Gravitational Redshift and Clock Comparison

Gravitational redshift is an endpoint comparison between clocks and the frequencies they measure. In a stationary spacetime, the time-translation Killing field supplies one conserved photon quantity, while each observer converts that quantity into a local frequency using a different clock normalization. This gives an exact stationary ratio before any weak-field expansion and avoids the misleading claim that a photon possesses one observer-independent energy that is gradually lost in flight.

Required background. We use The Einstein Equation to identify the metric as the gravitational field, Events, Intervals, and Minkowski Geometry for local frequency measurements, and The Newtonian Gravitational Field and Potential for the weak-potential expansion.

Helpful background. Tensor Fields and Natural Operations supports the endpoint contractions; Affine and Variational Geodesics supplies null geodesic transport; The Wave Equation and Characteristics connects wave crests to the geometric-optics frequency; Killing Fields and Isometry Groups supplies the Killing energy used in the stationary reduction.

Stationary observers and exact clock rates

Let $\xi^\mu$ be a timelike Killing field in the region of interest. Choose coordinates adapted to it, so $\xi=\partial_t$, and define its lapse along the stationary observer congruence by

$$N\equiv\sqrt{-\xi^\mu\xi_\mu}=\sqrt{-g_{tt}}.$$

An observer held on an integral curve of $\xi$ has four-velocity

$$u^\mu=\frac{\xi^\mu}{N}.$$

For that clock, $dx^i=0$ and hence

$$d\tau=N\,dt=\sqrt{-g_{tt}}\,dt.$$

This statement is exact wherever $\xi$ remains timelike. The observers need not be freely falling; in a gravitational field they generally require support forces to remain stationary. If two such clocks at emission and reception compare equal coordinate-time intervals defined by the stationary symmetry, then

$$\frac{d\tau_{\rm r}}{d\tau_{\rm e}} =\frac{N_{\rm r}}{N_{\rm e}}.$$

The normalization of $t$ rescales both $N$ values and cancels from the ratio.

Photon frequency as an endpoint contraction

Let $k^\mu$ be the future-directed geometric-optics wave vector, tangent to an affinely parametrized null ray. The wave phase fixes its physical normalization; if one starts from an arbitrarily normalized affine tangent, one frequency measurement fixes that remaining constant. The frequency ratio below is independent of this common scale. Killing’s equation and the geodesic equation imply

$$\frac{d}{d\lambda}(k_\mu\xi^\mu) =k^\nu k^\mu\nabla_{(\nu}\xi_{\mu)}=0.$$

Thus the Killing quantity $E_\xi=-k_\mu\xi^\mu$ is constant along the ray. The local angular frequency measured by any observer is the scalar

$$\omega=-k_\mu u^\mu.$$

For stationary observers, $\omega=E_\xi/N$, and therefore

$$\boxed{ \frac{\omega_{\rm r}}{\omega_{\rm e}} =\frac{N_{\rm e}}{N_{\rm r}} =\sqrt{\frac{-g_{tt}(\mathrm e)}{-g_{tt}(\mathrm r)}} }$$

and

$$1+z\equiv\frac{\omega_{\rm e}}{\omega_{\rm r}} =\frac{N_{\rm r}}{N_{\rm e}}.$$

For the exterior Schwarzschild geometry, whose derivation and domain are given on Deriving the Schwarzschild Metric, $N(r)=\sqrt{1-2GM/r}$. Two supported clocks outside the source therefore obey

$$\frac{\omega_{\rm r}}{\omega_{\rm e}} =\sqrt{\frac{1-2GM/r_{\rm e}}{1-2GM/r_{\rm r}}}.$$

The conserved energy and angular momentum used for Schwarzschild geodesics are derived from the same Killing-field mechanism in Reall 2013, § 12.2, printed pp. 134–135, Eqs. (12.7)–(12.13), PDF.

Weak-potential expansion

In a weak stationary field,

$$g_{tt}=-(1+2\Phi)+O(\Phi^2), \qquad N=1+\Phi+O(\Phi^2).$$

The Newtonian limit and the sign of this metric component are established in Reall 2013, § 8.2, printed pp. 80–83, especially Eqs. (8.26)–(8.35), PDF. Expanding the exact endpoint ratio gives

$$\frac{\omega_{\rm r}}{\omega_{\rm e}} =1+\Phi_{\rm e}-\Phi_{\rm r}+O(\Phi^2),$$

or, equivalently,

$$\frac{\Delta\omega}{\omega} \equiv\frac{\omega_{\rm r}-\omega_{\rm e}}{\omega_{\rm e}} \simeq\Phi_{\rm e}-\Phi_{\rm r}.$$

For light emitted deeper in an attractive potential, $\Phi_{\rm e}<\Phi_{\rm r}$, so the received frequency is smaller: the signal is redshifted. A lower stationary clock also accumulates less proper time per unit Killing time because $N$ is smaller. These are reciprocal aspects of the same endpoint normalization.

What is conserved and what is measured

$E_\xi$ is conserved only because the spacetime is stationary, and its value depends on the common normalization chosen for $\xi$. By contrast, $\omega=-k\cdot u$ is local and observer dependent. Two observers meeting the same photon with different velocities generally measure different frequencies even at the same event. It is therefore not invariantly meaningful to say that the photon has one local energy at all points and loses a definite amount between them.

The exact ratio also requires stationary emitter and receiver worldlines tangent to the same Killing field. Moving endpoints add Doppler factors. A time-dependent metric need not supply a conserved $E_\xi$, and multiple propagation paths may arrive at different events. Those cases remain well defined through the endpoint contractions $-k\cdot u$, but they are not captured by a ratio of $g_{tt}$ values alone. Nor does this evergreen result quote a current experimental bound; instrument modeling and dated constraints belong in an observational analysis.

Check your understanding

A signal is emitted by a supported clock at radius $r_{\rm e}$ and received by another at $r_{\rm r}>r_{\rm e}$ in Schwarzschild spacetime. Expand the exact ratio through first order in $GM/r$ and identify its sign.

Solution

Using $\sqrt{1-2GM/r}=1-GM/r+O((GM/r)^2)$,

$$\frac{\omega_{\rm r}}{\omega_{\rm e}} =1-\frac{GM}{r_{\rm e}}+\frac{GM}{r_{\rm r}}+O((GM/r)^2).$$

Because $r_{\rm r}>r_{\rm e}$, one has $GM/r_{\rm r}<GM/r_{\rm e}$, so the correction is negative and the received signal is redshifted. With $\Phi=-GM/r$, this is exactly $1+\Phi_{\rm e}-\Phi_{\rm r}$.

References

• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes (2013), § 8.2 and § 12.2, printed pp. 80–83 and 134–135. PDF.