# Eddington-Finkelstein coordinates, how to tell which is ingoing and which is outgoing?

The Eddington-Finkelstein coordinates in case of Schwarzschild metric are defined as

\begin{align} u&=t-r^*\\ v&=t+r^* \end{align}

where $$r^*=r+2GM\ln\left|\frac{r}{2GM}-1\right|$$

The question is that how to understand which one is ingoing and which one is outgoing. Why $v$ is ingoing and $u$ is outgoing?

$$du,dv$$ are light-like, i.e. they could, in principle, be viewed as some affine parameters of some light-rays. However, we will focus (in the spirit of the usual coordinate-nature analysis) on what is the nature of either $$u,v$$ constant. I.e., we want to know what is the nature of $$u,v$$ constant hypersurfaces and derive the nomenclature from this.
Now the question is what is the nature of congruence parametrized by $$u,v$$. Let us consider some finite $$t=t_0$$ and $$r_*=r_0$$. If we are investigating $$u,v$$ constant hypersurfaces, then $$t>t_0$$ means surely $$r_*>r_0$$ for $$u=t-r_*$$ constant. I.e., the $$u=const.$$ surface is farther out at a later time. Hence, the lightcone is interpreted as outgoing with respect to the centre $$r_*=0$$ for $$u=$$const. and we call the respective Finkelstein coordinate the outgoing coordinate.
The opposite is true for $$v=t+r_*$$. $$t>t_0$$ means surely $$r_* for $$v=$$ const., or that for $$v$$ constant we have a lightcone ingoing with respect to the centre $$r_*=0$$.