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Finally let us mention that the Fermat's principle looks superficially like the stationary action principle for the geodesic equation. Two comments. The first is about terminology. Space-time mixes time and space; it is the 4-dimensional "dimension" that we live in. Anyway, you need to be careful in talking about "time" or "distance" in space-time. And if you don't understand Relativity either flavor, depending on the discussion , then you probably should avoid using either term.

The other comment is about the least time path. Light in a vacuum travels at c. Nothing can travel faster. Often SR space time diagrams are represented as graphs with one spatial dimension often the x-direction, but sometimes as a projection of all three space-like distances from the origin and time as the vertical dimension. These paths are not allowed.

Your scenario would require such a disallowed path. Firstly, in connexion with optics, Fermat's principle is always and approximation: it defines an approximation, namely the first term in the WKB Approximation to the solution of either the quasi-time-harmonic Maxwell's equations or, in a more general setting, the Helmholtz equation. Here the WKB scale parameter is the wavelength, which is taken to be small compared to the features of a medium in question.

As explained beautifully in CR Drost's Answer , the intuition behind this is that Fermat's principle defines paths of stationary phase around which diffraction effects are added to get the complete solution to the problem and is extremely widely applicable without approximation, i. Fermat's principle will give, without approximation, the first term in the relevant WKB expansion in all the situations discussed below.

These include all special relativistic flat spacetime problems and static general relativistic ones too. The extremized quantity, or the optical Lagrangian, is the optical path length expressed as a phase difference between the ends of the ray path - in waves or radians, for example.

A careful reading and mulling over CR Drost's answer clearly shows this fact. So it is not probably not helpful to think of it as least time principle since, as in Michael Siefert's answer this is can be problematic in relativity. In contrast, the phase field of a steady state optical excitation in a medium is a scalar field i. In Special Relativity , one can see how the principle plays out in two different, relatively boosted inertial frames by looking at the steady-state optical field in question from the two different frames at the instant when the origins of their co-ordinate systems co-incide.

Let's put one of the frames at rest relative to the material mediums that the field is established in. Fermat's principle plays out in the wonted way in this frame. The relatively moving observer sees the medium in Lorentz-transformed co-ordinates. Maxwell's equations are still Lorentz covariant with the medium present, but the medium properties and the constitutive relationships transform radically.

Intuitively you can see this is so; the Lorentz Fitzgerald contraction changes the medium's optical density anisotropically. The upshot of all of this is that both observers calculate the same scalar phase field from their version of Maxwell equations and so a ray path is an extremal optical path length path in one frame if and only if it is an extremal path in the other. So we see that the Fermat principle gives us the same rays in both cases. Phase fronts are not needfully normal to the Poynting vectors.

This is the same situation as in an anisotropic crystal. In General Relativity we must be a little careful. The optical Fermat principle applies time-invariant mediums. Therefore, it cannot be applied at least I am not aware of any extension to nonstatic spacetime - or at least one without a timelike Killing field - with or without material mediums. This is because, in the first instance, Fermat's principle applies to time-harmonic electromagnetic fields, with pulses and the like being described by Fourier superpositions of time-harmonic solutions;.

But for a static, curved spacetime, the situation is similar to the special relativistic one. In fact, an empty, medium-less curved spacetime has the constitutive relationships Plebanski's constitutive equations, see J. Plebanski Phys. This observation is the starting point for the field of transformational optics : the use of metamaterial mediums to mimic propagation in the spatially curved part of static curved spacetime. For lenses embedded in a cosmological model, see Pyne and Birkinshaw [ ] who consider lenses that need not be thin and may be moving on a Robertson-Walker background with positive, negative, or zero spatial curvature.

Here Minkowski spacetime is taken as the background, and again the lenses need not be thin and may be moving.

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Throughout this review we restrict to the case that the light rays are freely propagating in vacuum, i. 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. In an arbitrary spacetime , g , what an observer at an event p O can see is determined by the lightlike geodesics that issue from p O into the past.

Their union gives the past light cone of p O. This is the central geometric object for lensing from the spacetime perspective. Every such direction defines a unique up to parametrization lightlike geodesic through p O , so may also be viewed as a subset of the space of all lightlike geodesics in cf. If U O is changed, this representation changes according to the standard aberration formula of special relativity. Thus, the past light cone of p O is the image of the map.

If we restrict to values of s sufficiently close to 0, the map 2 is an embedding, i. 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.

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. These pictures show light cones in spacetimes with symmetries, so their structure is rather regular. Ellis, Bassett, and Dunsby [ 99 ] estimate that such a light cone would have at least 10 22 caustics which are hierarchically structured in a way that reminds of fractals.

Past light cone in the Schwarzschild spacetime. One sees that the light cone wraps around the horizon, then forms a tangential caustic. In the picture the caustic looks like a transverse self-intersection because one spatial dimension is suppressed. There is no radial caustic. If one follows the light rays further back in time, the light cone wraps around the horizon again and again, thereby forming infinitely many tangential caustics which alternately cover the radius line through the observer and the radius line opposite to the observer.

The z coordinate is not shown, the vertical coordinate is time t. The light cone has no caustic but a transverse self-intersection cut locus. Note that the light cone is not a closed subset of the spacetime. The light rays which were blocked by the string in the non-transparent case now form a caustic. Taking the z-dimension into account, the caustic actually consists of two lightlike 2-manifolds fold surfaces that meet in a spacelike curve cusp ridge.

Each of the past-oriented lightlike geodesics that form the caustic first passes through the cut locus transverse self-intersection , then smoothly slips over one of the fold surfaces. 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.

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. Some observables, e. The above-mentioned observables, like redshift and luminosity distance, are then uniquely determined as functions of the observational coordinates. 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 , or in the Hamiltonian formalism, with the Hamiltonian. A non-trivial example where the solutions can be explicitly written in terms of elementary functions is the string spacetime of Section 5. This is true, e. 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. As a substitute for the source plane we choose a 3-dimensional submanifold with a prescribed ruling by timelike curves. We assume that is globally of the form , where the points of the 2-manifold label the timelike curves by which is ruled. These timelike curves are to be interpreted as the worldlines of light sources. We call any such 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 to. Clearly, non-invertibility of the lens map indicates multiple imaging. What one chooses for depends on the situation. Also, the surface of an extended light source is a possible choice for. Illustration of the exact lens map. For each , one follows the lightlike geodesic with this initial direction until it meets and then projects to.

The exact lens map was introduced by Frittelli and Newman [ ] and further discussed in [ 91 , 90 ]. The following global aspects of the exact lens map were investigated in [ ]. First, in general the lens map is not defined on all of because not all past-oriented lightlike geodesics that start at p O necessarily meet. Second, in general the lens map is multi-valued because a lightlike geodesic might meet several times. Third, the lens map need not be differentiable and not even continuous because a lightlike geodesic might meet tangentially.

In [ ], the notion of a simple lensing neighborhood is introduced which translates the statement that a deflector is transparent into precise mathematical language.

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This result applies, as a special case, to asymptotically simple and empty spacetimes see Section 3. For expressing the exact lens map in coordinate language, it is recommendable to choose coordinates x 0 , x 1 , x 2 , x 3 such that the source surface is given by the equation , with a constant , and that the worldlines of the light sources are x 0 -lines. In this situation the remaining coordinates x 1 and x 2 label the light sources and the exact lens map takes the form. This is the way in which the lens map was written in the original paper by Frittelli and Newman; see Equation 6 in [ ].

The exact lens map can also be used for testing the reliability of approximation techniques. In [ ] the authors find that the standard quasi-Newtonian approximation formalism may lead to significant errors for lensing configurations with two lenses. For the case at hand i.

At each point of , choose a lightlike direction orthogonal to that depends smoothly on the foot-point. You have to choose between two possibilities. 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 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 [ ]. For a modern review article see, e. A coordinate representation for a wave front can be given with the help of local coordinates u 1 , u 1 on.

This gives the wave front as the image of a map. Orthogonality to of the initial vectors assures that this submanifold is lightlike. Farther away from , 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. For the case that the wave front is a light cone with vertex p O , caustic points are said to be conjugate to p O 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. Along each generator, caustic points are isolated see Section 2. Hence, one may speak of the first caustic, the second caustic, and so on.


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At all points where the caustic is a manifold, it is either spacelike or lightlike. 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. Full details of this theory can be found in [ 11 ]. A caustic point of the wave front corresponds to a point where the differential of the projection from the Legendrian submanifold to has non-maximal rank.

For the case , 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. 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. Wave fronts that are locally stable in the sense of Arnold. Each picture shows the projection into 3-space of a wave-front, locally near a caustic point.

The projection is made along the integral curves of a timelike vector field. The qualitative features are independent of which timelike vector field is chosen. In addition to regular, i. The A k and D k notation refers to a relation to exceptional groups see [ 11 ]. The picture is taken from [ ]. Fold singularities of a wave front form a lightlike 2-manifold in spacetime, on a sufficiently small neighborhood of any fold caustic point. 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.

Along a cusp ridge, two fold surfaces meet tangentially. 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. Friedrich and Stewart [ ] 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 [ ] where the Lagrangian rather than the Legendrian formalism was used. However, the main result of this paper Theorem 4. 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.

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. Such a foliation can always be achieved locally, and in several spacetimes of interest even globally.

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. In this article, the author views wave fronts as subsets of the space of all lightlike geodesics in. General properties of this space are derived in earlier articles by Low [ , ] also see Penrose and Rindler [ ], volume II, where the space is treated in twistor language. Low considers, in particular, the case of a globally hyperbolic spacetime [ ]; he demonstrates the desired stability result for the intersections of a big wave front with Cauchy hypersurfaces see Section 3. As every point in an arbitrary spacetime admits a globally hyperbolic neighborhood, this local stability result is universal.

Instantaneous wave fronts of the light cone in the Schwarzschild spacetime. The instantaneous wave fronts wrap around the horizon and, after reaching the first caustic, have two caustic points each. If one goes further back in time than shown in the picture, the wave fronts another time wrap around the horizon, reach the second caustic, and now have four caustic points each, and so on.

Only one half of each instantaneous wave front and of the monopole is shown. When the wave front passes the monopole, a hole is pierced into it, then a tangential caustic develops. The caustic of each instantaneous wave front is a point, the caustic of the entire light cone is a spacelike curve in spacetime which projects to part of the axis in 3-space.

For each instantaneous wave front, the radial caustic is a cusp ridge. The radial caustic of the entire light cone is a lightlike 2-surface in spacetime which projects to a rotationally symmetric 2-surface in 3-space. Only one half of each instantaneous wave front is shown so that one can look into its interior. There is a transverse self-intersection cut locus but no caustic. Instantaneous wave fronts that have passed through the string have a caustic, consisting of two cusp ridges that meet in a swallow-tail point.

This caustic is stable see Section 2. Clearly, a big wave front is a one-parameter family of small 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. 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 [ ].

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 ]. For a point source whose worldline passes exactly through the caustic, the ray-optical treatment even gives an infinite brightness see Section 2. 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. 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 is usually done in terms of the optical scalars which were introduced by Sachs et al. Related background material on lightlike geodesic congruences can be found in many text-books see, e. In view of applications to lensing, a particularly useful exposition was given by Seitz, Schneider and Ehlers [ ]. In the following the basic notions and results will be summarized. We use the summation convention for capital indices A , B , … taking the values 1 and 2.

As usual, R denotes the curvature tensor, defined by. Equation 9 is the Jacobi equation. For discussing the geometry of infinitesimally thin bundles it is usual to introduce a Sachs basis , i. Apart from the possibility to interchange them, E 1 and E 2 are unique up to transformations. 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. With respect to a Sachs basis, the basis vector fields Y 1 and Y 2 of an infinitesimally thin bundle can be represented as.

Equation 9 implies that D satisfies the matrix Jacobi equation. The notation in Equation 18 is chosen in agreement with the Newman-Penrose formalism cf. As Y 1 , Y 2 , 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. Cross-section of an infinitesimally thin bundle. The Jacobi matrix D can be parametrized according to. 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.

The most interesting case for us is that of an infinitesimally thin bundle that issues from a vertex at an observation event p O into the past.

Introduction

If the Sachs basis is transformed according to Equations 13 , 14 and Y 1 and Y 2 are kept fixed, the Jacobi matrix changes according to. This demonstrates the important fact that the shape and the size of the cross-section of an infinitesimally thin bundle has an invariant meaning [ ]. Along each infinitesimally thin bundle one defines the deformation matrix S by.

This reduces the second-order linear differential equation 16 for D to a first-order non-linear differential equation for S ,. A change 13 , 14 of the Sachs basis affects the optical scalars according to and. If rewritten in terms of the optical scalars, Equation 23 gives the Sachs equations. However, as the shear appears in Equation 25 , conformal curvature indirectly influences focusing cf.

Penrose [ ]. With D written in terms of the shape parameters and S written in terms of the optical scalars, Equation 22 results in. 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 ]. As the optical tidal matrix R is symmetric, for any two solutions D 1 and D 2 of the matrix Jacobi equation 16 we have.

Thus, there are two types of infinitesimally thin bundles: those for which this constant is non-zero and those for which it is zero. For the definition of wave fronts see Section 2. This notion coincides exactly with the notion of caustic points, or conjugate points, of wave fronts as introduced in Section 2. The behavior of the optical scalars near caustic points can be deduced from Equation 27 with Equations 25 , A vertex is, in particular, a caustic point of multiplicity two. An infinitesimally thin bundle with a vertex must be non-twisting.

While any non-twisting infinitesimally thin bundle can be embedded in a wave front, an infinitesimally thin bundle with a vertex can be embedded in a light cone. Near the vertex, it has a circular cross-section. If D 1 has a vertex at s 1 and D 2 has a vertex at s 2 , the conservation law 28 implies. The method by which this law was proven here follows Ellis [ 97 ] cf. Schneider, Ehlers, and Falco [ ]. It was independently rediscovered in the s by Sachs and Penrose see [ , ]. The results of this section are the basis for Sections 2.

In this section we summarize various distance measures that are defined in an arbitrary spacetime. Some of them are directly related to observable quantities with relevance for lensing. The material of this section makes use of the results on infinitesimally thin bundles which are summarized in Section 2.

Then the affine parameter s itself can be viewed as a distance measure. This affine distance has the desirable features that it increases monotonously along each ray and that it coincides in an infinitesimal neighborhood of p O with Euclidean distance in the rest system of U O. The affine distance depends on the 4-velocity U O of the observer but not on the 4-velocity U S of the light source.

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It is a mathematically very convenient notion, but it is not an observable. It can be operationally realized in terms of an observer field whose 4-velocities are parallel along the ray. Then the affine distance results by integration if each observer measures the length of an infinitesimally short part of the ray in his rest system. However, in view of astronomical situations this is a purely theoretical construction. The notion of affine distance was introduced by Kermack, McCrea, and Whittaker [ ].

As an alternative distance measure one can use the travel time. This requires the choice of a time function, i. Such a time function globally exists if and only if the spacetime is stably causal; see, e. The travel time increases monotonously along each ray. Clearly, it depends neither on the 4-velocity U O of the observer nor on the 4-velocity U S of the light source. The travel time is not directly observable. However, travel time differences are observable in multiple-imaging situations if the intrinsic luminosity of the light source is time-dependent.

To illustrate this, think of a light source that flashes at a particular instant. In view of applications, the measurement of time delays is of great relevance for quasar lensing. In cosmology it is common to use the redshift as a distance measure. The redshift z is defined as. This general redshift formula is due to Kermack, McCrea, and Whittaker [ ]. The same proof can be found, in a more elegant form, in [ 41 ] and in [ ], p.

Equation 37 is in agreement with the Hamilton formalism for photons. Clearly, the redshift depends on the 4-velocity U O of the observer and on the 4-velocity U S of the light source. If the redshift is known for one observer field U , it can be calculated for any other U , according to Equation 37 , just by adding the usual special-relativistic Doppler factors. This shows that z is a reasonable distance measure only for special situations, e.


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In any case, the redshift is directly observable if the light source emits identifiable spectral lines. For the calculation of Sagnac-like effects, the redshift formula 37 can be evaluated piecewise along broken lightlike geodesics [ 23 ]. The notion of angular diameter distance is based on the intuitive idea that the farther an object is away the smaller it looks, according to the rule. The formal definition needs the results of Section 2. They have the following physical meaning. Images with odd parity show the neighborhood of the light source side-inverted in comparison to images with even parity.

The area distance D area is defined according to the idea. D area s 2 indeed relates, for a bundle with vertex at the observer, the cross-sectional area at the source to the opening solid angle at the observer. The area distance is observable for a light source whose true size is known or can be reasonably estimated. It is sometimes convenient to introduce the magnification or amplification factor. Note that in a multiple-imaging situation the individual images may have different affine distances.

Thus, the relative magnification factor of two images is not directly observable. This is an important difference to the magnification factor that is used in the quasi-Newtonian approximation formalism of lensing. One can derive a differential equation for the area distance or, equivalently, for the magnification factor as a function of affine distance in the following way. Then Equations 44 , 45 imply that.

In Minkowski spacetime, Equation 46 holds with equality. Hence, Equation 46 says that the gravitational field has a focusing, as opposed to a defocusing, effect. This is sometimes called the focusing theorem. The idea of defining distance measures in terms of bundle cross-sections dates back to Tolman [ ] and Whittaker [ ].

Originally, this idea was applied not to bundles with vertex at the observer but rather to bundles with vertex at the light source. It relates, for a bundle with vertex at the light source, the cross-sectional area at the observer to the opening solid angle at the light source. The physical meaning of the corrected luminosity distance is most easily understood in the photon picture.

Thus, D lum is the relevant quantity for calculating the luminosity apparent brightness of point-like light sources see Equation For this reason D lum is called the uncorrected luminosity distance. D lum depends on the 4-velocity U O of the observer and of the 4-velocity U S of the light source. This definition follows [ ]. Its relevance in view of cosmology was discussed in detail by Rosquist [ ]. D par can be measured by performing the standard trigonometric parallax method of elementary Euclidean geometry, with the observer at p O and an assistant observer at the perimeter of the bundle, and then averaging over all possible positions of the assistant.

Note that the method refers to a bundle with vertex at the light source, i. Averaging is not necessary if this bundle is circular. D par depends on the 4-velocity of the observer but not on the 4-velocity of the light source. To within first-order approximation near the observer it coincides with affine distance recall Equation For the potential obervational relevance of D par see [ ], and [ ], p.

In spacetimes with many symmetries, these quantities can be explicitly calculated see Section 4. Following Kristian and Sachs [ ], one often uses series expansions with respect to s. For statistical considerations one may work with the focusing equation in a Friedmann-Robertson-Walker spacetime with average density see Section 4.

The latter leads to the so-called Dyer-Roeder distance [ 86 , 87 ] which is discussed in several text-books see, e. For pre-Dyer-Roeder papers on optics in cosmological models with inhomogeneities, see the historical notes in [ ]. As overdensities have a focusing and underdensities have a defocusing effect, it is widely believed following [ ] that after averaging over sufficiently large angular scales the Friedmann-Robertson-Walker calculation gives the correct distance-redshift relation. For a spherically symmetric inhomogeneity, the effect on the distance-redshift relation can be calculated analytically [ ].

For thorough discussions of light propagation in a clumpy universe also see Pyne and Birkinshaw [ ], and Holz and Wald [ ]. This distortion is caused by the shearing effect of the spacetime geometry on light bundles. For the calculation of image distortion we need the material of Sections 2. In the terminology of Section 2. The complex quantity. It is also common to use other measures for the eccentricity, e. The initial conditions imply. This transformation corresponds to replacing the Jacobi matrix D by its inverse.

The latter condition means that is tangent to a principal null direction of the conformal curvature tensor see, e. At a point where the conformal curvature tensor is not zero, there are at most four different principal null directions. Hence, the distortion effect vanishes along all light rays if and only if the conformal curvature vanishes everywhere, i. This result is due to Sachs [ ]. An alternative proof, based on expressions for image distortions in terms of the exponential map, was given by Hasse [ ].

Instead of s, one may choose any of the distance measures discussed in Section 2. In spacetimes with sufficiently many symmetries, this function can be explicitly determined in terms of integrals over the metric function. This will be worked out for spherically symmetric static spacetimes in Section 4. A general consideration of image distortion and example calculations can also be found in papers by Frittelli, Kling and Newman [ , ].

This technique, which is of particular relevance in view of cosmology, dates back to Kristian and Sachs [ ] who introduced image distortion as an observable in cosmology. As outlined in [ 56 ], these patterns are closely related to what Penrose and Rindler [ ] call the fingerprint of the Weyl tensor.

At all observation events where the Weyl tensor is non-zero, the following is true. Distortion pattern. The pattern indicates the elliptical images of spherical objects to within lowest non-trivial order with respect to distance. The length of each line segment is a measure for the eccentricity of the elliptical image, the direction of the line segment indicates its major axis.

Contrary to the other Petrov types, for type N the pattern is universal up to an overall scaling factor. The picture is taken from [ 56 ] where the distortion patterns for the other Petrov types are given as well. The distortion effect is routinely observed since the mids in the form of arcs and radio rings see [ , , ] for an overview. In these cases a distant galaxy appears strongly elongated in one direction.

In the case of rings and long arcs, the entire bundle cannot be treated as infinitesimally thin, i. The rings that are actually observed show extended sources in situations close to rotational symmetry. The method is based on the assumption that there is no prefered direction in the universe, i.

Any deviation from a random distribution is to be attributed to a distortion effect, produced by the gravitational field of intervening masses. With the help of the quasi-Newtonian approximation, this method has been elaborated into a sophisticated formalism for determining mass distributions, projected onto the plane perpendicular to the line of sight, from observed image distortions. This is one of the most important astrophysical tools for detecting dark matter.

It has been used to determine the mass distribution in galaxies and galaxy clusters, and more recently observations of image distortions produced by large-scale structure have begun see [ 22 ] for a detailed review. From a methodological point of view, it would be desirable to analyse this important line of astronomical research within a spacetime setting. This should give prominence to the role of the conformal curvature tensor. Another interesting way of observing weak image distortions is possible for sources that emit linearly polarized radiation. This is true for many radio galaxies.

Polarization measurements are also relevant for strong-lensing situations; see Schneider, Ehlers, and Falco [ ], p. In this approximation, the polarization vector is parallel along each ray between source and observer [ 88 ] cf. We may, thus, use the polarization vector as a realization of the Sachs basis vector E 1. If the light source is a spheroidal celestial body e. It is to be emphasized that the deviation of the polarization direction from the elongation axis is not the result of a rotation the bundles under consideration have a vertex and are, thus, twist-free but rather of successive shearing processes along the ray.

Also, the effect has nothing to do with the rotation of an observer field. It is a pure conformal curvature effect. Related misunderstandings have been clarified by Panov and Sbytov [ , ]. The distortion effect on the polarization plane has, so far, not been observed. Panov and Sbytov [ ] have clearly shown that an effect observed by Birch [ 31 ], even if real, cannot be attributed to distortion.

Its future detectability is estimated, for distant radio sources, in [ ]. For calculating the brightness of images we need the definitions and results of Section 2. In particular we need the luminosity distance D lum and its relation to other distance measures. We begin by considering a point source worldline that emits isotropically with bolometric, i.

By definition of D lum , in this case the energy flux at the observer is. The magnitude m used by astronomers is essentially the negative logarithm of F ,. In Equation 52 , D lum can be expresed in terms of the area distance D area and the redshift z with the help of the general relation D lum can be explicitly calculated in spacetimes where the Jacobi fields along lightlike geodesics can be explicitly determined.

The resulting formulas are given in Section 4. D area together with the redshift determines D lum via Equation Such an explicit calculation is, of course, possible only for spacetimes with many symmetries. By Equation 48 , the zeros of D lum coincide with the zeros of D area , i. A wave-optical treatment shows that the energy flux at the observer is actually bounded by diffraction. In the quasi-Newtonian approximation formalism, this was demonstrated by an explicit calculation for light rays deflected by a spheroidal mass by Ohanian [ ] cf.

Quite generally, the ray-optical calculation of the energy flux gives incorrect results if, for two different light paths from the source worldline to the observation event, the time delay is smaller than or approximately equal to the coherence time. According to Fermat's Principle, the path taken by a ray of light from one point to another is such that the time is minimal for slight perturbations of the path.

Therefore, if we define a metric in the x,y space such that the metrical "distance" between any two infinitesimally close points is proportional to the time required by a photon to travel from one point to the other, then the paths of photons in this space will correspond to the geodesics. Since the refractive index n is a smooth continuous function of x and y, it can be regarded as constant in a sufficiently small region surrounding any particular point x,y.

Here the coordinate time t also serves as the absolute path length parameter. If, instead of x and y, we name our two spatial coordinates x 1 and x 2 where these superscripts denote indices, not exponents we can express equation 2 in tensor notation as. Note that in equation 3 we have invoked the usual summation convention. The contravariant form of the metric tensor, denoted by g uv , is the matrix inverse of 4. According to Fermat's Principle, the path of a light ray must be a geodesic path based on this metric.

As discussed in Section 5. Based on the metric of our two-dimensional optical space we have the eight Christoffel symbols. Inserting these into 5 gives the equations for geodesic paths, which define the paths of light rays in this region. Reverting back to our original notation of x,y for our spatial coordinates, the differential equations for ray paths in this medium of continuously varying refractive index are. These are the equations of motion for light based on the temporal metric approach. Expanding the derivative on the right side gives. Substituting this into the previous equation and factoring gives.

A similar derivation shows that 1b is equivalent to the geodesic equation 6b , so the two sets of equations of motion for light rays are identical. With these equations we can compute the locus of rays emanating from any given point in a medium with arbitrarily varying index of refraction. Of course, if the index of refraction is constant then the right hand sides of equations 6 vanish and the equations for light rays reduce to. For a less trivial case, suppose the index of refraction in this region is a linear function of the x parameter, i. In this case the equations of motion reduce to.

Figure 1. The correctness of the rays in Figure 1 are easily verified by noting that in a medium with n varying only in the horizontal direction it follows immediately from Snell's law that the product n sin q must be constant, where q is the angle which the ray makes with the horizontal axis. We can verify numerically that the rays shown in Figure 1, generated by the geodesic equations, satisfy Snell's Law throughout.

It is, however, possible for the index of refraction of a medium to be less than 1 for certain frequencies, such as x-rays in glass. This implies that the velocity of light exceeds c, which may seem to conflict with relativity. However, the "velocity of light" that appears in the denominator of the refractive index is actually the phase velocity, rather than the group velocity, and the latter is typically the speed of energy transfer and signal propagation.

The phenomenon of "anomalous dispersion" can actually result in a group velocity greater than c, but in all cases the signal velocity is less than or equal to c. Incidentally, these ray lines, in a medium with linearly varying index of refraction, are called catenary curves, which is the shape made by a heavy cable slung between two attachment points in uniform gravity. The general form of a catenary curve with vertical axis of symmetry is.

Integrating this gives y as a function of s, so we have the parametric equations. We can verify that the catenary represents the path of a light ray in a medium whose index of refraction varies linearly as a function of y by inserting these expressions for x, y, and n and their derivatives into equations of motion 1.