Fermat-type stationarity of closed light curves: extending the circular null-path/travel-time correspondence beyond stationary axisymmetric black-hole spacetimes

Gonera, Kosinski and Piwnik (2021) derived Hod's observation that the unique null circular geodesic of the Kerr black hole provides the fastest way, as measured by asymptotic observers, to circle the hole, starting from the Landau-Lifshitz form of Fermat's principle for stationary metrics: in a stationary spacetime, null geodesics make the coordinate travel time (the world time) stationary, and for closed spatial curves the stationarity survives variations among closed neighboring paths. For axisymmetric metrics admitting a certain reduction, this yields a one-to-one correspondence between circular null geodesics and circular null paths of stationary world time. The derivation is confined to stationary metrics of that reducible axisymmetric class and establishes only stationarity or local extremality. The open problem is to characterize this Fermat-type stationarity correspondence for closed light curves beyond that class: in non-stationary spacetimes, where the classical Kovner-Perlick arrival-time principle exists but says nothing about closed curves; in stationary metrics without the required reduction; and as regards the global extremal character and multiplicity of the relevant circular null paths. The general Fermat principle itself for non-stationary spacetimes is long settled; the open core is the closed-curve correspondence and its extremal structure.

Background

In flat optics, Fermat's principle states that light rays make the optical path length stationary. In general relativity the analogous statement requires care. Call a spacetime stationary if coordinates exist in which all components of the metric tensor are independent of the time coordinate $x^{0}$ (following Landau and Lifshitz's textbook, such a field is called a constant gravitational field, and $x^{0}$ is called the world time; this means time-independence of the metric, not a uniform Newtonian field). For such spacetimes the classical Landau-Lifshitz form of Fermat's principle holds: light rays (null geodesics, the curves of vanishing interval $ds^{2}=0$ followed by light) project to geodesics of an auxiliary Riemannian metric and one-form on three-dimensional space (the optical, or Fermat, geometry), and equivalently the null geodesics are exactly the curves for which the coordinate travel time integral $\int dx^{0}$ is stationary under variations of the spatial path with fixed endpoints.

Black-hole spacetimes can possess light rings: circular null geodesics on which light can orbit the hole at fixed radius (in static spherical spacetimes the surface foliated by them is called the photon sphere). Hod (2011) proved that, for black holes with such orbits, the null circular geodesic has the shortest orbital period, as measured by asymptotic observers far away in the asymptotically flat region, among all trajectories that circle the hole once, geodesic or not; he also conjectured a universal lower bound $T_{\infty}\geq 4\pi M$ on such periods in terms of the mass $M$. Hod (2018) gave a compact proof that the unique light ring of the spinning Kerr black hole, for each rotation sense, is likewise the fastest way to circle that hole.

Gonera, Kosinski and Piwnik (2021) explained the origin of Hod's Kerr result from the Landau-Lifshitz Fermat principle. Their key observation is that when the spatial path is closed, the boundary terms that normally require fixed endpoints cancel by periodicity, so the stationarity of the world time survives under variations among neighboring closed paths, whether or not they are light rays. For Kerr, and more generally for any stationary axisymmetric metric for which solving the null condition $ds^{2}=0$ allows the time derivative to be written as a function $dx^{0}/d\phi=F(r,\sin^{2}\theta,(d\theta/d\phi)^{2},(dr/d\phi)^{2})$ of the radius $r$, the polar variables and squared derivatives, they conclude a one-to-one correspondence between circular null geodesics and circular null paths along which the world time is stationary. They stress that the argument replaces Hod's global extremum by a local one, since a variational principle a priori yields only stationarity.

A parallel, older line concerns how far Fermat's principle reaches in general spacetimes. Kovner (1990) proposed, and Perlick (1990) proved, a general-relativistic Fermat principle valid in arbitrary spacetimes, stationary or not: fix an observation event and a timelike source worldline, and consider all lightlike curves from one to the other; the arrival time, read off as the parameter value at which the curve meets the source worldline, is a functional on this curve space whose critical points are exactly the light rays. The construction deliberately avoids any global time coordinate, which does not exist in a general spacetime. Perlick also established that such light rays are never local maxima of the arrival time: a ray with no conjugate point (a point where infinitesimally neighboring rays refocus, a standard notion of Lorentzian geometry) is a strict local minimum, while rays passing through conjugate points are saddles; this is the starting point of a Morse theory (the study of how critical points of a functional change as the topology of the underlying space forces them to exist). The stationary and conformally stationary special cases reduce to a purely spatial variational principle, surveyed by Perlick (2004), and Faraoni (1992) applied the general principle to gravitational lensing by explicitly non-stationary sources. What this framework does not contain is any statement about closed curves or circling times: the one-to-one correspondence of Gonera-Kosinski-Piwnik, and the fastest-way-to-circle interpretation, have no known analogue in non-stationary spacetimes, where in fact a canonical world time and a canonical asymptotic static observer are both absent.

On the third front, the existence and number of light rings in general stationary axisymmetric spacetimes has been clarified by topological methods: Cunha and Herdeiro (2020) proved that any stationary, axisymmetric, asymptotically flat spacetime with a non-extremal, topologically spherical Killing horizon (the lightlike surface of the black hole, on which the combined time-translation-plus-rotation symmetry direction itself becomes lightlike), assumed smooth and circular (circularity means the metric admits the standard reduction compatible with two commuting, mutually orthogonal Killing fields), possesses at least one light ring outside the horizon for each rotation sense, and that horizonless ultra-compact objects must possess an even number of non-degenerate light rings, one of which is stable; Cunha, Herdeiro and Novo (2024) extended the result to toroidal horizons and multi-black-hole systems. Uniqueness, however, fails in general: several light rings, including stable ones, can coexist, so the relevant open questions concern multiplicity structure rather than bare existence. Moreover, Mazharimousavi (2024), varying an explicit travel-time functional for a static spherical dirty black hole (a black hole surrounded by external matter fields), recovered the equivalence between the variational equation and the geodesic equation but found that the shortest circling time is then achieved on an almost-circular path of arbitrarily large radius, showing that the global-minimum character of the circular null path is genuinely metric-dependent.

Taken together, the literature leaves a coherent gap: the Fermat-type stationarity correspondence for closed light curves is established only for stationary metrics of a special reducible axisymmetric form, and only at the level of stationarity or local extremality; the general arrival-time principle for arbitrary spacetimes does not address closed curves; and the global extremal character and multiplicity of the stationary closed null paths are uncharacterized. The open problem asks for a resolution of exactly this gap.

Problem Statement

Extend and sharpen the Fermat-type stationarity correspondence for closed light curves established by Gonera, Kosinski and Piwnik, keeping one coherent objective: characterize, by theorem or by rigorous obstruction, the stationarity of the travel time of closed null paths around a compact gravitating system and its relation to circular null geodesics, beyond stationary metrics of the reducible axisymmetric class. The travel time at issue is the functional natural to each branch: the stationary world time in the stationary branches, and the Kovner-Perlick arrival time, or a chosen global time function, in the non-stationary one; each branch therefore characterizes one well-defined functional and its relation to closed null geodesics. The three directions below are independent sub-questions, each complete in itself and not jointly required: a resolution of any one of them by theorem or by rigorous obstruction resolves the task. Within a direction, each labeled sub-claim ((i-1), (i-2), (ii-1), (ii-2), (iii-1), (iii-2)) is likewise independently sufficient: settling any single one of them by theorem or by rigorous obstruction resolves that direction. Concretely: (i) Non-stationary spacetimes: (i-1) formulate and prove, or disprove by explicit example, an analogue of the correspondence in which the stationary-spacetime world time is replaced by the Kovner-Perlick arrival-time functional along a prescribed timelike observer worldline, or by a chosen global time function; (i-2) in particular determine whether null closed curves that are geodesics are exactly the stationary points of that functional among closed null curves, or whether generic time dependence obstructs any such statement. (ii) Beyond the reducible class: determine whether the one-to-one correspondence between circular null geodesics and circular null paths of stationary travel time extends (ii-1) to axisymmetric metrics that do not admit the reduction $dx^{0}/d\phi=F(r,\sin^{2}\theta,(d\theta/d\phi)^{2},(dr/d\phi)^{2})$ after solving $ds^{2}=0$, or (ii-2) to stationary metrics without axisymmetry, for which the correspondence is understood in its symmetry-free form, between closed null geodesics and closed null paths of stationary world time. (iii) Extremal character and multiplicity: (iii-1) establish conditions on the metric under which the closed-loop travel-time functional attains a local or global minimum at a circular null geodesic rather than a saddle, relating or contrasting with the conjugate-point criteria known for point-to-worldline rays; (iii-2) characterize the multiplicity structure of the relevant circular null paths, noting that uniqueness fails in general.

References

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