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On the classification of type D space–times
Joan Josep Ferrandoa) Departament d’Astronomia i Astrofı´sica, Universitat de Valencia, E-46100 Burjassot, Valencia, Spain Juan Antonio Sa´ ezb) Departament de Matematica Economico-Empresarial, Universitat de Valencia, E-46071 Valencia, Spain ~Received 5 November 2002; accepted 12 November 2003! We give a classification of the type D space–times based on the invariant differen- tial properties of the Weyl principal structure. Our classification is established using tensorial invariants of the Weyl tensor and, consequently, besides its intrinsic na- ture, 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– Nordstro¨m metric is accomplished. © 2004 American Institute of Physics. @DOI: 10.1063/1.1640795#
I. INTRODUCTION
Type D space–times have been widely considered in literature and we can point out not only the large number of known families of exact solutions but also the interest of these solutions from the physical point of view. Let us quote, for example, the Schwarszchild or the Kerr metrics which model the exterior gravitational field produced, respectively, by a nonrotating or a rotating spheri- cally symmetric bounded object. Or also the related metrics in the case of a charged object, the Reissner–Nordstro¨m or the Kerr–Newman solutions. However, although some classes of type D metrics have been considered taking into account algebraic properties of the Weyl eigenvalue or differential conditions on the null Weyl principal directions, a classification of the type D solutions involving all the first-order differential properties of the Weyl tensor geometry is a task which has not been totally accomplished yet. In this work we present this classification of the type D metrics and we show the role that it can play in studying geometric properties of known space–times, in looking for new solutions of Einstein equations or in offering new elements which allow us to give intrinsic and explicit characterizations of all these space–times. At an algebraic level, a type D Weyl tensor determines a complex scalar invariant, the eigen- value, and a 2 1 2 almost-product structure defined by its principal 2–planes. Some classes of type D metrics can be considered by imposing the real or imaginary nature of the Weyl eigenvalue. In this way we find the so-called purely electric or purely magnetic space–times. The purely electric character often appears as a consequence of usual geometric or physical restrictions.^1 This is the case of the static type D vacuum spacetimes found by Ehlers and Kundt,^2 or the Barnes degenerate perfect fluid solutions with shear-free normal flow.^3 On the other hand, some restrictions are
a!Electronic mail: [email protected] b!Electronic mail: [email protected]
JOURNAL OF MATHEMATICAL PHYSICS VOLUME 45, NUMBER 2 FEBRUARY 2004
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known on the existence of purely magnetic solutions.4,5^ A wide bibliography about Weyl-electric and Weyl-magnetic space–times can be found in a recent work^6 where these concepts have been generalized. The most usual approaches to look for exact solutions of the Einstein equations work in frames or local coordinates adapted to some outlined direction of the curvature tensor. For ex- ample, in the case of perfect fluid solutions or static metrics the 3 1 1 formalism adapted, respec- tively, to the fluid flow or to the normal timelike Killing vector can be useful. Sometimes one considers that some of the kinematic coefficients associated with the unitary vector are zero. This means that one is searching for new solutions belonging to a class of metrics that are defined by first-order differential conditions imposed on the curvature tensor. A similar situation appears when local coordinates adapted to the multiple Debever direction are considered when looking for algebraically special solutions. Indeed, if the hypotheses of the generalized Goldberg–Sachs theo- rem hold, the multiple Debever direction defines a shear-free geodesic null congruence. In this case, or when considering nondiverging or nontwisting restrictions on a Debever direction, we are imposing differential conditions on the Weyl tensor. It is worth pointing out that the kinematic coefficients associated with a unitary vector com- pletely determine the first-order differential properties of the 1 1 3 almost-product structure that it defines. Nevertheless, the conditions usually imposed on the two double Debever directions of a type D space–time do not cover all the differential properties of the principal 2 1 2 almost-product structure of the Weyl tensor exhaustively. The first goal of this work is to offer a classification of the type D metrics based on all the first-order differential properties of the principal structure, and to reinterpret under this view the usual conditions that can be found in the literature. This classi- fication is not based on the scalar invariants, but on tensorial invariants of the Weyl tensor. These invariants are well adapted to the generic type D metrics, where a Weyl canonical frame is not univocally determined, and where the eigenvalues and the 2 1 2 principal structure are the only invariants associated with the Weyl tensor. The ~proper! Riemannian almost-product structures have been classified according the invari- ant decomposition of their structure tensor,^7 and the classes have been interpreted in terms of the foliation, minimal and umbilical properties.^8 This classification can be generalized to the space– time structures by also considering the causal character of the planes.^9 Almost-product structures have shown their usefulness in studying the underlying geometry of some physical fields. The 1 1 3 structures are frequently used in relativity and sometimes the properties of a physical field can be expressed in terms of the kinematic properties of a unitary vector.10,11^ On the other hand, the 21 2 structure associated with a regular solution of Maxwell equations^12 is a basic concept in building the ‘‘already unified theory’’ for the electromagnetic field.^13 It has also allowed a geo- metric interpretation^14 of the Teukolsky–Press relations^15 used in analyzing incident electromag- netic waves on a Kerr black hole. In General Relativity we can also find almost-product structures attached to the geometric or physical properties of the spacetime. Indeed, some energy contents ~for example, in the Einstein– Maxwell or perfect fluid solutions! define underlying structures that restrict, via Einstein equa- tions, the Ricci tensor. On the other hand, the Weyl tensor also defines almost-product structures associated with its principal bivectors depending on the different Petrov types.^16 These structures determine the Weyl canonical frames.^17 In the type D case, only the principal structure is outlined. Until now we have mentioned two different ways of classifying the type D space–times: The first one is strictly algebraic and takes into account the real or imaginary character of the Weyl eigenvalues; the second one, which we will present here, involves differential conditions of the 21 2 principal structure, that is, on the Weyl eigenvectors. Nevertheless, there is a third natural manner to impose restrictions on the type D metrics: To take into account the relative position between the principal 2–planes and the gradient of the Weyl scalar invariants. This is a mixed classification, differential in the eigenvalues and algebraic in the principal structure, which affords 16 different classes of type D metrics. In this work we will show the marked relation that exists between this classification and the two previous ones. A classification of type D space–times taking into account the properties of the 2 1 2 principal
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Qv ~ x , y! 5 h ~ π vxvy! , ; x , y. ~ 1!
Let us consider the invariant decomposition of Qv into its antisymmetric part Av and its symmetric part Sv [ Sv^ T 1 (1/ p ) v ^ Tr Sv , where Sv^ T^ is a traceless tensor:
Qv 5 Av 1
p v ^ Tr Sv 1 Sv^ T^. ~ 2!
The plane V is foliation if, and only if, Av 5 0. In this case Qv 5 Sv and it coincides with the second fundamental form of the integral manifolds of the foliation V. 23 Moreover V is minimal, umbilical or geodesic if, and only if, Tr Sv 5 0, Sv^ T 5 0 or Sv 5 0, respectively. Then one can generalize these geometric concepts for plane fields which are not necessarily foliation: Definition 1 : A plane field V is said to be geodesic, umbilical or minimal if the symmetric part Sv of its (generalized) second fundamental form Qv satisfies, respectively , Sv 5 0, Sv^ T 50 or Tr Sv 5 0. The ~proper! Riemannian almost-product structures ( V , H ) have been classified taking into account the invariant decomposition ~ 2! of the tensors Qv and Q (^) h or, equivalently, according with the foliation, minimal, umbilical, or geodesic character of each plane.7,8^ Some of these properties have also been interpreted in terms of invariance along vector fields.^24 A generalization for the spacetime structures follows taking into account the causal character of the planes. We will say that a structure is integrable when both planes are foliation and we will say that it is minimal, umbilical or geodesic if both of the planes are so. This way, on an oriented space–time ( V 4 , g ) of signature ( 2111 ) we have generically 2 65 64 different classes of ~almost-product! structures depending on the first-order geometric properties. Nevertheless, when p 5 1, V is always an umbilical foliation and, consequently, only 16 possible classes exist. In this case Qv and Q (^) h depend on the kinematic coefficients associated with a unitary vector u , and the classes are defined by the vanishing or nonvanishing of the accelera- tion, rotation, shear, and expansion. Elsewhere this kinematical interpretation has been extended to the 2 1 2 space–time structures and, as a consequence, the Maxwell–Rainich equations have been expressed in terms of kinematical variables.^9 In order to be used in next sections, we now analyze the space–time 2 1 2 almost-product structures in detail by giving the characterization of their properties in terms of their canonical 2–form U , and by showing their relation with other usual approaches, the Newmann–Penrose and the self-dual formalisms. We also study the change of these properties for a conformal transfor- mation and we summarize some results about Maxwellian structures.
A. 2¿2 structures
In the case of a 2 1 2 space–time structure it is useful to work with the canonical unitary 2-form U , volume element of the time-like plane V. Then, the respective projectors are (^) v 5 U^2 and h 52 ( (^) * U ) 2 , where U^25 U 3 U 5 Tr 23 U ^ U and (^) * is the Hodge dual operator. The tensors Qv and Q (^) h determine the derivatives of the volume elements U and (^) * U by means of
πa U bl (^5) ~ Qv !am,[ b U^ ml] (^1) ~ Q (^) h !a[ b,^ m^ U l] m ,
πa* U bl (^5) ~ Q (^) h !am,[ b* U^ ml] (^1) ~ Qv !a[ b,^ m* U l] m. ~ 3!
Then, if we denote d 52 Tr π, a straightforward calculation leads to
d U 5 i ~Tr S h! U 22 ~ U , Av! d* U 5 i ~Tr Sv !* U 22 ~* U , A h !, ~ 4!
where 2( U , Av )^ m 5 U^ ab( Av ) abm^. So, the minimal and the foliation character of the planes can be
stated in terms of the projections of d U and d* U onto V and H. On the other hand, let us consider
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G ' 5 U ^ U 2 * U ^ * U 1 G ; h' 5 U ^ ˜^ * U 1 h, ~ 5!
Then, from expressions ~ 3! and ~ 4! we get
~ 2 π U 2 K! (^) lab 5 ~ Sv^ T^! (^) lm,[ a U^ mb] 1 ~ S (^) hT^! (^) lm,[ a * U^ mb] , ~ 7!
K [ i ~d U! G ' 2 i ~d* U !h' , ~ 8!
and so, the umbilicity of each plane is equivalent to the vanishing of the respective projections of the first member of ~ 7 !. We summarize these results in the following lemma: Lemma 1 : Let ( V , H ) be a 212 almost-product structure and let U be its canonical 2-form. Then, the following conditions hold :
(1) V ( resp. H ) is f oliation ⇔ i ( d U ) * U 5 0 ( resp. i ( d* U ) U 5 0);
(2) V ( resp. H ) is minimal ⇔ i ( d* U ) * U 5 0 ( resp. i ( d U ) U 5 0);
(3) V is umbilical ⇔ U 3 $ 2 π U 2 @ i ( d U ) G ' 2 i ( d* U ) h'# % 50
H is umbilical ⇔* U 3 $ 2 π U 2 @ i ( d U ) G ' 2 i ( d* U ) h'# % 5 0.
A 2 1 2 structure is also determined by the two null directions l 6 on the plane V. A family of
complex null bases $ l 1 , l 2 , m , m¯ % exists such that U 5 l 2 ∧ l 1. This family is fixed up to change
l 6 Å e^6 f l 6 , m Å e i^ u m. Then, conditions of lemma 1 can be interpreted in terms of the Newman– Penrose coefficients^25 as Lemma 2 : Let U 5 l 2 ∧ l 1 be the canonical 2 – form of a 212 structure. It holds :
(1) The plane V is umbilical iff k 505 n; (2) the plane H is umbilical iff l 505 s; (3) the plane V is minimal iff p ¯ 5 t; (4) the plane H is minimal iff r 1 ¯ r 505 m 1 m ¯ ; (5) the plane V is a foliation iff p ¯ 52 t; (6) the plane H is a foliation iff r 2 r ¯ 505 m 2 m ¯.
Taking into account the significance of the NP coefficients^25 this lemma implies that the umbilical nature of a 2 1 2 structure means that its principal directions l 6 define shear-free geodesic null congruences. The minimal or foliation character of the spacelike 2-plane have also a kinematical interpretation and state, respectively, that both principal directions are expansion-free or vorticity- free. Elsewhere^9 all the geometric properties have been interpreted in terms of kinematic coeffi- cients associated with every direction in a 2–plane ~not only the null ones! with respect to the other 2–plane. When both planes have a specific differential property, it is more convenient to introduce the self-dual unitary 2–form U[ (1/&) ( U 2 i* U ) associated with U. We have
2 Re@ i ~dU!U# 5 i ~d U! U 2 i ~d* U !* U [F ~ U !,
2 Im@ i ~dU!U# 52 i ~d U !* U 2 i ~d* U! U [C ~ U !. ~ 9!
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C. Maxwellian structures
A regular 2-form F takes the canonical expression F 5 e^ f^ @cos c U 1 sin c* U #, where U defines the 2 1 2 associated structure, f is the energetic index and c is the Rainich index. When F is
solution of the source-free Maxwell equations, d F 5 0, d* F 5 0, one says that U defines a Max-
wellian structure. In terms of the canonical elements ( U , f, c), Maxwell equations become:12,
df 5 F ~ U! [ i ~d U! U 2 i ~d* U !* U , ~ 14!
d c 5 C ~ U ! [ 2 i ~d U !* U 2 i ~d* U ! U. ~ 15!
Then, from ~ 14! and ~ 15! the Rainich theorem^12 follows: Lemma 7 : A unitary 2-form U defines a Maxwellian structure if, and only if, it satisfies :
dF (^) ~ U! 5 0; dC (^) ~ U! 5 0. ~ 16!
The Maxwell–Rainich equations ~ 14! and ~ 15! have a simple expression in the self-dual formal- ism. Indeed, the self-dual 2 –form F 5 (1/&) ( F 2 i* F ) may be written as F 5 e^ f^1 i^ c^ U. Then,
from Maxwell equations, dF 5 0, and taking into account that 2 U 25 g ,
d~f 1 i c! 52 i ~dU!U. ~ 17!
This last equation is equivalent to ~ 14! and ~ 15! if we take into account ~ 9 !. Moreover, from here we recover the complex version of ~ 16! easily
d i ~dU!U 5 0. ~ 18!
III. CLASSIFYING TYPE D SPACE–TIMES
The self–dual Weyl tensor W 5 12 ( W 2 i* W ) of a type D space–time takes the canonical expression^17
W 53 aU^ U 1 aG, ~ 19!
where a 52 Tr_W 3 /Tr_W 2 is the double eigenvalue and U is the self-dual principal 2–form. This principal 2–form defines a 2 1 2 almost-product structure which is called the principal structure of a type D space–time. In terms of the canonical 2–form U of the principal structure the self-dual 2–form U becomes U 5 (1/&) ( U 2 i* U ). So, at the algebraic level, a type D Weyl tensor only determines the complex scalar a and the principal structure U. Consequently, any generic classi- fication of the type D metrics must depend on these invariants associated with the Weyl tensor. The families of purely electric or purely magnetic type D spacetimes are defined, at first glance, by means of alternative conditions, namely, the nullity of the magnetic or the electric Weyl fields associated with an observer u. But, actually, they admit a simple intrinsic characterization in terms of the Weyl scalar invariant: The eigenvalue is real or imaginary.^5 In spite of these strong conditions, the family of Weyl-electric type D space–times contains quite interesting solutions. We can quote, for example, the static vacuum metrics^2 or the degenerate perfect fluids with shear-free normal flow.^3 All the type D silent universes are also known26,27^ as well as other families of purely electric type D perfect fluid solutions.28,30^ Nevertheless, few Weyl-magnetic type D solutions have been found,^29 and some restrictions about their existence are known. Indeed, there are not vacuum metrics with purely magnetic type D Weyl tensor.4,5^ The classification that we present below allows us to give an extension of this result in Sec. 5. On the other hand, the generalization of the purely electric or magnetic concepts to the spacelike or null directions does not afford new classes in the type D case.^6 But the purely electric or magnetic properties define very narrow subsets of the generic type D metrics because they impose one of the two real scalar invariants to be zero. The large family of known solutions of the Einstein equation recommends us to consider other classifications, based
658 J. Math. Phys., Vol. 45, No. 2, February 2004 J. J. Ferrando and J. A. Sa´ ez
on less restrictive properties, which afford new intrinsic elements that increase the knowledge of the metrics and permit their explicit characterization. Besides the intrinsic nature, the classification must be generic , that is, valid for the whole set of the type D metrics. Consequently, it will be independent of the energy content and it will have to be built on the intrinsic geometry associated with a type D Weyl tensor. The first classification that we propose is based on the geometric properties of the principal 2–planes, that is, it is induced by the geometric classification of the principal structure. Every principal 2–plane can be submitted or not to three properties, so 2 65 64 classes can be considered. Definition 2 : Taking into account the foliation, minimal, or umbilical character of each prin- cipal 2 – plane we distinguish 64 different classes of type D space – times. We denote the classes as D (^) lmnpqr^ , where the superscripts p , q , r take the value 0 if the time-like principal plane is, respectively, a foliation, a minimal or an umbilical distribution, and they take the value 1 otherwise. In the same way, the subscripts l , m , n collect the foliation, minimal or umbilical nature of the space-like plane. The most degenerated class that we can consider is D 000000 which corresponds to a type D product metric, and the most regular one is D 111111 which means that neither V nor H are foliation, minimal or umbilical distributions. We will put a dot in place of a fixed script ~1 or 0! to indicate the set of metrics that cover both possibilities. So, for example, the metrics of type D 11111 •^ are the union of the classes D 111111 and D 110111 ; or a metric is of type D (^0) • • •^ • •^ if the timelike 2–plane is a foliation. Taking into account lemma 1, every class is defined by means of first-order differential equations imposed on the canonical 2–form U. On the other hand, U can be written explicitly in terms of the Weyl tensor^17 and, consequently, every class admits an intrinsic and explicit charac- terization. The above classification depends on the derivatives of the principal 2–form U. An alternative classification at first order in the Weyl eigenvalues can also be considered by taking into account the four 1-forms defined by the principal 2–planes and the gradient of the modulus and the argument of the eigenvalue. So, we will have 2 45 16 classes.
Definition 3 : Let a 5 e
3 2 (^ r^1 i^ u)^ be the Weyl eigenvalue. Taking into account the relative
position between the gradients d u, d r and each principal 2 – plane we distinguish 16 different classes of type D space–times. We denote the classes D @ pq , rs # where p , q , r , s take the values 0 or 1 to indicate, respectively, that one of the 1-forms U ( d u), U ( d r), (^) * U ( d u), (^) * U ( d r) is zero or nonzero. The most degenerated class D@00;00# is occupied by the type D metrics with constant eigen- values, and the most general one D@11;11# by those type D space–times for which both, the modulus and the argument of the Weyl eigenvalue, have nonzero projection onto the principal planes. As above, a dot means that a condition is not fixed. So, for example, we write D@ 0 • ; • •# to indicate the type D metrics for which the argument of the eigenvalues have zero projection onto the timelike principal 2–plane. The type D metrics with constant modulus, d r 5 0, correspond to the classes D @ • 0; • (^0) #, and those with constant argument, d u 5 0, are the metrics of type D @ 0 • ;0 • #. This last family contains the Weyl-electric and the Weyl-magnetic space–times because a real or imaginary eigenvalue means that the argument takes the constant value 0, p or p/2, 3 p/2, respectively. In the next section we will show the marked relation between the two classifications given in definitions 2 and 3 when some usual restrictions are imposed on the Ricci tensor.
IV. TYPE D METRICS WITH ZERO COTTON TENSOR
The space–time Cotton tensor P is a vector valued 2–form which depends on the Ricci tensor as
P mn, b [π[ m Q n] b , 2 Q [ Ric 2 16 ~Tr Ric! g. ~ 20!
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V. SOME RESULTS ABOUT TYPE D „ M … 00 "" 00 SPACE–TIMES
Now, in this section, we restrict our study to the type D metrics with Maxwellian, integrable and umbilical structure, that is, those of type D~M! 00 • • 00. We can easily obtain a canonical form for these metrics. Indeed, lemma 6 states that the metric is conformal to a product one with a conformal factor determined by the potential of the closed 1–form F( U ). More precisely, the metric can be written as
g 5
V^2
@s AB^2 ~ x C^! dx A^ dx B 1 s i j^1 ~ x k^! dx i^ dx j^ #, ~ 25!
where V satisfies
2 d ln V 5 F ~ U! [ i ~d U! U 2 i ~d* U !* U. ~ 26!
Conversely, we can analyze the Petrov type of the metric ~ 25! by studying a product metric g˜ 5 s^2 1 s^1. Let X 2 and X 1 be the Gaussian curvatures of the arbitrary bidimensional metrics, s^2 and s^1 , hyperbolic and elliptic, respectively. The Gauss–Codazzi equations show that the Riemann and the Ricci tensors of ˜g are
~ 27!
So, the Weyl tensor of a product metric is Petrov-type O precisely when X 2 1 X 1 5 0, and then both curvatures are constant. On the other hand, when X 2 1 X 1 fi0, the space–time is type D. Moreover U determines the principal structure and the double eigenvalue is given by
a ˜ 52 16 ~ X 2 1 X 1 !. ~ 28!
So, we have Lemma 8 : Every 212 product metric s^2 1 s^1 is of type D ~ or O ) with real eigenvalues, and the double eigenvalue is given by (28), where X 2 and X 1 are the Gaussian curvatures of s^2 and s^1 , respectively. Moreover, it is of type O if, and only if , X 2 52 X 1 5 constant. A conformal transformation ˜g 5 V^2 g preserves the Petrov type and the Weyl eigenvalues change as a ˜ 5 V^22 a. Consequently, from Eq. ~ 26! and taking into account lemmas 1 and 8, we can conclude: Proposition 3 : A space – time is of type D ( M ) (^00) • • 00 if, and only if, there exist local coordinates such that the metric g takes the expression (25) with X 2 1 X 1 fi0, where X 2 and X 1 are the Gaussian curvatures of s^2 and s^1 , respectively. Moreover, it is of class D 010010 , D 010000 , D 000010 , or D 000000 if, and only if , s^2 ( d V)fi 0 fis^1 ( d V), s^1 ( d V) 50 fis^2 ( d V), s^1 ( d V)fi 05 s^2 ( d V), or d V 5 0, respectively. Furthermore, taking into account the expressions ~ 27! for the Ricci and ~ 28! for the eigenvalue of a product metric, and considering the change of these metric concomitants for a conformal transformation, we can state: Proposition 4 : The Weyl eigenvalue of the metric (25) is real and it is given by
a 52 16 V^2 ~ X 2 1 X 1 !. ~ 29!
The Ricci tensor of this metric is
Ric ~ g! 5
V
π d V 1 X 2 s^2 1 X 1 s^1 1 F
V
DV 2
V^2
˜g ~ d V, d V !G ˜g , ~ 30!
where π 5 πs 2 1 πs 1 is the connection of the product metric g˜ 5 s^2 1 s^1.
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Let us consider metrics with zero Cotton tensor again. If they have a constant argument, theorem 1 implies that the principal structure is integrable and so, the space–times are of type D~M! 00 • • 00. Consequently, from proposition 4 the Weyl tensor has real eigenvalues. So we can state: Theorem 2 : The Weyl eigenvalues of a type D space – time with zero Cotton tensor have constant argument if, and only if, they are real. This result generalizes a previous one by Hall^4 ~see also McIntosh et al.^5 !. He showed that there are no purely magnetic Type D vacuum metrics. But the purely magnetic case occurs when the eigenvalue argument is 32 u 56 p/2 , that is to say, a particular value of constant argument. So, from theorem 2 it follows: Corollary 1 : There is no purely magnetic Type D metric with zero Cotton tensor. This corollary shows that not only the purely magnetic vacuum solutions are forbidden, but also the Weyl-magnetic space–times with zero Cotton tensor. On the other hand the Hall result is also generalized in the sense that theorem 2 excludes all the constant arguments that differ from 0 or p. Although this approach could be of interest in studying the existence of purely magnetic type I solutions, the recent results on this subject have been obtained by using the 113 formalism.22,32, From the results above it is easy to recover the canonical form for the metrics with zero Cotton tensor and real Weyl eigenvalues. Indeed, expressions ~ 23! and ~ 26! show that the confor- mal factor and the Weyl eigenvalue are related by V^25 c^2 e^ r 5 c^2 a2/3, c being an arbitrary con- stant. On the other hand they also satisfy expression ~ 29! and, consequently, V coincides, up to a constant factor, with X 2 1 X 1. So we have Proposition 5 : Every type D metric with real eigenvalues and zero Cotton tensor may be written
g 5
~ X 2 1 X 1!^2
~s^2 1 s^1 !,
where s^2 5 s AB^2 ( x C ) dx A^ dx B , s^1 5 s i j^1 ( x k ) dx i^ dx j , are two arbitrary bidimensional metrics , s^2 hyperbolic and s^1 elliptic, with Gaussian curvatures X 2 and X 1 , respectively. This canonical expression was obtained in a previous work^18 where it was used to integrate the Einstein vacuum equations, in this way getting an intrinsic algorithm to identify every A, B, and C-metric of Ehlers and Kundt.^2 In the following section, starting from the propositions 3 and 4 we present a similar study for the charged counterpart of these vacuum solutions.
VI. ALIGNED EINSTEIN–MAXWELL SOLUTIONS OF TYPE D (^00) "" 00
If ( v , h ) is the principal structure of the Weyl tensor, the aligned Einstein–Maxwell solutions satisfy
Ric ~ g! 5 x ~ v 2 h! 5 k ~s^2 2 s^1 !, ~ 31!
where the second equality is satisfied for the type D 00 • • 00 metrics as a consequence of proposition 3: x 5 kV^2 , s^2 5 V^2 v , s^1 5 V^2 h. Moreover, as the principal structure is integrable, it is Maxwell- ian and the associated Rainich index is a constant. So, ~ 31! is a necessary and sufficient condition for the metric ~ 25! to be an aligned solution of the Einstein–Maxwell equations. Taking into account the expression ~ 30! for the Ricci tensor, condition ~ 31! becomes
V 5 l 2 ~ x A^! 1 l 1 ~ x i^ !, ~ 32!
πdle 5 b (^) ese, ~ 33!
V^2 6 ~ X 2 1 X 1! 1 V ~b 2 1 b 1! 5 s^2 ~dl 2 ,dl 2! 1 s^1 ~dl 1 ,dl 1 !, ~ 34!
d b (^) e 1 X edle 5 0. ~ 35!
662 J. Math. Phys., Vol. 45, No. 2, February 2004 J. J. Ferrando and J. A. Sa´ ez
tions that we have given in previous sections could be written explicitly in terms of metric concomitants because U can be determined from the Weyl tensor.^17 Nevertheless, as a consequence of the Bianchi identities some of the above conditions can be satisfied identically taking into account the properties of the Ricci tensor. This is the case of vacuum metrics: As Ric 5 0 implies the nullity of the Cotton tensor, the principal planes always define an umbilical and Maxwellian structure as a consequence of the results in Sec. IV. Actually we want to characterize aligned Einstein–Maxwell solutions that are conformal to a product metric. So, the Weyl tensor must have real eigenvalues and the principal planes are the eigenspaces of the Ricci tensor, that is,
W 53 a ~ U ^ U (^2) * U ^ (^) * U! 1 a G , Ric 5 x ~ v 2 h !, ~ 40!
where v 5 U^2 , h (^52) * U^2. Then, taking into account the expressions in Sec. II about 2 1 2 almost- product structures, a straightforward calculation shows that the Bianchi identities ~ 21! can be written
~ 3 a 12 x! Qv 5 v ^ h ~ d a !; ~ 3 a 22 x! Q (^) h 5 h ^ v ~ d a !, ~ 41!
v ~ d x! 22 x i ~d U! U 5 0; h ~ d x! 12 x i ~d* U !* U 5 0. ~ 42!
From these expressions we find that, under the scalar restriction (3 a) 2 fi(2 x) 2 , the properties of the structure follow just by imposing that the Weyl and the Ricci tensor take expressions ~ 40 !. On the other hand, the case (3 a) 25 (2 x) 2 leads to the exceptional metrics considered by Pleban´ski and Hacyan.^34 Nevertheless, it can easily be shown that (3 a) 2 fi(2 x) 2 for the solutions recovered in the subsection above. So we get the following characterization: Lemma 10 : The charged counterpart of the A , B , and C - metrics are the only aligned Einstein – Maxwell solutions of type D with real eigenvalues that satisfy (3 a) 2 fi(2 x) 2 , a and x being, respectively, the Weyl and the Ricci eigenvalues. Elsewhere,^18 conditions for g to be of type D with real eigenvalues have been given in terms of Weyl concomitants. In order to impose the Ricci tensor to take the form ~ 40! we can use the algebraic Rainich conditions.^12 But if the Weyl tensor is of type D with real eigenvalues, a part of these Rainich conditions hold identically when we impose the aligned restriction. From these considerations and lemma 10 we have: Theorem 3 : The A , B , and C Einstein – Maxwell solutions can be characterized by conditions
afi0; S^21 S 5 0; Ric ~ x , x! >0,
Tr Ric 5 0, S @ Ric # 1 Ric 5 0; ~ 3 a!^22 ~ 2 x!^2 fi0.
This theorem offers an intrinsic and explicit description of the aligned Einstein–Maxwell solutions of type D 00 • • 00. Now we look for an intrinsic and explicit way to identify every metric of this family, that is, to distinguish the A (^) i , B (^) i , and C charged metrics. In a first step we must discriminate between the classes D 00 mp^00 and, as a consequence of proposition 3, this depends on the nullity of the vectors v ( d V) and h ( d V). But the expression ~ 29! for the Weyl eigenvalue and lemma 9 imply that, equivalently, the vectors v ( d a) and h ( d a) determine these properties. So, the same scheme as in the vacuum case^18 can be used to distinguish between the classes. The last step to obtain the intrinsic and explicit characterization of the solutions is to get an invariant that provides the sign of the bidimensional curvature when this is constant. A straight- forward calculation shows that if X e is constant, then
X e V^25 ve [ 19 ~d ln~a 1 x !!^222 a 2 ex. ~ 43!
664 J. Math. Phys., Vol. 45, No. 2, February 2004 J. J. Ferrando and J. A. Sa´ ez
So, we have a characterization of the Einstein–Maxwell A , B , and C -metrics, and we recover the type D static vacuum solutions making x 5 0. Theorem 4 : Let g be an aligned Einstein – Maxwell solution of type D 00 • • 00 (characterized in theorem 3). Let us take the metric concomitants
M [* W ~ d a,•, d a,•! N [ S ~ d a,•, d a,• !,
and let x be an arbitrary unitary timelike vector. Then ,
~i! g is a charged C - metric if, and only if , M fi0; ~ii! g is a charged A - metric if, and only if , M 50 and 2 N ( x , x ) 1 trN .0. Furthermore, it is of type A 1 , A 2 or A 3 if v 1 .0, v 1 , 0 or v 1 5 0, respectively, where v 1 [ 19 (d ln( a 1 x))^222 a 2 x; ~iii! g is a charged B - metric if, and only if , M 50 and 2 N ( x , x ) 1 trN ,0. Furthermore, it is of type B 1 , B 2 or B 3 if v 2 .0, v 2 , 0 or v 2 5 0, respectively, where v 2 [ 19 (d ln( a 1 x))^222 a 1 x.
This theorem provides an algorithm to identify, in the set of all metrics, the charged counterpart of the Ehlers and Kundt^2 vacuum solutions. The particular case of the A 1 metrics corresponds to a charged spherically symmetric spacetime, that is, to the Reissner–Nordstro¨m solution. In this case the metric takes the form ~ 38! with X 5 1, and the mass and the charge are related with the constants C and D , respectively. Moreover, these constants can be given in terms of Weyl and Ricci invariants. Then, from the last theorem and previous subsection it follows: Theorem 5 : Let Ric [ Ric ( g ) and W [ W ( g ) be the Ricci and the Weyl tensors of a spacetime metric g , and let us take the metric concomitants :
a[ 2 ~ 121 Tr W^3 !1/3, x[ 2 12 ~Tr Ric^2 !1/2, v[ 19 g ~ d ln a, d ln a! 22 a 2 x, ~ 44!
The necessary and sufficient conditions for g to be the Reissner – Nordstro¨m metric are
afi0, S^21 S 5 0, Ric ~ x , x (^)! >0,
Tr Ric 5 0, S @ Ric # 1 Ric 5 0, ~ 3 a!^22 ~ 2 x!^2 fi0,
M 5 0, 2 N ~ x , x! 1 trN .0, v.0,
where x is an arbitrary unitary time-like vector. Moreover, the mass m and the electric charge e are given, respectively, by m 5 ( a 1 x)/ v3/2^ and e^252 x/ v^2 , and the timelike Killing vector by j (^5) @A v(3 a 12 x)#^21 @ N ( x )/A N ( x , x )#.
C. A summary in algorithmic form
Finally, in order to emphasize the algorithmic nature of our results, we present them as a flow diagram that identifies, among all metrics, every A, B, or C Einstein–Maxwell solution ~in the following flow chart we denote them A * , B * , and C * -metrics!. The exceptional metrics studied by Pleban´ski are also identified and they are denoted Exc -metrics. This operational algorithm involves an arbitrary unitary timelike vector, x , and some metric concomitants that may be ob- tained from the components of the metric tensor g in arbitrary local coordinates: The invariants a,
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