# Metric expansion of space and Newton's second law

I wanted to ask the following question: If the expansion of the universe is really accelerating, does that mean a certain force applied on the universe? (According to Newton's second law)

What kind of force is that?

• Newton's second law says that a force is required to accelerate a mass. A spacetime coordinate is not something with an associated mass value, so by Newton's laws, no force is required – Jim Jun 25 '14 at 13:32
• That said, Newtonian mechanics is not what we would normally use to describe this phenomenon – Jim Jun 25 '14 at 13:34
• Cosmological expansion is of the intergalactical distance; at smaller scale, various forces that lead to formation of structures preserve distances. – auxsvr Jun 25 '14 at 20:54

There are three different effects to be distinguished here:

(1) The universe is expanding. Let object A be at rest with respect to the average motion of the matter in its neighborhood of the universe. We describe A as being at rest relative to the "Hubble flow." Let object B also be at rest relative to the Hubble flow, at some cosmological distance $x$ from A. (We define $x$ in principle by using a chain of rulers, each of them at rest relative to the Hubble flow.) At a later time $t$, we will observe that $x$ has increased, $dx/dt>0$, where $t$ is time measured by a clock at rest relative to the Hubble flow.

(2) We might naively expect gravitational attraction to decelerate the expansion, so that $d^2x/dt^2<0$, where $t$ is the time measured on a clock that is at rest relative

(3) The expansion did in fact used to be decelerating, but is currently accelerating, $d^2x/dt^2>0$.

In Newtonian terms, number 1 doesn't require any special explanation other than Newton's first law.

Number 2 actually can't be explained using Newton's second law, because the universe is approximately homogeneous, so all gravitational forces on an object should cancel out by symmetry. The argument that gravity should decelerate the expansion is simply wrong in the Newtonian context; in the context of GR, it's wrong for that reason and also wrong for the reason that Newtonian physics doesn't apply at cosmological scales.

Number 3 is similar to number 2, but makes even less sense in terms of Newtonian intuition. Another way to see that acceleration doesn't come from a force is that in the distant future, the universe will go on accelerating at a constant exponential growth rate, even though matter will have been so diluted that there will be essentially no gravitational interaction of matter with matter. The acceleration of cosmological expansion can be thought of as a repulsion of space for itself, not a repulsion of matter for other matter.

Although Jim and Alfred Centauri have brought up coordinates, coordinates aren't really relevant here. The numbers $x$ and $t$ I defined above are not just coordinates, they're measurements performed using clocks and rulers that are in a preferred state of motion (at rest relative to the Hubble flow).

• Welcome back, Ben. – Alfred Centauri Jun 25 '14 at 21:51

does that mean a certain force applied on the universe?

No, the acceleration of the metric expansion of space is not acceleration in the sense of Newton's 2nd law

$$\vec F = m \vec a$$

where, in the above, the acceleration $\vec a$ is understood to be the 2nd time rate of change of the position of the object with mass $m$

$$\vec a = \frac{d^2\vec x}{dt^2}$$

The acceleration of the metric expansion of space is an entirely different notion of acceleration.

Essentially, when space is expanding, objects in space can have constant (co-moving) coordinates and, yet, the distance between the objects increases with time. This may seem contradictory but, in fact, it isn't.

The spatial metric tells us, roughly, the change in distance associated with a change in spatial coordinate. It the spatial metric is expanding, two objects with constant coordinate differences will have non-constant, increasing distance between them.

And, if the rate of the increase in distance is itself increasing, we say the metric expansion of space is accelerating.

But, importantly, this does not require that the objects in space themselves are accelerated in the sense of Newton's 2nd law.

Finally, keep in mind that Newton's laws are merely an approximation and we need Einstein's Special and General relativistic mechanics to adequately describe what we observe in the cosmos.