Can Two Stars in a Static Universe Ever Collide Due to Gravitational Influence?

[Holocene]

Gravitational influence?

Suppose for a senond that the only objects in the entire univesre are 2 stars, both about as massive as the sun.

Also, this univesre is completely static. It is not expanding.

The stars, at least initially, are not moving relative to each other.

Under these conditions, about how much distance would need to be between the stars so as to ensure that they never collide?

Or does gravitational influence never deminish to zero, no matter how far away you get?

 

[granpa]

gravity=1/d^2
it equals zero at infinity

 

[Holocene]

gravity=1/d^2
it equals zero at infinity

So 2 grains of sand, 50 Billion light years apart, will eventually collide, even under static conditions?

That is pretty interesting when you think about it.

 

[Oberst Villa]

So 2 grains of sand, 50 Billion light years apart, will eventually collide, even under static conditions?

That is pretty interesting when you think about it.

Thinking of all this Pioneer Anomaly and MOND mumbo jumbo, I would not bet my life that the equation given by granpa is really correct under such extreme conditions as you specified. A few decades later we may be smarter - or maybe not... All we can say now is yes, there is wide agreement that the 1/r^2 relation is correct and yes, the 2 grains of sand will eventually collide (but don't hold your breath for it...)

 

[DaveC426913]

Thinking of all this Pioneer Anomaly and MOND mumbo jumbo, I would not bet my life that the equation given by granpa is really correct under such extreme conditions as you specified.

Well, 1] even if there is any substance to the Pioneer anomaly, then they still will collide. At worst, we might have the delay wrong.

2] It will take more than one anomaly to overturn centuries of verification and countless confirming experiment after countless confirming experiment of gravity's effect.

 

[granpa]

So 2 grains of sand, 50 Billion light years apart, will eventually collide, even under static conditions?

That is pretty interesting when you think about it.

yes, its actually very interesting. the gravitational field presumably like all fields is quantized both spatially and in intensity (I think that's called second quantization or something more or less like that) so what happens when the gravitational field from a particle diminishes to the level of one quanta?

 

[joris20]

So 2 grains of sand, 50 Billion light years apart, will eventually collide, even under static conditions?

That is pretty interesting when you think about it.

unless you define static conditions for me, i guess those grains of sand would just explode because theyre in a vacuum... or am i wrong here?

 

[DaveC426913]

unless you define static conditions for me, i guess those grains of sand would just explode because theyre in a vacuum... or am i wrong here?

Grains of sand explode in vacuum?? Awesome!

:uhh:

 

[MeJennifer]

The stars, at least initially, are not moving relative to each other.

I have a hard-time seeing how this configuration is a valid GR solution.

 

[tony873004]

The amount of time will be

$$t \approx 2\pi\sqrt{\frac{(0.5 d)^{3}}{G(m_{1}+m_{2})}}$$

where d is the distance that separates them, and m₁ and m₂ are their masses.

 

[Count Iblis]

So 2 grains of sand, 50 Billion light years apart, will eventually collide, even under static conditions?

If the grains of sand have a small angular velocity relative to each other, they'll miss. The farther away they are relative to each other, the smaller that angular velocity can be for the grains to hit each other. Now, the initial state will be some so-called coherent state in which the initial position is described by some wavefunction that corresponds to both position and momentum being in some small uncertainty interval.

So, you can simply use the uncertainty relation to estimate the maximum distance beyond which it becomes impossible to let the grains hit each other with reasonable probability.

 

[joris20]

but even if they mis they could re-attract each other and have a "second chance"

etc. etc.

 

[DaveC426913]

but even if they mis they could re-attract each other and have a "second chance"

etc. etc.

No. They are in orbit. If they didn't collide on the first pass, they never will. There's nothing to change their paths from the orbit.

In fact, one orbit will bring them back to their initial starting point, 50Gly apart and stopped wrt each other.


BTW, this whole problem makes the assumption that the universe is of infinite age (which, I guess, if there are only two particles in it, is pretty much a given) because these passes will take longer than our current universe's likely age.

 

[MeJennifer]

No. They are in orbit. If they didn't collide on the first pass, they never will. There's nothing to change their paths from the orbit.

That is simply not true under general relativity. The orbit will decay slowly.

 

[Oberst Villa]

That is simply not true under general relativity. The orbit will decay slowly.

Yes, but really really really slowly. They will only come close to each other every [too lazy to put the numbers in tony873004's equation] billion years, and even during this short encounter the loss of energy due to gravitational radiation will be soooooo tiny.

 

[joris20]

well were tqlking about 'in the end' so technically, we have eternity which seems long enough :p

 

[Count Iblis]

Take two grains of sand a distance of r (say about 50 billion lightyears) apart at rest. The mass of a grain of sand is m
(say 10^-6 Kg) . The potential energy is:


E = G m^2/r

For the system to be in a bound state, the momentum of the grains must not exceed P, given by:

P^2/m = G m^2/r ----------->


P = Sqrt[G m^3/r]



If the intital state is given by some wavepacket then in momentum space the spread must be significantly less than P. The uncertainty relation then gives a spread of the wavefunction in configuration space of:

Delta X = hbar/2 Sqrt[r/(G m^3)]

This is about 140 nm for the two dust grains placed 50 billion lightyears apart. I don't think it is realistic to assume that this wavefunction won't decohere into a statistical mixture of wavefunctions with much sharper defined positions.