Collineations of a symmetric 2-covariant tensor : Ricci collineations

The infinitesimal transformations that leave invariant a two-covariant symmetric tensor are studied. The interest of these symmetry transformations lays in the fact that this class of tensors includes the energy-momentum and Ricci tensors. We find that in most cases the class of infinitesimal generators of these transformations is a finite dimensional Lie algebra, but in some cases exhibiting a higher degree of degeneracy, this class is infinite dimensional and may fail to be a Lie algebra. As an application, we study the Ricci collineations of a type B warped spacetime.


I. INTRODUCTION
The interest in the study of symmetries in General Relativity is long-standing.][3][4] In the past 20 years, there has been an steady interest in curvature collineations, Ricci collineations and even matter (Einstein) collineations, [5][6][7][8] to quote a few.Their infinitesimal counterparts, namely collineation fields, are characterized by the vanishing of the Lie derivative of the curvature tensor (resp., the Ricci or the energy-momentum tensor).Collineation fields are thus an extension of the aforementioned Killing fields and affine fields in that every Killing vector field is an affine vector filed which in turn is a curvature collineation field and also a Ricci and a matter collineation field.However, it is well known that collineation fields present new features.Indeed, contrarily to the case of Killing and affine fields, the class C of curvature (resp., Ricci and matter) collineation fields is a real vector space which may be infinite dimensional; this is due to the dependence on arbitrary functions, which also results in the fact that a collineation field needs not to be smooth and, as a consequence, C might not be a Lie algebra. 7e shall here concentrate in infinitesimal Ricci collineations, but our results are also relevant in the study of curvature collineations because any of them is necessarily a Ricci collineation too. 9ontrarily to what is done in most recent literature on the subject, our approach does not use the spacetime metric from which the Ricci tensor is derived; we rather study collineations of a general symmetric 2-covariant tensor because, in our view, paying attention to the metric is rather hindering than helpful.
Properly the results derived here do not apply to the energy-momentum tensor because, from a physical viewpoint, it is rather a (1,1)-tensor.Indeed, it is the 4-current density-hence, contravariant-of 4-momentum, which is a covariant magnitude.
Given a 4-manifold M and a smooth field of symmetric 2-covariant tensors T, we shall concentrate on finding the class C T of vector fields X such that L X T = 0 and try to find out whether the number of dimensions of C T is finite, whether X is smooth and whether C T is a Lie algebra.
The answer to these questions depend, but not exclusively, on the rank of T. Particularly, if the rank is 4, T itself can be taken as a non-degenerate metric tensor, the collineation equation is actually a) Electronic mail: pitu.llosa@ub.edu0022-2488/2013/54(7)/072501/13/$30.00 C 2013 AIP Publishing LLC 54, 072501-1 a Killing equation and, as it is well known, 1 C T is a subalgebra of the Lie algebra of smooth vector fields, X (M), and dim C T ≤ 10.
For rank T < 4, we come across an assorted casuistry which depends not only on the rank of T but also on the derivatives of T. We aim to set up a classification of the tensor fields T according to its class C T of collineation fields.The first variable to consider is the rank and different methods and techniques are suitable for different ranks, e.g., for rank three tensors the method is more similar to that used in studying the Killing fields whereas techniques imported from symplectic mechanics are best suited for rank one tensors.Although it could seem whimsical, the order in which the different ranks are presented here is dictated by their progressive degree of difficulty.
The classification we obtain only holds in a local sense and is rather simple if only the generic, i.e., less degenerate, cases are considered.However as the "degree of degeneracy" (in a sense that will be understood along the way) increases, an intricated mess of cases and subcases arises.We close the paper with an application to type B warped spacetimes.

II. COLLINEATION FIELDS
Let T be a 2-covariant symmetric smooth tensor field on a 4-manifold M. A T-collineation field (shortly, a collineation field) is a vector field X such that Notice that the definition requires that X is of class we have that if X and Y are collineation fields of class C 2 at least, then [X, Y] is also a collineation field. 7he case rank T = 4 having been discussed, and finished off, in the Introduction, we shall assume that rank T = m < 4, constant.Therefore, in the neighbourhood of every p ∈ M, it exists as a base of 1 M, {φ a } a = 1. . . 4 , such that with η αβ = diag(+1 r . . .+1, −1 s . . .−1) , r + s = m.(As there is no distinguished metric it is pointless to wonder whether this base is orthonormal or not.)Greek indices run from 1 to m, upper case indices A, B, . . .run from m + 1 to 4, latin indices a, b, . . .run from 1 to 4 and the summation convention is always understood unless the contrary is explicitly stated.The 1-forms φ α being independent, we have that φ 1 ∧ . . .∧ φ m = 0 and the set of 1-forms Let us now expand the Lie derivatives of any φ ν as Including this and (2) it easily follows that Eq. ( 1) is equivalent to that is, the matrix M ν β is an so(r, s)-valued function on M. Any two T-frames, {φ α } α = 1. . .m and { φα } α=1...m , are connected through an η-orthogonal transformation: i.e., R α β is a field of O(r, s) matrices.For the sake of brevity, we shall refer hereafter to these transformations as T-rotations.

III. COLLINEATIONS OF A RANK 3 TENSOR
If rank T = 3, it is obvious that T is holonomous and local charts exist such that the expressions (5) hold.We write the collineation field as X = Z + f ∂ 4 , where Z = Z α ∂ α is tangential to the submanifolds y 4 = constant and f is a function.
As T 4a = 0, Eq. ( 1) amounts to whose components 4a and αβ are respectively, where K αβ := 1 2 ∂ 4 T αβ and ∇ is the Levi-Civita connection for the non-degenerate metric T αβ on the hypersurfaces y 4 = constant.The second of these equations looks like a nonhomogeneous Killing equation (parametrized with y 4 ) and the question is: does it admit solutions Z α that do not depend on y 4 for some appropriate f?
If K αβ = 0, the answer is obviously yes, because it reduces to a Killing equation in 3 dimensions.The collineation field is then X = Z + f ∂ 4 , where f is arbitrary and Z is a Killing vector for the non-degenerate metric T αβ in each 3-submanifold y 4 constant.
If K αβ = 0, things are not so simple.The second of Eq. ( 7) implies that Their integrability conditions imply new equations on αβ and f.These can be derived by means of the Lie derivative of a connection (see Ref. 2, Sec.I.4) which after a little algebra yields On their turn, these equations on κλ produce new integrability conditions that involve L Z R νμλκ and, as the latter has the kind of symmetries of a Riemann tensor in three effective dimensions, they amount to one of its traces, that is, ) Then, similarly as in the theory of Killing vectors (see Ref. 1, Chap.8), using a relation analogous to (9) for general tensors-see Ref. 2, Eq.(I.4.9)-we obtain the hierarchy of integrability conditions So far we have analysed the integrability conditions derived from the commutation of ∇ μ and ∇ ν .Let us now apply the commutation relations for ∇ ν and ∂ 4 to Eq. ( 7); we obtain that (Were there more than one independent matrix M α β fulfilling the above trace equalities, it would result in constraints connecting Z α and μβ .) Substituting then this f in Eqs. ( 7), (10), and (12), we obtain a closed partial differential system on Z α and μν .If it is integrable, each solution is parametrized by six real numbers, namely Z α (0) and μν (0).The above mentioned hierarchy of integrability conditions then act as constraints on these parameters and the number of dimensions of the collineation algebra C T is at most six.
If, on the contrary, K μ β = bK μ β , then Eq. ( 13) implies that which allows to derive ḟ as a linear function of Z α , μν , and f.Now, applying ∂ 4 to both sides of Eq. ( 10) and including the commutation relations for ∇ μ and ∂ 4 , after some algebra we arrive at where W λ μκ is a linear function of Z α and μν .In many cases, this permits to obtain f |α as a unique linear function of Z α , μβ , and f.This happens whenever the linear map K λα K μκ − K μβ K β κ T αλ is injective and the right-hand side W λ μκ fulfills some compatibility conditions, that amount to some linear constraints on Z α , μβ , and f.This expression for f |α as a linear function of Z α , μβ , and f, together with ( 8), ( 10), ( 12) and ( 14), yields a partial differential system on the variables Z α , μν , and f.If it is integrable, each solution is parametrized by the seven real numbers Z α (0), μν (0), and f(0), and the collineation algebra C T has at most seven dimensions.
The highly degenerate cases in which Eq. ( 15) cannot be solved for f |α require further analysis.

IV. COLLINEATIONS OF A RANK 1 TENSOR
If rank T = 1, then it exists φ ∈ 1 M such that T = ± φ ⊗ φ, and the collineation condition (1) is equivalent to L X φ = 0, which means that, locally, a function f exists such that This is a linear system on X whose compatibility depends on f and on the class of the differential form φ. We shall need the following corollary of Darboux theorem-see Ref. 10, Theorem VI.4.1.
A remark on notation is appropriate: hereon a stroke means partial derivative, so v |a := ∂ a v := ∂v/∂x a ; particularly in canonical coordinates (q i , p j ), with i, j = 1, 2. Writing now X and df in canonical coordinates, Eq. (16b) amounts to Then, including this and (17), we obtain that Eq. (16a) amounts to with φ i := ψ |i + e i p i and φ i := ψ |i .According to the values of e 1 and e 2 , different cases are possible, which we shall analyse separately.
In this case, the class of the differential form φ is 4 (see Ref. 10, Section VI.1.3)and Darboux theorem states more precisely that canonical local charts exist such that ψ = 0, that is, φ = p 1 dq 1 + p 2 dq 2 .Equation (18) then implies that where { , } is the Poisson bracket for dφ.Including this, Eq. (19) becomes which, by Euler theorem, means that f(q i , p j ) is an homogeneous function of the first degree in the variables p j .The general collineation field is thus X = −{ f, -} , where f ∈ 0 M is a solution of (20).

V. COLLINEATIONS OF A RANK 2 TENSOR
In what follows, it will be helpful to consider the 2-forms dφ α and the exterior products A T-rotation (4) is defined by a matrix R α β (ζ ) which depends on a function ζ and we have that where Ṙα Now, let ∈ 4 M be a volume tensor ( = 0) and define l α by ϒ α = l α .The relation (23) implies that lα := ±R α β l β and, as a consequence, Therefore, unless η αβ = diag(1, − 1) and ϒ 1 = ϒ 2 , we can always perform a T-rotation such that one of the exterior products ϒ α vanishes (we can label the 1-forms φ β so that this is ϒ 1 ) and T can be classified in one of the following types: [Notice that Type 2.N only occurs if η αβ has no defined sign.]

A. Type 2.I.a
A little algebra allows to proof the following.
As rank T = 2, Eq. ( 3) reads Now, as ϒ 1 = 0 and ϒ 2 = 0, it follows that b = 0 which, substituted in (27) yields On its turn, this implies that L X dφ α = 0 which, including Eq. ( 25), leads to Summarizing, if T is type 2.I.a, first we find the canonical base {φ a } a = 1. . . 4 and its dual base {Y a } a=1... 4 .Then the collineation equations supplemented with their integrability conditions amount to L X φ a = 0 which, writing X = X a Y a and If this partial differential system is integrable, each solution is parametrized by the values X b 0 at one point.Therefore, the dimension of the collineation algebra for type 2.I.a tensors is at most 4.
The integrability conditions of (30) put some further constraints on the parameters X b 0 .These integrability conditions are obtained by taking the exterior derivative and read L X dφ a = 0 or, in terms of the coefficients C a bc , XC a bc = 0, (31) which eventually implies that This is an infinite homogeneous linear system on the parameters X b 0 .Provided that its rank is not greater than 4, the codimension of the collineation algebra for type 2.I.a tensors is precisely this rank, otherwise T admits no collineation fields.
The differential form φ 4 is determined up to the gauge transformation, φ 4 = φ 4 + mφ 3 , where m is an arbitrary function.

Case 2.I.b.1:
), and the integrability conditions for Eq.(34) imply that L X dφ α = 0, which, with a little algebra, lead to Equations ( 34) and (35) can then be unified as which is equivalent to , where as before X = X a Y a and C a bc are the commutation coefficients in this base.Due to the occurrence of the unknown function f, this partial differential system is not in closed form.However, in most generic cases the integrability conditions could help to determine f.
The integrability condition for Eq. ( 36), a = 4, yields with ), which closes the differential system (36).If it is integrable, then the solution depends on the four real parameters X a 0 , which are subject to the hierarchy of constraints that follow from the full integrability conditions of the system (36), and dim C T ≤ 4.
If, on the contrary, φ 3 ∧ dφ 3 = 0, after some elaboration we arrive at where F acb are some coefficients that do not depend on X e .Then, provided that the left-hand side does not vanish, we can derive f = f(X a ), which closes the partial differential system (37), and therefore dim C T ≤ 4.
We do not analyse here the highly non-generic case that neither Eq. (37) nor Eq. ( 38) can be solved for f, which would require further study.
Case 2.I.b.0:If v α = 0, then it follows from (33) that The integrability of the latter implies that r = 0, i.e., dφ 1 = 0, and locally a function y exists such that φ 1 = dy.Equation (34) then implies that Xy = C, constant, and two cases must be considered depending on whether s does vanish or not.
where all U a c vanish, except Let {Y a } be the dual base of { φa }, and write d φa = − 1 2 Ĉa cb φc ∧ φb , and X = X a Y a .Equation ( 50) is then a partial differential system on the unknowns X a and b which, due to the presence of f, is not in closed form, but the integrability conditions may help to determine it.Indeed, by a similar technique as for case 2.I.b.1 in Subsection V B, we can easily conclude that, whenever φ3 ∧ d φ3 = 0 or φ3 ∧ d Ĉ4 4β φβ = 0, C T is a Lie algebra and dim C T ≤ 5. Like then, we do not consider the residual nongeneric subcase when f cannot be derived from the integrability conditions of Eq. (50).
For subtypes 2.N.01 and 2.N.00, we have to take into account that

VI. APPLICATION
As an application, we study the Ricci collineations of 2 + 2 type B warped spacetimes ds 2 = g AB (x C )dx A dx B + h 2 (x C )g i j (x k )dx i dx j with A, B = 3, 4 , i, j. . .= 1, 2.
The non-vanishing components of the Ricci tensor are where F := 1 2 R 2 − D A D A h 2 , and R 1 (x B ) and R 2 (x k ) respectively are the Ricci scalars for the 2-dimensional metrics g AB and g ij .
If F • det(R AB ) = 0, then the Ricci tensor has rank 4 and the Ricci collineations conform a Lie algebra whose dimension is at most ten.This case has been completely solved in Ref. 11 and we shall confine ourselves to the case when the rank is less than 4, which will serve as a test for the power of our approach.

A. The case R AB = 0
Apart from trivial case F = 0 (and Ric = 0), we have that, due to the low number of dimensions, coordinates can be chosen so that Ric = ±e f η i j dx i ⊗ dx j , η i j = diag(1, σ ) with σ = ± 1 and f := log |F|.
The Ricci collineation condition then implies that with X i := η ij X j , whence it follows that where U(x b ) is a solution of η ij ∂ i ∂ j U = 0 and In case that ∂ A f = 0, the latter is a constraint connecting the two components X B , whereas if ∂ A f = 0, it yields a further constraint on U(x b ).

B. The case F = 0 and rank R AB = 1
The Ricci tensor has rank 3 and coordinates exist such that Ric = H (x C ) dx 3 ⊗ dx 3 ± e f η i j dx i ⊗ dx j and the results obtained in Sec.III apply, with the non-vanishing components of T αβ and K αβ given by T i j = ±e f η i j , T 33 = H , However, due to the simplicity of this particular case we can go further, x 3 can be chosen so that H = ± 1 and Eq. ( 7) reads det P αβ = r − (s + t)/2, α, β = 1, 2, and the differential forms φ α + 2 are determined up to the gauge transformation φ α+2 = L α ν φ ν+2 where L α ν is a SL(2)-valued function.The value of det(P αβ ) is T-frame dependent.Indeed, by a T-rotation the volume forms αβ transform according to (24) and, including that R α ν = cosh ζ sinh ζ sinh ζ cosh ζ , and that Υ1 = e ζ ϒ 1 , we finally arrive at e 3ζ t + s − 2r = (t + s − 2r ) + 2(P 24 + P 14 ) ζ 3 − 2(P 23 + P 13 ) ζ 4 , (42) where dζ = ζ a φ a .If det(P αβ ) = 0, then the coefficients of ζ 3 and ζ 4 cannot vanish simultaneously and the latter equation can be used to obtain a T-frame in which det( Pαβ ) = 0.

2K i j = ±∂ 4
f e f η i j , 2K 33 = ∂ 4 H , (a) If ∂ 4 H = ∂ 4 f = 0,then it belongs to the subtype 3.0 and the class of Ricci collineations has an infinite number of dimensions.