In response to a student question from Barro's lecture:

There are actually two concepts in growth economics- absolute convergence and conditional convergence. The idea of absolute convergence is that there is one value of per capita GDP that all countries are growing toward (an equilibrium level if you will). Alternatively, you can think of this as poorer countries growing faster than richer countries no matter what, so that eventually they will all come to the same steady state level. Conditional convergence, on the other hand, postulates that there is a unique steady state level for each country. This level depends on things like natural resources, institutions and government frameworks in place, etc. If these are not present to a sufficient degree, the country will have a very low steady state and/or may not converge at all. Alternatively, you can think about convergence to steady state (and catching up to rich countries) is conditional on these things being present. With absolute convergence, a country with a lower GDP would be growing faster, since it has further to go to achieve steady state. With conditional convergence, a country that it further from its own steady state would be growing faster, and it slows down as it approaches this steady state. I don't know that the distinction between absolute and conditional convergence was explicitly made in lecture. The point is that being further away from steady state implies faster growth.

There are a lot of analogies that I could make here. My current favorite is Newton's law of cooling. My physics is a bit (read, very) rusty, but I seem to remember that if you take an object of one temperature and drop it into a substance of another temperature, it heats/cools quickly at first, since the temperature differential between the two substances is high, and then as the substance gets closer in temperature to that of its surroundings, it heats/cools more slowly. The process is described by a differential equation of the form dT/dt = k(T-T*) or something similar (again, rusty), which looks not too different from what you saw in lecture.