The remaining conditions

In order that our ansatz (6.23) for the scalars wi gives rise to a spacetime solution corresponding to a five-dimensional black string, we need to ensure that the integration constants satisfy the model-dependent conditions (6.25), which we recall correspond to the Hamiltonian constraint and thewi equations of motion.

Our strategy here is to look for configurations of charges which allow us to construct regular black string solutions in a relatively model-independent fashion.

To see how this might work, let us focus on the Hamiltonian constraint, which we rewrite in terms of ¯pi, the ‘horizon charge’ associated with the outer horizon:

ˆ

gij(H) ¯pip¯j−2chi∞p¯j−p˜ip˜j

= 0. (6.34)

Completing the square, this becomes

ˆ gij(H) ¯pi−chi∞ ¯ pj −chj∞ = ˆgij(H) c2h∞i hj∞+ ˜pip˜j . (6.35)

If we consider a diagonal model, i.e. choose a Hesse potential H(h) such that ˆgij

solved by ¯ pi−chi_∞=± q c2_(hi ∞)2+ (˜pi)2, (6.36) fori= 0, . . . , n(5)_V . This determines the integration constants (‘horizon charges’) ¯pi, or equivalently pi = ¯pi−2chi∞, in terms of the quantitieshi∞ and ˜pi. The sign in (6.36) should be chosen such that the line element (6.30) is ‘regular’ in the sense of having no metric singularities outside the event horizon. In particular, we should require that theHi_{(ρ) do not vanish for anyρ >2c. Writing}

Hi(ρ) =hi∞+

pi ρ =

(ρ−2c)hi∞+ ¯pi

ρ ,

we see that the harmonic function remains non-zero for ρ > 2c provided sign(hi∞) = sign(¯pi_{). Hence, if h}i

∞ is positive we should choose the + sign in (6.36), whereas if it is negative we should choose the− sign. One can show, moreover, that with this sign choice the functionsHi_{(ρ) remain non-zero up to the inner horizon at ρ= 0.}

Hence, for diagonal models we have identified a class of solutions depending on 2n(5)_V + 2 independent parameters: then(5)_V + 1 magnetic charges ˜pi; then(5)_V + 1 asymp- totic values of the scalar fields hi∞ subject to the constraint H(h∞) = 1; and the non-extremality parameterc.

One method of constructing solutions for a completely general class of models is to impose the condition (6.36) and determine what restrictions the off-diagonal components of (6.35) put on the various parameters. First, we define βi ≡

sign(hi_∞)pc2_(hi

∞)2+ (˜pi)2. Then substituting (6.36) into (6.35) we find ˆ

gij(H) c2hi∞h∞j + ˜pip˜j −βiβj

= 0. (6.37)

Certainly for diagonal elements, i.e. i = j, the LHS of this expression vanishes iden- tically. For off-diagonal elements, i6=j, a model-independent way of satisfying (6.37) can be found by imposing the stronger condition that

for each i, j with ˆgij(H)6= 0, which can be shown to be equivalent to

hi∞p˜j−hj∞p˜i= 0.

Hence, the ratios

hi_∞ hj∞ = p˜ i ˜ pj ≡µ i j,

are constant. In the generic case, where all elements of ˆgij(H) are non-zero, this implies

that allhi∞and ˜piare proportional to each other, and can be written simply as multiples of, say, h0∞ and ˜p0. In particular, we write hi∞ = µih0∞, ˜pi = µip˜0. Note that h0∞ is not itself an independent constant, but should be chosen such that H(h∞) = 1. Furthermore, the ansatz (6.36) fixes ¯pi=µip¯0, and so all of the functionsHi_{(ρ) should}

be proportional toH0_{(ρ). This in turn tells us that H(H)∝(H}0₎3_{, and so the scalar} fields (6.31) are constant and equal to their asymptotic values.

Going back to the five-dimensional line element (6.30), we see that in this case the solution takes the form

ds2₍₅₎= 1 H0 −W dt 2_+dy2 + (H0)2 dρ2 W +ρ 2_dΩ2 2 ,

which is simply the Reissner-Nordstr¨om (RN) black string [88]. Hence, for the generic case, our ansatz (6.23) with (6.36) produces the magnetically-charged five-dimensional RN black string.

Between these two extremes (‘diagonal’ models vs. generic models) there are those for which ˆgij and its derivatives admit a block decomposition. In this case, the ansatz

(6.36) restricts each of the integration constants within a block to be proportional to one another. For k blocks we therefore obtain a solution depending on k harmonic functions, as in [47, 48].

It turns out, however, that there exist a large class of models for which (6.36) is not the most general ansatz one can make [43]. In particular, suppose that we can find some constant matrixRij 6=±δji which leaves ˆgij invariant, i.e.

as matrices. Then we can solve (6.35) by setting ¯ pi−chi∞=±Rij q c2_(hj ∞)2+ (˜pj)2, (6.38) where the sign is chosen such that sign(hi

∞) = sign(¯pi), and by taking the integra- tion constants to again be proportional to each other within each block of the metric. However, we now see that having all ofhi_∞ and ˜pi proportional to each other does not necessarily imply that all of the ¯pi should be proportional, and hence that the scalar fields hi(ρ) need not be constant.

Hence, for models admitting an ‘R-matrix’, we can find solutions depending on a reduced number of integration constants but which nevertheless have non-constant scalar fields.