The Kerr-star spacetime is the extension over the horizons and in the negative radial region of the Kerr spacetime. Despite the presence of closed timelike curves below the inner horizon, we prove that the timelike geodesics cannot be closed in the Kerr-star spacetime. Since the existence of closed null geodesics was ruled out by the author in Sanzeni [arXiv:2308.09631v3 (2024)], this result shows the absence of closed causal geodesics in the Kerr-star spacetime.

1.
Andrews
,
G. E.
,
Askey
,
R.
, and
Roy
,
R.
,“
Special functions
,” in
Encyclopedia of Mathematics and its Applications
(
Cambridge University Press
,
1999
).
2.
Boyer
,
R. H.
and
Lindquist
,
R. W.
, “
Maximal analytic extension of the Kerr metric
,”
J. Math. Phys.
8
(
2
),
265
281
(
1967
).
3.
Boyer
,
R. H.
and
Price
,
T. G.
, “
An interpretation of the kerr metric in general relativity
,”
Math. Proc. Cambridge Philos. Soc.
61
(
2
),
531
534
(
1965
).
4.
Carter
,
B.
, “
Complete analytic extension of the symmetry axis of Kerr’s solution of Einstein’s equations
,”
Phys. Rev.
141
,
1242
1247
(
1966
).
5.
Carter
,
B.
, “
Global structure of the Kerr family of gravitational fields
,”
Phys. Rev.
174
,
1559
1571
(
1968
).
6.
Chandrasekhar
,
S.
,
The Mathematical Theory of Black Holes
(
Oxford: Clarendon Press
,
1983
).
7.
Chandrasekhar
,
S.
and
Wright
,
J. P.
, “
The geodesics in Gödel’s universe
,”
Proc. Natl. Acad. Sci. U. S. A.
47
(
3
),
341
347
(
1961
).
8.
Chruściel
,
P.
,
Maliborski
,
M.
, and
Yunes
,
N.
, “
Structure of the singular ring in Kerr-like metrics
,”
Phys. Rev. D
101
,
104048
(
2020
).
9.
de Felice
,
F.
, “
Equatorial geodesic motion in the gravitational field of a rotating source
,”
Nuovo Cimento B
57
,
351
(
1968
).
10.
Gödel
,
K.
, “
An example of a new type of cosmological solutions of Einstein’s field equations of gravitation
,”
Rev. Mod. Phys.
21
,
447
450
(
1949
).
11.
Kapec
,
D.
and
Lupsasca
,
A.
, “
Particle motion near high-spin black holes
,”
Classical Quantum Gravity
37
(
1
),
015006
(
2019
).
12.
Kerr
,
R. P.
, “
Gravitational field of a spinning mass as an example of algebraically special metrics
,”
Phys. Rev. Lett.
11
,
237
238
(
1963
).
13.
Kundt
,
W.
, “
Trägheitsbahnen in einem von Gödel angegebenen kosmologischen modell
,”
Z. Phys.
145
,
611
620
(
1956
).
14.
Nolan
,
B.
, “
Causality violation without time-travel: Closed lightlike paths in Gödel’s universe
,”
Classical Quantum Gravity
37
(
8
),
085007
(
2020
).
15.
O’Neill
,
B.
,
The Geometry of Kerr Black Holes
,
Ak Peters Series
(
Taylor & Francis
,
1995
).
16.
Sanzeni
,
G.
, “
Non existence of closed and bounded null geodesics in Kerr spacetimes
,” arXiv:2308.09631v3 (
2024
).
17.
Schwarzschild
,
K.
,“
Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie
,” in
Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften
(
Berlin: Deutsche Akademie der Wissenschaften zu Berlin
,
1916
), pp.
189
196
.
18.
Teo
,
E.
, “
Spherical photon orbits around a Kerr black hole
,”
Gen. Relativ. Gravitation
35
,
1909
1926
(
2003
).
19.
Walker
,
M.
and
Penrose
,
R.
, “
On quadratic first integrals of the geodesic equations for type {22} spacetimes
,”
Commun. Math. Phys.
18
,
265
274
(
1970
).
20.
Wilkins
,
D.
, “
Bound geodesics in the Kerr metric
,”
Phys. Rev. D
5
,
814
822
(
1972
).
You do not currently have access to this content.