To use M Quack's example, imagine that you have a spring in your hand with a weight attached. The spring has some damping and the you can move your hand around in a spatially varying gravitational field.
You know that the equilibrium position of the spring depends on the local gravitational field. You also know that because of the damping the spring will reach this equilibrium position in a time that is roughly 1/(decay rate) provided you hold your hand still.
But suppose your hand does move slowly. A question you could ask is, how slowly should your hand move so that the spring is always in local equilibrium. It's reasonable to suppose that you would want a large decay rate compared to the timescale of hand motion so that you are effectively sitting in one place for much longer than it takes to reach equilbrium.
However, I'm also not sure this is precisely what this book is talking about. Looking at page 150 just above section 6.3, the book states that [itex] \omega_{rec} \ll \Gamma [/itex] which implies that [itex] 1/\Gamma \ll 1/\omega_{rec} [/itex]. However, just below that they state the internal timescale [itex] 1/\Gamma [/itex] should be MUCH SLOWER than the external timescale [itex] 1/\omega_{rec} [/itex]. I would interpret much slower to mean the internal time scale is longer than the external timescale, which gives the opposite inequality. Perhaps I misread or misunderstood or perhaps they meant much shorter?
