Distance expansion and escape velocity ( thought experiment )

In summary, the question of whether the Hubble law recession rate exceeds the classic escape speed for a pair of kilogram masses placed slightly over one light second apart in open space was explored. It was found that at a distance of 400,000,000 meters, the Hubble expansion of 9.2 x 10-10 meters per second exceeds the escape speed, resulting in the balls continuing to separate. This is equivalent to the distance from Earth to the Moon. However, at a distance of 300,000 kilometers, the expansion rate does not exceed the escape speed, and the balls would eventually begin to fall towards each other.
  • #1
marcus
Science Advisor
Gold Member
Dearly Missed
24,775
792
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.
 
Last edited:
Space news on Phys.org
  • #2


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.
 
Last edited:

1. What is distance expansion and how does it relate to escape velocity?

Distance expansion is the phenomenon in which the space between objects in the universe is increasing over time. Escape velocity is the minimum speed an object needs to reach in order to break free from the gravitational pull of a larger body, such as a planet or star. The two concepts are related because as the distance between objects in the universe expands, the escape velocity needed to break free from their gravitational pull also increases.

2. How does distance expansion affect the motion of objects in the universe?

Distance expansion does not directly affect the motion of objects in the universe. Instead, it affects the space between objects, causing them to move farther apart over time. However, this change in distance can indirectly impact the motion of objects, as the gravitational pull between them decreases with increasing distance.

3. Can distance expansion be observed in real time?

No, distance expansion occurs on a very large scale and takes place over billions of years. It is not something that can be observed in real time, but rather can only be observed by looking at the changes in the positions and distances of objects over long periods of time.

4. Is there a limit to how fast objects can move due to distance expansion?

No, there is no limit to how fast objects can move due to distance expansion. The expansion of space itself does not impose a speed limit on objects, and the velocity of objects can still be affected by other forces, such as gravity.

5. How does escape velocity change with the expansion of space?

As the distance between objects in the universe increases due to distance expansion, the escape velocity required to break free from their gravitational pull also increases. This is because the gravitational pull between objects weakens with increasing distance, requiring more energy to overcome it and reach escape velocity.

Similar threads

Replies
15
Views
2K
Replies
12
Views
3K
Replies
2
Views
1K
Replies
16
Views
792
  • Cosmology
Replies
14
Views
1K
Replies
4
Views
1K
Replies
33
Views
3K
Replies
13
Views
8K
Replies
17
Views
5K
  • Special and General Relativity
Replies
4
Views
1K
Back
Top