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Class. Quantum Grav. 14 (1997) 129–138. Printed in the UK PII: S0264-9381(97)73270-
A covariant determination of the Weyl canonical frames in
Petrov type I spacetimes
Joan Josep Ferrando†§ and Juan Antonio S´aez‡ † Departament d’Astronomia i Astrof´ısica, Universitat de Valencia, E-46100 Burjassot, Valencia, Spain ‡ Departament d’Economia Financera i Matematica, Universitat de Valencia, E-46010 Val`encia, Spain
Received 1 April 1996, in final form 24 September 1996
Abstract. A covariant algorithm is given to obtain principal 2-forms, Debever null directions and canonical frames associated with Petrov type I Weyl tensors. The relationship between these Weyl elements is explained, and their explicit expressions depending on Weyl invariants are obtained. These results are used to determine a cosmological observer in type I universes, and their usefulness in spacetime intrinsic characterization is shown.
PACS numbers: 0240H, 0420C
1. Introduction
Algebraic classification of the Weyl tensor has been considered from two different points of view. The approach initiated by Petrov [1] studies the eigenvalue and eigenvector problem for the Weyl tensor regarded as an endomorphism on the 2-form space. This classification was completed by G´eh´eniau [2] and Bel [3] considering, not only the number of independent invariant subspaces, but also the eigenvalue multiplicity. In this framework we find in a natural way the notion of principal 2-form which was analysed widely by Bel [4] for the different algebraic types. An alternative viewpoint [5, 6] consists in the study of the relative positions between the ‘light cones’ determined on the 2-form space by the canonical metric and the Weyl tensor as a quadratic form. From this angle, which is equivalent to the spinorial approach [7], the Weyl classification implies studying the roots of a fourth degree algebraic equation with complex coefficients. A Debever null direction of the Weyl tensor corresponds to every root of this equation [8, 9, 7]. Both approaches show a richness and variety of directions and 2-planes associated with the Weyl tensor. These geometric elements are generically different from either viewpoint, but the more degenerate the Weyl tensor is, the more common these elements become. Indeed, the relation between the Debever null directions and the principal 2-forms is complex, and only the multiple Debever directions are fundamental directions of null principal 2-forms [4]. In the most regular case, Petrov type I, a Debever direction never coincides with a principal direction of a principal 2-form, and their relation depends strongly on the four Weyl scalar invariants.
§ E-mail: [email protected]
0264-9381/97/010129+10$19.50 ©c 1997 IOP Publishing Ltd 129
130 J J Ferrando and J A S´aez
Following the work of Sachs [10], where he gave the hierarchy of equations for the multiplicity of the Debever directions, and the publication of the d’Inverno and Russell- Clark [11] algorithm, the null direction approach became popular; since then the geometry of principal 2-forms has seldom been considered in the literature. Therefore, although the geometric richness of both points of view were underlined and widely studied in pioneering papers by Penrose [7], Bel [4] and Debever [6] some years ago, until now the relation between them has not been sufficiently analysed for the type I case. However, when considering this subject we can quote the papers by Tr¨umper [12] and Narain [13], or those more recent ones by McIntosh et al [14] and Bonanos [15]. In the latter, a method is given to determine Weyl canonical frames from null Debever directions. In the other three works, we can find expressions for Debever directions in terms of an orthonormal canonical frame in several particular cases: purely electric, purely magnetic and when the four Debever directions span a 3-plane. Here we extend these results about Petrov type I and present a geometric interpretation of them. We show that the Debever directions are the principal directions of two characteristic 2-forms associated to the Weyl tensor. We give a covariant expression for these 2-forms in terms of principal 2-forms and a complex invariant scalar. Furthermore we also determine principal 2-forms in a covariant way. These results provide an algorithm to determine the Debever directions. The covariant determination of the Ricci eigenvectors and its causal character [16] is a necessary tool in the characterization of spacetimes obeying their energy content. In particular its usefulness has been shown in building a Rainich theory for the thermodynamic perfect fluid [17]. In a similar way, in addition to knowing its classification and giving algorithms to distinguish every case, a complete algebraic study of the Weyl tensor implies knowing the covariant determination of the Weyl eigenvectors, i.e. of the principal 2-forms. A complete analysis of this subject, considering all Petrov types, has been developed elsewhere [18, 19]. Here we only present the results for type I and we use them to determine the Weyl canonical frames in an alternative way to the Bonanos method [15]. The aforementioned paper departs from the Debever directions, whereas here we give covariant expressions, in terms only of the Weyl tensor, using principal 2-forms. The usefulness of Weyl canonical frames in the metric equivalence problem is well known. In the algebraically special cases there is a close relationship between canonical frames and Debever directions, but in the algebraically general one its relation involves Weyl scalar invariants and, as a consequence, canonical frame determination is more difficult. Here, we give a geometric interpretation of this relationship and offer an algorithm which allows us to obtain canonical frames directly without solving any algebraic equation. The paper is organized as follows. In section 2 we introduce the formalism used in the paper and recapitulate some properties about 2-forms, in particular the covariant way used to determine their principal directions. We also point out the bijection and give its covariant expression between orthonormal frames on the tangent space and orthonormal bases on the self-dual 2-form space. In section 3 we summarize the characterization and the canonical expressions of a type I Weyl tensor, and present the Weyl concomitants that enable us to obtain the three orthonormal principal 2-forms. From them, and using the results in section 2, we give a covariant algorithm to determine the Weyl canonical frames. The connection between principal 2-forms and Debever directions is analysed in section 4. We give a clear interpretation in the 2-form space framework, and comment on the relationship between our approach and the previous results quoted above. Finally, in section 5 we apply our results to determine a cosmological observer in Petrov
132 J J Ferrando and J A S´aez
We know explicitly {Ui } in terms of {e (^) α }; we will now determine covariant expressions which give {e (^) α } in terms of {Ui }. The associated real 2-form, Ui = Re(
2 Ui ), verifies U (^) i × U (^) i = − e 0 ⊗ e 0 + e (^) i ⊗ e (^) i. Then, it is easy to show that
e 0 ⊗ e 0 = (^12)
g −
∑^3
i= 1
U (^) i × U (^) i
≡ P 0. (5)
Thus, we arrive at the following
Lemma 1. If {eα } is an oriented and orthochronous orthonormal frame and Ui = e 0 ∧ ei , then the term {Ui }, Ui = √^12 (Ui − i ∗ U (^) i ), is an oriented frame on the self-dual 2-forms space. This is a one-to-one map and its inverse is given by:
e 0 =
−P 0 (x) √ P 0 (x, x)
; ei = Ui (e 0 )
with
P 0 ≡ 12
g −
∑^3
i= 1
Ui × U (^) i
and where x is an arbitrary future-pointing vector.
3. Principal 2-forms and canonical frames
The algebraic classification of the Weyl tensor W can be obtained [2] by studying the linear map defined by the self-dual Weyl tensor W = 12 (W − i ∗ W ) on the self-dual 2-form space. In terms of its complex scalar invariants, a = tr W^2 , b = tr W^3 , the characteristic equation reads
x^3 − 12 ax − 13 b = 0 (6)
and their roots are
αk = βe
2 πk 3 i
a 6 β
e−^
2 πk 3 i ,
with
β = 3
b +
b^2 −
a^3 6
The Weyl tensor is Petrov type I if (6) has three different roots {α (^) i }, which is equivalent to the condition 6b^2 6 = a^3. In this case an orthonormal frame {Ui }, built up with eigenvectors of W, exists: they are the principal 2-forms of the Weyl tensor [4]. Then, the self-dual Weyl tensor takes the canonical expression
W = −
∑^3
i= 1
α (^) i Ui ⊗ Ui. (7)
Now we consider the determination of the principal 2-forms {Ui } in terms of the Weyl tensor. From the characteristic equation,
∏^3
i= 1
(W − α (^) i G) = 0 ,
Weyl canonical frames in Petrov type I spacetimes 133
it follows that (W − α (^) i G) (Pi (X )) = 0 for every self-dual 2-form X , where
Pi =
j 6 =i
(W − αj G).
Thus Pi (X ) belongs to the eigenspace corresponding to the eigenvalue αi , that is to say, Pi is the projection map on this eigenspace. In consequence, we are lead to the following
Proposition 1. Let W be the self-dual Weyl tensor of a Petrov type I spacetime. The principal 2-form Ui corresponding to the eigenvalue αi may be obtained as
Ui =
Pi (X ) √ −P^2 i (X , X )
with Pi = W^2 + α (^) i W + (α^2 i − 12 a)G, and where X is an arbitrary self-dual 2-form such that Pi (X ) 6 = 0.
This proposition provides a covariant algorithm to obtain the principal 2-forms of a type I Weyl tensor. They are given by (8) and are determined up to sign and permutation. Thus, we can consider 24 oriented eigenframes {Ui }: for every permutation, the sign of two of them gives us four possibilities, the third being given by (4). The one-to-one map defined in lemma 1 associates an oriented and orthochronous orthonormal frame {e (^) α } to every oriented eigenframe {Ui }: they are the 24 canonical frames of the Weyl tensor [4, 21]. Then lemma 1 and proposition 1 offer us a covariant algorithm to determine these canonical frames.
Corollary 1. The Weyl canonical frames {e (^) α } of a Petrov type I spacetime may be determined to be
e 0 =
−P 0 (x) √ P 0 (x, x)
; ei = Ui (e 0 ) (9)
with
P 0 ≡
g −
∑^3
i= 1
U (^) i × U (^) i
, Ui = Re(
2 Ui )
and Ui , the principal 2-forms given in proposition 1, and where x is an arbitrary future- pointing vector.
Although there are 24 canonical frames, they define four unique orthogonal directions: one timelike principal direction and three spacelike principal directions.
4. Principal 2-forms and Debever directions
The ‘light cones’ defined by the metric G and the self-dual Weyl tensor W cut, generically, on four null directions of the self-dual 2-form space. Each one defines a null direction on the spacetime, usually called Weyl principal null direction or Debever direction [9]. These characteristic directions may be determined by solving a fourth degree algebraic equation with coefficients given by the components of W in a complex null frame. Then, an alternative approach to the Weyl algebraic classification follows analysing the multiplicity of the roots of this equation [9]. Type I appears as the case where four simple Debever directions exist. Depending on their multiplicity, these vectors satisfy an equation of the Sachs [10] hierarchy; for a simple Debever vector l this equation can be written as:
(l ∧ W(l; l) ∧ l) (^) αβγ δ ≡ −l[α Wβ]λμ[γ l (^) δ]l λ^ l μ^ = 0. (10)
Weyl canonical frames in Petrov type I spacetimes 135
It is easy to show the consistency between our expressions (15) and the results obtained by Debever in his early papers [6, 9]. Notice that in the above study we have taken i = 3 but we could equally have opted for any other eigenvalue. For each selection a different pair of unitary 2-forms like (14) exists, with Debever directions as principal directions: we call them Debever 2-forms. Thus, there are six Debever 2-forms, in accordance with all the possible pairs that may be considered with the four Debever directions of a type I spacetime. Now let us pose the inverse problem: the determination of principal 2-forms and canonical frames using Debever vectors. Bonanos has given a geometric interpretation of the relationship between Debever vectors and canonical frames [15]. Here we present an alternative approach using our results, and offer a simple algorithm to calculate canonical frames in terms of Debever vectors. Let l+ be the four Debever vectors normalized in such a way that (l+, l−) = −2,
and consider the Debever 2-forms V = √^12 (l (^) − ∧ l+ − i ∗ (l− ∧ l+)). Then, taking into account proposition 2, the bisectors of V are eigendirections of the Weyl tensor. Therefore, it follows
Proposition 3. Let V = √^12 (l (^) − ∧ l+ − i ∗ (l− ∧ l+)) be two Debever 2-forms of a Petrov type I spacetime. Principal 2-forms {Ui } may be calculated as
U 1 =
V+ + V−
1 − (V+, V−)
U 2 =
V+ − V−
1 + (V+, V−)
U 3 = i
2 U 1 × U 2.
Once the principal 2-forms have been determined by (16), we can use (9) to calculate the canonical frames. However, a more straightforward calculation follows reversing expressions (15). To sum up, we can state
Corollary 3. Let k (^) a , a = 1 , 2 , 3 , 4, be the Debever vectors of a Petrov type I spacetime. Let us normalize them as follows:
l 1 =
κ 42 κ 43 κ 12 κ 13
k 1 , l 2 =
κ 41 κ 43 κ 21 κ 23
k 2 ,
l 3 =
κ 41 κ 42 κ 31 κ 32
k 3 , l 4 = k 4
where κ (^) ab = (ka , kb ). Then, the timelike and the spacelike principal directions may be obtained as
e 0 ∝
∑^4
a= 1
l (^) a , ei ∝ l 4 −
∑^3
a= 1
(− 1 )δ^ ia^ l (^) a (18)
Finally, from our above analysis we will obtain the known results for type I degenerate cases (see [14] and references therein). The four Debever vectors span a three-dimensional space if, and only if, the real part of the Debever 2-forms (14) satisfy V+ ∧V− = 0, i.e. when (V+, V−) = cos 2 is real. So, (14) implies that the ratio between every two eigenvalues is real or, equivalently, the scalar invariant M = a
3 b^2 −^ 6 is a positive real or infinity (I (M
or I (M∞) in the Arianrhod et al classification [22]). In this case the eigenvalues may be strictly ordered by their modulus, and is real (ψ = 0) if we take α 3 as the shortest
136 J J Ferrando and J A S´aez
eigenvalue. As a consequence, (15) implies that the orthogonal to e 3 subspace contains the four Debever directions. Therefore, we can conclude
Corollary 4. In a Petrov type I spacetime, the four Debever vectors span a three-dimensional space iff M = a
3 b^2 −^ 6 is a positive real or infinity. This space is the orthogonal subspace to the spacelike principal direction associated with the shortest eigenvalue.
Thus we have recovered some previous results on this subject [14, 23]. Moreover, the known expressions for the Debever vectors in the degenerate type I cases [12, 13, 14] may also be easily deduced by taking ψ = 0 in (15).
5. Type I spacetimes admitting isotropic radiation
The spacetimes admitting isotropic radiation with respect to a vorticity-free observer have been studied elsewhere [24], and their value for modelling the present universe has also been widely pointed out. These spacetimes may be characterized [25] as those admitting a timelike hypersurface-orthogonal conformal Killing vector or, in terms of the unitary vector field u, by conditions:
ω = 0 , σ = 0 , d( u˙ − 13 θu) = 0 (19)
ω, σ , θ and u˙ being, respectively, the rotation vector, the shear, the expansion and the acceleration of u. Therefore, the two first conditions (19) state that u defines a shear-free and vorticity-free congruence, and so the Weyl tensor has real eigenvalues and is of Petrov type I, D or 0 [12]. In type I, u is the unitary vector in the timelike principal direction, which will be called the Weyl principal observer. We summarize some of these known results [25, 12] in a lemma.
Lemma 2. In a Petrov type I spacetime, one has the following three equivalent statements.
(i) A hypersurface-orthogonal timelike conformal Killing vector exists. (ii) A solution of Liouville equation, isotropic for a vorticity-free observer, exists. (iii) The Weyl principal observer satisfies conditions (19).
In these spacetimes the Weyl principal observer keeps some of the main properties of the Robertson–Walker cosmological observer, in particular, that of observing isotropic radiation. Consequently it could play a similar role in building and interpreting type I cosmological models. Even, in generic type I universes, we can consider a privileged cosmological observer : the Weyl principal one. Thus, the interest in determining the timelike principal direction of a Petrov type I spacetime is plainly evident, a question that we have solved in corollary 1. Let us go back to the spacetimes considered in lemma 2. The three statements characterize them, but only the third gives us an invariant characterization because it asserts conditions on an invariant direction of the Weyl tensor. But this characterization is only useful after using our corollary 1 in order to calculate the timelike principal direction in terms of the Weyl tensor, and so obtain an intrinsic characterization of these spacetimes. Indeed, conditions (19) may be easily written as equations on the projector tensor h = u ⊗ u + g:
ω = 0 , σ = 0 ⇐⇒ hλα hμβ ∇λ hνμ = hαβ h λμ∇λ hνμ (20) d( u˙ − 13 θu) = 0 ⇐⇒ d[4h(δh) − δh] = 0 (21)
138 J J Ferrando and J A S´aez
[24] Ferrando J J, Morales J A and Portilla M 1992 Phys. Rev D 46 578 [25] Ehlers J, Geren P and Sachs J 1968 J. Math. Phys. 9 1344 [26] Treciokas R and Ellis G F R 1971 Commun. Math. Phys. 23 1 [27] Ferrando J J, Morales J A and Portilla M in preparation
