We give a classification of the type D space–times based on the invariant differential properties of the Weyl principal structure. Our classification is established using tensorial invariants of the Weyl tensor and, consequently, besides its intrinsic nature, it is valid for the whole set of the type D metrics and it applies on both, vacuum and nonvacuum solutions. We consider the Cotton-zero type D metrics and we study the classes that are compatible with this condition. The subfamily of space–times with constant argument of the Weyl eigenvalue is analyzed in more detail by offering a canonical expression for the metric tensor and by giving a generalization of some results about the nonexistence of purely magnetic solutions. The usefulness of these results is illustrated in characterizing and classifying a family of Einstein–Maxwell solutions. Our approach permits us to give intrinsic and explicit conditions that label every metric, obtaining in this way an operational algorithm to detect them. In particular a characterization of the Reissner–Nordström metric is accomplished.

1.
M.
Trümper
,
J. Math. Phys.
6
,
584
(
1965
).
2.
J. Ehlers and W. Kundt, in Gravitation: An Introduction to Current Research, edited by L. Witten (Wiley, New York, 1962).
3.
A.
Barnes
,
Gen. Relativ. Gravit.
4
,
105
(
1973
).
4.
G. S.
Hall
,
J. Phys. A
6
,
619
(
1973
).
5.
C. B. G.
McIntosh
,
R.
Arianrhod
,
S. T.
Wade
, and
C.
Hoenselaers
,
Class. Quantum Grav.
11
,
1555
(
1994
).
6.
J. J.
Ferrando
and
J. A.
Sáez
,
Class. Quantum Grav.
19
,
2437
(
2002
).
7.
A. M.
Naveira
,
Rend. Circ. Mat. Palermo
3
,
577
(
1983
).
8.
O.
Gil-Medrano
,
Rend. Circ. Mat. Palermo
32
,
315
(
1983
).
9.
B. Coll and J. J. Ferrando, Almost-Product Structures in Relativity in Recent Developments in Gravitation, Proceeding of the “ Relativistic Meeting-89” (World Scientific, Singapore, 1990), p. 338.
10.
B.
Coll
and
J. J.
Ferrando
,
Gen. Relativ. Gravit.
21
,
1159
(
1989
).
11.
B.
Coll
and
J. J.
Ferrando
,
J. Math. Phys.
31
,
1020
(
1990
).
12.
G. Y.
Rainich
,
Trans. Am. Math. Soc.
27
,
106
(
1925
).
13.
C. W.
Misner
and
J. A.
Wheeler
,
Ann. Phys. (N.Y.)
2
,
525
(
1957
).
14.
B.
Coll
,
F.
Fayos
, and
J. J.
Ferrando
,
J. Math. Phys.
28
,
1075
(
1987
).
15.
S. A.
Teukolsky
and
W. H.
Press
,
Astrophys. J.
193
,
443
(
1974
).
16.
L.
Bel
,
Cah. de Phys.
16
,
59
(
1962
)
L.
Bel
, [Engl. Transl.
Gen. Relativ. Gravit.
32
,
2047
(
2000
)].
17.
J. J.
Ferrando
,
J. A.
Morales
, and
J. A.
Sáez
,
Class. Quantum Grav.
18
,
4939
(
2001
).
18.
J. J.
Ferrando
and
J. A.
Sáez
,
Class. Quantum Grav.
15
,
1323
(
1998
).
19.
J. J. Ferrando and J. A. Sáez, On typeDspacetimes in Some topics on General Relativity and gravitational radiation, Proceedings of the Spanish Relativity Meeting–96 (Frontieres, Paris, 1997), p. 209.
20.
W.
Kinnersley
,
J. Math. Phys.
10
,
1195
(
1969
).
21.
J. J. Ferrando and J. A. Sáez (in preparation).
22.
J. J. Ferrando and J. A. Sáez, Class. Quantum Grav. (submitted), gr-qc/031189.
23.
B.
Reinhart
,
J. Diff. Geom.
12
,
619
(
1977
).
24.
A.
Montesinos
,
Mich. Math. J.
33
,
31
(
1983
).
25.
D. Kramer, H. Stephani, M. MacCallum, and E. Herlt, Exact Solutions of Einstein’s field equations (VEB, Berlin, 1980).
26.
M.
Mars
,
Class. Quantum Grav.
16
,
3245
(
1999
).
27.
P.
Szekeres
,
Commun. Math. Phys.
41
,
55
(
1975
).
28.
C. B.
Collins
,
J. Math. Phys.
25
,
995
(
1984
).
29.
C.
Lozanovski
and
M.
Aarons
,
Class. Quantum Grav.
16
,
4075
(
1999
).
30.
A.
Barnes
and
R. R.
Rowlingson
,
Class. Quantum Grav.
6
,
949
(
1989
).
31.
J. J.
Ferrando
and
J. A.
Sáez
,
Gen. Relativ. Gravit.
35
,
1191
(
2003
).
32.
J. J.
Ferrando
and
J. A.
Sáez
,
Class. Quantum Grav.
20
,
2835
(
2003
).
33.
N.
Van der Bergh
,
Class. Quantum Grav.
20
,
L165
(
2003
).
34.
J. F.
Plebański
and
S.
Hacyan
,
J. Math. Phys.
20
,
1004
(
1979
).
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