Odd-Number-Theorem
As Odd -number theorem, the statement is called, stating that in a gravitational lens situation always occurs an odd number of images of a radiation source (eg a star ). Some of the theorem involves the claim that the number of images that are observed with the orientation of the source, the number of mirror images observed surpasses exactly by one.
The "odd -number theorem " is formulated in the literature under various conditions. These can be, for example quasi -Newtonian approximations or the requirement of special space-times. May also be additional implicit conditions are included on what to look for in a comparison of different formulations of the theorem in particular.
The evidence of the "odd -number theorem " use a variety of methods. In between Lorentz model, among others, Morse theory and arguments are used with the mapping degree, where - are considered different functions and variational principles - just as in quasi -Newtonian considerations.
It is often observed an even number of images
Some papers analyze possible reasons why, even in situations where the "odd -number theorem " applies, an even number is observed in multiple images. This is for example the case when the source is located behind the lens, images are too weak or multiple images can not be resolved. There are not only for existing real lens systems, the brightness of the images predicted and investigated possible superpositions of the images, but there are also general such forecasts employed. These can be found in 1998 Giannoni and Lombardi, which take into account the absorption of a quasi -Newtonian thin lens. For this purpose they use Morse theory, which has been developed by Giannoni, Masiello and Piccione, in 1995, Kovner and the principle also applies in the Lorentz model between.
In quasi -Newtonian considerations point-like lenses resulting conditions under which the image number is even. Such a statement is proved by Schneider, Ehlers and Falco S. 175 as a modification of its quasi -Newtonian formulation of the "odd -number theorem " for a thin, extended, transparent lens with finite mass in a plane and limited deflection angles. Petters also gets under typical quasi -Newtonian assumptions conditions for an even number of images.
Results so far
In pre-relativistic models that consider non-curved spaces, it follows that each radiation source is exactly one to watch. Nevertheless, assuming a curvature of space, the existence of multiple images is intuitive and clear. Also, the " odd -number theorem " is plausible (see McKenzie and Schneider, Ehlers, Falco ) and it requires closer consideration to find situations where it is not satisfied. ( Such a situation exists in non-transparent cosmic strings. ) It is not known whether the theorem can be proved with weaker than the present-day conditions. There are essentially two approaches to proof: using the Morse theory and the Lorentz geometry, the mapping degree.
Petters shows the "odd -number theorem " using Morse theory and by considering a quasi -Newtonian time difference function. It leads to the conclusion of its deliberations, in which the lens is transparent and non-singular.
Burke used 1980, a quasi -Newtonian argument, which uses the mapping degree. This he considers the difference between the two vector fields on the lens plane which are defined by the directions in which the source to the lens plane or the observer would be seen. The number of images seen by the observer of a source is equal to the number of zeros of this difference vector field. Is the angle of diffraction limited, then the difference vector field on the exterior of the lens plane is radial and the index theorem of Poincaré - Hopf delivers an odd number of images with n = n- 1.
Also with the mapping degree argues Lombardi in 1998 within the quasi -Newtonian model for non- thin lenses and without the space-time must be stationary.
McKenzie examined in 1984, the "odd -number theorem " as first lorentzsch. It applies Uhlenbecksche Morse theory in globally hyperbolic space-times. The consideration of his space-times must in addition to the global hyperbolicity meet strong requirements on the topology of the paths rooms which consist of certain paths within the space-time. ( These conditions he demands to do his morse theoretical considerations can. )
The previously mentioned theorems ( Schneider, Ehlers and Falco, by Petters, of Burke and of McKenzie ) include the statement that the number of images with the orientation of the source, the number of mirror images observed surpasses exactly by one.
Meanwhile, the occurrence of an odd number of images shown morse theory for globally hyperbolic space-times, if they are contractible meet ( otherwise occur an infinite number of images), and certain technical conditions. Since asymptotically simple and empty spacetimes are globally hyperbolic and contractible, this proof also applies to them. The proof can be tracked at Perlick. The Morse theory used in this case is supplied by Giannoni, Masiello and Piccione.
A Lorentzian geometric proof, which uses the mapping degree, Perlick delivers 2001. This is so far the most common evidence of those who use the mapping degree, and in this defined " simple lens- ligand environments " applies.
Global hyperbolic spaces are simply not necessary linsende environments, such as in asymptotic de Sitter space-times can be seen. Simply linsende environments are themselves generally not globally hyperbolic.
So far it could be shown either with the Lorentz geometric proof of Perlick, nor with the morse theory in globally hyperbolic space-times that the number of images with the orientation of the source, the number of mirror images observed surpasses exactly by one. The resulting proof attempts in this asymptotic simple and empty space-times of Perlick, and Kozameh, Lamberti and Reula are incomplete.
Concrete calculations of the number of images as well as the orientations of the images can be found for special space-times in many publications. These provide both examples where the conditions of the "odd -number theorem " are met, as well as examples where this is not the case and in which actually yields a straight image number. In connection with the methods considered here, we refer to concrete examples on the work of Perlick.