The first observation of a ‘gravitational
lensing’ effect was made when the deflection of star light by our
Sun was verified during a Solar eclipse in 1919. Today, the list of
observed phenomena includes the following:
Multiple
quasars.
The gravitational field of a galaxy (or a cluster of galaxies)
bends the light from a distant quasar in such a way that the
observer on Earth sees two or more images of the quasar.
Rings.
An extended light source, like a galaxy or a lobe of a galaxy, is
distorted into a closed or almost closed ring by the gravitational
field of an intervening galaxy. This phenomenon occurs in
situations where the gravitational field is almost rotationally
symmetric, with observer and light source close to the axis of
symmetry. It is observed primarily, but not exclusively, in the
radio range.
Arcs.
Distant galaxies are distorted into arcs by the gravitational field
of an intervening cluster of galaxies. Here the situation is less
symmetric than in the case of rings. The effect is observed in the
optical range and may produce “giant luminous arcs”, typically of a
characteristic blue color.
Microlensing.
When a light source passes behind a compact mass, the focusing
effect on the light leads to a temporal change in brightness
(energy flux). This microlensing effect is routinely observed since
the early 1990s by monitoring a large number of stars in the bulge
of our Galaxy, in the Magellanic Clouds and in the Andromeda
galaxy. Microlensing has also been observed on quasars.
Image distortion by weak
lensing.
In cases where the distortion effect on galaxies is too weak for
producing rings or arcs, it can be verified with statistical
methods. By evaluating the shape of a large number of background
galaxies in the field of a galaxy cluster, one can determine the
surface mass density of the cluster. By evaluating fields without a
foreground cluster one gets information about the large-scale mass
distribution.
Observational aspects of gravitational lensing
and methods of how to use lensing as a tool in astrophysics are the
subject of the Living Review by Wambsganss [343]. There the reader
may also find some notes on the history of lensing.
The present review is meant as complementary to
the review by Wambsganss. While all the theoretical methods
reviewed in [343] rely on
quasi-Newtonian approximations, the present review is devoted to
the theory of gravitational lensing from a spaectime perspective,
without such approximations. Here the terminology is as follows:
“Lensing from a spacetime perspective” means that light propagation
is described in terms of lightlike geodesics of a
general-relativistic spacetime metric, without further
approximations. (The term “non-perturbative lensing” is sometimes
used in the same sense.) “Quasi-Newtonian approximation” means that
the general-relativistic spacetime formalism is reduced by
approximative assumptions to essentially Newtonian terms (Newtonian
space, Newtonian time, Newtonian gravitational field). The
quasi-Newtonian approximation formalism of lensing comes in several
variants, and the relation to the exact formalism is not always
evident because sometimes plausibility and ad-hoc assumptions are
implicitly made. A common feature of all variants is that they are
“weak-field approximations” in the sense that the spacetime metric
is decomposed into a background (“spacetime without the lens”) and
a small perturbation of this background (“gravitational field of
the lens”). For the background one usually chooses either Minkowski
spacetime (isolated lens) or a spatially flat Robertson-Walker
spacetime (lens embedded in a cosmological model). The background
then defines a Euclidean 3-space, similar to Newtonian space, and
the gravitational field of the lens is similar to a Newtonian
gravitational field on this Euclidean 3-space. Treating the lens as
a small perturbation of the background means that the gravitational
field of the lens is weak and causes only a small deviation of the
light rays from the straight lines in Euclidean 3-space. In its
most traditional version, the formalism assumes in addition that
the lens is “thin”, and that the lens and the light sources are at
rest in Euclidean 3-space, but there are also variants for “thick”
and moving lenses. Also, modifications for a spatially curved
Robertson-Walker background exist, but in all variants a
non-trivial topological or causal structure of spacetime is
(explicitly or implicitly) excluded. At the center of the
quasi-Newtonian formalism is a “lens equation” or “lens map”, which
relates the position of a “lensed image” to the position of the
corresponding “unlensed image”. In the most traditional version one
considers a thin lens at rest, modeled by a Newtonian gravitational
potential given on a plane in Euclidean 3-space (“lens plane”). The
light rays are taken to be straight lines in Euclidean 3-space
except for a sharp bend at the lens plane. For a fixed observer and
light sources distributed on a plane parallel to the lens plane
(“source plane”), the lens map is then a map from the lens plane to
the source plane. In this way, the geometric spacetime setting of
general relativity is completely covered behind a curtain of
approximations, and one is left simply with a map from a plane to a
plane. Details of the quasi-Newtonian approximation formalism can
be found not only in the above-mentioned Living Review [343
], but also in the
monographs of Schneider, Ehlers, and Falco [299
] and Petters,
Levine, and Wambsganss [275
].
The quasi-Newtonian approximation formalism has
proven very successful for using gravitational lensing as a tool in
astrophysics. This is impressively demonstrated by the work
reviewed in [343]. On the other hand,
studying lensing from a spacetime perspective is of relevance under
three aspects:
Didactical.
The theoretical foundations of lensing can be properly formulated
only in terms of the full formalism of general relativity. Working
out examples with strong curvature and with non-trivial causal or
topological structure demonstrates that, in principle, lensing
situations can be much more complicated than suggested by the
quasi-Newtonian formalism.
Methodological.
General theorems on lensing (e.g., criteria for multiple imaging,
characterizations of caustics, etc.) should be formulated within
the exact spacetime setting of general relativity, if possible, to
make sure that they are not just an artifact of approximative
assumptions. For those results which do not hold in arbitrary
spacetimes, one should try to find the precise conditions on the
spacetime under which they are true.
Practical.
There are some situations of astrophysical interest to which the
quasi-Newtonian formalism does not apply. For instance, near a
black hole light rays are so strongly bent that, in principle, they
can make arbitrarily many turns around the hole. Clearly, in this
situation it is impossible to use the quasi-Newtonian formalism
which would treat these light rays as small perturbations of
straight lines.
The present review tries to elucidate all three aspects. More precisely, the following subjects will be covered:
This introduction ends with some notes on
subjects not covered in this
review:
Wave optics.
In the electromagnetic theory, light is described by wavelike
solutions to Maxwell’s equations. The ray-optical treatment used
throughout this review is the standard high-frequency approximation
(geometric optics approximation) of the electromagnetic theory for
light propagation in vacuum on a general-relativistic spacetime
(see, e.g., [225], § 22.5 or [299
], Section 3.2).
(Other notions of vacuum light rays, based on a different
approximation procedure, have been occasionally
suggested [217], but will
not be considered here. Also, results specific to spacetime
dimensions other than four or to gravitational theories other than
Einstein’s are not covered.) For most applications to lensing the
ray-optical treatment is valid and appropriate. An exception, where
wave-optical corrections are necessary, is the calculation of the
brightness of images if a light source comes very close to the
caustic of the observer’s light cone (see Section 2.6).
Light propagation in
matter.
If light is directly influenced by a medium, the light rays are no
longer the lightlike geodesics of the spacetime metric. For an
isotropic non-dispersive medium, they are the lightlike geodesics
of another metric which is again of Lorentzian signature. (This
“optical metric” was introduced by Gordon [143]. For a
rigourous derivation, starting from Maxwell’s equation in an
isotropic non-dispersive medium, see Ehlers [88].) Hence, the
formalism used throughout this review still applies to this
situation after an appropriate re-interpretation of the metric. In
anisotropic or dispersive media, however, the light rays are not
the lightlike geodesics of a Lorentzian metric. There are some
lensing situations where the influence of matter has to be taken
into account. For instance., for the deflection of radio signals by
our Sun the influence of the plasma in the Solar corona (to be
treated as a dispersive medium) is very well measurable. However,
such situations will not be considered in this review. For light
propagation in media on a general-relativistic spacetime,
see [269] and
references cited therein.
Kinetic
theory.
As an alternative to the (geometric optics approximation of)
electromagnetic theory, light can be treated as a photon gas, using
the formalism of kinetic theory. This has relevance, e.g., for the
cosmic background radiation. For basic notions of
general-relativistic kinetic theory see, e.g., [89]. Apart from some
occasional remarks, kinetic theory will not be considered in this
review.
Derivation of the
quasi-Newtonian formalism.
It is not satisfacory if the quasi-Newtonian formalism of lensing
is set up with the help of ad-hoc assumptions, even if the latter
look plausible. From a methodological point of view, it is more
desirable to start from the exact spacetime setting of general
relativity and to derive the quasi-Newtonian lens equation by a
well-defined approximation procedure. In comparison to earlier such
derivations [299, 294, 303
] more recent effort
has led to considerable improvements. For lenses embedded in a
cosmological model, see Pyne and Birkinshaw [283
] who consider lenses
that need not be thin and may be moving on a Robertson-Walker
background (with positive, negative, or zero spatial curvature).
For the non-cosmological situation, a Lorentz covariant
approximation formalism was derived by Kopeikin and
Schäfer [185]. Here Minkowski
spacetime is taken as the background, and again the lenses need not
be thin and may be moving.