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Optical Geometry?
Communicated by Notices Associate Editor Chikako Mese
Optical geometry is a differential geometrical approach to study gravitational lensing, that is, the gravitational deflection of light which gives rise to some of the most stunning phenomena in astronomy. An important recent breakthrough was the first detection of gravitational lensing near black holes (see Fig. 1, more on this below).
Gravitational lensing near supermassive black holes in the centers of galaxy M87 (left) and our own Milky Way (right).

In Einstein’s general relativity, gravity is described geometrically in terms of the curvature of spacetime, which is a 4-dimensional manifold endowed with a pseudo-Riemannian metric of Lorentzian signature (say, Then the trajectory of a freely falling particle corresponds to a geodesic ). in spacetime, and one can distinguish between timelike geodesics with for massive particles and null geodesics with for massless particles. The latter applies to light and hence to gravitational lensing. In fact, the first observation of this effect by Eddington’s 1919 eclipse expeditions provided crucial corroborating evidence for general relativity.
Gravitational lensing formalisms
Different approaches can be employed to study this effect mathematically:
- (1)
Spacetime lensing. Since gravitational lensing is fundamentally determined by the properties of null geodesics, foundational work is usually carried out directly in the 4-dimensional spacetime of general relativity. This includes extensions of the so-called odd number theorem to spacetime and, notably, the general relativistic version of Fermat’s principle, a variational principle of stationary time defining light rays (cf.
Per00 ).- (2)
Standard formalism. Certain limits applicable in astronomy yield a quasi-Newtonian thin lens approximation in 3-dimensional space. This widely used standard formalism turns out to be mathematically rich in its own right, involving, for instance, applications of Morse theory, ADE singularity theory, Lefschetz fixed point theory, and complex harmonic analysis. See
Pet10 for some of the beautiful results arising within this approach.- (3)
Optical geometry is a third approach which we shall focus on in this note. It employs a 3-dimensional space defined such that its geodesics are spatial light rays. Thus, optical geometry uses a 3-dimensional spatial geometry rather than 4-dimensional spacetime, as the standard formalism. But instead of approximating general relativity from the outset, optical geometry remains exact and allows the use of differential geometry. Applications of the Gauss-Bonnet theorem and isoperimetry will be discussed here.
Classifying optical metrics
The subject of optical geometry, also known as optical reference geometry or Fermat geometry in the literature, has a long history. This includes pioneering work by Abramowicz et al. on using optical geometry to understand inertial forces in general relativity (e.g.,
Recall that a spacetime is stationary if there is timelike Killing vector field and thus a chart with time and spatial coordinates in which the line element of the spacetime metric , takes the form
with time-independent metric components, where is a spatial Riemannian metric, a spatial one-form, and a conformal factor. If is also hypersurface-orthogonal, the spacetime becomes static and there is a chart such that
that is, without mixed time-space components.
Now, for null curves in spacetime, and so we obtain a line element for in 3-dimensional space defining the optical metric. Indeed, by Fermat’s principle in general relativity, null geodesics in spacetime project to geodesics of the optical metric in space, that is, to spatial light rays. Therefore, a stationary spacetime has an optical metric given by
Clearly, this is not the line element of a Riemannian metric but of a so-called Finsler metric of Randers type. Finsler metrics constitute a larger class of geometries that contain Riemannian metrics as special case.Footnote1 These are defined by non-negative functions on the slit tangent bundle (excluding the zero section) which are positively homogeneous of degree one in the vectors and subject to a convexity condition. The latter states that the Hessian of with respect to the vectors,
As such, they are also called Riemann-Finsler metrics. In fact, one can define more general Finsler metrics that contain pseudo-Riemannian metrics as special case. For example, Lorentz-Finsler metrics are studied as spacetime structures of certain modified theories of gravity beyond general relativity.
be positive definite. Then by Euler’s theorem on homogeneous functions, one finds that the squared length of vectors is in fact
as in the Riemannian case—but deceptively so! The point is that can depend on the vectors themselves, not just on the points in the underlying manifold. So Finsler geometry admits an anisotropic, or direction-dependent, metric. In the Randers case, it turns out that these conditions are satisfied provided that in other words, the one-form , is ‘sufficiently small’ relative to the Riemannian metric (see, e.g.,
So it is interesting to note that, by restricting ourselves to such 3-dimensional spatial geometries, Finsler metrics can arise from pseudo-Riemannian spacetime metrics. An important example from general relativity is the Kerr metric of rotating black holes and its Kerr-Randers optical geometry. This breaks down on the surface where corresponding to the boundary of the ergo-region which plays an important role in the Penrose process for energy extraction from black holes.
In the following, however, we will focus on the more simple special case of a static spacetime, that is, Its optical geometry is now seen to be a 3-dimensional space with line element .
provided that Thus, our optical geometry specializes from a Randers metric to an ordinary Riemannian metric in the static case. To illustrate, let us proceed with the following instructive .
By Birkhoff’s theorem, there is a unique spatially spherically symmetric spacetime solving the vacuum field equations of general relativity, called the Schwarzschild solution. It is static, aymptotically flat, characterized by a mass parameter and describes a nonrotating black hole with the event horizon at , where Its optical geometry is thus defined outside the event horizon with Riemannian metric .
in Schwarzschild spherical polar coordinates By spherical symmetry, it suffices to consider the equatorial plane . as a totally geodesic submanifold (with boundary). Then the isometric embedding of in 3-dimensional Euclidean space is shown in Fig. 2a. Hence, geodesics of this surface correspond to spatial light rays which are gravitationally lensed by a Schwarzschild black hole. The thick blue geodesic around the waist at in Fig. 2a represents a special circular light ray. Restoring the third dimension, this is the photon sphere, a closed timelike hypersurface that is totally null geodesic, and gravitational lensing in its vicinity can give rise to characteristic dark patches as in Fig. 1. The physical reality involving black hole accretion disks is, of course, more complicated (cf.
By inspection of the extrinsic geometry of in Fig. 2a, one sees immediately that the two principal curvatures have opposite sign, so the Gaussian curvature in the optical geometry is negative everywhere. Indeed, a direct computation yields outside the event horizon. Thus, the geodesic deviation equationFootnote2 implies that light rays must locally diverge everywhere. But this seems rather surprising—how is it possible that light rays reconverge at the observer to form gravitationally lensed images?
Considering a connecting vector field of neighboring geodesics on recall that the geodesic deviation equation becomes , whence divergence for , .
(a) Isometric embedding of the Schwarzschild equatorial plane in the optical geometry. The blue circle marks the black hole photon sphere. (b) Cone inside (red) for Dido’s isoperimetric theorem in Schwarzschild optical geometry.


The Gauss-Bonnet method
This apparent paradox is resolved by the nontrivial topology of our surface To see this, it is helpful to recall the Gauss-Bonnet theorem: .
A surface with Euler characteristic Gaussian curvature , and piecewise-smooth boundary , with exterior jump angles and geodesic curvature obeys ,
Now to apply Gauss-Bonnet to a typical gravitational lensing situation, suppose a massive object (‘lens’) deflects light rays from a light source to an observer in a surface as shown in Fig. ,3. If is a Schwarzschild black hole as described above, then has a hole with the circular light ray at the photon sphere as inner boundary, say, and thus set with for Schwarzschild and for a regular such as a star. Then since all boundaries are geodesic with Gauss-Bonnet implies ,
showing that, for lensed images with , can occur only if Thus, the nontrivial topology of the Schwarzschild optical geometry is essential for multiple imaging to occur! It is easy to see how this argument using Gauss-Bonnet can be generalized to obtain other topological constraints for the existence of multiple images. Indeed, similar constraints can also been found using Morse theory in the standard formalism mentioned above. .
Gravitational lensing in optical geometry.

But there is more: we can also apply Gauss-Bonnet to derive the gravitational deflection angle itself. For a totally geodesic surface in Riemannian optical geometry excluding the lens and bounded by a light ray from a source to an observer in the asymptotically flat region (i.e., far away from ),
Conceptually, it is quite astonishing that the deflection toward can be obtained by integrating away from it, and this again underscores the partially global nature of gravitational lensing. Also, since is bounded by the deflected light ray, this formula can be solved iteratively to find For example, approximating the ray by an undeflected straight line . where is the impact parameter, we can easily compute the leading order in for Schwarzschild, One can also implement fluids to model stars or galaxies acting as . using the Tolman-Oppenheimer-Volkoff equation in general relativity, and details can be found in ,
Moreover, this Gauss-Bonnet method can be extended to the optical Finsler metric, for instance by constructing a suitable osculating Riemannian metric. This is an early approach to Finsler geometry developed by Nazım, a student of Carathéodory. One can also apply Gauss-Bonnet in a different spatial geometry to find the deflection angle for finite positions of and as proposed by Asada’s group (see, e.g., ,
Isoperimetric considerations
Finally, a few words about isoperimetry. Since the arc length in optical geometry measures time, and time delay between images is an important observable in gravitational lensing, it may be interesting to investigate isoperimetric properties in optical geometry: these may help establish general constraints on time delays, possibly involving angles between images. While finding such constraints is still an open problem in gravitational lensing, some isoperimetric results in optical geometry have already been found.
The basic isoperimetric problem of finding the simple closed curve minimizing perimeter length for a given area (or, maximizing enclosed area for a given perimeter length) in planar Euclidean geometry is, of course, solved by the circle—a result traditionally attributed to Dido using a string fashioned from a bull’s hide in the foundation myth of Carthage (cf. Virgil, Aeneid I, 365–8, see also
In Schwarzschild optical geometry, light rays bounding solutions of the isoperimetric problem must lie on the photon sphere.
By showing that the curve minimizes length within the homology class of piecewise-smooth curves bounding, with the area , this result follows since the only circular light rays are on the photon sphere. Inspired by Bray’s work on the Penrose inequality, the proof uses a cone glued in for comparison (cf. Fig. ,2b) and Hopf’s maximum principle. This result can also be extended to the Reissner-Nordström solution, that is, a nonrotating charged black hole, and again to relativistic fluids with Tolman-Oppenheimer-Volkoff. Using curve shortening flow, an isoperimetric inequality applicable to optical geometry with a negative upper bound on the Gaussian curvature can also be derived. The interested reader is referred to
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Credits
Figure 1 is courtesy of ESO, CC-BY-4.0.
Figures 2 and 3 and author photo are courtesy of Marcus C. Werner.