black scholes - Why gamma and theta have opposite signs?

I saw some textbooks use B-S equation to explain why gamma and theta have opposite signs in most of the cases. For example, John Hull's classic book.

The explanation is, first write B-S equation in terms of greeks:

$\frac{\partial V}{\partial t}+rS\frac{\partial V}{\partial S}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2}=rV$

$\Theta+rS\Delta+\frac{1}{2}\sigma^2S^2\Gamma=rV$

$\Theta+\frac{1}{2}\sigma^2S^2\Gamma=r(V-S\Delta)$

Do we need to assume r=0, in order to draw the conclusion that gamma and theta have opposite signs?