# Universe Expansion and two tennis balls

Clear the universe of all matter except for two tennis balls. Place the two tennis balls in the same inertial frame 1 Mpc apart.

• Are the tennis balls getting further apart?
• Will the tennis balls remain in the same inertial frame?

EDIT: Don't assume the balls are massless, but please ignore the gravitational attraction two tennis balls separated at 1 Mpc would exhibit on one another. (Previously, I had asked to assume the tennis balls were massless. I couldn't switch to a strike out font, so I just removed it entirely.)

• This question is a lot deeper than it at first appears. Whether "matter" includes "dark energy" is the most trivial part. More interesting to ask are "how does $\Omega_k$ change if all other $\Omega$'s are zeroed?" and "can we really talk about global inertial frames?" Also hysteresis: what's the difference between an empty spacetime and a recently emptied space? – user10851 Mar 23 '13 at 1:49
• Is it simpler to just put the tennis balls in our universe relatively far away from anything else? Any idea on the answer in either case? – aepryus Mar 23 '13 at 2:03
• @ChrisWhite: if I remember correctly, the Weyl tensor of the robertson-walker spacetime is zero. If you zero out the mass density, then the spacetime has to go Riemann flat, which means that you'll have to get $k=0$. There are probably subtle issues that would arise with the exact way you do the gluing, but a finite subset of that spacetime will definitely be Minkowski. – Jerry Schirmer Mar 23 '13 at 2:22
• It isn't really possible to put two things in the same inertial frame while also putting them one mega-parsec apart. Inertial frames are local things, but (if we assume the universe is expanding) the space-time in between the two tennis balls is not flat, and the concept of an inertial frame doesn't apply. – Nathaniel Mar 23 '13 at 7:00
• I just wanted people to ignore the gravitational attraction between them. At 1 Mpc it would be pretty small anyway. At any rate, I think it's a fairly common thought experiment technique. (Frictionless, ignore air resistance, point particles, pendulums with 0 mass strings, etc) – aepryus Mar 24 '13 at 0:51

In the abscence of dark energy, yes (and ignoring the balls' infinitesimal gravitational attraction). The background of the universe will then be special relativity, which predicts that the two geodesics traced by the balls will remain forever parallel.

• Are they initially moving apart? Their relative velocity will never change, at least in the most obvious, sensible time slicing. – Jerry Schirmer Mar 22 '13 at 23:14
• You're not reading it wrong, but you're missing Jerry's point. The reason the universe is expanding is because it is filled with dark energy, i.e. there is a nonzero cosmological constant. By "clear the universe of all matter" it was assumed that dark energy should also be removed. Remove the DE and space doesn't expand. Leave it in and it will, albeit at a different rate than we observe because the normal matter in the universe helps to "counteract" the effect of DE. – Jold Mar 23 '13 at 0:49
• So, universe expansion only occurs because of dark energy and will not occur in its absence? Does dark energy have any other impact on the universe other than its expansion? – aepryus Mar 23 '13 at 0:52
• Well, since we have no idea what "dark energy" actually is, that's a tough question to answer. – Jold Mar 23 '13 at 0:53
• One cannot sit here discussing massless things at rest. Anything massless has to move with the velocity of light, otherwise it would have no energy to do anything, even geodesics. – anna v Mar 24 '13 at 5:35

After pondering this for a day and reading http://en.wikipedia.org/wiki/Metric_expansion_of_space over and over again, I think I know the answer to my questions.

• Are the tennis balls getting further apart? - Yes
• Will the tennis balls remain in the same inertial frame? - Yes

Conceptually, let's say that the universe is not expanding at all. We place the two tennis balls 1 Mpc apart, both stationary to one another. The momentum of the system is zero.

Now instantly, we change the Hubble Constant to 67.8 km/s/Mpc. The two tennis balls are now moving away from one another at 67.8 km/s (and of course, as they move away from one another that rate increases with distance).

Theoretically, the momentum: p = mv is now not equal to 0. But, momentum has always been a relative notion and in the context of space expansion I suspect p = m(v-HD) and so in this case still equal to 0.

If we now switch the Hubble Constant back to 0, even though the two tennis balls were moving apart at 67.8 km/s, they would instantly stop moving relative to one another. Their momentum remaining 0 throughout the experiment.

When I was thinking about this one thing that was perhaps helpful was thinking of the universe as a checkerboard. My tennis balls are checkers on opposite ends of the board; checkers that aren't moving from their current space. In this scenario, "expansion of the universe" means to decrease the size of the spaces of the grid on the checkerboard.

When I start out, I can ask how many spaces apart are the checkers, which could be 7 for example. Now I divide each space into 4 sections. The checkers haven't moved, but they are now 14 spaces apart. I divide the spaces again and now without moving the checkers are 28 spaces apart.

They are getting further apart, not because they are moving but because of the "metric expansion of space".

At any rate, if any and all of this is wrong please let me know. All comments greatly appreciated.

• You cannot have at rest a massless anything. It is moving with the velocity of light if it is massless. The geodesics appear because of their energy. – anna v Mar 24 '13 at 5:29
• @annav: I don't know that you dinged this answer, but did you have a problem with it beside the "massless" thing? – aepryus Mar 24 '13 at 8:47
• The massless thing is the lynch pin. Something with 0 mass has 0 energy, it is non existent unless, like the photon it has energy h*nu and one has the relativistic energy equation. It is like talking about fairies and ghost physics. One cannot separate General Relativity from Special Relativity when one posist zero mass. – anna v Mar 24 '13 at 9:32
• @annav Fair enough. Now that I have removed the massless qualifier, any comment on the question or this answer? – aepryus Mar 24 '13 at 9:36
• you have to edit your answer for me to remove the negative. I am not proficient on such modeling so I will have to think if I agree or not with your statements here . A simple grammatical edit will do so I can remove the -1. – anna v Mar 24 '13 at 9:40