The odd number theorem is a theorem in strong gravitational lensing which comes directly from differential topology .
The theorem states that the number of multiple images produced by a bounded transparent lens must be odd .
The gravitational lensing is a thought to mapped from what's known as image plane to source plane following the formula :
M
:
(
u
,
v
)
↦
(
u
′
,
v
′
)
{\displaystyle M:(u,v)\mapsto (u',v')}
Argument
If we use direction cosines describing the bent light rays , we can write a vector field on
(
u
,
v
)
{\displaystyle (u,v)}
V
:
(
s
,
w
)
{\displaystyle V:(s,w)}
However, only in some specific directions
V
0
:
(
s
0
,
w
0
)
{\displaystyle V_{0}:(s_{0},w_{0})}
D
=
δ
V
=
0
|
(
s
0
,
w
0
)
{\displaystyle D=\delta V=0|_{(s_{0},w_{0})}}
Poincaré–Hopf theorem
χ
=
∑
index
D
=
constant
{\displaystyle \chi =\sum {\text{index}}_{D}={\text{constant}}}
The index of sources and sinks is +1, and that of saddle points is −1. So the Euler characteristic equals the difference between the number of positive indices
n
+
{\displaystyle n_{+}}
n
−
{\displaystyle n_{-}}
χ
=
n
+
−
n
−
=
1
{\displaystyle \chi =n_{+}-n_{-}=1}
N
=
n
+
+
n
−
=
2
n
−
+
1
{\displaystyle N=n_{+}+n_{-}=2n_{-}+1}
Morse theory of null geodesics .
References
Chwolson, O. (1924). "Über eine mögliche Form fiktiver Doppelsterne". Astronomische Nachrichten (in German). 221 (20). Wiley: 329– 330. Bibcode :1924AN....221..329C . doi :10.1002/asna.19242212003 . ISSN 0004-6337 . Burke, W. L. (1981). "Multiple Gravitational Imaging by Distributed Masses" . The Astrophysical Journal . 244 . IOP Publishing: L1. Bibcode :1981ApJ...244L...1B . doi :10.1086/183466 ISSN 0004-637X . McKenzie, Ross H. (1985). "A gravitational lens produces an odd number of images". Journal of Mathematical Physics . 26 (7). AIP Publishing: 1592– 1596. Bibcode :1985JMP....26.1592M . doi :10.1063/1.526923 . ISSN 0022-2488 . Kozameh, Carlos; Lamberti, Pedro W.; Reula, Oscar (1991). "Global aspects of light cone cuts". Journal of Mathematical Physics . 32 (12). AIP Publishing: 3423– 3426. Bibcode :1991JMP....32.3423K . doi :10.1063/1.529456 . ISSN 0022-2488 . Lombardi, Marco (1998-01-20). "An application of the topological degree to gravitational lenses". Modern Physics Letters A . 13 (2). World Scientific Pub Co Pte Lt: 83– 86. Bibcode :1998MPLA...13...83L . doi :10.1142/s0217732398000115 . ISSN 0217-7323 . Wambsganss, Joachim (1998). "Gravitational Lensing in Astronomy" . Living Reviews in Relativity . 1 (1): 12. arXiv :astro-ph/9812021 Bibcode :1998LRR.....1...12W . doi :10.12942/lrr-1998-12 PMC 5567250 PMID 28937183 . Schneider, P.; Ehlers, J.; Falco, E. E. (1999). Gravitational Lenses" . Astronomy and Astrophysics Library. Springer. ISBN 9783540665069 Giannoni, Fabio; Lombardi, Marco (1999). "Gravitational lenses: Odd or even images?". Classical and Quantum Gravity . 16 (6): 1689– 1694. Bibcode :1999CQGra..16.1689G . doi :10.1088/0264-9381/16/6/303 . S2CID 250827307 . Frittelli, Simonetta; Newman, Ezra T. (1999-04-28). "Exact universal gravitational lensing equation". Physical Review D . 59 (12): 124001. arXiv :gr-qc/9810017 Bibcode :1999PhRvD..59l4001F . doi :10.1103/physrevd.59.124001 . ISSN 0556-2821 . S2CID 248125 . Perlick, Volker (1999). "Gravitational Lensing from a Geometric Viewpoint". Einstein's Field Equations and Their Physical Implications: Selected Essays in Honour of Jürgen Ehlers . Lecture Notes in Physics. Vol. 540. pp. 373– 425. doi :10.1007/3-540-46580-4_6 . ISBN 978-3-540-67073-5 Perlick, Volker (September 2004). "Gravitational lensing from a spacetime perspective" . Living Reviews in Relativity . 7 (1): 9. arXiv :1010.3416 Bibcode :2004LRR.....7....9P . doi :10.12942/lrr-2004-9 PMC 5255571 PMID 28179867 .
