Proper Time, Redshift, Deflection, and Time Delay

Proper time and a measured photon frequency are scalars once the clock worldline, emitter, receiver, and light ray are fixed. Deflection, magnification, and time delay need more structure: an observer screen, a reference direction or unlensed model, and sometimes a thin-lens approximation. This card keeps those choices visible so that coordinate quantities are not mistaken for observables.

Clock readings and frequency ratios

For a timelike worldline $x^a(\lambda)$, an ideal clock records

$$\tau_2-\tau_1 =\int_{\lambda_1}^{\lambda_2} \sqrt{-g_{ab}\frac{dx^a}{d\lambda}\frac{dx^b}{d\lambda}}\,d\lambda, \qquad u^a=\frac{dx^a}{d\tau}, \qquad u^au_a=-1.$$

Acceleration changes the elapsed proper time by changing the worldline; the ideal-clock model contains no additional instantaneous acceleration term. Finite response time, acceleration sensitivity, environmental shifts, and readout conventions belong to the physical clock model, not to the definition of $\tau$. See the complete treatment in Proper Time and the Clock Hypothesis. The displayed geometric definitions are collected in Reall 2013, § 3.2, printed pp. 35–36.

If a future-directed photon has tangent or wave covector $k_a$, an observer with four-velocity $u^a$ measures the positive angular frequency

$$\omega=-k_au^a.$$

Thus one ray emitted at $e$ and received at $r$ has the invariant ratio

$$1+z\equiv\frac{\omega_e}{\omega_r} =\frac{(-k\mathbin{\cdot}u)_e}{(-k\mathbin{\cdot}u)_r}.$$

This endpoint formula is the general observable. A split into “Doppler” and “gravitational” factors is not unique until an observer field, transport rule, or spacetime symmetry has been chosen. For example, if $\xi^a$ is timelike and Killing along the ray and the emitter and receiver follow $u^a=\xi^a/\sqrt{-\xi^2}$, then $E_\xi=-k\mathbin{\cdot}\xi$ is conserved and

$$1+z=\frac{\sqrt{-\xi^2_r}}{\sqrt{-\xi^2_e}}.$$

Measured Energy, Momentum, and Frequency derives $-k\mathbin{\cdot}u$, while Redshift Between Emission and Reception develops the endpoint and stationary formulas. The weak-field specialization and its domain are checked in Will 2014, § 2.1.3, printed pp. 12–14.

One observer can also assign radar time and distance to an event connected to the worldline by a unique outgoing-and-returning null-path pair. If the corresponding clock readings are $\tau_-$ and $\tau_+$, then

$$\tau_{\rm rad}=\frac{\tau_++\tau_-}{2}, \qquad R_{\rm rad}=\frac{\tau_+-\tau_-}{2}.$$

These assignments are local and operational, but not automatically global. They become multivalued when several null rays return and fail where no return ray exists. Perlick 2008, §§1–4, PDF pp. 1–12 gives the construction and compares its domain with other observer-adapted coordinates.

Angles and the exact lens map

At reception, decompose the ray relative to the observer as

$$k^a=\omega_o(u_o^a+n^a), \qquad n^an_a=1, \qquad n^au_{o a}=0.$$

The two-dimensional screen orthogonal to both $u_o^a$ and $n^a$ has projector

$$S_{ab}=g_{ab}+u_{o a}u_{o b}-n_an_b.$$

An angular deflection is therefore a comparison in a declared observer rest space. Given a reference direction $n_{\rm ref}^a$ for the same observer, one may define $\cos\alpha=n\mathbin{\cdot}n_{\rm ref}$. Changing the observer, tetrad, or reference ray changes $\alpha$; a coordinate slope by itself is not the deflection observable.

For a narrow null bundle, choose oriented screen bases at observer and source. The Jacobi map $\mathcal D^A{}_B$ sends an infinitesimal observed angular separation to the physical connecting vector at the source,

$$\xi_s^A=\mathcal D^A{}_B\,\delta\theta_o^B, \qquad D_A=\sqrt{\left\lvert\det\mathcal D\right\rvert}.$$

Relative to a declared reference spacetime or unlensed bundle, the signed magnification is

$$\mu_{\rm signed} =\frac{\det\mathcal D_{\rm ref}}{\det\mathcal D}.$$

Its sign records image parity, whereas observed flux magnification uses its absolute value under the usual geometric-optics and photon-conservation assumptions. At a caustic, $\det\mathcal D=0$: parity can change and the geometric-optics magnification diverges, signaling failure of the approximation rather than infinite measured power. Under the reciprocity hypotheses, luminosity and angular-diameter distance obey $D_{\rm lum}=(1+z)^2D_A$. These constructions and their hypotheses are given in Perlick 2004, §§2.1–2.4. The canonical derivation is Light Bundles and Area Distance.

Thin-lens potential and differential time delay

The following formulas are not the exact Jacobi map. Assume weak and approximately stationary gravity, small angles, geometric optics, a thin lens at redshift $z_{\rm L}$, and angular-diameter distances $D_{\rm L}$, $D_{\rm S}$, and $D_{\rm LS}$. For a dimensionless lensing potential $\psi(\boldsymbol\theta)$, define

$$\boldsymbol\beta =\boldsymbol\theta-\boldsymbol\nabla_{\!\theta}\psi, \qquad \mathcal A_{ij} =\frac{\partial\beta_i}{\partial\theta_j} =\delta_{ij}-\partial_i\partial_j\psi, \qquad \mu_{\rm signed}=\frac{1}{\det\mathcal A}.$$

The corresponding arrival-time surface, up to an additive constant, is

$$t(\boldsymbol\theta;\boldsymbol\beta) =(1+z_{\rm L})\frac{D_{\rm L}D_{\rm S}}{D_{\rm LS}} \left[ \frac12\left\lVert\boldsymbol\theta-\boldsymbol\beta\right\rVert^2 -\psi(\boldsymbol\theta) \right].$$

Images are stationary points of this surface; a stationary arrival time is not necessarily a minimum. Observable delays are differences between stationary paths or repeated signals, not the additive coordinate travel time. Multiple lens planes, strong-field winding, time-dependent lenses, plasma, diffraction, absorption, or a caustic require a more complete propagation model. The spacetime and variational foundations are summarized in Perlick 2004, §§3.1–3.4 and developed in Fermat Principles in Stationary Spacetimes.

For a static point mass in the weak Schwarzschild regime, the leading checks are

$$\widehat\alpha=\frac{4GM}{b}, \qquad \Delta t_{\rm Shapiro} =2GM\ln\!\left( \frac{r_e+r_r+R}{r_e+r_r-R} \right)\gt0,$$

where $b$ is the impact parameter, $r_e$ and $r_r$ are the endpoint distances from the mass, and $R$ is their leading Euclidean separation. The second expression compares one-way propagation with the flat-space time for that geometry. Both temporal and spatial metric perturbations contribute. See Will 2014, §§4.1.1–4.1.2, pp. 40–44 and the full derivation in Light Deflection and Shapiro Delay.

Compact checks and failure boundaries

Situation	Required result or warning
Inertial observer in Minkowski spacetime	Proper time reduces to the usual clock reading, radar distance to half the round-trip interval, and an unlensed Jacobi map gives the flat area distance.
Stationary emitter and receiver	The Killing energy cancels from the frequency ratio; without a timelike symmetry, that decomposition is unavailable.
Weak Schwarzschild field	Recover positive logarithmic Shapiro delay and leading deflection $4GM/b$; a time-dilation-only argument misses the spatial-curvature contribution.
Caustic or conjugate point	The Jacobi determinant vanishes, parity may flip, and geometric-optics magnification is not finite.
Changed observer or reference ray	Frequency, screen angle, and the quoted deflection generally change even though the same null geodesic is used.

The chapter Observers and Measurement in Curved Spacetime gives the common observer framework for the clock and frequency entries.

References

• Perlick, Volker. “Gravitational Lensing from a Spacetime Perspective.” Living Reviews in Relativity 7 (2004): 9. DOI. Open PDF.
• Perlick, Volker. “On the Radar Method in General-Relativistic Spacetimes.” In Lasers, Clocks and Drag-Free Control. Springer, 2008. DOI. Open PDF.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes, 2013. Open PDF.
• Will, Clifford M. “The Confrontation between General Relativity and Experiment.” Living Reviews in Relativity 17 (2014): 4. DOI. Open PDF.