= Gravitational redshift
{title2=$\nu_{\mathrm r}/\nu_{\mathrm e}=N_{\mathrm e}/N_{\mathrm r}$}
{wiki}
In a static spacetime with $g_{tt}=-N^2$, a photon has conserved <Killing energy> $E=-p_t$. A static observer with four-velocity $N^{-1}\partial_t$ measures frequency proportional to $E/N$. Hence light emitted at one lapse and received at another obeys the displayed ratio. For $N=1+\Phi/c^2+O(\Phi^2/c^4)$ it gives the weak-field shift $(\nu_{\mathrm r}-\nu_{\mathrm e})/\nu_{\mathrm e}=(\Phi_{\mathrm e}-\Phi_{\mathrm r})/c^2$. This local clock effect does not require light-ray bending.
An upward <photon> has lower received frequency when compared with the receiver's local clock: $\Phi_{\mathrm r}>\Phi_{\mathrm e}$ makes the shift negative. The corresponding upper clock runs faster in a comparison using the same static time coordinate. These are reciprocal descriptions of <gravitational redshift> and <gravitational time dilation>, not opposite predictions. A locally measured <speed of light> remains $c$.
Back to article page
