# How can the expansion of space be considered a force?

I have been looking for a long while now for an explanation as to why the expansion of space does not apply on small scales. I have gone through StackExchange - specifically, this post - and Quora; however, none of the sources adequately answer the question, some saying that the model of expansion of space does not describe small-scale behaviour and others saying the expansion of space can be viewed as a 'force', which forces such as gravity and weak force easily overcome. The latter is not only ambiguous but also extremely counter-intuitive: take, for example, a situation with a moon orbiting a planet; as space expands, the distance from the moon to the planet increases linearly and the speed decreases linearly. However, since the gravitational 'force' decreases quadratically with increasing distance, the gravitational force decreases more quickly than the speed decreases. This means that, at some point, the moon's velocity will surpass the ever-decreasing escape velocity for its position and will fly off the planet's orbit into space. So, my ultimate questions are:

1. In what regard can the expansion of space be viewed as a force?

2. How does that explain the fact that the elementary forces are not affected, including the aforedescribed scenario?

• Dozens of variations on this question have been asked on this site. Many incorrect answers have been upvoted. For an analysis that answers your question, see Cooperstock, Faraoni, and Vollick, "The influence of the cosmological expansion on local systems," arxiv.org/abs/astro-ph/9803097v1 , or my own summary, physics.stackexchange.com/questions/70047/… . The linked question, "Why does space expansion not expand matter?" is of the form "Why is X true?," where X is false. The correct question would be why the effect is unmeasurably small. – Ben Crowell Jun 11 '17 at 20:57
• So, then, it IS expanding matter - just the effect is negligible? If that is true, a question arises: at the early stages of the universe, everything was very tightly packed together. This means that, if the expansion of space was expanding matter, we would be able to see the edges of the things in the universe approximately at the same place very close to us. Of course, this is not the case; where am I wrong? – Max Jun 11 '17 at 21:30
• – Count Iblis Jun 11 '17 at 21:34

## 1 Answer

1. The expansion of space is gravity (e.g. it is the left hand side of the Friedmann equations....the information about how space expands is contained in the scale factor $a$). That being said, I say that the expansion of space is not a force, in the same manner that gravity is not a force from the point of view of general relativity.

2. When one solves the Friedmann equations, one will usually make simplifying assumptions about how matter is distributed throughout the universe (such as assuming that the matter in the universe is a perfectly uniform fluid). Obviously, this is not the case for the universe on the scales of solar system and galaxies, but is a pretty close assumption for the universe at large. If we consider the moon orbiting the earth, the size of spacetime (e.g. the scale factor) should be affected by this motion. The reason this is not considered in calculations of orbits is that (1): on small scales, the expansion is so tiny that it would have no noticeable effect. To consider this point, think about the Hubble parameter $H_o\sim 70$ km/s*Mpc. At a distance of 1 pc away, the expansion would be $\sim 7$ cm/s. (And this is still >3 LY away.). The expansion of spacetime for an object at solar system scales is so tiny (compared to the characteristic velocities of those objects in their orbits) that it can be neglected. (2): the Friedmann Equations themselves would be a mess. I think I once saw a paper that claimed to calculate the scale factor for a universe that is empty aside from a single spherically symmetric mass, but that is all I have ever seen on the topic (basically the equations do get nasty fast).

• Indeed, gravity is not a force itself, but, as was said in another post regarding the expansion of space, force is needed to 'resist' gravity and remain stationary with respect to the object imposing the gravity, so in some regard, it could be viewed as a force. I agree that the same argument could be made about the expansion of space; however, for the very reason that gravity is NOT a force, it should not be able to resist the expansion of space. More specifically, if the distance from a moon to a planet increases by a factor of $a$ due to the expansion of space, the gravitational 'force' – Max Jun 11 '17 at 20:37
• will be weakened by a factor of $a^2$, while the speed of the moon only decreases by a factor of $a$, which means the gravity can't hold off the moon in any scenario. I do appreciate that this factor $a$ is negligible at small scales, but, looking at the universe as a whole, if the expansion of space was also expanding stars, planets, etc, we would now see the edges all the other galaxies very close to us, as the galaxies would be so big. So I am not only talking about small scales here. – Max Jun 11 '17 at 20:50
• I don't understand the point that you're trying to make. The expansion of spacetime is gravity. From the GR viewpoint, where $G_{\mu\nu}=M_{PL}^{-2}T_{\mu\nu}$, $G_{\mu\nu}$ is gravity (in that it contains the information about spacetime's curvature) and $T_{\mu\nu}$ is the distribution of matter (what I suppose you may call the 'source' of gravity). – Bob Jun 12 '17 at 4:12