Fermat Principles in Stationary Spacetimes

In a stationary spacetime, the null condition can be solved for coordinate travel time and converted into a variational principle on space. The resulting integrand is generally a Randers norm: a Riemannian optical length plus a one-form encoding the stationary shift. Its stationary curves are the spatial projections of null geodesics. “Stationary” is essential—an actual ray need not minimize arrival time, especially after a conjugate point.

Required background. Classical Fields and Their Actions supplies variational equations; The Wave Equation and Characteristics supplies the eikonal interpretation of rays; Affine Connections and Covariant Derivatives supplies null geodesics and metric differentiation.

Helpful background. Events, Intervals, and Minkowski Geometry reviews future null directions; Tensor Fields and Natural Operations reviews one-forms and coordinate changes; Timelike Congruences and Spatial Projection helps interpret the stationary observer family.

Stationary metric and optical data

Let $K=\partial_t$ be a timelike Killing field on the region of interest, and choose stationary coordinates so $\partial_t g_{\mu\nu}=0$. Write

$$ds^2 =g_{tt}dt^2+2g_{ti}dt\,dx^i+g_{ij}dx^i dx^j, \qquad V^2\equiv-g_{tt}>0.$$

For a future-directed null displacement with increasing $t$, solving $ds^2=0$ gives

$$dt=F(x,dx) =\sqrt{a_{ij}dx^i dx^j}+b_i dx^i,$$

where

$$a_{ij} =\frac{g_{ij}}{V^2} +\frac{g_{ti}g_{tj}}{V^4}, \qquad b_i=\frac{g_{ti}}{V^2}.$$

For $F$ to be an ordinary positive, strongly convex Randers norm on every nonzero spatial direction, $a_{ij}$ must be positive definite and

$$a^{ij}b_i b_j<1.$$

These conditions hold in a standard stationary region where $K$ is timelike and the chosen $t={\rm constant}$ slices are spacelike. At an ergosurface $K$ becomes null, $V^2$ vanishes, and this reduction fails. A coordinate choice with nonspacelike slices can instead yield only a conic travel-time function, not the global Randers structure assumed below.

Arrival time and its stationary curves

For two fixed spatial endpoints and a piecewise smooth spatial path $x^i(s)$, define

$$T[x]=\int_{s_1}^{s_2} F(x,\dot x)\,ds =\int_{s_1}^{s_2} \left( \sqrt{a_{ij}\dot x^i\dot x^j} +b_i\dot x^i \right)ds.$$

The functional is invariant under orientation-preserving reparametrization because $F$ is homogeneous of degree one in $\dot x$. Its critical curves, lifted with $dt=Fds$, are future null geodesics connecting the initial event to the stationary endpoint worldline. The spacetime statement is that arrival time is stationary on null trial curves, not necessarily minimal; Perlick 2010, §2.9, pp. 30–32, PDF states the general arrival-time principle and explains how conjugate points turn critical rays into saddles.

If the curve is parametrized by $a_{ij}\dot x^i\dot x^j=1$, the same unparametrized paths follow from

$$L=\frac12a_{ij}\dot x^i\dot x^j+b_i\dot x^i.$$

Their Euler–Lagrange equation is

$$\frac{D^{(a)}\dot x^i}{ds} =a^{i\ell} \left(\partial_\ell b_j-\partial_j b_\ell\right) \dot x^j.$$

Thus $db$ bends projected rays like a magnetic field. It describes a Coriolis-type effect associated with the chosen stationary observer flow; it can encode gravitational frame dragging, but it can also be nonzero for rotating observers in flat spacetime. It is not an electromagnetic force on light or, by itself, evidence of curvature. The stationary decomposition and its optical one-form are derived in Perlick 2010, §4.2, pp. 40–42, PDF.

Static limit and coordinate freedom

For a static chart with $g_{ti}=0$, one has $b=0$ and

$$F(x,dx)=\sqrt{a_{ij}dx^i dx^j}, \qquad a_{ij}=\frac{g_{ij}}{V^2}.$$

Projected light rays are geodesics of this Riemannian optical metric. More generally, if $b=d\psi$ on the domain, its contribution to $T$ is the fixed endpoint term $\psi(x_2)-\psi(x_1)$, so it does not change the stationary paths. This also checks the transformation $t\mapsto t+\psi(x)$: it shifts $b$ by an exact one-form while leaving the spacetime rays invariant.

A conformal rescaling $g_{\mu\nu}\mapsto\Omega^2g_{\mu\nu}$ leaves both $a_{ij}$ and $b_i$ unchanged, provided $\Omega$ respects the stationary decomposition locally. This matches the conformal invariance of unparametrized null geodesics. Redshift and area distance are not similarly invariant, because they also use affine normalization and observer clocks.

Local principle, global qualifications, and wave limits

An instructive flat-space check uses rotating cylindrical coordinates with constant $\Omega\ne0$ and inertial azimuth $\bar\phi=\phi+\Omega t$. The metric becomes

$$ds^2=-(1-\Omega^2r^2)dt^2 +2\Omega r^2dt\,d\phi+dr^2+r^2d\phi^2+dz^2.$$

Inside $r\lt|\Omega|^{-1}$, the stationary observers are timelike and

$$b_\phi=\frac{\Omega r^2}{1-\Omega^2r^2}, \qquad a^{ij}b_i b_j=\Omega^2r^2\lt1.$$

Here $db\ne0$ although the Riemann tensor is zero. At the light cylinder $r=|\Omega|^{-1}$, the chosen rotating Killing field becomes null and the optical reduction fails; the underlying Minkowski spacetime is regular. This separates a failure of the observer-based construction from a curvature singularity.

The Randers reduction requires stationary metric coefficients and endpoints at rest with respect to the stationary flow. In a genuinely time-dependent spacetime there is no time-independent spatial $F$ of this form. A spacetime arrival-time principle can still be formulated using null trial curves ending on a timelike worldline, but it cannot be replaced naively by a three-dimensional optical length.

Several stationary rays may join the same endpoints. Before the first conjugate point a ray can be a local minimum; after one it is generally a saddle. Caustics therefore affect the second variation and produce multiple stationary paths rather than invalidating the first-variation equation. Global minimality additionally needs causal, completeness, convexity, and endpoint hypotheses that are not consequences of stationarity alone.

Finally, this construction describes vacuum geometric-optics rays. Its electromagnetic use assumes $\varepsilon=\lambda_{\rm red}/L\ll1$; omitted field corrections are normally relative $O(\varepsilon)$ away from caustics but become nonuniform near them. Diffraction chooses a finite wave pattern where ray arrival times coalesce. In a dispersive medium the travel functional depends on frequency and may no longer be Randers; group and phase rays must then be distinguished.

Exercise

For a static metric

$$ds^2=-V(x)^2dt^2+h_{ij}(x)dx^i dx^j,$$

derive the optical metric and recover straight rays when $V=1$ and $h_{ij}=\delta_{ij}$.

Solution

The null condition gives

$$V^2dt^2=h_{ij}dx^i dx^j.$$

For a future-directed displacement,

$$dt=\sqrt{\frac{h_{ij}}{V^2}dx^i dx^j}.$$

Hence the travel time is the length functional of

$$a_{ij}=V^{-2}h_{ij}.$$

When $V=1$ and $h_{ij}=\delta_{ij}$, $T=\int\sqrt{\delta_{ij}\dot x^i\dot x^j}\,ds$ is Euclidean length. Its stationary curves with fixed endpoints are straight lines, and the null lift satisfies $dt=d\ell$.

References

• Perlick, Volker. “Gravitational Lensing from a Spacetime Perspective.” Living Reviews in Relativity 7 (2004): 9; revised arXiv version, 2010. DOI. Open PDF.