Colliding Branes and Formation of Spacetime Singularities

We construct a class of analytic solutions with two free parameters to the five-dimensional Einstein field equations, which represents the collision of two timelike 3-branes. We study the local and global properties of the spacetime, and find that spacelike singularities generically develop after the collision, due to the mutual focus of the two branes. Non-singular spacetime can be constructed only in the case where both of the two branes violate the energy conditions.

Recently, Maeda and his collaborators numerically studied the collision of two branes in a five-dimensional bulk, and found that the formation of a spacelike singularity after the collision is generic [6]. This is a very important result, as it implies that a low-energy description of colliding branes breaks down at some point, and a complete predictability is lost, without the complete theory of quantum gravity. Similar conclusions were obtained from the studies of two colliding orbifold branes [7]. However, lately it was argued that, from the point of view of the higher dimensional spacetime where the low effective action was derived, these singularities are very mild and can be easily regularised [8].
In this paper, we present a class of analytic solutions to the five-dimensional Einstein field equations, which represents the collision of two timelike 3-branes in a fivedimensional vacuum bulk, and show explicitly that a spacelike singularity always develops after the collision due to the mutual focus of the two branes, when both of them satisfy the energy conditions. If only one of them satisfies the energy conditions, spacetime singularities also always exist either before or after the collision. Non-singular spacetimes can be constructed but only in the case where both of the two branes violate the energy conditions. Specifically, the paper is organized as follows: in the next section we present such solutions, and study their local and global properties, while in Section III we give our main conclusions and remarks. * Electronic address: Andreas˙Tziolas@baylor.edu † Electronic address: Anzhong˙Wang@baylor.edu

II. COLLIDING TIMELIKE 3-BRANES
Let us consider the solutions, 3,4), and with a, b and A 0 being arbitrary constants, and H(x) the Heavside function, defined as Without loss of generality, we assume a = −b and A 0 > 0. Then, it can be shown that the corresponding spacetime is vacuum, except on the hypersurfaces t = ay and t = −by, where the non-vanishing components of the Einstein tensor are given by where δ(x) denotes the Dirac delta function. As to be explained below, with the proper choice of the free parameters a and b, on each of these two hypersurfaces the spacetime represents a 3-brane filled with a perfect fluid. The normal vector to the surfaces t − ay = 0 and t + by = 0 are given, respectively, by for which we find Thus, in order to have these surfaces be timelike, we must choose a and b such that Introducing the timelike vectors u A and v A along each of the two 3-branes by From the five-dimensional Einstein field equations, G AB = κT AB , we obtain (2.11) are unit vectors, defined as X 2, 3, 4; a = u, v), and .
(2.12) Therefore, the solutions in the present case represent the collision of two timelike 3-branes, moving along, respectively, the line t − ay = 0 and the line t + by = 0. Each of the two 3-branes supports a perfect fluid. They approach each other as t increases, and collide at point (t, y) = (0, 0), and then move apart. Depending on the specific values of the free parameters a and b, we have three distinguishable cases: (a) a, b < −1; (b) a > 1, b < −1; and (c) a, b > 1. The case a < −1, b > 1 can be obtained from Case (b) by exchanging the two free parameters. In the following let us consider them separately.
The two 3-branes approach each other from t = −∞ and collide at (t, y) = (0, 0). Due to their gravitational mutual focus, the spacetime ends up at a spacelike singularity on the hypersurface A0 + (a + b)t = 0 in Region IV , denoted by the horizontal dashed line. The spacetime is also singular along the line A0 − |a|(t − |b|y) = 0 (A0 − |b|(t + |a|y) = 0) in Region III (II), which is parallel to the 3-brane located on the hypersurface t+by = 0 (t−ay = 0).
In this subcase, from Eq.(2.12) we can see that the perfect fluids on both of the two branes satisfy all the three energy conditions, weak, strong, and dominant [9]. To study the solutions further, we divide the spacetime into four regions, Region I: t < 0, t/|b| < y < t/|a|, Region II: y > 0, −|a|y < t < |a|y, Region III: y < 0, |b|y < t < −|a|y, and Region IV: t > 0, −t/|a| < y < t/|b|, as shown in Fig. 1, with the two 3-branes as their boundaries, where we denote them, respectively, as, Along the hypersurface Σ v , we find Exchanging the free parameters a and b we can get the corresponding expressions for the brane located on the hypersurface t − ay = 0. From these expressions and Eq.(2.12) we can see that the two 3-branes come from t = −∞ with constant energy densities and pressures, for which the spacetime on each of the branes is Minkowski. After they collide at the point (t, y) = (0, 0), they focus each other, wherė a v,u (τ ) < 0, and finally end up at a singularity where a v,u (τ ) = 0, denoted, respectively, by the point A and B in Fig. 1.
The spacetime outside the two 3-branes are vacuum, and the function A(t, y) is given by From this expression we can see that the spacetime is Minkowski in Region I and the function A(t, y) vanishes on the hypersurfaces Fig. 1. These hypersurfaces actually represent the spacetime singularities. This can be seen clearly from the Kretschmann scalar, (2.16) The above analysis shows clearly that, when the matter fields on the two branes satisfy the energy conditions, due to their mutual gravitational focus, a spacelike singularity is always formed after the collision. This is similar to the conclusion obtained by Maeda and his collaborators [6].
In this case, Eq.(2.12) shows that the perfect fluid on the brane t = ay satisfies all the three energy conditions, while the one on the brane t = −by does not. To study these solutions further, it is found convenient to consider the two cases a > |b| > 1 and |b| > a > 1 separately.
Case 2.1) a > |b| > 1: In this case, the two colliding branes divide the whole spacetime into the following four regions, I: t = < ay, y < 0, < |b|y, y > 0, II: y < 0, ay < t < |b|y, III: y > 0, |b|y < t < ay, The spacetime on the 3-brane located on the hypersurface t − ay = 0 is flat before the collision, but starts to expand as au(η) ∝ (η + η0) 1/2 after the collision. This 3-brane is free of any kind of spacetime singularities. as shown in Fig. 2. Then, we find that Clearly, the spacetime is again Minkowski in Region I, but the function A(t, y) now vanishes only on the hypersurfaces A 0 + (a − |b|) t = 0 in Region IV , and A 0 − |b| (t − ay) = 0 in Region II, denoted by the dashed lines in Fig. 2. Similar to the last case, the Kretschmann scalar blows up on these surfaces, so they actually represent the spacetime singularities. As a result, the region A 0 /|b| + ay < t < −A 0 /(a − |b|), y < 0, denoted by D in Fig. 2, is not part of the whole spacetime. In Region III we have A(t, y) > 0, and no any kind of spacetime singularities appears in this region. Along the hypersurface t+ by = 0, the metric takes the same form as that given by Eq.(2.13) but now with where t = t s ≡ −A 0 /(a − |b|) corresponds to τ = τ s and Thus, in this case the 3brane located on the hypersurface t + by = 0 starts to expand from the singular point τ = τ s and collides with the other incoming 3-brane at the point (t, y) = (0, 0). After the collision, the 3-brane transfers part of its energy to the one moving along the hypersurface t−ay = 0, so that its energy density and pressure remain constant, and whereby the spacetime on this 3-brane becomes Minkowski.
Along the hypersurface t − ay = 0, the metric takes the form of Eq.(2.20) but now with Thus, in the present case the brane located on the hypersurface t − ay = 0 comes from t = −∞ with energy density and pressure ρ u = −3p u ∝ (η 0 − η) −1 , which satisfies all the three energy conditions. The spacetime on this brane is non-flat before the collision and becomes flat after the collision.
Along the line t + by = 0, the metric takes the same form as that given by Eq.(2.13) but now with (2.24) where t = t s ≡ A 0 /(|b| − a) corresponds to τ = τ s and . Thus, in this case the 3-brane located on the hypersurface t + by = 0 moves in from t = −∞ and has constant energy density and pressure before the collision. After the collision, it collapses to a singularity at τ = τ s .
In this subcase, from Eq.(2.12) we can see that both of the two branes violate all the three energy conditions [9]. Dividing the spacetime into the following four regions, II: y > 0, −by < t < ay, III: y < 0, ay < t < −by, as shown in Fig. 4, we find that which are non-zero in the whole spacetime. Thus, in the present case the spacetime is free of any kind of spacetime singularities, and flat in Region I. Before the collision the two branes move in from t = −∞ all with constant energy density and pressure. After the collision, their energy densities and pressures all decrease like τ −1 , while the spacetime on these two branes is expanding like a(τ ) ∝ τ 1/2 , where τ is the proper time on each of the two branes, and a(τ ) their expansion factor.

III. CONCLUSIONS
In this paper, we have studied the collision of branes and the formation of spacetime singularities. We have constructed a class of analytic solutions to the fivedimensional Einstein field equations, which represents such a collision, and found that when both of the two 3-branes satisfy the energy conditions, a spacelike singularity is always developed after the collision, due to their mutual gravitational focus. This is consistent with the results obtained by Maeda and his collaborators [6].
When only one of the two branes satisfies the energy It is free of any kind of spacetime singularities in the whole spacetime, including the two hypersurfaces of the 3-branes. The two 3-branes all come from t = −∞ with constant energy density and pressure. They remain so until the moment right before collision. After the collision, the spacetime on each of the 3-branes is expanding like a(τ ) ∝ τ 1/2 , while their energy densities and pressures decrease like ρ = −3p ∝ τ −1 .
conditions, the other brane either starts to expands from a singular point [cf. Fig. 2], or comes from t = −∞ and then focuses to a singular point after the collision [cf. Fig. 3]. However, if both of the two colliding 3branes violate the weak energy condition, no spacetime singularities exist at all in the whole spacetime. Before the collision, the two branes approach each other in a flat background with constant energy densities and pressures. After they collide at (t, y) = (0, 0), they start to expand as a(τ ) ∝ τ 1/2 , where a(τ ) denotes their expansion factor, and τ their proper time. As the branes are expanding, their energy densities and pressures decrease as ρ, p ∝ τ −1 , in contrast to that of ρ, p ∝ τ −2 in the four-dimensional FRW model. As argued in [8], these singularities may become very mild when the five-dimensional spacetime is left to higher dimensional spacetimes, ten dimensions in string theory and eleven in M-Theory, a question that is under our current investigation.

ACKNOWLEDGMENTS
The financial assistance from the vice provost office for research at Baylor University is kindly acknowledged.