Equation in a Susskind video

I was watching a video of Leonard Susskind giving a lecture at Stanford (I forget which one) and he started talking about the cosmological constant and how that, on the scale of galactic clusters, it causes them to accelerate away from each other and gave the equation Force = - (spring constant) * position + (cosmological constant) * position, noting that you would notice its effects unless the distances were very large.

Is he describing an actual equation or is he merely presenting an analogy? I was also under the impression that it wasn’t that galaxies were moving away from each other, but rather that space itself is expanding, which is important because the expansion of spacetime isn’t bound by c and the movement of galaxies is.

Can the Dope help me grok this?


It’s not that bad of an equation, though of course it’s a little oversimplified. The idea is, yes, that space itself expands. This means that the distance between two clusters increases with time. Moreover, the “acceleration” of this distance at any moment (properly speaking, the second derivative of the distance with respect to time) is proportional to the distance between them at that moment . Thus, “acceleration” is proportional to distance. But in Newtonian mechanics, force is proportional to acceleration. So if we were sitting in one of these clusters, we would see the other cluster run away from us just as though it was experiencing a repulsive force whose magnitude was proportional to its distance from us.

In other words, it’s a bit of a fudge; Susskind is using the “acceleration” due to the expansion of space itself and then applying Newtonian mechanics to say “this cluster behaves like a force is acting on it”. But it’s not that awful of an analogy, and we often use such “fictitious forces” in general relativity anyway. (Example: gravity.) For a popular-level lecture, it’s perfectly acceptable.

Here is the video, BTW.

If he was using spring forces as his comparison, then the analogy falls apart horribly, since spring forces also increase with distance. His expression can be simplified to (cosmological constant - spring constant)*distance, the sign of which depends only on whether the cosmological constant or spring constant is greater, and does not depend on distance at all. In the real world, of course, what we’re usually comparing the cosmological constant to is not galaxy-spanning springs, but ordinary gravity, which gets weaker with distance, and so the comparison holds.