special relativity - Klein gordon field and positive/negative energy solutions

I my course we calculated the Klein-Gordon field:

$$\phi(x)= \int \frac{d^3k}{(2 \pi)^3}\frac{1}{2k_0} ~ \left[a(\vec{k})e^{-ik.x}+b^*(\vec{k})e^{i kx}\right]$$

We said that the part $a(\vec{k})e^{-ikx}$ correspond to the positive energies solutions, and the part $b^*(\vec{k})e^{i kx}$ to the negative ones. (We are not doing quantum mechanics here, $a(\vec{k})$ and $b^*(\vec{k})$ are numbers).

Why do we say this?