# Two observers initially at rest in the expanding Universe

The standard question asked about the expansion of the Universe sounds roughly like: "Does my pencil expands, together with expansion of the Universe?"

And the standard answer sounds roughly like: "No, expansion of the Universe works at cosmological scales -- everything must be homogeneous and isotropic and be interacting through gravity only. Your pencil is a very non-homogeneous low-scale thing that is bound by electromagnetic forces. So no."

I've noticed that I'm not the only one who feels that the answer is kind of dodging the substance of the question. What I really want to know is if this "space expansion" really "pushes stuff apart" or not. So I wanted do "distill" this idea into a thought experiment that formulates this intuition into a precise setup.

So, here is the setup:

• We have two non-interacting observers $A$ and $B$ in the expanding Universe.
• The distance (say, proper distance) between them is large enough to consider the Universe to be homogeneous and isotropic.
• We make sure that at the start of the experiment the observers do not move with respect to each other (again, in a sense that the proper distance between them is not changing). We can ensure this with, say, requiring no redshift of light signals between them. (You can propose some more intricate Einstein-light-ray-synchronization procedure for that.)

I think this setup captures the substance of the question quite well. If space really expands, then it is natural to expect the observers $A$ and $B$ to start moving apart from each other. If that doesn't happen, on the other hand, then it doesn't sound like "expanding space" at all.

So the question is: what would be the strict and formal solution for the setup above?

• the pencil does not expand but the space "between" the "atoms" is expanding the same way it is expanding between galaxies, you just cannot observe it because the electromagnetic attraction counteracts it – Adam Sep 30 '16 at 18:21
• this may be of interest to you arxiv.org/abs/1312.7797 – Adam Sep 30 '16 at 18:29
• This asks again, in different words, the same question as in your previous question answer at a related question at physics.stackexchange.com/questions/282511/…. You didn't like my answer there so now you are asking the same thing. So, this is a duplicate. – Bob Bee Sep 30 '16 at 22:57
• Anyway, your error is in saying that two particles are at rest with respect to each to start with, i.e. No redshift. You need to say the particles are at rest in the comoving frame. Same arrow as before – Bob Bee Sep 30 '16 at 23:04

Space and matter

In the standard picture, the reason that space doesn't expand on small scales is not that "it does expand, but gravitational or electromagnetic forces keeps pulling matter together again with a stronger force than the expansion can pull them apart". Rather, space and matter is tied together through gravity:

An initial and still uncertain mechanism called inflation made space expand. The matter (or rather energy at that time) followed along, but its mutual gravitational attraction counteracts the expansion. In some places, there is enough matter that expansion is counteracted completely, keeping matter in these regions from receding from each other. The relevant physical scale for this is galaxy groups.

On the other hand, in some regions there is so little matter that expansion is faster than the average. These regions are the huge voids which are virtually free of galaxies.

Resisting the Hubble flow

Anyway, back to your question: If $A$ and $B$ are $d$ Mpc apart, then the galaxies in the vicinity of $A$ will recede from the galaxies in the vicinity of $B$ with velocity $v = H_0 d$, where $H_0$ is the Hubble constant. How can we ensure that $A$ and $B$ start out with no initial velocity wrt. each other? We can either 1) put $A$ on a spaceship that flies in the direction toward $B$ with velocity $v_\mathrm{B} = v$, as measured by a "normal" observer $A'$ which is at rest wrt. some galaxy close to $A$, or 2) we can do the same with $B$, or 3) we can make both $A$ and $B$ fly, e.g. both at $v_\mathrm{A} = v_\mathrm{B} = v/2$ (where $v_\mathrm{A}$ and $v_\mathrm{B}$ are directed toward each other).

This puts an upper limit on how far $A$ and $B$ can be from each other: Since it's impossible to travel through space faster than $c$, the maximum distance is (just under) $d_\mathrm{max} = 2c/H_0 \simeq 8.8\,\mathrm{Gpc}^\dagger$.

Kinematics in an expanding coordinate system

So, what happens to a spacehip traveling through expanding space at a high velocity? We can investigate this using comoving coordinates, i.e. the coordinate system that expands along with the Universe. In this coordinate system, all galaxies lie approximately still (save for their small "peculiar" velocities; of the order a few 100 km s–1). $A$ and $B$, on the other hand, while their velocity wrt. each other vanish in physical coordinates (by construction of your experiment), they have a non-zero velocity in comoving coordinates. Nevertheless, unless they spend energy to accelerate their spaceships — and this can still only bring them up to a max velocity of (almost) $c$ — their velocity will decrease in comoving coordinates. If in the beginning the fly at (almost) $c$, and if for simplicity we assume that galaxies lie evenly distributed with one galaxy for every 1 Mpc, they will initially pass one galaxy every three million years. However, it will take more and more time to get from one galaxy to the next, since space expands.

In other words, a particle with a velocity through an expanding space will slowly lose kinetic energy, analogous to the redshifting of light.

So, the answer to your question is: If $A$ and $B$ start out sufficiently close to each other, they will eventually meet. But if they start out too far, or if they fly too slowly, they will decelerate and asymtotically go toward rest, ending up being dragged along with the Hubble flow.

$^\dagger$Ignoring minor complications like the fast heating of a spaceship traveling through the interstellar medium at a velocity (almost) the speed of light.

• Thank you very much for an extended explanation. Still, I'm missing a piece in it -- namely, "they will eventually meet" doesn't really follow from your previous statements. Can you give me a reference or the calculation itself (don't need to dumb it down -- I can handle GR formalism). – Kostya Oct 1 '16 at 9:09
• @Kostya: By that statement I simply meant if they start out so close to each other that they will meet before cosmological expansion can "overtake" their motion. But for this to be possible, they do need to accelerate. If they just maintain their original velocity (in physical coords), they will never meet. – pela Oct 1 '16 at 11:07
• But for instance, if a spaceship departs today 4.4 Gpc away in our direction with velocity (almost) $c$, it will be at rest wrt. us (since 4.4 Gpc is the distance at which Hubble expansion equals the speed of light). If it starts out farther away than 5 Gpc, it will never be able to reach us, but if its nearer than this, and it keeps accelerating to maintain a velocity close to $c$, it will reach us before $t=\infty$. – pela Oct 1 '16 at 11:09
• Lets consider non-accelerating observers. Am l right that you are saying that the observers' proper distance will start to increase in my setup? – Kostya Oct 1 '16 at 12:31
• @Kostya and pela. May I suggest you both a very interesting thesis work. It shows the geodesics, that Kostya asks for in the chapter 3. – Tziolkovski Oct 1 '16 at 18:52

There is some merit to the idea of spatial expansion: FLRW spacetime is conformally flat, leading to a natural notion of freely falling 'rest frames' given by the Hubble flow. The proper distance at constant cosmological time between any points 'at rest' increases, so the space between them is said to expand. This is especially instructive in case of finite universes where this is accompanied by an increase in total spatial volume.

However, what is problematic is that you have to be careful not to forget that despite conformal flatness and a preferred spatial slicing, we're still dealing with a general-relativistic model with nonzero curvature instead of Minkowski space.

For example, your idea about establishing zero proper motion between observers by looking at redshift does not work: Zero redshift means zero relative velocity as evaluated by parallel transport along the light path, which is different from zero change in proper distance at constant cosmological time$^\dagger$.

Now regarding your setup, in an expanding universe particles that start out with zero proper velocity are moving towards each other if you take the comoving perspective. Whether this means that they will meet or be pulled apart beforehand (in terms of proper distance) cannot be answered in general as this depends on the initial distance as well as the time evolution of the scale factor. For example, figure 3.1 of the master's thesis linked by Tziolkovski shows one case where the particles never meet, and three cases where they do. In all of the cases, the proper distance ends up increasing, but in the last three cases only after the particles have moved past each other.

As far as a formal solution is concerned, let's see how far we can get without too much effort.

First, the peculiar velocities decrease according to $$|v_\text{pec}| = \frac 1 {\sqrt{1 + \frac {a^2}{{\pi_0}^2}}}$$ where $\pi_0 = \text{const}$.

Given a particle at initial distance $d_0$ from the origin and initial proper velocity $v_0 = 0$, this yields proper velocities $$v = Hd - \frac 1 {\sqrt{ 1 + \frac {a^2}{{a_0}^2} \left( \frac 1 {(H_0\,d_0)^2} - 1 \right) }}$$

Now, let's look at de Sitter spacetime specifically with $a(t) = e^{Ht}$ and assume small initial recession velocities $H_0\,d_0 \ll 1$.

Taylor expansion yields $$v(t) \approx H\left( d(t) - d_0\,e^{-Ht} \right)$$ which is solved by $$d(t) = d_0\,\cosh{Ht}$$

$^\dagger$ However, what you can do is split proper velocities into recession and peculiar velocities, which corresponds to a factoring of the frequency shift into cosmological and peculiar Doppler shift. In contrast, if you use the generic approach of parallel transport along the light path, the frequency shift will be wholly Doppler in nature.

• Maybe I wasn't clear in my setup -- I wanted to set an initial condition where $A$ and $B$ are at zero velocity w.r.t. each other. Lets say I want the derivative of proper distance to be zero at $t=t_0$. I want to know how will they move. I'd like to see the GR solution to that -- two geodesics in the FLRW that satisfy my initial condition. Can you refer me to such a solution? – Kostya Oct 1 '16 at 18:08
• your setup was clear, but the procedure you proposed to create it (light signals with zero redshift) doesn't work – Christoph Oct 2 '16 at 16:24
• I also added another paragraph; still too lazy to figure out the detailed solution, though ;) – Christoph Oct 2 '16 at 16:32
• Did you have a chance to look at that chapter 3 ? – Kostya Oct 2 '16 at 19:32
• @Kostya: I rewrote my answer; hopefully, the points I was trying to make are more clear now – Christoph Oct 3 '16 at 10:20