Transverse Gravitational Redshift

These two independently derived equations predict a relativistic redshift between two ideal clocks at identical gravitational potential separated by a constant distance $(d)$. The general formula (1) includes an elliptic integral of the second kind $(E\;|2)$. Here, $b$ is the effective radius of the Earth ($\sim\!6371 km$) and $d$ the horizontal distance between two clocks at the altitude $b$. Eq. (2) is similar to the approximate formula $z \approx gh/c^{2}$ for the radial Einstein shift as both formulas are valid for small separation distances and both are readily derived from first principles. For a clock separation of 500 $km$ at sea level, the two formulas yield the same prediction $(z \sim10^{-12})$ to an accuracy of $\pm1\times10^{-15}.$ For 50 $km$, these formulas yield the same prediction $(z \sim10^{-14})$ to an accuracy of $\pm1\times10^{-19}.$ This empirical prediction can be tested using modern atomic clocks and frequency transfer by fiber-optic cable. \begin{equation} z = \sec\left( \frac{\sqrt{\frac{8GM\cos\phi}{bc^{2}}} \; E\left(\frac{\phi}{2}|2 \right)}{\sqrt{\cos\phi}} \left.\vphantom{\rule{0mm}{10mm}}\right]^{+\phi}_{-\phi} \right)-1 \hspace{10mm} \left[\phi = \frac{d}{2b}\right] \end{equation} \begin{equation} z = \frac{\Delta f}{f} \approx \frac{4GM}{bc^{2}} \sin^{2} \left(\frac{d}{2b}\right) \hspace{33 mm} \left[d \ll b\right] \end{equation}