# Gravitational redshift discrepancy?

I want to compute the redshift of a signal emitted by a static observer in $$r=R_1$$, $$\phi=\phi_1$$and recieved by another static observ at $$r=R_2$$, $$\phi=\phi_2$$ with $$R_2>R_1$$, in Schwarzschild metric. So i determined it in two different manners obtaing different results. First i considered the metric for a static observer

$$ds^2=-(1-\frac{2m}{r})dt^2=-d\tau^2$$ $$dt=\frac{d\tau_1}{(1-\frac{2m}{R_1})^{1/2}}=\frac{d\tau_2}{(1-\frac{2m}{R_2})^{1/2}}$$ So results

$$\frac{\lambda_2}{\lambda}=\frac{(1-\frac{2m}{R_2})^{1/2}}{(1-\frac{2m}{R_1})^{1/2}}$$

Instead using the simmetry under timereversal of the metric we have

$$\frac{dt}{d\tau}(1-\frac{2m}{r})=constant$$ $$dt=\frac{d\tau_1}{(1-\frac{2m}{R_1})}=\frac{d\tau_2}{(1-\frac{2m}{R_2})}$$

Giving

$$\frac{\lambda_2}{\lambda}=\frac{(1-\frac{2m}{R_2})}{(1-\frac{2m}{R_1})}$$ What i'm doing wrong?