CCNV Space-Times as Potential Supergravity Solutions

It is of interest to study supergravity solutions preserving a non-minimal fraction of supersymmetries. A necessary condition for supersymmetry to be preserved is that the spacetime admits a Killing spinor and hence a null or timelike Killing vector field. Any spacetime admitting a covariantly constant null vector field ($CCNV$) belongs to the Kundt class of metrics, and more importantly admits a null Killing vector field. We investigate the existence of additional non-spacelike isometries in the class of higher-dimensional $CCNV$ Kundt metrics in order to produce potential solutions that preserve some supersymmetries.


Introduction
Supersymmetric supergravity solutions are of interest in the context of the AdS/CFT conjecture, the microscopic properties of black hole entropy, and in a search for a deeper understanding of string theory dualities.For example, in five dimensions solutions preserving various fractions of supersymmetry of N = 2 gauged supergravity have been studied.The Killing spinor equations imply that supersymmetric solutions preserve 2, 4, 6 or 8 of the supersymmetries.The AdS 5 solution with vanishing gauge field strengths and constant scalars preserves all of the supersymmetries.Half supersymmetric solutions in gauged five dimensional supergravity with vector multiplets possess two Dirac Killing spinors and hence two time-like or null Killing vectors.These solutions have been fully classified , using the spinorial geometry method, in [1].Indeed, in a number of supergravity theories [2], in order to preserve some supersymmetry it is necessary that the spacetime admits a Killing spinor which then yields a null or timelike Killing vector from its Dirac current.Therefore, a necessary (but not sufficient) condition for supersymmetry to be preserved is that the spacetime admits a null or timelike Killing vector (KV).
In this short communication we study supergravity solutions preserving a non-minimal fraction of supersymmetries, by discussing the existence of additional KVs in the class of higher-dimensional Kundt spacetimes admitting a covariantly constant null vector (CCNV ) [3].CCNV spacetimes belong to the Kundt class because they contain a null KV which is geodesic, nonexpanding, shear-free and non-twisting.The existence of an additional KV puts constraints on the metric functions and the KV components.KVs that are null or timelike locally or globally (for all values of the coordinate v) are of particular importance.As an illustration we present two explicit examples.
A constant scalar invariant (CSI) spacetime is a spacetime such that all of the polynomial scalar invariants constructed from the Riemann tensor and its covariant derivatives are constant [4].The V SI spacetimes are CSI spacetimes for which all of these polynomial scalar invariants vanish.The subset of CCNV spacetimes which are also CSI or V SI are of particular interest.Indeed, it has been shown previously that the higher-dimensional V SI spacetimes with fluxes and dilaton are solutions of type IIB supergravity [5].A subset of Ricci type N V SI spacetimes, the higher-dimensional Weyl type N pp-wave spacetimes, are known to be solutions in type IIB supergravity with an R-R five-form or with NS-NS form fields [6,7].In fact, all Ricci type N V SI spacetimes are solutions to supergravity and, moreover, there are V SI spacetime solutions of type IIB supergravity which are of Ricci type III, including the string gyratons, assuming appropriate source fields are provided [5].It has been argued that the V SI supergravity spacetimes are exact string solutions to all orders in the string tension.Those V SI spacetimes in which supersymmetry is preserved admit a CCNV .Higher-dimensional V SI spacetime solutions to type IIB supergravity preserving some supersymmetry are of Ricci type N, Weyl type III(a) or N [8].It is also known that AdS d × S (D−d) spacetimes are supersymmetric CSI solutions of IIB super-gravity.There are a number of other CSI spacetimes known to be solutions of supergravity and admit supersymmetries [4], including generalizations of AdS × S [9], of the chiral null models [6], and the string gyratons [10].Some explicit examples of CSI CCNV Ricci type N supergravity spacetimes have been constructed [11].

Kundt metrics and CCNV spacetimes
A spacetime possessing a CCNV, ℓ, is necessarily of higher-dimensional Kundt form.Local coordinates (u, v, x e ) can be chosen, where ℓ = ∂ v , so that the metric can be written [12] where the metric functions are independent of the light-cone coordinate v.
A Kundt metric admitting a CCNV is CSI if and only if the transverse metric g ef is locally homogeneous [4].(Due to the local homogeneity of g ef a coordinate transformation can be performed so that the m ie in eqn.(2) below are independent of u.)This implies that the Riemann tensor is of type II or less [12].If a CSI-CCNV metric satisfies R ab R ab = 0 then the metric is V SI, and the Riemann tensor will be of type III, N or O and the transverse metric is flat (i.e., g ef = δ ef ).The constraints on a CSI CCNV spacetime to admit an additional KV are obtained as subcases of the cases analyzed below where the transverse metric is a locally homogeneous.

Additional isometries
Let us choose the coframe {m a } where m i e m if = g ef and m ie m e j = δ ij .The frame derivatives are given by The KV can be written as X = X 1 n + X 2 ℓ + X i m i .A coordinate transformation can be made to eliminate Ŵ3 in (1) and we may rotate the frame in order to set X 3 = 0 and X m = 0 [3].X is now given by ( Henceforth it will also be assumed that the matrix m ie is upper-triangular.The Killing equations can then be written as: which imply and where B ij = m ie,u m e j , W i = m e i Ŵe , and . Further information can be found by taking the Killing equations and applying the commutation relations, which leads to two cases; (1) Using equation ( 6) and the definition of F 2 from (5), we have that H and W m are given in terms of these two functions (where g ′ ≡ dg du ) (ii) c 1 = 0, X 1 = 1; F 2,u = 0, and H and W n are In either case, the only requirement on the transverse metric is that it be independent of u.The arbitrary functions in this case are F 2 and the functions arising from integration.Subcase 1.2: F 3 = 0.The transverse metric is now determined by m nr,u = −m nr,3 x e ) (i = 1, 2) are arbitrary functions, H is given by and W n is determined by H may be written as The only equation for W n is (iii) X 1 = 0: There are two further subcases depending upon whether m 33,r = 0 or not, whence we may further integrate to determine the transverse metric.
If we assume that F 1,3 = 0 and F 1 is independent of x r : Thus m 33 (u, x 3 ) is entirely defined by F 1 .We may solve for H and the W n : F 3 is of the form: There are differential equations for F 2 in terms of the arbitrary functions F 1 (u, x 3 ) and A 6 (u, x r ).These solutions are summarized in Table 2 in [3].Killing Lie Algebra: There are three particular forms for the KV in those CCNV spacetimes admitting an additional isometry: To determine if these spacetimes admit even more KVs we examine the commutator of X with ℓ in each case.In case (A), [X A , ℓ] = 0 and in case B [X B , ℓ] = −ℓ, and thus there are no additional KVs.In the most general case However, this will always be spacelike since Non-spacelike isometries: Let us consider the set of CCNV spacetimes admitting an additional non-spacelike KV, so that If the KV field is non-spacelike for all values of v, then D 3 (X 1 ) must vanish and X 1 is constant.Therefore, various subcases discussed above are excluded.
In the remaining cases In the timelike case, the subcases with X 1 = 0 are no longer valid since F 2 3 < 0. In the case that X is null and c 2 = 0 we can rescale n so that 2F 2 = F 2 3 .We can then integrate out the various cases: If F 3 = 0, F 2 must vanish as well and X = n.The remaining metric functions are now (logm 33 ) ,u = D 2 (logF 3 ).If c 2 = 0, F 2 must be constant, and the KV is a scalar multiple of ℓ and can be disregarded.The remaining cases are just a repetition of the above with added constraints.The CSI CCNV spacetimes admitting KVs which are non-spacelike for all values of v are the subcases of the above cases where the transverse space is locally homogenous.

Explicit examples I:
We first present an explicit example for the case where X 1 = u and F 3 = 0. Assuming that F 3 (u, x i ) = ǫum 33 and ǫ is a nonzero constant, we obtain and the transverse metric is thus given by We have the algebraic solution where F 2 (u, x i ) is an arbitrary function and H is given by where A is an arbitrary function and S is given by Furthermore, the solution for Ŵn , n = 4, . . ., N is Ŵn (u, where B n are arbitrary functions and T n is given by In this example, the KV and its magnitude are given by Clearly, the causal character of X will depend on the choice of F 2 (u, x i ), and for any fixed (u, x i ) X is timelike or null for appropriately chosen values of v.Moreover, (32) is an example of case (B); therefore the commutator of X and ℓ gives rise to a constant rescaling of ℓ and, in general, there are no more KVs.The additional KV is only timelike or null locally (for a restricted range of coordinate values).However, the solutions can be extended smoothly so that the KV is timelike or null on a physically interesting part of spacetime.For example, a solution valid on u > 0, v > 0 (with F 2 < 0), can be smoothly matched across u = v = 0 to a solution valid on u < 0, v < 0 (with F 2 > 0), so that the KV is timelike on the resulting coordinate patch.
As an illustration, suppose the m 3s are separable as follows and F 2 has the form where the p s are constants and h s , g arbitrary functions.Thus, from (28) and hence from (27) Last, equation (30) gives

II:
A second example corresponding to the distinct subcase where X 1 = 1 and assuming F 3 (u, x i ) = ǫm 33 gives the same solutions (26) for the transverse metric (although, in this case, the additional KV is globally timelike or null).In addition, we have where H(u, x i ), F 2 (x 3 − ǫu, x n ) and f (x i ) are arbitrary functions.Last, the metric functions Ŵn are Ŵn (u, x i ) = u L n (z, x 3 − ǫu + ǫz, x m )dz + E n (x 3 − ǫu, x m ), ( 39) with E n arbitrary and L n given by L n (u, x 3 , x m ) = H ,n + ǫ H ,3n du + f ,n .
The KV and its magnitude is Since F 2 and m 33 have the same functional dependence there always exists F 2 such that X is everywhere timelike or null.The KV (41) is an example of case (A) and thus X and ℓ commute and hence no additional KVs arise.For instance, suppose H = H(x 3 − ǫu, x n ) and f is analytic at x 3 = 0 (say) then ( 38) and (39) simplify to give This explicit solution is an example of a spacetime admitting 2 global null or timelike KVs, and is of importance in the study of supergravity solutions preserving a non-minimal fraction of supersymmetries.