Rigorous results are given to the effect that a transparent gravitational lens produces an odd number of images. Suppose that p is an event and T the history of a light source in a globally hyperbolic space‐time (M,g). Uhlenbeck’s Morse theory of null geodesics is used to show under quite general conditions that if there are at most a finite number n of future‐directed null geodesics from T to p, then M is contractible to a point. Moreover, n is odd and 1/2 (n−1) of the images of the source seen by an observer at p have the opposite orientation to the source. An analogous result is noted for Riemannian manifolds with positive definite metric.
REFERENCES
1.
P.
Young
, J. E.
Gunn
, J.
Kristian
, J. B.
Oke
, and J. A.
Westphal
, Astrophys. J.
241
, 507
(1980
).2.
P.
Young
, R. S.
Deverill
, J. E.
Gunn
, J. A.
Westphal
, and J.
Kristian
, Astrophys. J.
224
, 723
(1981
).3.
D. W.
Weedman
, R. J.
Weymann
, R. F.
Green
, and T. M.
Heckman
, Astrophys. J. Lett.
255
, L5
(1982
).4.
C. R.
Lawrence
, D. P.
Schneider
, M.
Schmidt
, C. L.
Bennett
, J. N.
Hewitt
, B. F.
Burke
, E. L.
Turner
, and J. E.
Gunn
, Science
223
, 46
(1984
).5.
S. G. Djorgovski and H. Spinrad (to appear).
6.
7.
8.
V. Guillemin and A. Pollack, Differential Topology (Prentice‐Hall, Englewood Cliffs, NJ, 1974).
9.
10.
11.
J. Milnor, Morse Theory (Princeton U.P., Princeton, NJ, 1963).
12.
S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space‐Time (Cambridge U.P., London, 1973).
13.
The formal definition of a Cauchy surface is given in Ref. 12, p. 205.
14.
Ref. 10, Theorem 2, p. 71.
15.
R. H. McKenzie, B. Sc. (Honors) thesis, Australian National University, 1982, p. 30.
16.
Ref. 10, Theorem 4.5, p. 85.
17.
Ref. 10, Theorem 3, p. 73.
18.
J. F. Adams, Infinite Loop Spaces (Princeton U.P., Princeton, NJ, 1978), p. 8.
19.
Reference 15, p. 146.
20.
21.
S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces (Academic, New York, 1978), p. 59.
22.
Reference 11, p. 96.
23.
An intuitive explanation is given in Ref. 15, p. 153.
24.
25.
26.
P.
Young
, J. E.
Gunn
, J.
Kristian
, J. B.
Oke
, and J. A.
Westphal
, Astrophys. J.
244
, 736
(1981
).27.
J. E. Gunn, “Gravitational Lenses in Astrophysics,” in Tenth Texas Symposium on Relativistic Astrophysics, edited by R. Ramaty and F. C. Jones (N.Y. Acad. Sci., New York, 1981), p. 287.
28.
R. Geroch and G. T. Horowitz, “Global Structure of Spacetimes,” in General Relativity: An Einstein Centenary Survey, edited by S. W. Hawking and W. Israel (Cambridge U.P., London, 1979), p. 253.
29.
The formal definition is given in Ref. 28, p. 241.
30.
A proof is given in Ref. 12, p. 223.
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