We now turn our attention to the remainder of the three-dimensional field content, namely the 2n(5)V + 3 scalar fields (wi,ζ˜i, ξ). Again, we impose spherical symmetry at the level of the equations of motion, so that they only depend on the coordinateτ. The goal is to solve the remaining field equations coming from (6.6) in order to determine the three-dimensional field configuration, and try to relate this geometrically to the scalar manifoldS.
6.4.1 Determining ξ
We first consider the equation of motion for the scalarξ. This is simply ¨ξ = 0 and can be solved by ξ(τ) = aτ +b for some integration constants a, b. Since ξ appears as a metric degree of freedom, we expect these constants to be fixed by considering suitable boundary conditions for the five-dimensional line element. In particular, we require that the line element becomes flat in the asymptotic (τ → 0) limit, and has a finite (non-zero) horizon area per unit length in the near horizon (τ → ∞) limit.
Looking at (6.7) in the τ → 0 limit, we see that both σ and ξ should tend to zero to ensure transverse asymptotic flatness, which fixesb= 0.
The horizon of the black string has topologyR×S2, with theRfactor corresponding
to the spatial worldvolume of the string. In order to have a finite horizon size per unit length of the string, we need the coefficients of both theS2 andRfactors in the metric
to be finite in the limitτ → ∞.
The S2 factor appearing in the five-dimensional line element can be read off by taking the τ → ∞ limit of (6.12) and plugging it into (6.7) to find
ds2(5) ⊃(2c)2e−4σ(τ)−2(a+c)τdΩ22.
In order that this integrates up to give a finite non-zero result as τ → ∞, the scalar fieldσ(τ) must have the asymptotic expansion
2σ(τ) = 2σhor−(a+c)τ as τ → ∞.
Substituting this into the part of (6.7) containing the worldvolume directions we have
ds2(5)⊃e2σhor−cτ−
1e−aτ(dx0)2−2eaτ(dx4)2
.
Precisely which factor corresponds to the spatial part of the horizon depends on whether we are considering a time-space (1 = 1, 2 = −1) or space-time (1 = −1, 2 = 1) reduction of the five-dimensional theory. For the former, wherex4 is a spatial direction, the requirement of finite (non-zero) horizon size imposes a = c, while for the latter,
where x0 is a spatial direction, we must takea=−c. Hence, in general, a=1c, and we find
ξ(τ) =1cτ. (6.17)
We can thus understand in what sense the scalarξ encodes the non-extremality of the spacetime solution.
We have seen, then, that the physical requirements that our metric describes an asymptotically flat black string with finite horizon area per unit length determines exactly the profile of ξ in terms of the parameter c, thereby reducing the number of independent integration constants by 2.
6.4.2 Determining ζ˜i
We now move on to the equations of motion for the scalars ˜ζi dual to the three-
dimensional field strengths. From (6.6) we find
d dτ ˆ gij(w)ζ˙˜j = 0,
which integrates to give
˙˜
ζi= ˆgij(w)˜pj, (6.18)
where the ˜pi are integration constants proportional to the magnetic charge of the so- lution under the five-dimensional gauge fields Ai
ˆ
µ. To see this, we use (6.8) to relate
the three-dimensional scalars (6.18) to the non-zero components of the five-dimensional field strength. In particular, we find
Fθϕi =−√1
2p˜
isinθ. (6.19)
Note that constant shifts in the scalars ˜ζi simply correspond to gauge transformations
of the five-dimensional gauge fields, and so further integrating (6.18) would not give rise to additional physical degrees of freedom.
(6.13), which now becomes 3 4c 2−gˆ ij(w) w˙iw˙j −p˜ip˜j = 0. (6.20) 6.4.3 Determining wi
Finally, the equations of motion for the scalarswi read, after using (6.18) and the fact that the metric ˆgij is Hessian,
ˆ gij(w) ¨wj+ 1 2∂igˆjk ˙ wjw˙k−p˜jp˜k= 0. (6.21)
Due to the explicit dependence of (6.21) on ˆgij and its derivatives, it is difficult to solve
(6.21) in a model-independent way. For now we content ourselves with finding a class of explicit solutions which always contains the standard Reissner-Nordstr¨om black string. Following [49] we first contract (6.21) with wi, and use homogeneity of the metric
ˆ
gij, along with the Hamiltonian constraint (6.20), to arrive at
ˆ
gij(w)wi( ¨wj−c2wj) = 0. (6.22)
We can obtain our class of universal solutions by setting ¨wj−c2wj = 0, from which we obtain
wi(τ) =Aicosh(cτ) +B
i
c sinh(cτ), (6.23)
where the Ai and Bi are integration constants, and we have chosen the prefactors for later convenience. We can find relations between the integration constants by ensuring that the solutions (6.23) still satisfy the Hamiltonian constraint (6.20) and the full equations of motion (6.22), which become, respectively,
ˆ gij c2AiAj−BiBj + ˜pip˜j = 0, and ∂kgˆij c2AiAj−BiBj+ ˜pip˜j = 0.
It is convenient at this point to introduce the quantities
in terms of which the Hamiltonian constraint and equations of motion become
ˆ
gij(pip¯j−p˜ip˜j) = 0, ∂kˆgij(pip¯j −p˜ip˜j) = 0. (6.25)
It will turn out when we look at the full five-dimensional solution that the pi,p¯i
encode the behaviour of the five-dimensional scalar fields at the inner and outer horizons respectively.
Writing the ansatz (6.23) in terms of the radial coordinate ρ defined in (6.14) we have wi(ρ) = Ai+p i ρ W−12 :=Hi(ρ)W− 1 2, (6.26)
where W is given in (6.16), and the functions Hi(ρ) are harmonic in the three-
dimensional transverse space.
This completes the determination of the three-dimensional instanton solution. To recap, the three-dimensional line element is given in terms of the affine coordinate τ
by (6.12), and in terms of the isotropic radial coordinate ρ by (6.15). The scalar field
ξ is fixed in terms of the parameter c by (6.17) or, using (6.16), by
ξ(ρ) =−1
21logW(ρ). (6.27)
The scalar fields ˜ζi satisfy (6.18) and encode the magnetic charges of the five-
dimensional solution via (6.19): in this sense we have already ‘dimensionally lifted’ these fields. Finally, the scalar fieldswi are given in terms ofτ by (6.23) and in terms of ρ by (6.26). However, the wi as they stand still depend on 2(n(5)V + 1) undeter- mined integration constants, which should be fixed in terms of the ‘physical charges’ of the solution. Given the relations (6.25) this will necessarily have to be achieved in a model-dependent fashion.