= Gravitational time dilation
{title2=$d\tau=N\,dt,\quad\Delta\tau/\tau\simeq\Delta\Phi/c^2$}
{wiki}
In a static <spacetime>, a clock held at a fixed position records <proper time> $d\tau=N\,dt$ for metric time component $-N^2c^2dt^2$. Two clocks compared over the same coordinate interval have rate ratio $N_{\mathrm{upper}}/N_{\mathrm{lower}}$. Each clock ticks normally in its own laboratory; the difference appears when their readings or transmitted frequencies are compared. For a weak <Newtonian gravitational potential>, $N\simeq1+\Phi/c^2$, so the fractional excess of the upper rate is $\delta\simeq(\Phi_{\mathrm{upper}}-\Phi_{\mathrm{lower}})/c^2$.
Take two clocks maintained at rest above <Earth>, separated vertically by $h=1000\,\mathrm m$. Approximate the local <gravitational acceleration> as constant, $g=9.81\,\mathrm{m\,s^{-2}}$, neglect rotation, and use the exact <speed of light> $c=299792458\,\mathrm{m\,s^{-1}}$ from https://physics.nist.gov/cgi-bin/cuu/Value?c[NIST's constants database]. Then $\Delta\Phi\simeq gh=9810\,\mathrm{m^2\,s^{-2}}$ and
$$
\delta\simeq\frac{gh}{c^2}=1.09151\times10^{-13},\qquad
\Delta\tau\simeq\delta(86400\,\mathrm s)=9.43064\times10^{-9}\,\mathrm s.
$$
The upper clock gains about $9.4$ nanoseconds during one $86400$-second interval on the lower clock. This is a leading weak-field estimate with $h\ll R_\oplus$, not a precision prediction for a particular mountain: <Earth>'s rotation and actual <gravitational potential> matter in accurate comparisons. The effect is experimentally accessible. Chou, Hume, Rosenband and Wineland raised an optical clock by $0.33\,\mathrm m$ and measured a fractional rate change $(4.1\pm1.6)\times10^{-17}$; the same formula with $g=9.80\,\mathrm{m\,s^{-2}}$ predicts $3.60\times10^{-17}$. See their https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=905055[2010 paper, Optical Clocks and Relativity, equation (2) and figure 3]{ref}. Such comparisons make clocks probes of <gravitational potential> as well as instruments for timekeeping.
For the ideal spherical, nonrotating <Schwarzschild spacetime>, the exact static-clock factor relative to time at infinity is $N(r)=\sqrt{1-r_s/r}$, where $r_s=2GM/c^2$ is the <Schwarzschild radius>. Its weak-field expansion is $1-r_s/(2r)$. The figure compares these factors only in a vacuum exterior with $r>r_s$; for a material body, the radius must also exceed its surface radius. It does not extend the exterior formula through the body's interior. See https://davidtong.org/pdfs/teaching/general-relativity/gr.pdf#page=240[Tong's lecture notes, section 6.1 on Schwarzschild exterior geometry].
\Image[/gravitational-clock-comparison.png]
{title=Clock gain at 1 km above Earth and exact versus weak-field Schwarzschild exterior clock factors}
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