Odd number theorem

The title of this article is ambiguous. Further meanings are listed under Odd Number Theorem (disambiguation) .

Schematic explanation of the gravitational lensing effect. The light from the red galaxy in the background is deflected by the gravity of the white galaxy in the foreground.

The odd number theorem describes an effect when observing celestial bodies. It says that in a gravitational lens situation there is always an odd number of images of a radiation source (for example a star). In part, the theorem includes the assertion that the number of images observed with the orientation of the source exceeds the number of mirror-inverted images by exactly one.

The “odd number theorem” is formulated in the literature under various conditions. These can be, for example, quasi-Newtonian approximation assumptions or the prerequisite for special spacetime . Often, implicit conditions are also included, which is particularly important when comparing different formulations of the theorem.

The proofs of the “odd number theorem” use different methods. In the Lorentzian model , among other things, Morse theory and arguments about the degree of mapping are used, whereby - just like in quasi-Newtonian considerations - different functions and variation principles are considered.

An even number of images is often observed

Multiple images of the quasar QSO 2237 + 0305, known as the Einstein Cross, created by gravitational lensing .

Some papers analyze possible reasons why even in situations where the "odd number theorem" holds, an even number of multiple images is observed. This is the case, for example, when the source is behind the lens, images are too weak or several images cannot be resolved. The brightness of the images is not only predicted for existing real lens systems and possible superimpositions of the images are examined, but general such prognoses are also made. Such can be found in 1998 by Giannoni and Lombardi, who take into account the absorption of a quasi-Newtonian thin lens. To do this, they use Morse code that has been developed by Giannoni, Masiello and Piccione since 1995 with Kovner's principle and that also applies to Lorentz's model.

In quasi-Newtonian considerations of point lenses, conditions arise under which the number of images is even. Such a statement is proven by Schneider, Ehlers and Falco p. 175 as a modification of their quasi-Newtonian formulation of the "Odd Number Theorem" for a thin, extended, transparent lens with finite mass in one plane and limited deflection angles. Under also typical quasi-Newtonian assumptions, Petters receives conditions for an even number of images.

Results so far

In pre-relativistic models, which consider uncurved spaces, it follows that each radiation source can be observed exactly once. Nevertheless, assuming a spatial curvature, the existence of multiple images is intuitive and clear. The "odd number theorem" is also plausible (see McKenzie and Schneider, Ehlers , Falco) and more detailed considerations are required in order to find situations in which it is not fulfilled. (Such a situation exists with opaque cosmic strings .) It is not known whether the theorem can be proven under weaker than today's conditions. There are essentially two approaches to proof: the Morse theoretical and the Lorentz geometric, which uses degrees of mapping.

Petters shows the "odd number theorem" using Morse code and considering a quasi-Newtonian time difference function. It arises as a consequence of his considerations that the lens is transparent and not singular.

In 1980 Burke used a quasi-Newtonian argument that made use of the degree of mapping. To do this, he considers the difference between the two vector fields on the lens plane, which are given by the directions in which the source or the observer would be seen on the lens plane. The number of images of a source seen by the observer is equal to the number of zeros of this difference vector field. If the angle of diffraction is limited, the difference vector field on the outside of the lens plane is radial and the Poincaré-Hopf index theorem delivers an odd number of images with n ₊ = n _- +1.

Lombardi also argued with the degree of mapping in 1998 within the quasi-Newtonian model for non-thin lenses and without space-time having to be stationary. Stationary means that there is a time-like, future - oriented killing vector field everywhere .

In 1984 McKenzie was the first Lorentzsch to investigate the "odd number theorem". He uses Uhlenbeck's morse theory in global hyperbolic spacetime. In addition to global hyperbolicity , the space- times considered by him must meet strong requirements for the topology of the path spaces , which consist of certain paths within space-time. (He demands these prerequisites in order to be able to make his Morseoretical considerations.)

The theorems mentioned so far (by Schneider, Ehlers and Falco, von Petters, von Burke and von McKenzie) contain the statement that the number of images with the orientation of the source exceeds the number of mirror-inverted images by exactly one.

In the meantime, the occurrence of an uneven number of images has been shown in Morse theory for global hyperbolic spacetime, provided that these can be contracted (otherwise an infinite number of images will result) and meet certain technical conditions. Since asymptotically simple and empty spacetimes are globally hyperbolic and contractible, this proof also applies to them. The proof can be traced back to Perlick. The Morse theory used here is supplied by Giannoni, Masiello and Piccione.

Perlick 2001 provides a Lorentz geometric proof that uses the degree of mapping. This is the most general proof of those who use the degree of mapping to date, and is valid in "simple lensing environments" defined for this purpose.

Globally hyperbolic spaces are not necessarily simply lensing environments, as can be seen from asymptotic de-sitter spacetimes. Single lens environments, in turn, are generally not globally hyperbolic.

So far it has not been possible to show, neither with Perlick's Lorentz geometric proof nor with Morse’s theoretical in global hyperbolic spacetime, that the number of images with the orientation of the source exceeds the number of mirror-inverted images by exactly one. The attempts to prove this in asymptotic simple and empty spacetime by Perlick and by Kozameh, Lamberti and Reula are incomplete.

Concrete calculations of the number of images as well as the orientation of the images can be found for special spacetime in many publications. These provide examples in which the requirements of the "odd number theorem" are met, as well as examples in which this is not the case and in which there is actually an even number of images. In connection with the methods considered here, reference is made to the work of Perlick for specific examples.

literature

• F. Giannoni, M. Lombardi: Gravitational lenses: odd or even images? In: Classical and Quantum Gravity . tape 16 , no. 6 , 1999, p. 1689-1694 , doi : 10.1088 / 0264-9381 / 16/6/303 .
• F. Giannoni, A. Masiello, P. Piccone: A Variational Theory for Light Rays in Stably Causal Lorentzian Manifolds: Regularity and Multiplicity Results . In: Communications in Mathematical Physics . tape 187 , no. 2 , 1997, p. 375-415 , doi : 10.1007 / s002200050141 .
• F. Giannoni, A. Masiello, P. Piccione: A Morse Theory for Light Rays on Stably Causal Lorentzian Manifolds . In: Annales de l'institut Henri Poincaré (A) Physique théorique . tape 69 , no. 4 , 1998, pp. 359-412 ( abstract and PDF ).
• I. Kovner: Fermat principle in arbitrary gravitational fields . In: Astrophysical Journal, Part 1 . tape 351 , 1990, pp. 114-120 , doi : 10.1086 / 168450 .
• P. Schneider, J. Ehlers, EE Falco: Gravitational Lenses . Springer, 1992, ISBN 978-3-662-03758-4 ( limited preview at Springer ).
• AO Petters: Morse theory and gravitational microlensing . In: Journal of Mathematical Physics . tape 33 , no. 5 , 1992, pp. 1915–1931 , doi : 10.1063 / 1.529667 ( abstract and PDF ).
• WL Burke: Multiple Gravitational Imaging by Distributed Masses . In: The Astrophysical Journal . tape 244 , 1981, pp. L1 , doi : 10.1086 / 183466 ( abstract and PDF ).
• M. Lombardi: An Application of the Topological Degree to Gravitational Lenses . In: Modern Phys. Lett . A, no. 13 , 1998, pp. 83–86 ( PDF from the Gravity Research Foundation ).
• Ross H. McKenzie: A gravitational lens produces an odd number of images . In: Journal of Mathematical Physics . tape 26 , 1985, pp. 1592-1596 , doi : 10.1063 / 1.526923 .
• K. Uhlenbeck: A Morse theory for geodesics on a Lorentz manifold . In: Topology . tape 14 , no. 1 , 1975, p. 69-90 , doi : 10.1016 / 0040-9383 (75) 90037-3 .
• Volker Perlick: Gravitational Lensing from a Geometric Viewpoint . In: BG Schmidt (Ed.): Einstein's Field Equations and Their Physical Implications . Lecture Notes in Physics, No. 540 . Springer, 2000, ISBN 3-540-67073-4 , pp. 373-425 , doi : 10.1007 / 3-540-46580-4_6 (Selected Essays in Honor of Jürgen Ehlers).
• Volker Perlick: Global properties of gravitational lens maps in a Lorentzian manifold setting . In: Commun. Math. Phys . tape 220 , 2001, p. 403-428 , doi : 10.1007 / s002200100450 ( abstract and PDF ).
• Volker Perlick: Gravitational lensing in asymptotically simple and empty spacetimes . In: Annals of Physics . tape 9 , 2000, pp. SI-139-142 .
• C. Kozameh, PW Lamberti, O. Reula: Global aspects of light cone cuts . In: Journal of Mathematical Physics . tape 32 , no. 12 , 1991, pp. 3423-3426 ( abstract ).

Individual evidence

1. ↑ Helmut Fischer, Helmut Kaul: Mathematics for Physicists Volume 3: Calculus of Variations - Differential Geometry - Mathematical Foundations of General Relativity . Springer-Verlag, 2017, ISBN 3-662-53969-1 , pp. 344 ( limited preview in Google Book search).