ag.algebraic geometry - Is there any rational curve on an Abelian variety?

Over $mathbb C$ you can argue as follows.
Suppose you have a morphism $mathbb P^1(mathbb C) to A $ ($A$= abelian variety ). Since $mathbb P^1(mathbb C) $ is simply connected , the morphism lifts to the universal cover of $A$, affine space $mathbb C^n$. But since $mathbb P^1(mathbb C)$ is complete and connected, the lift to affine space must be constant and hence the original morphism is constant too.

The answers by Charles, Felipe and jvp are better because they work over arbitrary fields, but since the argument just given is so ridiculously elementary (introductory topology), I thought it might still be of some interest ( also it works in the holomorphic category if $A$ is a complex torus, maybe not algebraic).