Gravitational redshift (${\nu }_{\mathrm{r}}/{\nu }_{\mathrm{e}}=N_{\mathrm{e}}/N_{\mathrm{r}}$)

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 $c$.

For stationary emitter and observer in the same static patch, repeated rays between fixed radii have the same coordinate travel delay. Since a stationary clock obeys $d\tau =\sqrt{\alpha }dt$, the received-to-emitted proper-period ratio is $\sqrt{{\alpha }_O/{\alpha }_E}$. At a central observer and emitter radius $r_E$, this is $(1-r_E^2/r_S^2{)}^{-1/2}$. Holding an emitter stationary requires unbounded proper acceleration as it approaches the horizon; the limiting redshift formula does not describe a timelike static worldline on the horizon itself.

A static emitter at radius $R>2GM/c^2$ and a static receiver at infinity compare photon wavelengths with $1+z=(1-2GM/(R{c}^2){)}^{-1/2}$. Conservation of the time-translation Killing energy and the static clock rate $d\tau =\sqrt{1-2GM/(r{c}^2)}dt$ prove the relation. Motion of the emitter adds a Doppler factor; a rotating stellar exterior may require a different metric.

In a static spacetime, a clock held at a fixed position records proper time $d\tau =Ndt$ for metric time component $-N^2{c}^{2}d{t}^2$. 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, $N\simeq 1+\Phi /c^2$, 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, $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$, neglect rotation, and use the exact speed of light $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$ from NIST's constants database. Then $\Delta \Phi \simeq gh=9810{\mathrm{m}}^2{\mathrm{s}}^{-2}$ andThe 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 $(4.1\pm 1.6)\times 10^{-17}$; the same formula with $g=9.80\mathrm{m}{\mathrm{s}}^{-2}$ predicts $3.60\times 10^{-17}$. See their ^[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 Tong's lecture notes, section 6.1 on Schwarzschild exterior geometry.

For spherical weak-field gravity, a stationary clock at radius $r_g>R$ gains the displayed proper time relative to a surface clock over coordinate interval $t$. A geostationary orbit has $r_g=(GM/{\Omega }_{\oplus }^2{)}^{1/3}$. This is the potential contribution only. Combining it with orbital and surface rotational special-relativistic terms requires a consistent reference frame and must not be silently included in a gravitational-only comparison.

A radial infalling geodesic in Schwarzschild spacetime has $dt/ds=E/f$ and $dr/ds=-\sqrt{E^2-f}$, where $f=1-2M/r$. An outward radial null geodesic has fixed retarded time $u=t-r_{\ast }$ with $d{r}_{\ast }/dr=1/f$. Consequently $du/ds=(E+\sqrt{E^2-f})/f$. This factor stretches a short emitted pulse's duration, and its reciprocal multiplies the received frequency at infinity. Near horizon crossing $s_H$, it becomes $du/ds\sim 4M/(s_H-s)$, including both gravitational redshift and the emitter's Doppler shift. The local short-pulse approximation requires its emitted duration to be much smaller than $s_H-s$.

// Target: general-relativity.bigb

The experiment compares gamma-ray emission and resonant absorption at different heights in Earth's field, using the Mössbauer effect to resolve the small gravitational redshift. Its weak-field result is the frequency shift $\Delta \nu /\nu =-gH/c^2$ for an upward ray over height $H$. Any metric theory with the corresponding static lapse predicts this leading effect; the test alone does not distinguish general relativity from the weak-field prediction of Nordstrom theory of gravitation.