Constructing more non-singular Sp-branes

In this note, we construct an array of non-singular Sp (where p=D-4) branes in arbitrary $D$ dimensions starting from static solutions of black $p$ brane. These solutions carry nontrivial time dependent profiles of dilaton and associated form fields. We also study other non-singular time dependent configurations. These are obtained from proper analytic continuations of non-extremal diholes.

Non-singular time dependent solutions are very rare in gravity theories, be it supergravity or string theory. Therefore, the recent constructions of non-singular Sp-brane solutions [1] have attracted much attention [2]. Sp branes are time dependent gravitational configurations with p longitudinal directions [3]. However, most of these solutions are singular * . In a recent paper [1], non-singular S0 brane solutions are constructed by applying double-Wick rotations [1] to known black diholes [6]. Analytic continuation of these solutions give exact, time dependent, non-singular S0-brane in four dimensions. These solutions are non-singular in the sense that, all the conical singularities, which were there in the static solutions of black diholes, can be pushed off to the complex plane of ρ and t † . These solutions often lead to accelerating and bouncing cosmologies. In this note, we generalise these non-singular solutions by analytically continuning p branes of [7]. We construct (D − 4) dimensional Sp-brane solutions in D dimensions. These are the solutions of D dimensional maximal supergravity theories and hence can be embedded in string theories. We also construct other non-singular time dependent configurations by analytically continuing the non-extremal diholes of [8], [9]. This letter is organised as follows. First, we construct the non-singular Sp branes by Wick rotating the solutions of [7]. Here, we give various plots to expose the non-singular nature of the solutions. Subsequently, we analyse the double analytic continuation of non-extremal generalisation of black dihole which are different from [1], following the solutions given in [8], [9]. This allow us to construct non-singular S0 branes.
A general class of non-extremal black p brane solutions in arbitrary D dimensional space-time was constructed in [7]. The center of the brane sits along the transverse z axis. We take the linear superposition (unlike the non-linear superposition in the case of non-extremal black dihole) of the N individual p-brane potentials to get the potential of N number of branes along the z axis. As non-extremal branes exert forces on each other, the superposition of these branes is a tricky issue. This can be achieved however, if many of them are organised on a circle. This has been discussed in [7].
We start with the general supergravity action in D dimensions of the form Here φ is the dilaton field, and F (p+2) is the antisymmetric (p + 2) form field strength. The constant a denotes the dilaton coupling to the form field F (p+2) . From this action we can construct a static solution which corresponds to N (D − 4) branes organised along a common transverse axis. Below, we present the solution; for more detail we refer the reader to the original paper [7].
r n +r n − k n r n +r n + k n , (3) * Construction of S-branes through double Wick rotations are discussed in [4], but all these solutions suffer from singularities. For earlier construction of S brane solutions, see [5] † ρ is the radial direction of cylindrical polar coordinate system and t is the time coordinate.
In (2) τ , x i are the p + 1 world volume coordinates and ρ, z, θ are the transverse coordinates.
Branes are of masses k n = 2M n (for nth brane) and centered at points z n = nb (that is several branes are arranged periodically along the symmetry axis as discussed earlier). r n andr n are the distances from the brane center to the field point. Here we would like to make some comments on according to Ref. [10]. If we consider black hole (p = 0) solution in pure Einstein-Maxwell theory where there is no dilaton coupling in the action, then a = 0 and ∆ = 1 ‡ . We get back the array of standard neutral black hole solution by putting c = 0 (in general c can take values other than zero). Analytic continuation then gives S0-branes of [1]. For different values of a, we get different supergravity theories. In this note we will focus on the solutions with single scalar φ D , but there can be other solutions with more than one non-zero scalar field excited.
The double analytic continuation, τ → iy and z → it, changes the solutions to a time dependent one.
r n +r n − k n r n +r n + k n , (11) ‡ for more detail see [11] This is a non-singular Sp (p = D − 4) brane, as can be seen from the figures below. Here, we plot various metric components for N = 2 and ∆ = 4 (for M n < b). This value of ∆ appears in the ten dimensional type IIA supergravity theory where the dilaton coupling a to the eight form field strength is 3 2 ; see for example [12]. their common transverse direction. Here we give the values of e −2Ũ 0 and e −2K 0 following [13] : We are therefore able to construct non-singular Sp-branes by analytically continuing certain static black brane solutions. For particular choices of the dilaton coupling a, these solutions can be embedded in string theory.
In the next part of this note, we construct some new S0 brane configurations starting from nonextremal black dihole solutions. These non-extremal black dihole solutions are given in [8], [9]. With the choice of the parameters (e.g. mass, charge, angular momentum and shifts along the z axis of each hole) the axisymmetric solution written in cylindrical coordinate, is where A, B and C are given as The constant g in the expression of e 2K was determined in [9] by analysing proper location of the conical singularity along z axis. One can realize this singularity as a strut between the holes and also outside as a string suspended from infinity. g, computed in [9], is given by r ± ,r ± and κ ± are given by Black holes are lying along the symmetric z axis (where ρ = 0) centered at the points, z = κ + .
Here, 2κ + is the distance between the center of the two black holes. This distance is controlled by the parameters m, k and q. κ − is the non-extremal parameter in the solutions. In the extremal limit (m = q, κ − = 0) the solutions go back to the standard black dihole solutions. In order to obtain the extremal solution, one has to go to a suitable coordinate system [9]. Metric components diverges when κ − → 0 in the present coordinate system. Black hole horizons in this case also lie along the z axis at −κ + − κ − ≤ z ≤ −κ + + κ − and κ + − κ − ≤ z ≤ κ + + κ − . In the extremal limit, we get degenerate horizons like black dihole as described in [1]. r ± ,r ± in (19) are the radial distances to the point of observation from the inner and the outer edge of the black holes respectively.