Lensing in conformally stationary spacetimes

4.2 Lensing in conformally stationary spacetimes

Conformally stationary spacetimes are models for gravitational fields that are time-independent up to an overall conformal factor. (The time-dependence of the conformal factor is important, e.g., if cosmic expansion is to be taken into account.) This is a reasonable model assumption for many, though not all, lensing situations of interest. It allows describing light rays in a 3-dimensional (spatial) formalism that will be outlined in this section. The class of conformally stationary spacetimes includes spherically symmetric and static spacetimes (see Sections 4.3) and axisymmetric stationary spacetimes (see Section 4.4). Also, conformally flat spacetimes (see Section 4.1) are conformally stationary, at least locally. A physically relevant example where the conformal-stationarity assumption is not satisfied is lensing by a gravitational wave (see Section 5.11).

By definition, a spacetime is conformally stationary if it admits a timelike conformal Killing vector field $W$ . If $W$ is complete and if there are no closed timelike curves, the spacetime must be a product, $^ ℳ ≃ ℝ × ℳ$ with a (Hausdorff and paracompact) 3-manifold $^ ℳ$ and $W$ parallel to the $ℝ$ -lines [147 ]. If we denote the projection from $ℳ$ to $ℝ$ by $t$ and choose local coordinates $x = (x1, x2,x3)$ on $^ℳ$ , the metric takes the form

with $μ, ν,...= 1, 2,3$ . The conformal factor $e2f$ does not affect the lightlike geodesics apart from their parametrization. So the paths of light rays are completely determined by the metric $μ ν ˆg = ˆgμν(x)dx dx$ and the one-form $ˆϕ = ˆϕμ(x )dxμ$ which live on $^ℳ$ . The metric $ˆg$ must be positive definite to give a spacetime metric of Lorentzian signature. We call $f$ the redshift potential, $ˆg$ the Fermat metric and $ˆϕ$ the Fermat one-form. The motivation for these names will become clear from the discussion below.

If $ˆϕμ = ∂μh$ , where $h$ is a function of $x = (x1, x2,x3)$ , we can change the time coordinate according to $t ↦− → t + h (x )$ , thereby transforming $ˆϕμdxμ$ to zero, i.e., making the surfaces $t = constant$ orthogonal to the $t$ -lines. This is the conformally static case. Also, Equation (61 ) includes the stationary case ( $f$ independent of $t$ ) and the static case ( $ˆϕμ = ∂μh$ and $f$ independent of $t$ ).

In Section 2.9 we have discussed Kovner’s version of Fermat’s principle which characterizes the lightlike geodesics between a point (observation event) $pO$ and a timelike curve (worldline of light source) $γS$ . In a conformally stationary spacetime we may specialize to the case that $γS$ is an integral curve of the conformal Killing vector field, parametrized by the “conformal time” coordinate $t$ (in the past-pointing sense, to be in agreement with Section 2.9). Without loss of generality, we may assume that the observation event $pO$ takes place at $t = 0$ . Then for each trial path (past-oriented lightlike curve) $λ$ from $pO$ to $γS$ the arrival time is equal to the travel time in terms of the time function $t$ . By Equation (61 ) this puts the arrival time functional into the following coordinate form

where $ℓ$ is any parameter along the trial path, ranging over an interval $[ℓ1,ℓ2]$ that depends on the individual curve. The right-hand side of Equation (62

) is a functional for curves in $ℳ^$ with fixed end-points. The projections to $^ℳ$ of light rays are the stationary points of this functional. In general, the right-hand side of Equation (62

) is the length functional of a Finsler metric. In the conformally static case $ˆ ϕμ = ∂ μh$ , the integral over $ˆ μ ϕμ(x)dx ∕d ℓ$ is the same for all trial paths, so we are left with the length functional of the Fermat metric $ˆg$ . In this case the light rays, if projected to $ℳ^$ , are the geodesics of $ˆg$ . Note that the travel time functional (62

) is invariant under reparametrization; in the terminology of classical mechanics, it is a special case of Maupertuis’ principle. It is often convenient to switch to a parametrization-dependent variational principle which, in the terminology of classical mechanics, is called Hamilton’s principle. The Maupertuis principle with action functional (62

) corresponds to Hamilton’s principle with a Lagrangian

(see, e.g., Carathéodory [52 ], Sections 304 – 307). The pertaining Euler–Lagrange equations read

where $ˆΓ νστ$ are the Christoffel symbols of the Fermat metric $ˆg$ . The solutions admit the constant of motion

which can be chosen equal to 1 for each ray, such that $ℓ$ gives the $ˆg$ -arclength. By Equation (62

), the latter gives the travel time if $ϕˆ= 0$ . According to Equation (64

), the Fermat two-form

exerts a kind of Coriolis force on the light rays. This force has the same mathematical structure as the Lorentz force in a magnetostatic field. In this analogy, $ˆ ϕ$ corresponds to the magnetic (vector) potential. In other words, light rays in a conformally stationary spacetime behave like charged particles, with fixed charge-to-mass ratio, in a magnetostatic field $ωˆ$ on a Riemannian manifold $(Mˆ ,ˆg)$ .

Fermat’s principle in static spacetimes dates back to Weyl [347 ] (cf. [207 , 318 ]). The stationary case was treated by Pham Mau Quan [276 ], who even took an isotropic medium into account, and later, in a more elegant presentation, by Brill [42 ]. These versions of Fermat’s principle are discussed in several text-books on general relativity (see, e.g., [226 , 115 , 311 ] for the static and [199 ] for the stationary case). A detailed discussion of the conformally stationary case can be found in [265 ]. Fermat’s principle in conformally stationary spacetimes was used as the starting point for deriving the lens equation of the quasi-Newtonian apporoximation formalism by Schneider [296 ] (cf. [298 ]). As an alternative to the name “Fermat metric” (used, e.g., in [115 , 311 , 265 ]), the names “optical metric” (see, e.g., [140 , 79 ]) and “optical reference geometry” (see, e.g., [4 ]) are also used.

In the conformally static case, one can apply the standard Morse theory for Riemannian geodesics to the Fermat metric $ˆg$ to get results on the number of $gˆ$ -geodesics joining two points in space. This immediately gives results on the number of lightlike geodesics joining a point in spacetime to an integral curve of $W = ∂ t$ . Completeness of the Fermat metric corresponds to global hyperbolicity of the spacetime metric. The relevant techniques, and their generalization to (conformally) stationary spacetimes, are detailed in a book by Masiello [219 ]. (Note that, in contrast to standard terminology, Masiello’s definition of a stationary spacetime includes the assumption that the hypersurfaces $t = constant$ are spacelike.) The resulting Morse theory is a special case of the Morse theory for Fermat’s principle in globally hyperbolic spacetimes (see Section 3.3). In addition to Morse theory, other standard methods from Riemannian geometry have been applied to the Fermat metric, e.g., convexity techniques [138 , 139 ].

If the metric (61 ) is conformally static, $ˆϕμ(x) = ∂μh (x)$ , and if the Fermat metric is conformal to the Euclidean metric, $ˆgμν(x) = n (x )2δμν$ , the arrival time functional (62 ) can be written as

where $ℓ$ is Euclidean arclength. Hence, Fermat’s principle reduces to its standard optics form for an isotropic medium with index of refraction $n$ on Euclidean space. As a consequence, light propagation in a spacetime with the assumed properties can be mimicked by a medium with an appropriately chosen index of refraction. This remark applies, e.g., to spherically symmetric and static spacetimes (see Section 4.3) and, in particular, to the Schwarzschild spacetime (see Section 5.1). The analogy with ordinary optics in media has been used for constructing, in the laboratory, analogue models for light propagation in general-relativistic spacetimes (see [243 ]).

Extremizing the functional (67 ) is formally analogous to Maupertuis’ principle for a particle in a scalar potential on flat space, which is discussed in any book on classical mechanics. Dropping the assumption that the Fermat one-form is a differential, but still requiring the Fermat metric to be conformal to the Euclidean metric, corresponds to introducing an additional vector potential. This form of the optical-mechanical analogy, for light rays in stationary spacetimes whose Fermat metric is conformal to the Euclidean metric, is discussed, e.g., in [7 ].

The conformal factor $e2f$ in Equation (61 ) does not affect the paths of light rays. However, it does affect redshifts and distance measures (recall Section 2.4). If $g$ is of the form (61 ), for every lightlike geodesic $λ$ the quantity $g(˙λ,∂t)$ is a constant of motion. This leads to a particularly simple form of the general redshift formula (36 ). We consider an arbitrary lightlike geodesic $s ↦→ λ(s)$ in terms of its coordinate representation $s ↦→ (t(s),x1(s),x2(s),x3(s))$ . If both observer and emitter are at rest in the sense that their 4-velocities $U O$ and $U S$ are parallel to $W = ∂ t$ , Equation (36 ) can be rewritten as

This justifies calling $f$ the redshift potential. It is shown in [150

] that there is a redshift potential for a congruence of timelike curves in a spacetime if and only if the timelike curves are the integral curves of a conformal Killing vector field. The notion of a redshift potential or redshift function is also discussed in [74 ]. Note that Equation (68

) immediately determines the redshift in conformally stationary spacetimes for any pair of observer and emitter. If the 4-velocity of the observer or of the emitter is not parallel to $W = ∂t$ , one just has to add the usual special-relativistic Doppler factor.

Conformally stationary spacetimes can be characterized by another interesting property. Let $W$ be a timelike vector field in a spacetime and fix three observers whose worldlines are integral curves of $W$ . Then the angle under which two of them are seen by the third one remains constant in the course of time, for any choice of the observers, if and only if $W$ is proportional to a conformal Killing vector field. For a proof see [150 ].