains of sand at a light-year is approx 10^-55 newtons - it would be ten thousand times the age of the universe so far before they had closed the gap by one solitary metre presuming a standing start.

#### MikeS

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##### Would two grains of sand find each other in an otherwise empty Universe?
« Reply #4 on: 25/04/2011 17:22:51 »
Hi Guys,

I would like to approach this from a different angle.

Its possible to deduce from E = mc2 that it is mass that gives a direction to the arrow of time and energy that gives it a rate of flow.  Rt = √E/m  where Rt is the rate of flow of time, E is energy and m is mass.

Its perhaps reasonable to assume the net energy of the universe to be zero.  Let energy be a positive ‘force’ and gravity be a negative ‘force’.  Energy being the ‘force’ that causes the expansion of the universe.  (No cosmological constant required) Gravity being the ‘force trying to make the universe collapse.

For all intents and purposes we can ignore the square root sign, so Rt = E/m
A universe that contains energy is going to produce pair particles and some will survive mutual annihilation.  For the universe to only contain two grains of sand it can not contain much energy.

The rate of flow of time is the energy of the universe divided by its mass.  Even taking into account the above, the energy of the universe is likely to be large in comparison to its mass.  Therefore, the rate of flow of time will be fast.

The speed of light is allowed to be a constant because the rate of flow of time is a variable. Now a universe that contains little energy is going to be very small and the rate of flow of time fast.

Now without an input of ‘new’ energy there will be no expansion of the universe.  If it is energy that causes the expansion, then entropy will do the opposite.  We eventually end up with a very small universe, probably of a steady fixed size with a fast rate of flow of time.

The two sand particles might find each other a lot quicker than we think.

Mike

#### yor_on

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##### Would two grains of sand find each other in an otherwise empty Universe?
« Reply #5 on: 25/04/2011 19:33:10 »
Mike, this sub-forum is mainly for answering the questions from a 'main stream' view point. I say mainly because sometime it splits into, more or less, widening definitions :) so it can be hard to see the difference at times.

Still, if you want to propose a new theory, feel free to use  'New Theories'
It's made for just that purpose, and some of the most interesting discussions, and arguments, are there too.

#### The Naked Scientists Forum

##### Would two grains of sand find each other in an otherwise empty Universe?
« Reply #5 on: 25/04/2011 19:33:10 »