4.3 Lensing in spherically symmetric and static spacetimes

The class of spherically symmetric and static spacetimes is of particular relevance in view of lensing, because it includes models for non-rotating stars and black holes (see Sections 5.1, 5.2, 5.3), but also for more exotic objects such as wormholes (see Section 5.4), monopoles (see Section 5.5), naked singularities (see Section 5.6), and Boson or Fermion stars (see Section 5.7). Here we collect the relevant formulas for an unspecified spherically symmetric and static metric. We find it convenient to write the metric in the form
2f(r)( 2 2 2 2( 2 2 2)) g = e − dt + S(r) dr + R (r) dπœ— + sin πœ—d φ . (69 )
As Equation (69View Equation) is a special case of Equation (61View Equation), all results of Section 4.2 for conformally stationary metrics apply. However, much stronger results are possible because for metrics of the form (69View Equation) the geodesic equation is completely integrable. Hence, all relevant quantities can be determined explicitly in terms of integrals over the metric coefficients.

Redshift and Fermat geometry.
Comparison of Equation (69View Equation) with the general form (61View Equation) of a conformally stationary spacetime shows that here the redshift potential f is a function of r only, the Fermat one-form ˆ Ο• vanishes, and the Fermat metric ˆg is of the special form

ˆg = S(r)2dr2 + R(r)2 (dπœ—2 + sin2 πœ—dφ2 ). (70 )
By Fermat’s principle, the geodesics of ˆg coincide with the projection to 3-space of light rays. The travel time (in terms of the time coordinate t) of a lightlike curve coincides with the ˆg-arclength of its projection. By symmetry, every ˆg-geodesic stays in a plane through the origin. From Equation (70View Equation) we read that the sphere of radius r has area 2 4πR (r) with respect to the Fermat metric. Also, Equation (70View Equation) implies that the second fundamental form of this sphere is a multiple of its first fundamental form, with a factor − R ′(r) (R(r)S (r ))− 1. If
R ′(rp) = 0, (71 )
the sphere r = rp is totally geodesic, i.e., a ˆg-geodesic that starts tangent to this sphere remains in it. The best known example for such a light sphere or photon sphere is the sphere r = 3m in the Schwarzschild spacetime (see Section 5.1). Light spheres also occur in the spacetimes of wormholes (see Section 5.4). If R ′′(r ) < 0 p, the circular light rays in a light sphere are stable with respect to radial perturbations, and if ′′ R (rp) > 0, they are unstable like in the Schwarschild case. The condition under which a spherically symmetric static spacetime admits a light sphere was first given by Atkinson [13Jump To The Next Citation Point]. Abramowicz [1] has shown that for an observer traveling along a circular light orbit (with subluminal velocity) there is no centrifugal force and no gyroscopic precession. Claudel, Virbhadra, and Ellis [59Jump To The Next Citation Point] investigated, with the help of Einstein’s field equation and energy conditions, the amount of matter surrounded by a light sphere. Among other things, they found an energy condition under which a spherically symmetric static black hole must be surrounded by a light sphere. A purely kinematical argument shows that any spherically symmetric and static spacetime that has a horizon at r = rH and is asymptotically flat for r → ∞ must contain a light sphere at some radius between rH and ∞ (see Hasse and Perlick [152Jump To The Next Citation Point]). In the same article, it is shown that in any spherically symmetric static spacetime with a light sphere there is gravitational lensing with infinitely many images. Bozza [37Jump To The Next Citation Point] investigated a strong-field limit of lensing in spherically symmetric static spacetimes, as opposed to the well-known weak-field limit, which applies to light rays that come close to an unstable light sphere. (Actually, the term “strong-bending limit” would be more appropriate because the gravitational field, measured in terms of tidal forces, need not be particularly strong near an unstable light sphere.) This limit applies, in particular, to light rays that approach the sphere r = 3m in the Schwarzschild spacetime (see [39Jump To The Next Citation Point] and, for illustrations, Figures 15View Image, 16View Image, and 17View Image).

Index of refraction and embedding diagrams.
Transformation to an isotropic radius coordinate &tidle;r via

S(r)dr d&tidle;r -------= --- (72 ) R (r ) r&tidle;
takes the Fermat metric (70View Equation) to the form
( ) ˆg = n(&tidle;r)2 dr&tidle;2 + &tidle;r2(dπœ—2 + sin2πœ—d φ2) (73 )
R (r) n(&tidle;r) = --&tidle;r--. (74 )
On the right-hand side r has to be expressed by &tidle;r with the help of Equation (72View Equation). The results of Section 4.2 imply that the lightlike geodesics in a spherically symmetric static spacetime are equivalent to the light rays in a medium with index of refraction (74View Equation) on Euclidean 3-space. For arbitrary metrics of the form (69View Equation), this result is due to Atkinson [13Jump To The Next Citation Point]. It reduces the lightlike geodesic problem in a spherically symmetric static spacetime to a standard problem in ordinary optics, as treated, e.g., in [213], §27, and [198], Section 4. One can combine this result with our earlier observation that the integral in Equation (67View Equation) has the same form as the functional in Maupertuis’ principle in classical mechanics. This demonstrates that light rays in spherically symmetric and static spacetimes behave like particles in a spherically symmetric potential on Euclidean 3-space (cf., e.g., [104Jump To The Next Citation Point]). If the embeddability condition
2 ′ 2 S(r) ≥ R (r) (75 )
is satisfied, we define a function Z (r) by
∘ -------------- Z′(r) = S(r)2 − R ′(r)2. (76 )
Then the Fermat metric (70View Equation) reads
2 2( 2 2 2) 2 ˆg = (dR (r)) + R (r) dπœ— + sin πœ—d φ + dZ (r). (77 )
If restricted to the equatorial plane πœ— = πβˆ•2, the metric (77View Equation) describes a surface of revolution, embedded into Euclidean 3-space as
(r,φ ) ↦→ (R (r)cosφ, R(r) sin φ,Z (r)). (78 )
Such embeddings of the Fermat geometry have been visualized for several spacetimes of interest (see Figure 11View Image for the Schwarzschild case and [159Jump To The Next Citation Point] for other examples). This is quite instructive because from a picture of a surface of revolution one can read the qualitative features of its geodesics without calculating them. Note that Equation (72View Equation) defines the isotropic radius coordinate uniquely up to a multiplicative constant. Hence, the straight lines in this coordinate representation give us an unambiguously defined reference grid for every spherically symmetric and static spacetime. These straight lines have been called triangulation lines in [6263], where their use for calculating bending angles, exactly or approximately, is outlined.

Light cone.
In a spherically symmetric static spacetime, the (past) light cone of an event p O can be written in terms of integrals over the metric coefficients. We first restrict to the equatorial plane πœ— = π βˆ•2. The ˆg-geodesics are then determined by the Lagrangian

( ( )2 ( )2 ) β„’ = 1- S(r)2 dr- + R (r )2 dφ- . (79 ) 2 dβ„“ dβ„“
For fixed radius value rO, initial conditions
r(0) = r , dr(0) = cos-Θ-, O dβ„“ S (rO ) dφ sin Θ (80 ) φ(0) = 0, ---(0) = ------ d β„“ R (rO)
determine a unique solution r = r(β„“,Θ ), φ = Ο•(β„“,Θ) of the Euler–Lagrange equations. Θ measures the initial direction with respect to the symmetry axis (see Figure 6View Image). We get all light rays issuing from the event r = rO, φ = 0, πœ— = πβˆ•2, t = tO into the past by letting Θ range from 0 to π and applying rotations around the symmetry axis. This gives us the past light cone of this event in the form
( ) tO − β„“ | r(β„“,Θ )sin Ο•(β„“,Θ )cos Ψ| (β„“,Ψ,Θ ) ↦−→ |( |) . (81 ) r(β„“,Θ )sinΟ•(β„“,Θ )sinΨ r(β„“,Θ) cosΟ•(β„“,Θ )
Ψ and Θ are spherical coordinates on the observer’s sky. If we let tO float over ℝ, we get the observational coordinates (4View Equation) for an observer on a t-line, up to two modifications. First, t O is not the same as proper time τ; however, they are related just by a constant,
dτ ----= e−f(rO ). (82 ) dtO
Second, β„“ is not the same as the affine parameter s; along a ray with initial direction Θ, they are related by
ds f(r(β„“,Θ)) ---= e . (83 ) d β„“
The constants of motion
( )2 ( )2 2dφ- 2 dr- 2 dφ- R(r) dβ„“ = R (rO)sin Θ, S(r) dβ„“ + R (r) dβ„“ = 1 (84 )
give us the functions r(β„“,Θ ), Ο•(β„“,Θ ) in terms of integrals,
∫ ...r(β„“,Θ) β„“ = ∘-----R-(r)S-(r)dr------, (85 ) rO... R (r)2 − R (rO)2sin2Θ ∫ ...r(β„“,Θ) Ο• (β„“,Θ) = R (r )sinΘ -----∘-----S(r)dr-----------. (86 ) O rO... R (r) R (r)2 − R (rO)2sin2Θ
Here the notation with the dots is a short-hand; it means that the integral is to be decomposed into sections where r(β„“,Θ ) is a monotonous function of β„“, and that the absolute value of the integrals over all sections have to be added up. Turning points occur at radius values where R (r) = R (rO) sin Θ and R ′(r) ⁄= 0 (see Figure 9View Image). If the metric coefficients S and R have been specified, these integrals can be calculated and give us the light cone (see Figure 12View Image for an example). Having parametrized the rays with ˆg-arclength (= travel time), we immediately get the intersections of the light cone with hypersurfaces t = constant (“instantaneous wave fronts”); see Figures 13View Image, 18View Image, and 19View Image.

Exact lens map.
Recall from Section 2.1 that the exact lens map [122Jump To The Next Citation Point] refers to a chosen observation event pO and a chosen “source surface” 𝒯. In general, for 𝒯 we may choose any 3-dimensional submanifold that is ruled by timelike curves. The latter are to be interpreted as wordlines of light sources. In a spherically symmetric and static spacetime, we may take advantage of the symmetry by choosing for 𝒯 a sphere r = rS with its ruling by the t-lines. This restricts the consideration to lensing for static light sources. Note that all static light sources at radius rS undergo the same redshift, log(1 + z) = f(rS) − f(rO). Without loss of generality, we place the observation event pO on the 3-axis at radius r O. This gives us the past light cone in the representation (81View Equation). To each ray from the observer, with initial direction characterized by Θ, we can assign the total angle Φ(Θ ) the ray sweeps out on its way from rO to rS (see Figure 6View Image). Φ(Θ ) is given by Equation (86View Equation),

∫ ...rS Φ(Θ ) = R (rO )sinΘ -----∘-----S(r)dr-----------, (87 ) rO... R (r) R (r)2 − R (rO)2sin2Θ
where the same short-hand notation is used as in Equation (86View Equation). Φ (Θ ) is not necessarily defined for all Θ because not all light rays that start at rO may reach rS. Also, Φ(Θ ) may be multi-valued because a light ray may intersect the sphere r = rS several times. Equation (81View Equation) gives us the (possibly multi-valued) lens map
( r sin Φ(Θ )cos Ψ) ( S ) (Ψ,Θ ) ↦−→ rS sin Φ(Θ )sinΨ . (88 ) rScos Φ(Θ )
It assigns to each point on the observer’s sky the position of the light source which is seen at that point. Φ (Θ) may take all values between 0 and infinity. For each image we can define the order
i(Θ) = min {m ∈ β„• ||Φ (Θ ) < m π} , (89 )
which counts how often the ray has met the axis. The standard example where images of arbitrarily high order occur is the Schwarzschild spacetime (see Section 5.1). For many, though not all, applications one may restrict to the case that the spacetime is asymptotically flat and that rO and rS are so large that the spacetime is almost flat at these radius values. For a light ray with turning point at rm (Θ), Equation (87View Equation) can then be approximated by
∫ ∞ S (r)dr Φ(Θ ) = 2 ----∘------2---------2---2--. (90 ) rm(Θ) R(r) R(r) − R(rO ) sin Θ
If the entire ray remains in the region where the spacetime is almost flat, Equation (90View Equation) gives the usual weak-field approximation of light bending with Φ(Θ ) close to π. However, Equation (90View Equation) does not require that the ray stays in the region that is almost flat. The integral in Equation (90View Equation) becomes arbitrarily large if rm (Θ ) comes close to an unstable light sphere, R ′(rp) = 0 and R ′′(rp) > 0. This situation is well known to occur in the Schwarzschild spacetime with rp = 3m (see Section 5.1, in particular Figures 9View Image, 14View Image, and 15View Image). The divergence of Φ (Θ) is always logarithmic [37Jump To The Next Citation Point]. Virbhadra and Ellis [336Jump To The Next Citation Point] (cf. [338Jump To The Next Citation Point] for an earlier version) approximately evaluate Equation (90View Equation) for the case that source and observer are almost aligned, i.e., that Φ(Θ ) is close to an odd multiple of π. This corresponds to replacing the sphere at rS with its tangent plane. The resulting “almost exact lens map” takes an intermediary position between the exact treatment and the quasi-Newtonian approximation. It was originally introduced for the Schwarzschild metric [336Jump To The Next Citation Point] where it approximates the exact treatment remarkably well within a wide range of validity [118Jump To The Next Citation Point]. On the other hand, neither analytical nor numerical evaluation of the “almost exact lens map” is significantly easier than that of the exact lens map. For situations where the assumption of almost perfect alignment cannot be maintained the Virbhadra–Ellis lens equation must be modified (see [70Jump To The Next Citation Point]; related material can also be found in [38Jump To The Next Citation Point]).
View Image

Figure 6: Illustration of the exact lens map in spherically symmetric static spacetimes. The picture shows a spatial plane. The observation event (dot) is at r = rO, static light sources are distributed at r = r S. Θ is the colatitude coordinate on the observer’s sky. It takes values between 0 and π. Φ (Θ ) is the angle swept out by the ray with initial direction Θ on its way from rO to rS. It takes values between 0 and ∞. In general, neither existence nor uniqueness of Φ (Θ) is guaranteed for given Θ. A similar picture is in [271Jump To The Next Citation Point]. Distance measures, image distortion and brightness of images.
For calculating image distortion (see Section 2.5) and the brightness of images (see Section 2.6) we have to consider infinitesimally thin bundles with vertex at the observer. In a spherically symmetric and static spacetime, we can apply the orthonormal derivative operators ∂Θ and sin Θ ∂Ψ to the representation (81View Equation) of the past light cone. Along each ray, this gives us two Jacobi fields Y1 and Y2 which span an infinitesimally thin bundle with vertex at the observer. Y1 points in the radial direction and Y 2 points in the tangential direction (see Figure 7View Image). The radial and the tangential direction are orthogonal to each other and, by symmetry, parallel-transported along each ray. Thus, in contrast to the general situation of Figure 3View Image, Y1 and Y2 are related to a Sachs basis (E1, E2) simply by Y1 = D+E1 and Y2 = D − E2. The coefficients D+ and D − are the extremal angular diameter distances of Section 2.4 with respect to a static observer (because the (Ψ, Θ )-grid refers to a static observer). In the case at hand, they are called the radial and tangential angular diameter distances. They can be calculated by normalizing Y1 and Y2,

∘ -------------------------- f(r(β„“,Θ)) 2 2 2 D+ (β„“,Θ ) = e R(rO )cosΘ R (r(β„“,Θ)) − R (rO)sin Θ ∫ ...r(Θ,β„“) S (r)R(r)dr × βˆ˜----------------------3, (91 ) rO... R (r)2 − R (rO)2sin2Θ sinΟ• (β„“,Θ ) D − (β„“,Θ ) = ef(r(β„“,Θ))R (r(β„“,Θ ))----------. (92 ) sin Θ
These formulas have been derived first for the special case of the Schwarzschild metric by Dwivedi and Kantowski [84Jump To The Next Citation Point] and then for arbitrary spherically symmetric static spacetimes by Dyer [85Jump To The Next Citation Point]. (In [85Jump To The Next Citation Point], Equation (92View Equation) is erroneously given only for the case that, in our notation, f(r) e R(r) = r.) From these formulas we immediately get the area distance ∘ -------- Darea = |D+D − | for a static observer and, with the help of the redshift z, the luminosity distance Dlum = (1 + z )2Darea (recall Section 2.4). In this way, Equation (91View Equation) and Equation (92View Equation) allow to calculate the brightness of images according to the formulas of Section 2.6. Similarly, Equation (91View Equation) and Equation (92View Equation) allow to calculate image distortion in terms of the ellipticity πœ€ (recall Section 2.5). In general, πœ€ is a complex quantity, defined by Equation (49View Equation). In the case at hand, it reduces to the real quantity πœ€ = D − βˆ•D+ − D+ βˆ•D −. The expansion πœƒ and the shear σ of the bundles under consideration can be calculated from Kantowski’s formula [17284Jump To The Next Citation Point],
Λ™ D ± = (πœƒ ± σ) D ±, (93 )
to which Equation (27View Equation) reduces in the case at hand. The dot (= derivative with respect to the affine parameter s) is related to the derivative with respect to β„“ by Equation (83View Equation). Evaluating Equations (91View Equation, 92View Equation) in connection with the exact lens map leads to quite convenient formulas, for static light sources at r = rS. Setting r(β„“,Θ ) = rS and Ο•(β„“,Θ ) = Φ(Θ ) and comparing with Equation (87View Equation) yields (cf. [271Jump To The Next Citation Point])
∘ ----------------------- D+ (Θ ) = ef(rS) R(rS)2 − R (rO)2sin2Θ Φ′(Θ ), (94 ) D − (Θ ) = ef(rS)R (rS)sinΦ (Θ ). (95 )
These formulas immediately give image distortion and the brightness of images if the map Θ β†¦→ Φ (Θ ) is known.
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Figure 7: Thin bundle around a ray in a spherically symmetric static spacetime. The picture is purely spatial, i.e., the time coordinate t is not shown. The ray is contained in a plane, so there are two distinguished spatial directions orthogonal to the ray: the “radial” direction (in the plane) and the “tangential” direction (orthogonal to the plane). For a bundle with vertex at the observer, the radial diameter of the cross-section gives the radial angular diameter distance D+, and the tangential diameter of the cross-section gives the tangential angular diameter distance D−. In contrast to the general situation of Figure 3View Image, here the angle χ is zero (if the Sachs basis (E ,E ) 1 2 is chosen appropriately). Recall that D+ and D − are positive up to the first caustic point.

Caustics of light cones.
Quite generally, the past light cone has a caustic point exactly where at least one of the extremal angular diameter distances D+, D − vanishes (see Sections 2.2, 2.3, and 2.4). In the case at hand, zeros of D+ are called radial caustic points and zeros of D − are called tangential caustic points (see Figure 8View Image). By Equation (92View Equation), tangential caustic points occur if Ο•(β„“,Θ ) is a multiple of π, i.e., whenever a light ray crosses the axis of symmetry through the observer (see Figure 8View Image). Symmetry implies that a point source is seen as a ring (“Einstein ring”) if its worldline crosses a tangential caustic point. By contrast, a point source whose wordline crosses a radial caustic point is seen infinitesimally extended in the radial direction. The set of all tangential caustic points of the past light cone is called the tangential caustic for short. In general, it has several connected components (“first, second, etc. tangential caustic”). Each connected component is a spacelike curve in spacetime which projects to (part of) the axis of symmetry through the observer. The radial caustic is a lightlike surface in spacetime unless at points where it meets the axis; its projection to space is rotationally symmetric around the axis. The best known example for a tangential caustic, with infinitely many connected components, occurs in the Schwarzschild spacetime (see Figure 12View Image). It is also instructive to visualize radial and tangential caustics in terms of instantaneous wave fronts, i.e., intersections of the light cone with hypersurfaces t = constant. Examples are shown in Figures 13View Image, 18View Image, and 19View Image. By symmetry, a tangential caustic point of an instantaneous wave front can be neither a cusp nor a swallow-tail. Hence, the general result of Section 2.2 implies that the tangential caustic is always unstable. The radial caustic in Figure 19View Image consists of cusps and is, thus, stable.

View Image

Figure 8: Tangential and radial caustic points. Tangential caustic points, D − = 0, occur on the axis of symmetry through the observer. A (point) source at a tangential caustic point is seen as a (1-dimensional) Einstein ring on the observer’s sky. A point source at a radial caustic point, D+ = 0, appears “infinitesimally extended” in the radial direction.

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