Distance expansion and escape velocity ( thought experiment )

[marcus]

Distance expansion and escape velocity ("thought experiment")

If you care to, it would help if you would check my arithmetic. I may have made one or more mistakes. Thanks to anyone who can show the conclusion here is wrong.

The question came up if you have a pair of kilogram masses and you place them each at CMB rest slightly over one light second apart, then does their Hubble law recession rate exceed their classic escape speed?

It is imagined that you do this out in "open" space, away from any galaxies, groups of galaxies, superclusters etc. So that even though the force of attraction between the pair of kilogram balls is unimaginably weak all the other forces can be neglected.

As time permits I will show some work indicating how I came by the answer, but for the moment to make a long story short, I found that the Hubble law expansion of 400,000,000 meters even though unimaginably slow does exceed the escape speed. So the two balls would continue to separate and not eventually start to fall towards each other.

That distance is roughly the distance from the Earth to the Moon.

 

[marcus]

Maybe if I made a mistake I'll find it myself as I type in some work. Everybody should know that the Hubble parameter is the reciprocal of a time (called the "Hubble time") and therefore it is a frequency. The Hubble frequency.

Indeed if you type "71 km/s per Mpc " into the googlebox, ending with a space or equal sign, it will tell you "2.3 x 10^-18 Hz".

So imagine you have a proper distance between two objects at CMB rest and the distance is 400,000,000 meters. How fast is this distance expanding? Just multiply by the Hubble frequency. 9.2 x 10^-10 meters per second.

From a Newtonian center of mass perspective each has kinetic energy .5*(4.6 x 10^-10)² joules.
So the combined kinetic energy is (4.6 x 10^-10)² joules. In other words, 2.12 x 10^-19 joules.

If I am not mistaken that more than cancels the Newtonian gravitational potential of a pair of kilo balls 400,000 kilometers apart.

Namely 6.67 x 10^-11/400,000,000 = 1.67 x 10^-19 joules.

So at 400,000 kilometers expansion rate exceeds escape speed. A similar calculation shows that at 300,000 kilometers it does not. The balls would begin by getting farther from each other but would eventually begin to fall together.