1 Moogukora

## How To Write A Lensing Essay

Gravitational Lensing - by Ricky Leon Murphy:

Introduction
A Gravitational Lens
The Gravity Lens in Use
Gravity Lens and Dark Matter - Microlensing
Gravity Lens and Dark Matter - Weak Lensing
Gravity Lens and Dark Matter - Strong Lensing
Summary
References
Web Sites
Image Credits

Back to Cosmology

Introduction:

The breakthrough of Einstein’s General and Special Theories of Relativity give us a much clearer picture of some of the observed and theoretical processes within the Universe. Much of his theories have already been confirmed by observation including the bending of light waves by a massive object. Such proof was witnesses as stars near the Sun were shown to shift positions – observed during a solar eclipse. On a much larger scale, massive objects like black holes and brown dwarfs also bend distant light rays as do galaxies and galaxy clusters. When using massive objects like galaxies and galaxy clusters to examine the bending of these light rays, the gravity of these objects acts like a lens. This effect is called gravitational lensing and has proven very effective in observing some of the most exotic phenomenon such as exoplanets and quasars. Even more remarkable is the use of gravitational lensing to detect and map dark matter regions surrounding galaxies and galaxy clusters. By using a gravity lens, the detection of dark matter has been confirmed and is providing valuable data for cosmologists to help mold the theories involving the constituents and origins of dark matter.

A Gravitational Lens:

The presence of mass within space-time creates a curve (or depression) in the fabric of space-time. The common analogy of this is the rubber sheet.

 Figure 1. Figure 1 shows an example of the rubber sheet, with a mass – that is invisible in this example – that has created a depression within space-time. As light waves (indicated in blue) pass close to the curved space, its path is altered resulting is a bending of light. For the lens to work properly, the source of the light must be in

the line of sight to the observer with the massive object in between.  This basic two dimensional lens demonstrates a correlation between length and angles based on the radius of influence by the massive object.

 Figure 2.

 Figure 2 is a graphical example of this two dimensional lens. The point L is the massive object while point S1 is the distant object. S is the apparent position of the object to the observer; O. S2 is ignored in this example. This equation (right) demonstrates the basic properties of the lens effect in figure 2 (the following examples: Wambsgauss, 2001).

It is important to state that several correlations exist with the basic lens equation that carries over to real world examples. The length to the lenses object is correlated to the distance to the lensing object (L) by the following equation:

This directly translates to a correlation to the angles involved:

.

This is an important realization as this gives astronomers a tool for measuring the strength of a gravity lens with the benefit of helping to determine the distance as well.

As a summary, the above example can be put together to form the Einstein Radius, the radius of influence by the lensing object:

.

So what would a gravity lens look like for a familiar object?

Figure 3.

The above image demonstrates what an invisible mass would do if placed between us and the Mona Lisa. The point source is tiny with a mass of the planet Saturn. Notice the obvious circular effect of the lens – the radius. Also notice the tiny nose and the small arc of the mouth within the radius. The overall image is also bloated – spread out as a result of the lensing. This is a typical effect of the gravity lens.

The Gravity Lens in use:

On September 13, 1990, the Hubble Space Telescope issued a press release of what is probably the most famous of all images of a gravity lens:

Figure 4.

This particular arrangement is called the Einstein Cross. The four points of the cross are a lenses distant quasar, 8 billion light years away. The center if the cross is the lensing body, a galaxy “only” 400 million light years away (http://hubblesite.org/newscenter/newsdesk/archive/releases/1990/20/image/a).

The gravity lens is used to study a variety of phenomenon, and is separated into three groups (http://astron.berkeley.edu/~jcohn/lens.html ):

• Strong Lensing
• Weak Lensing
• Microlensing

Strong lensing is the result of a lensing object splitting the lensed object into separate distinct images – like our Einstein Cross example above. Strong lensing can also produced a large number of arcs as well. The usual targets for strong lensing are clusters of galaxies. By studying the strength of the lens, the astronomers learn about the mass distribution throughout the cluster.

Weak lensing is defined by arcs of the lensed object by a lensing object.

Figure 5.

This Hubble Space Telescope image of galaxy cluster Abell 2218 shows an example of weak lensing, the arcs of distant galaxies lensed by the cluster. Studying weak gravitational lensing is very useful in the study and detection of dark matter.

Microlensing is also a gravity effect, but not as pronounced as strong or weak lensing. In this case, the lensing mass is a MACHO (Massive Compact Halo Object) – black holes, white dwarfs, brown dwarfs. The result of a microlens is a momentary increase in brightness of a distant object. A type of baryonic dark matter (although only a small percentage of dark matter) are MACHO’s, so mapping the distribution of these objects is useful in the study of dark matter (Silk, 1999). Microlensing has also been effective in the detection of exoplanets as well, but that is another story.

The Gravity Lens and Dark Matter - Microlensing:

For the duration of this paper, we will focus on microlensing and weak lensing and its effectiveness in the detection and study of dark matter (although a brief entry of strong lensing will occur later on). The nature of dark matter is such that direct observation is not possible but direct detection is possible. We know of its existence by the nature of the gravity lens as well as rotation curves of spiral galaxies. Careful study of gravitational lensing helps astronomers determine the type of dark matter in existence as well as mapping its distribution – the gravitational lens is the fundamental test in the nature of dark matter (Metcalf and Silk, 1999). While the scope of this paper is not to determine what dark matter is – MACHO’s, Cold Dark Matter, Warm Dark Matter or other form of non-baryonic dark matter – but to determine its ability to affect a gravitational lens. The fundamental test for dark matter is using the gravity lens to map the distribution of this material through a modified form of the Einstein Radius equation:

.

This equation may seem rather complicated, but according to Metcalf and Silk, 1999 this equation is used for a point source such as a supernova. The implication is that a mass of an exploding star is known so with the strength of the lensed supernova (granted one is in the right spot), the mass and radius of the lens determines the nature of the lensing material – in the case of Metcalf and Silk, 1999, either MACHO’s and/or interacting elementary particles (possibly neutrinos).

What is unique about microlensing is its ability to pinpoint sources for direct detection. For example, very bright stars in distant galaxies can be lensed by local massive objects in order to determine the mass of the lens source. A relic massive black hole is presumed to be an example of a MACHO (MAssive Compact Halo Object), and very strong lensing of an individual star was detected to show single objects that collectively give a total universal mass density of 0.4 – which is determined by the probability of microlensing events (Turner and Umemura, 1997). Two gravitational microlensing surveys were performed to map out the distribution of MACHO’s near the galactic bulge and the Large Magellanic Cloud (LMC). By continuous telescope searches from two locations, it was possible to perform real-time spectroscopic data on these microlensing events (Alcock et al., 1996). The spectroscopic data would determine the nature of the lensing object. Because the galactic bulge and LMC were chosen, the likelihood of MACHO detection was more likely (Alcock et al., 1996). The equation for determining the Einstein Radius for the MACHO survey is a bit different:

.

This variation of the equation is almost elegantly simple and has some added benefits:

• If we know the distance (D) to the lensed object (ls) and the lens (l), we can determine mass
• If we know the mass of the lens as well as distance, we can determine distance to the lensed object

all of which to a good approximation.

In the case of this specific MACHO survey, the lens is predominantly brown dwarf stars. While technically a MACHO (brown dwarfs reside in the halo and it is a compact object), it is unlikely that brown dwarfs contribute any significant mass to dark matter. The figure above (http://www.llnl.gov/str/June03/Cook.html) demonstrates how a microlens works. It can also create tiny arcs, but more commonly the intensity of the lensed object increases over a short time. Luminosity changed in the point source (like a distant star) can also be used to determine the strength of the lens. This is done by comparing brightness levels before and after the lensing event. A simple method is compare the mass of the actual star, then compare with a mass of a star of equal brightness of the lensed star – that is, the increased brightness compared to an actual star of the same brightness.

A second major MACHO microlensing survey was performed, this time acquiring spectroscopic and B-V data on more sources – including binary stars. While the equations and techniques were similar to the first, the benefit of this survey also allowed for the mass of distant stars (if unknown) to be determined based on binary and B-V data (Alcock et al., 2000). Again, the primary lens of this survey is also brown dwarf stars.

The Gravity Lens and Dark Matter – Weak Lensing:

The most common type of gravitational lensing is that of the weak lens; weak gravitational lensing results in tangential and radial arcs surrounding and within the lens. The lens source is usually a galaxy or a cluster of galaxies. The weak lensing varies greatly from microlensing:

• The mathematics involved in weak lensing can be a bit too much to swallow for the scope of this paper, so will not be included
• The Einstein Radius equation and its variation are not used in weak lensing
• Direct mass measurements of individual objects are not possible

The weak gravitational lensing is used instead to map the overall distribution of dark matter within the halo of galaxies, or within the spaces between galaxies in a galaxy cluster. The probability of arcs present in any given galaxy or galaxy cluster helps to determine the overall mass of dark matter within the lens as well as constrain the mass density of the Universe (Cooray, 1999).

Figure 7: A beautiful example of weak lensing.

By using a variety of computer simulations and know CDM variables, it is suggested that lensing of quasars by nearby dark matter filled halos and statistically evaluating the size and number of arcs created by the halo of the quasar will constrain the mass density of the Universe (Li and Ostriker, 2002) which is a recurrent theme of weak gravitational lensing and dark matter.

To take this a step further, it is also proposed that the variations in count of radial and tangential arcs will provide a more accurate distribution of dark matter (Oguri et al., 2001).

Figure 8.

The image above shows the difference between a radial arc and a tangential arc. The problem is that determining the mass profile of dark matter in a halo using these methods relies on variations of N-Body simulations and comparisons to current CDM theories. However the first observational test for these models came in the form of comparing these test situations with observation data from the Jodrell-Bank VLA Astrometric Survey and the Cosmic Lens All-Sky Survey (Zhang, 2004). The observed data was in good agreement with the proposed simulations. The result is a Universe with a mass density of:

.

Other observational tests include study of dark matter halos using surveys like the Suprime-Cam 2 Square Degree Field (Miyazaki et al., 2002).

With the current observation fitting theoretical models, study of dark matter using weak gravitational lensing is most effective at studying dark matter directly (Waerbeke et al., 2002).

By studying weak lensing in a variety of galaxies with ellipticities (elliptical galaxies included) between 0.5 and 3.5, the distribution of dark matter has been shown by observations from the Canada-France-Hawaii Telescope (CFHT) to be correlated (Waerbeke et al., 2002). This helps to solidify our current value of the mass density of the Universe.

So what does all this mean exactly?

The studying of weak gravitational lensing has:

• Provided valuable data to determine the mass density of the Universe
• Proved that dark matter is real
• Dark matter resides in the halos of galaxies, and is distribution is determined by the number of arcs present in the lens
• Dark matter also resides within galaxy clusters and its distribution is also determined by the number of arcs present
• Elliptical galaxies also contain dark matter

While weak gravitational lensing observations and simulations are valuable for cosmology, it still does not answer the question of what are the constituents of dark matter.

The Gravity Lens and Dark Matter – Strong Lensing:

Much of the attention in gravitational lensing has been centered on microlensing and weak lensing, but there have been some use of the strong lens in helping to solve the dark matter mystery. By evaluating the strong lensing phenomenon, astronomers can look to the early Universe dominated mostly by quasars. The Cold Dark Matter (CDM) model relies on the gravitational lens data from quasars and the computer models demonstrate that distortions of arcs from distant quasars as well as secondary reflections will help correlate the redshift of a quasar (Matsubara, 2000). Such a correlation was found when the four lensed image (the Einstein Cross) of a quasar was evaluated for statistical variations in the broad-line and narrow-line (BLR, NLR) data; however, these statistical analysis is used primarily for tweaking the CDM model to determine when the galaxy substructure in the early Universe occurred (Metcalf et al., 2004). This correlation of redshift and lensing strength was found just recently so further surveys are needed to collect this valuable data.

Summary:

Since dark matter is a major constituent of matter in the Universe, the detection and measure of dark matter using lensing techniques has proved vital in determining the overall density parameter of our Universe. Such implications can reach deep into the field of cosmology as the density parameter also determines the value of the Hubble constant as well as the overall shape of our Universe.

Based on what has been measure by gravitational lensing, the mass density of the Universe has been established at:

(Li and Ostriker, 2002)(Miyazaki et al., 2002)(Waerbeke et al., 2002).

In addition, the mass density of the Universe has been constrained to:

(Cooray, 1999)

which means if the value of mass density changes, it should not be higher than 0.62. This does not take into account the contribution of dark energy, which is not covered here.

The future of gravitational lensing will require additional direct detection of dark matter. One proposed project is the detection of massive compact objects in other galaxies (called MASCO’s). By using yet another variation of the Einstein Radius equation, it is believed that a survey using optical and radio (VLBI) maps of other galactic halo that the distribution of these compact objects will determine the nature of these objects (Inoue and Chiba, 2003). Such a survey could add valuable data to the Cold Dark Matter (CDM) model. Up to date data on the current MACHO projects can be found on The MACHO Project website: http://wwwmacho.anu.edu.au/. Project OGLE (Optical Gravitational Lensing Experiment) is an ongoing project to collect real-time data on MACHO’s near our own galactic center: http://bulge.astro.princeton.edu/~ogle/.

While many of these projects are either ongoing or proposed, there is no clear answer as to what dark matter really is; but gravitational lensing is providing most of the much needed valuable data to help solve this puzzle.

References:

Alcock, C. et al. “Real-Time Detection and Multisite Observations of Gravitational Microlensing.” The Astrophysical Journal, 463:L67-L70, June 1, 1996.

Alcock, C. et al. “Binary Microlensing Events from the Macho Project.” The Astrophysical Journal, 541:270-297, September 20, 2000.

Cooray, Asantha. “An Upper Limit on Ωm Using Lensed Arcs.” The Astrophysical Journal, 524:504-509, October 20, 1999.

Inoue, Kaiki Taro and Masashi Chiba. “Direct Mapping of Massive Compact Objects in Extragalactic Dark Halos.” The Astrophysical Journal, 591:L83-L86, July 10, 2003.

Li, Li-Xin and Jeremiah Ostriker. “Semianalytical Models for Lensing by Dark Halos. I. Splitting Angles.” The Astrophysical Journal, 566:652-666, February 20, 2002.

Matsubara, Takahiko. “The Gravitational Lensing in Redshift-Space Correlation Functions of Galaxies and Quasars.” The Astrophysical Journal, 537:L77-L80, July 10, 2000.

Metcalf, R. Benton and Joseph Silk. “A Fundamental Test of the Nature of Dark Matter.” The Astrophysical Journal, 519:L1-L4, July 1, 1999.

Metcalf, R. Benton, et al. “Spectroscopic Gravitational Lensing and Limits on the Dark Matter Substructure in Q2237+0305.” The Astrophysical Journal, 607:43-59, May 20, 2004.

Miyazaki, Satoshi, et al. “Searching for Dark Matter Halos in the Suprime-Cam 2 Square Degree Field.” The Astrophysical Journal, 580:L97-L100, December 1, 2002.

Oguri, Masamune; Taruya, Atsushi and Yasushi Suto. “Probing the Core Structure of Dark Halos with Tangential and Radial Arc Statistics.” The Astrophysical Journal, 559:572-583, October 1, 2001.

Silk, Joseph. A Short History of the Universe. Scientific American Library, New York. 1999.

Turner, Edwin and Masayuki Umemura. “Very Strong Microlensing of Distant Luminous Stars by Relic Massive Black Holes.” The Astrophysical Journal, 483:603-607, July 10, 1999.

Waerbeke, L. Van, et al. “Detection of Correlated Galaxy Ellipticities from CFHT Data: First Evidence for Gravitational Lensing by Large-Scale Structures.” Astronomy and Astrophysics Pre-Print, April 28, 2002.

Wambsgauss, Joachim. “Gravitational Lensing in Astronomy.” Living Reviews in Relativity, Internet: http://relativity.livingreviews.org/Articles/lrr-1998-12/. 2001.

Zhang, Tong-Jie. “Gravitational Lensing by Dark Matter Halos with Nonuniversal Density Profiles.” The Astrophysical Journal, 602:L5-L8, February 10, 2004.

Websites:

Hubble Press Release: http://hubblesite.org/newscenter/newsdesk/archive/releases/1990/20/image/a

Gravitational Lensing: http://astron.berkeley.edu/~jcohn/lens.html

The MACHO Project: http://wwwmacho.anu.edu.au/

OGLE: http://bulge.astro.princeton.edu/~ogle/

The Hubble Newsdesk – Gravitational Lens: http://hubblesite.org/newscenter/newsdesk/archive/releases/category/exotic/gravitational lens/

Image Credits (in order of appearance):

Figure 1: http://astron.berkeley.edu/~jcohn/lens.html

Figure 2: http://relativity.livingreviews.org/Articles/lrr-1998-12/

Figure 3: http://www.mpa-garching.mpg.de/Lenses/museum.en/index.html

Figure 4: http://hubblesite.org/newscenter/newsdesk/archive/releases/1990/20/image/a

Figure 5: http://hubblesite.org/newscenter/newsdesk/archive/releases/1995/14/image/a

Figure not labeled: http://www.llnl.gov/str/June03/Cook.html

Figure 7: http://hubblesite.org/newscenter/newsdesk/archive/releases/2001/32/image/b

Figure 8: http://www2.iap.fr/LaboEtActivites/ThemesRecherche/Lentilles/arcs/ms2137.html

By a spacetime we mean a 4-dimensional manifold $${\mathcal M}$$ with a (C, if not otherwise stated) metric tensor field g of signature (+, +, +, −) that is time-oriented. The latter means that the non-spacelike vectors make up two connected components in the entire tangent bundle, one of which is called “future-pointing” and the other one “past-pointing”. Throughout this review we restrict to the case that the light rays are freely propagating in vacuum, i.e., are not influenced by mirrors, refractive media, or any other impediments. The light rays are then the lightlike geodesics of the spacetime metric. We first summarize results on the lightlike geodesics that hold in arbitrary spacetimes. In Section 3 these results will be specified for spacetimes with conditions on the causal structure and in Section 4 for spacetimes with symmetries.

### 2.1 Light cone and exact lens map

In an arbitrary spacetime ($${\mathcal M}$$, g), what an observer at an event pO can see is determined by the lightlike geodesics that issue from pO into the past. Their union gives the past light cone of pO. This is the central geometric object for lensing from the spacetime perspective. For a point source with worldline γS, each past-oriented lightlike geodesic λ from pO to γS gives rise to an image of γS on the observer’s sky. One should view any such λ as the central ray of a thin bundle that is focused by the observer’s eye lens onto the observer’s retina (or by a telescope onto a photographic plate). Hence, the intersection of the past light cone with the world-line of a point source (or with the world-tube of an extended source) determines the visual appearance of the latter on the observer’s sky.

In mathematical terms, the observer’s sky or celestial sphere$$({{\mathcal S}_{\rm{O}}})$$ can be viewed as the set of all lightlike directions at pO. Every such direction defines a unique (up to parametrization) lightlike geodesic through pO, so $$({{\mathcal S}_{\rm{O}}})$$ may also be viewed as a subset of the space of all lightlike geodesics in ($${\mathcal M},g$$) (cf. [209]). One may choose at pO a future-pointing vector UO with g(UO, UO) = −1, to be interpreted as the 4-velocity of the observer. This allows identifying the observer’s sky $${{\mathcal S}_{\rm{O}}}$$ with a subset of the tangent space $${T_{P{\rm{O}}}}{\mathcal M}$$,

$${{\mathcal S}_{\rm{O}}} \simeq \{w \in {T_{p{\rm{O}}}}{\mathcal M}\vert g(w,w) = 0\,\,{\rm{and}}\,\,g(w,{U_{\rm{O}}}) = 1\}.$$

(1)

If UO is changed, this representation changes according to the standard aberration formula of special relativity. By definition of the exponential map exp, every affinely parametrized geodesic s ↦ τ(s) satisfies $$\lambda (s) = \exp (s\dot \lambda (0))$$. Thus, the past light cone of pO is the image of the map

$$(s,w) \mapsto \exp (sw),$$

(2)

which is defined on a subset of $$]0,\infty [ \times {{\mathcal S}_{\rm{O}}}$$. If we restrict to values of s sufficiently close to 0, the map (2) is an embedding, i.e., this truncated light cone is an embedded submanifold; this follows from the well-known fact that exp maps a neighborhood of the origin, in each tangent space, diffeomorphically into the manifold. However, if we extend the map (2) to larger values of s, it is in general neither injective nor an immersion; it may form folds, cusps, and other forms of caustics, or transverse self-intersections. This observation is of crucial importance in view of lensing. There are some lensing phenomena, such as multiple imaging and image distortion of (point) sources into (1-dimensional) rings, which can occur only if the light cone fails to be an embedded submanifold (see Section 2.8). Such lensing phenomena are summarized under the name strong lensing effects. As long as the light cone is an embedded submanifold, the effects exerted by the gravitational field on the apparent shape and on the apparent brightness of light sources are called weak lensing effects. For examples of light cones with caustics and/or transverse self-intersections, see Figures 12, 24, and 25. These pictures show light cones in spacetimes with symmetries, so their structure is rather regular. A realistic model of our own light cone, in the real world, would have to take into account numerous irregularly distributed inhomogeneities (“clumps”) that bend light rays in their neighborhood. Ellis, Bassett, and Dunsby [99] estimate that such a light cone would have at least 1022 caustics which are hierarchically structured in a way that reminds of fractals.
For calculations it is recommendable to introduce coordinates on the observer’s past light cone. This can be done by choosing an orthonormal tetrad (e0, e1, e2, e3) with e0 = −UO at the observation event pO. This parametrizes the points of the observer’s celestial sphere by spherical coordinates (Ψ, Θ),

$$w = \sin \Theta \cos \Psi \,{e_1} + \sin \Theta \sin \Psi \,{e_2} + \cos \Theta \,{e_3} + {e_0}.$$

(3)

In this representation, map (2) maps each (s, Ψ Θ) to a spacetime point. Letting the observation event float along the observer’s worldline, parametrized by proper time τ, gives a map that assigns to each (s, Ψ, Θ, τ) a spacetime point. In terms of coordinates x = (x0, x1, x2, x3) on the spacetime manifold, this map is of the form

$${x^i} = {F^i}(s,\Psi ,\Theta ,\tau),\quad \quad i = 0,1,2,3.$$

(4)

It can be viewed as a map from the world as it appears to the observer (via optical observations) to the world as it is. The observational coordinates (s, Ψ, Θ, τ) were introduced by Ellis [98] (see [100] for a detailed discussion). They are particularly useful in cosmology but can be introduced for any observer in any spacetime. It is useful to consider observables, such as distance measures (see Section 2.4) or the ellipticity that describes image distortion (see Section 2.5) as functions of the observational coordinates. Some observables, e.g., the redshift and the luminosity distance, are not determined by the spacetime geometry and the observer alone, but also depend on the 4-velocities of the light sources. If a vector field U with g(U, U) = −1 has been fixed, one may restrict to an observer and to light sources which are integral curves of U. The above-mentioned observables, like redshift and luminosity distance, are then uniquely determined as functions of the observational coordinates. In applications to cosmology one chooses U as tracing the mean flow of luminous matter (“Hubble flow”) or as the rest system of the cosmic background radiation; present observations are compatible with the assumption that these two distinguished observer fields coincide [32].

Writing map (4) explicitly requires solving the lightlike geodesic equation. This is usually done, using standard index notation, in the Lagrangian formalism, with the Lagrangian $${\mathcal L} = {1 \over 2}{g_{ij}}(x){{\dot x}^i}{{\dot x}^j}$$, or in the Hamiltonian formalism, with the Hamiltonian $${\mathcal H} = {1 \over 2}{g^{ij}}(x){p_i}{p_j}$$. A non-trivial example where the solutions can be explicitly written in terms of elementary functions is the string spacetime of Section 5.10. Somewhat more general, although still very special, is the situation that the lightlike geodesic equation admits three independent constants of motion in addition to the obvious one gij(x)pipj = 0. If, for any pair of the four constants of motion, the Poisson bracket vanishes (“complete integrability”), the lightlike geodesic equation can be reduced to first-order form, i.e., the light cone can be written in terms of integrals over the metric coefficients. This is true, e.g., in spherically symmetric and static spacetimes (see Section 4.3).

Having parametrized the past light cone of the observation event pO in terms of (s, w), or more specifically in terms of (s, Ψ, Θ), one may set up an exact lens map. This exact lens map is analogous to the lens map of the quasi-Newtonian approximation formalism, as far as possible, but it is valid in an arbitrary spacetime without approximation. In the quasi-Newtonian formalism for thin lenses at rest, the lens map assigns to each point in the lens plane a point in the source plane (see, e.g., [299, 275, 343]). When working in an arbitrary spacetime without approximations, the observer’s sky $${{\mathcal S}_{\rm{O}}}$$ is an obvious substitute for the lens plane. As a substitute for the source plane we choose a 3-dimensional submanifold $${\mathcal T}$$ with a prescribed ruling by timelike curves. We assume that $${\mathcal T}$$ is globally of the form $${\mathcal Q} \times {\mathbb R}$$, where the points of the 2-manifold $${\mathcal Q}$$ label the timelike curves by which $${\mathcal T}$$ is ruled. These timelike curves are to be interpreted as the worldlines of light sources. We call any such $${\mathcal T}$$ a source surface. In a nutshell, choosing a source surface means choosing a two-parameter family of light sources.

The exact lens map is a map from $${{\mathcal S}_{\rm{O}}}$$ to $${\mathcal Q}$$. It is defined by following, for each $$w \in {{\mathcal S}_{\rm{O}}}$$, the past-pointing geodesic with initial vector ω until it meets $${\mathcal T}$$ and then projecting to $${\mathcal Q}$$ (see Figure 1). In other words, the exact lens map says, for each point on the observer’s celestial sphere, which of the chosen light sources is seen at this point. Clearly, non-invertibility of the lens map indicates multiple imaging. What one chooses for $${\mathcal T}$$ depends on the situation. In applications to cosmology, one may choose galaxies at a fixed redshift z = zS around the observer. In a spherically-symmetric and static spacetime one may choose static light sources at a fixed radius value r = rS. Also, the surface of an extended light source is a possible choice for $${\mathcal T}$$.

The exact lens map was introduced by Frittelli and Newman [123] and further discussed in [91, 90]. The following global aspects of the exact lens map were investigated in [270]. First, in general the lens map is not defined on all of $${{\mathcal S}_{\rm{O}}}$$ because not all past-oriented lightlike geodesics that start at pO necessarily meet $${\mathcal T}$$. Second, in general the lens map is multi-valued because a lightlike geodesic might meet $${\mathcal T}$$ several times. Third, the lens map need not be differentiable and not even continuous because a lightlike geodesic might meet $${\mathcal T}$$ tangentially. In [270], the notion of a simple lensing neighborhood is introduced which translates the statement that a deflector is transparent into precise mathematical language. It is shown that the lens map is globally well-defined and differentiable if the source surface is the boundary of such a simple lensing neighborhood, and that for each light source that does not meet the caustic of the observer’s past light cone the number of images is finite and odd. This result applies, as a special case, to asymptotically simple and empty spacetimes (see Section 3.4).

For expressing the exact lens map in coordinate language, it is recommendable to choose coordinates (x0, x1, x2, x3) such that the source surface $${\mathcal T}$$ is given by the equation $${x^3} = x_{\rm{S}}^3$$, with a constant $$x_{\rm{S}}^3$$, and that the worldlines of the light sources are x0-lines. In this situation the remaining coordinates x1 and x2 label the light sources and the exact lens map takes the form

$$(\Psi, \Theta) \mapsto ({x^1},{x^2}).$$

(5)

It is given by eliminating the two variables s and x0 from the four equations (4) with $${F^3}(s,\Psi, \Theta, \tau) = x_{\rm{S}}^3$$ and fixed τ. This is the way in which the lens map was written in the original paper by Frittelli and Newman; see Equation (6) in [123]. (They used complex coordinates ($$\eta, \bar \eta$$) for the observer’s celestial sphere that are related to our spherical coordinates (Ψ, Θ) by stereographic projection.) In this explicit coordinate version, the exact lens map can be succesfully applied, in particular, to spherically symmetric and static spacetimes, with x0 = t, x1 = ϕ, x2 = ϑ, and x3 = r (see Section 4.3 and the Schwarzschild example in Section 5.1). The exact lens map can also be used for testing the reliability of approximation techniques. In [184] the authors find that the standard quasi-Newtonian approximation formalism may lead to significant errors for lensing configurations with two lenses.

### 2.2 Wave fronts

Wave fronts are related to light rays as solutions of the Hamilton-Jacobi equation are related to solutions of Hamilton’s equations in classical mechanics. For the case at hand (i.e., vacuum light propagation in an arbitrary spacetime, corresponding to the Hamiltonian $${\mathcal H} = {1 \over 2}{g^{ij}}(x){p_i}{p_j})$$, a wave front is a subset of the spacetime that can be constructed in the following way:
1. 1.

Choose a spacelike 2-surface S that is orientable.

2. 2.

At each point of $${\mathcal S}$$, choose a lightlike direction orthogonal to $${\mathcal S}$$ that depends smoothly on the foot-point. (You have to choose between two possibilities.)

3. 3.

Take all lightlike geodesics that are tangent to the chosen directions. These lightlike geodesics are called the generators of the wave front, and the wave front is the union of all generators.

Clearly, a light cone is a special case of a wave front. One gets this special case by choosing for $${\mathcal S}$$ an appropriate (small) sphere. Any wave front is the envelope of all light cones with vertices on the wave front. In this sense, general-relativistic wave fronts can be constructed according to the Huygens principle.

In the context of general relativity the notion of wave fronts was introduced by Kermack, McCrea, and Whittaker [180]. For a modern review article see, e.g., Ehlers and Newman [93].

A coordinate representation for a wave front can be given with the help of (local) coordinates (u1, u1) on $${\mathcal S}$$. One chooses a parameter value s0 and parametrizes each generator λ affinely such that $$\lambda ({s_0}) \in {\mathcal S}$$ and $$\dot \lambda ({s_0})$$ depends smoothly on the foot-point in $${\mathcal S}$$. This gives the wave front as the image of a map

$$(s,{u^1},{u^2}) \mapsto {F^i}(s,{u^1},{u^2}),\quad \quad i = 0,1,2,3.$$

(6)

For light cones we may choose spherical coordinates, (u1 = Ψ, u2 = Θ), (cf. Equation (4) with fixed τ). Near s = s0, map (6) is an embedding, i.e., the wave front is a submanifold. Orthogonality to $${\mathcal S}$$ of the initial vectors $$\dot \lambda ({s_0})$$ assures that this submanifold is lightlike. Farther away from $${\mathcal S}$$, however, the wave front need not be a submanifold. The caustic of the wave front is the set of all points where the map (6) is not an immersion, i.e., where its differential has rank < 3. As the derivative with respect to s is always non-zero, the rank can be 3 − 1 (caustic point of multiplicity one, astigmatic focusing) or 3 − 2 (caustic point of multiplicity two, anastigmatic focusing). In the first case, the cross-section of an “infinitesimally thin” bundle of generators collapses to a line, in the second case to a point (see Section 2.3). For the case that the wave front is a light cone with vertex pO, caustic points are said to be conjugate to pO along the respective generator. For an arbitrary wave front, one says that a caustic point is conjugate to any spacelike 2-surface in the wave front. In this sense, the terms “conjugate point” and “caustic point” are synonymous. Along each generator, caustic points are isolated (see Section 2.3) and thus denumerable. Hence, one may speak of the first caustic, the second caustic, and so on. At all points where the caustic is a manifold, it is either spacelike or lightlike. For instance, the caustic of the Schwarzschild light cone in Figure 12 is a spacelike curve; in the spacetime of a transparent string, the caustic of the light cone consists of two lightlike 2-manifolds that meet in a spacelike curve (see Figure 25).
Near a non-caustic point, a wave front is a hypersurface S = constant where S satisfies the Hamilton-Jacobi equation

$${g^{ij}}(x){\partial _i}S(x){\partial _j}S(x) = 0.$$

(7)

In the terminology of optics, Equation (7) is called the eikonal equation.
At caustic points, a wave front typically forms cuspidal edges or vertices whose geometry might be arbitrarily complicated, even locally. If one restricts to caustics which are stable against perturbations in a certain sense, then a local classification of caustics is possible with the help of Arnold’s singularity theory of Lagrangian or Legendrian maps. Full details of this theory can be found in [11]. For a readable review of Arnold’s results and its applications to wave fronts in general relativity, we refer again to [93]. In order to apply Arnold’s theory to wave fronts, one associates each wave front with a Legendrian submanifold in the projective cotangent bundle over $${\mathcal M}$$ (or with a Lagrangian submanifold in an appropriately reduced bundle). A caustic point of the wave front corresponds to a point where the differential of the projection from the Legendrian submanifold to $${\mathcal M}$$ has non-maximal rank. For the case $$\dim ({\mathcal M}) = 4$$, which is of interest here, Arnold has shown that there are only five types of caustic points that are stable with respect to perturbations within the class of all Legendrian submanifolds. They are known as fold, cusp, swallow-tail, pyramid, and purse (see Figure 2). Any other type of caustic is unstable in the sense that it changes non-diffeomorphically if it is perturbed within the class of Legendrian submanifolds.

Fold singularities of a wave front form a lightlike 2-manifold in spacetime, on a sufficiently small neighborhood of any fold caustic point. The second picture in Figure 2 shows such a “fold surface”, projected to 3-space along the integral curves of a timelike vector field. This projected fold surface separates a region covered twice by the wave front from a region not covered at all. If the wave front is the past light cone of an observation event, and if one restricts to light sources with worldlines in a sufficiently small neighborhood of a fold caustic point, there are two images for light sources on one side and no images for light sources on the other side of the fold surface. Cusp singularities of a wave front form a spacelike curve in spacetime, again locally near any cusp caustic point. Such a curve is often called a “cusp ridge”. Along a cusp ridge, two fold surfaces meet tangentially. The third picture in Figure 2 shows the situation projected to 3-space. Near a cusp singularity of a past light cone, there is local triple-imaging for light sources in the wedge between the two fold surfaces and local single-imaging for light sources outside this wedge. Swallow-tail, pyramid, and purse singularities are points where two or more cusp ridges meet with a common tangent, as illustrated by the last three pictures in Figure 2.

Friedrich and Stewart [118] have demonstrated that all caustic types that are stable in the sense of Arnold can be realized by wave fronts in Minkowski spacetime. Moreover, they stated without proof that, quite generally, one gets the same stable caustic types if one allows for perturbations only within the class of wave fronts (rather than within the larger class of Legendrian submanifolds). A proof of this statement was claimed to be given in [150] where the Lagrangian rather than the Legendrian formalism was used. However, the main result of this paper (Theorem 4.4 of [150]) is actually too weak to justify this claim. A different version of the desired stability result was indeed proven by another approach. In this approach one concentrates on an instantaneous wave front, i.e., on the intersection of a wave front with a spacelike hypersurface $${\mathcal C}$$. As an alternative terminology, one calls the intersection of a (“big”) wave front with a hypersurface $${\mathcal C}$$ that is transverse to all generators a “small wave front”. Instantaneous wave fronts are special cases of small wave fronts. The caustic of a small wave front is the set of all points where the small wave front fails to be an immersed 2-dimensional submanifold of $${\mathcal C}$$. If the spacetime is foliated by spacelike hypersurfaces, the caustic of a wave front is the union of the caustics of its small (= instantaneous) wave fronts. Such a foliation can always be achieved locally, and in several spacetimes of interest even globally. If one identifies different slices with the help of a timelike vector field, one can visualize a wave front, and in particular a light cone, as a motion of small (= instantaneous) wave fronts in 3-space. Examples are shown in Figures 13, 18, 19, 27, and 28. Mathematically, the same can be done for non-spacelike slices as long as they are transverse to the generators of the considered wave front (see Figure 30 for an example). Turning from (big) wave fronts to small wave fronts reduces the dimension by one. The only caustic points of a small wave front that are stable in the sense of Arnold are cusps and swallow-tails. What one wants to prove is that all other caustic points are unstable with respect to perturbations of the wave front within the class of wave fronts, keeping the metric and the slicing fixed. For spacelike slicings (i.e., for instantaneous wave fronts), this was indeed demonstrated by Low [210]. In this article, the author views wave fronts as subsets of the space $${\mathcal N}$$ of all lightlike geodesics in ($${\mathcal M},g$$). General properties of this space $${\mathcal N}$$ are derived in earlier articles by Low [208, 209] (also see Penrose and Rindler [262], volume II, where the space $${\mathcal N}$$ is treated in twistor language). Low considers, in particular, the case of a globally hyperbolic spacetime [210]; he demonstrates the desired stability result for the intersections of a (big) wave front with Cauchy hypersurfaces (see Section 3.2). As every point in an arbitrary spacetime admits a globally hyperbolic neighborhood, this local stability result is universal. Figure 28 shows an instantaneous wave front with cusps and a swallow-tail point. Figure 13 shows instantaneous wave fronts with caustic points that are neither cusps nor swallow-tails; hence, they must be unstable with respect to perturbations of the wave front within the class of wave fronts.

It is to be emphasized that Low’s work allows to classify the stable caustics of small wave fronts, but not directly of (big) wave fronts. Clearly, a (big) wave front is a one-parameter family of small wave fronts. A qualitative change of a small wave front, in dependence of a parameter, is called a “metamorphosis” in the English literature and a “perestroika” in the Russian literature. Combining Low’s results with the theory of metamorphoses, or perestroikas, could lead to a classsification of the stable caustics of (big) wave fronts. However, this has not been worked out until now.

Wave fronts in general relativity have been studied in a long series of articles by Newman, Frittelli, and collaborators. For some aspects of their work see Sections 2.9 and 3.4. In the quasi-Newtonian approximation formalism of lensing, the classification of caustics is treated in great detail in the book by Petters, Levine, and Wambsganss [275]. Interesting related mateial can also be found in Blandford and Narayan [33]. For a nice exposition of caustics in ordinary optics see Berry and Upstill [28].

A light source that comes close to the caustic of the observer’s past light cone is seen strongly magnified. For a point source whose worldline passes exactly through the caustic, the ray-optical treatment even gives an infinite brightness (see Section 2.6). If a light source passes behind a compact deflecting mass, its brightness increases and decreases in the course of time, with a maximum at the moment of closest approach to the caustic. Such microlensing events are routinely observed by monitoring a large number of stars in the bulge of our Galaxy, in the Magellanic Clouds, and in the Andromeda Galaxy (see, e.g., [226] for an overview). In his millennium essay on future perspectives of gravitational lensing, Blandford [34] mentioned the possibility of observing a chosen light source strongly magnified over a period of time with the help of a space-born telescope. The idea is to guide the spacecraft such that the worldline of the light source remains in (or close to) the one-parameter family of caustics of past light cones of the spacecraft over a period of time. This futuristic idea of “caustic surfing” was mathematically further discussed by Frittelli and Petters [128].

### 2.3 Optical scalars and Sachs equations

For the calculation of distance measures, of image distortion, and of the brightness of images one has to study the Jacobi equation (= equation of geodesic deviation) along lightlike geodesics. This is usually done in terms of the optical scalars which were introduced by Sachs et al. [172, 292]. Related background material on lightlike geodesic congruences can be found in many text-books (see, e.g., Wald [341], Section 9.2). In view of applications to lensing, a particularly useful exposition was given by Seitz, Schneider and Ehlers [303]. In the following the basic notions and results will be summarized.

#### 2.3.1 Infinitesimally thin bundles

Let s ↦ λ(s) be an affinely parametrized lightlike geodesic with tangent vector field $$K = \dot \lambda$$. We assume that λ is past-oriented, because in applications to lensing one usually considers rays from the observer to the source. We use the summation convention for capital indices A, B, … taking the values 1 and 2. An infinitesimally thin bundle (with elliptical cross-section) along λ is a set

$${\mathcal B} = \{{c^A}{Y_A}\vert {c^1},{c^2} \in \mathbb{R},\,\quad {\delta _{AB}}{c^A}{c^B} \leq 1\}.$$

(8)

Here δab denotes the Kronecker delta, and Y1 and Y2 are two vector fields along λ with

$${\nabla _K}{\nabla _K}{Y_A} = R(K,{Y_A},K),$$

(9)

such that Y1(s), Y2(s), and K(s) are linearly independent for almost all s. As usual, R denotes the curvature tensor, defined by

$$R(X,Y,Z) = {\nabla _X}{\nabla _Y}Z - {\nabla _Y}{\nabla _X}Z - {\nabla _{[X,Y]}}Z.$$

(11)

Equation (9) is the Jacobi equation. It is a precise mathematical formulation of the statement that “the arrow-head of Ya traces an infinitesimally neighboring geodesic”. Equation (10) guarantees that this neighboring geodesic is, again, lightlike and spatially related to λ.

#### 2.3.2 Sachs basis

For discussing the geometry of infinitesimally thin bundles it is usual to introduce a Sachs basis, i.e., two vector fields E1 and E2 along λ that are orthonormal, orthogonal to $$K = \dot \lambda$$, and parallelly transported,

$$g({E_A},{E_B}) = {\delta _{AB}},\quad g(K,{E_A}) = 0,\quad {\nabla _K}{E_A} = 0.$$

(12)

Apart from the possibility to interchange them, E1 and E2 are unique up to transformations

$${\tilde E_1} = \cos \alpha {E_1} + \sin \alpha {E_2} + {a_1}K,$$

(13)

$${\tilde E_2} = - \sin \alpha {E_1} + \cos \alpha {E_2} + {a_2}K,$$

(14)

where a, a1, and a2 are constant along λ. A Sachs basis determines a unique vector field U with g(U, U) = −1 and g(U, K) = 1 along λ that is perpendicular to E1, and E2. As K is assumed past-oriented, U is future-oriented. In the rest system of the observer field U, the Sachs basis spans the 2-space perpendicular to the ray. It is helpful to interpret this 2-space as a “screen”; correspondingly, linear combinations of E1 and E2 are often refered to as “screen vectors”.

#### 2.3.3 Jacobi matrix

With respect to a Sachs basis, the basis vector fields Y1 and Y2 of an infinitesimally thin bundle can be represented as

$${Y_A} = D_A^B{E_B} + {y_A}K.$$

(15)

The Jacobi matrix$$D = (D_A^B)$$ relates the shape of the cross-section of the infinitesimally thin bundle to the Sachs basis (see Figure 3). Equation (9) implies that D satisfies the matrix Jacobi equation where an overdot means derivative with respect to the affine parameter s, and

$$R = \left({\begin{array}{*{20}c} {{\Phi _{00}}} & 0 \\ 0 & {{\Phi _{00}}} \\ \end{array}} \right) + \left({\begin{array}{*{20}c}{- {{\rm Re}} ({\psi _0})} & {{{\rm Im}} ({\psi _0})} \\ {{{\rm Im}} ({\psi _0})} & {{{\rm Re}} ({\psi _0})} \\ \end{array}} \right)$$

(17)

is the optical tidal matrix, with

$${\Phi _{00}} = - {1 \over 2}{\rm{Ric}}(K,K),\quad {\psi _0} = - {1 \over 2}C({E_1} - i{E_2},K,{E_1} - i{E_2},K).$$

(18)

Here Ric denotes the Ricci tensor, defined by Ric(X, Y) = tr(R(·, X, Y)), and C denotes the conformal curvature tensor (= Weyl tensor). The notation in Equation (18) is chosen in agreement with the Newman-Penrose formalism (cf., e.g., [54]). As Y1, Y2, and K are not everywhere linearly dependent, det(D) does not vanish identically. Linearity of the matrix Jacobi equation implies that det(D) has only isolated zeros. These are the “caustic points” of the bundle (see below).

#### 2.3.4 Shape parameters

The Jacobi matrix D can be parametrized according to

$$D = \left({\begin{array}{*{20}c}{\cos \psi} & {- \sin \psi} \\ {\sin \psi} & {\cos \psi} \\ \end{array}} \right)\, \, \left({\begin{array}{*{20}c}{{D_ +}} & 0 \\ 0 & {D_-} \\ \end{array}} \right)\, \, \left({\begin{array}{*{20}c}{\cos \chi} & {\sin \chi} \\ {- \sin \chi} & {\cos \chi} \\ \end{array}} \right).$$

(19)

Here we made use of the fact that any matrix can be written as the product of an orthogonal and a symmetric matrix, and that any symmetric matrix can be diagonalized. Note that, by our definition of infinitesimally thin bundles, D+ and D are non-zero almost everywhere. Equation (19) determines D+ and D up to sign. The most interesting case for us is that of an infinitesimally thin bundle that issues from a vertex at an observation event pO into the past. For such bundles we require D+ and D to be positive near the vertex and differentiable everywhere; this uniquely determines D+ and D everywhere. With D+ and D fixed, the angles χ and ψ are unique at all points where the bundle is non-circular; in other words, requiring them to be continuous determines these angles uniquely along every infinitesimally thin bundle that is non-circular almost everywhere. In the representation of Equation (19), the extremal points of the bundle’s elliptical cross-section are given by the position vectors

$${Y_ +} = \cos \psi {Y_1} + \sin {Y_1} \simeq {D_ +}(\cos \chi {E_1} + \sin \chi {E_2}),$$

(20)

$${Y_ -} = - \sin \psi {Y_1} + \cos \psi {Y_2} \simeq {D_ -}(- \sin \chi {E_1} + \cos \chi {E_2}),$$

(21)

where ≃ means equality up to multiples of K. Hence, |D+| and |D| give the semi-axes of the elliptical cross-section and χ gives the angle by which the ellipse is rotated with respect to the Sachs basis (see Figure 3). We call D+, D, and χ the shape parameters of the bundle, following Frittelli, Kling, and Newman [121, 120]. Instead of D+ and D one may also use D+D and D+/D. For the case that the infinitesimally thin bundle can be embedded in a wave front, the shape parameters D+ and D have the following interesting property (see Kantowski et al. [173, 84]). $${{\dot D}_ +}/{D_ +}$$ and $${{\dot D}_ -}/{D_ -}$$ give the principal curvatures of the wave front in the rest system of the observer field U which is perpendicular to the Sachs basis. The notation D+ and D, which is taken from [84], is convenient because it often allows to write two equations in the form of one equation with a ± sign (see, e.g., Equation (27) or Equation (93) below). The angle χ can be directly linked with observations if a light source emits linearly polarized light (see Section 2.5). If the Sachs basis is transformed according to Equations (13, 14) and Y1 and Y2 are kept fixed, the Jacobi matrix changes according to $${{\tilde D}_ \pm} = {{\tilde D}_ \pm},\tilde {\mathcal X} = {\mathcal X} + \alpha, \psi = \psi$$. This demonstrates the important fact that the shape and the size of the cross-section of an infinitesimally thin bundle has an invariant meaning [292].

#### 2.3.5 Optical scalars

Along each infinitesimally thin bundle one defines the deformation matrixS by

$${{\dot D}} = {{DS}}.$$

(22)

This reduces the second-order linear differential equation (16) for D to a first-order non-linear differential equation for S,

$${{\dot S}} + {{SS}} = {{R}}.$$

(23)

It is usual to decompose S into antisymmetric, symmetric-tracefree, and trace parts,

$${{S}} = \left({\begin{array}{*{20}c} 0 & \omega \\ {- \omega} & 0 \\ \end{array}} \right) + \left({\begin{array}{*{20}c}{{\sigma _1}} & {{\sigma _2}} \\ {{\sigma _2}} & {- {\sigma _1}} \\ \end{array}} \right) + \left({\begin{array}{*{20}c}\theta & 0 \\ 0 & \theta \\ \end{array}} \right).$$

(24)

This defines the optical scalars ω (twist), φ (expansion), and (σ1, σ2) (shear). One usually combines them into two complex scalars ϱ = φ + and σ = σ1 + 2. A change (13, 14) of the Sachs basis affects the optical scalars according to $$\tilde \varrho = \varrho$$ and $$\tilde \sigma = {e^{- 2i\alpha}}\sigma$$. Thus, ϱ and |σ| are invariant. If rewritten in terms of the optical scalars, Equation (23) gives the Sachs equations

$$\dot \varrho = - {\varrho ^2} - \vert \sigma {\vert ^2} + {\Phi _{00}},$$

(25)

$$\dot \sigma = - \sigma (\varrho + \bar \varrho) + {\psi _0}.$$

(26)

One sees that the Ricci curvature term Φ00 directly produces expansion (focusing) and that the conformal curvature term ψ0 directly produces shear. However, as the shear appears in Equation (25), conformal curvature indirectly influences focusing (cf. Penrose [260]). With D written in terms of the shape parameters and S written in terms of the optical scalars, Equation (22) results in

$${\dot D_ \pm} - i\dot \chi {D_ \pm} = (\rho \pm {e^{2i\chi}}\sigma){D_ \pm}.$$

(27)

Along λ, Equations (25, 26) give a system of 4 real first-order differential equations for the 4 real variables ϱ and σ; if ϱ and σ are known, Equation (27) gives a system of 4 real first-order differential equations for the 4 real variables D±, χ, and ψ. The twist-free solutions (g real) to Equations (25, 26) constitute a 3-dimensional linear subspace of the 4-dimensional space of all solutions. This subspace carries a natural metric of Lorentzian signature, unique up to a conformal factor, and was nicknamed Minikowski space in [20].

#### 2.3.6 Conservation law

As the optical tidal matrix R is symmetric, for any two solutions D1 and D2 of the matrix Jacobi equation (16) we have

$${{{\dot D}}_1}{{D}}_2^T - {{{D}}_1}{{\dot D}}_2^T = {\rm{constant}},$$

(28)

where ()T means transposition. Evaluating the case D1 = D2 shows that for every infinitesimally thin bundle

$$\omega {D_ +}{D_ -} = {\rm{constant}}.$$

(29)

Thus, there are two types of infinitesimally thin bundles: those for which this constant is non-zero and those for which it is zero. In the first case the bundle is twisting (ω ≠ 0 everywhere) and its cross-section nowhere collapses to a line or to a point (D+ = 0 and D = 0 everywhere). In the second case the bundle must be non-twisting (ω = 0 everywhere), because our definition of infinitesimally thin bundles implies that D+ = 0 and D = 0 almost everywhere. A quick calculation shows that ω = 0 is exactly the integrability condition that makes sure that the infinitesimally thin bundle can be embedded in a wave front. (For the definition of wave fronts see Section 2.2.) In other words, for an infinitesimally thin bundle we can find a wave front such that λ is one of the generators, and Y1 and Y2 connect λ with infinitesimally neighboring generators if and only if the bundle is twist-free. For a (necessarily twist-free) infinitesimally thin bundle, points where one of the two shape parameters D+ and D vanishes are called caustic points of multiplicity one, and points where both shape parameters D+ and D vanish are called caustic points of multiplicity two. This notion coincides exactly with the notion of caustic points, or conjugate points, of wave fronts as introduced in Section 2.2. The behavior of the optical scalars near caustic points can be deduced from Equation (27) with Equations (25, 26). For a caustic point of multiplicty one at s = s0 one finds

$$\theta (s) = {1 \over {2(s - {s_0})}}(1 + {\mathcal O}(s - {s_0})),$$

(30)

$$\vert \sigma (s)\vert = {1 \over {2(s - {s_0})}}(1 + {\mathcal O}(s - {s_0})),$$

(31)

By contrast, for a caustic point of multiplicity two at s = s0 the equations read (cf. [303])

$$\theta (s) = {1 \over {s - {s_0}}} + {\mathcal O}(s - {s_0}),$$

(32)

$$\sigma (s) = {1 \over 3}{\psi _0}({s_0})(s - {s_0}) + {\mathcal O}({(s - {s_0})^2}).$$

(33)