# Colliding Branes and Formation of Spacetime Singularities

###### Abstract

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.

###### pacs:

98.80.Cq, 98.80.-k, 04.40.Nr## I Introduction

Branes in string/M-Theory are fundamental constituents strings , and of particular relevance to cosmology JST02 ; branes . These substances can move freely in bulk, collide, recoil, reconnect, and whereby form a brane gas in the early universe branegas , or create an ekpyrotic/cyclic universe cyc . Understanding these processes is fundamental to both string/M-Theory and their applications to cosmology.

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 MT04 . 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 chen06 . 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 LFT06 .

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 five-dimensional 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.

## Ii Colliding timelike 3-Branes

Let us consider the solutions,

(2.1) |

where , , and

(2.2) | |||||

with and being arbitrary constants, and the Heavside function, defined as

(2.3) |

Without loss of generality, we assume and . Then, it can be shown that the corresponding spacetime is vacuum, except on the hypersurfaces and , where the non-vanishing components of the Einstein tensor are given by

(2.4) | |||||

where denotes the Dirac delta function. As to be explained below, with the proper choice of the free parameters and , on each of these two hypersurfaces the spacetime represents a 3-brane filled with a perfect fluid.

The normal vector to the surfaces and are given, respectively, by

(2.5) |

for which we find

(2.6) |

Thus, in order to have these surfaces be timelike, we must choose and such that

(2.7) |

Introducing the timelike vectors and along each of the two 3-branes by

(2.8) |

where

(2.9) | |||||

we find . From the five-dimensional Einstein field equations, , we obtain

(2.10) |

where

(2.11) |

are unit vectors, defined as , and

(2.12) |

Therefore, the solutions in the present case represent the collision of two timelike 3-branes, moving along, respectively, the line and the line . Each of the two 3-branes supports a perfect fluid. They approach each other as increases, and collide at point , and then move apart. Depending on the specific values of the free parameters and , we have three distinguishable cases: (a) ; (b) ; and (c) . The case can be obtained from Case (b) by exchanging the two free parameters. In the following let us consider them separately.

### ii.1

In this subcase, from Eq.(II) we can see that the perfect fluids on both of the two branes satisfy all the three energy conditions, weak, strong, and dominant HE73 . To study the solutions further, we divide the spacetime into four regions, , , , and , as shown in Fig. 1, with the two 3-branes as their boundaries, where we denote them, respectively, as, and .

Along the hypersurface , we find

(2.13) | |||||

where

(2.14) |

with , and . Exchanging the free parameters and we can get the corresponding expressions for the brane located on the hypersurface . From these expressions and Eq.(II) we can see that the two 3-branes come from with constant energy densities and pressures, for which the spacetime on each of the branes is Minkowski. After they collide at the point , they focus each other, where , and finally end up at a singularity where , denoted, respectively, by the point and in Fig. 1.

The spacetime outside the two 3-branes are vacuum, and the function is given by

(2.15) |

From this expression we can see that the spacetime is Minkowski in Region and the function vanishes on the hypersurfaces in Region , in Region , and in Region , denoted by the dashed lines in 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 MT04 .

### ii.2

In this case, Eq.(II) shows that the perfect fluid on the brane satisfies all the three energy conditions, while the one on the brane does not. To study these solutions further, it is found convenient to consider the two cases and separately.

Case : In this case, the two colliding branes divide the whole spacetime into the following four regions,

I: | (2.17) | ||||

II: | |||||

III: | |||||

IV: |

as shown in Fig. 2. Then, we find that

(2.18) |

Clearly, the spacetime is again Minkowski in Region , but the function now vanishes only on the hypersurfaces in Region , and in Region , 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 , denoted by in Fig. 2, is not part of the whole spacetime. In Region we have , and no any kind of spacetime singularities appears in this region.

Along the hypersurface , the metric takes the same form as that given by Eq.(2.13) but now with

(2.19) |

where corresponds to and to , with , and . Thus, in this case the 3-brane located on the hypersurface starts to expand from the singular point and collides with the other incoming 3-brane at the point . After the collision, the 3-brane transfers part of its energy to the one moving along the hypersurface , so that its energy density and pressure remain constant, and whereby the spacetime on this 3-brane becomes Minkowski.

Along the hypersurface , the metric takes the form

(2.20) | |||||

where

(2.21) |

where corresponds to and . Thus, in the present case the brane located on the hypersurface comes from with constant energy density and pressure , which satisfies all the three energy conditions. The spacetime on this brane is flat before the collision. After the collision, the spacetime of the brane starts to expand as without the big-bang type of singularities. The expansion rate is the same as that of a radiation-dominated universe in Einstein’s theory of 4D gravity, where . But its energy density and pressure now decreases as , in contrast to in Einstein’s 4D gravity HE73 .

Case : In this case, the two colliding branes divide the whole spacetime into the four regions,

I: | (2.22) | ||||

II: | |||||

III: | |||||

IV: |

as shown in Fig. 3.

Following a similar analysis as we did in the last subcase one can show that the spacetime now is singular on the half lines in Region , and in Region , denoted by the dashed lines in Fig. 3.

Along the hypersurface , the metric takes the form of Eq.(2.20) but now with

(2.23) |

where corresponds to with . Thus, in the present case the brane located on the hypersurface comes from with energy density and pressure , 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 , the metric takes the same form as that given by Eq.(2.13) but now with

(2.24) |

where corresponds to and to , with . Thus, in this case the 3-brane located on the hypersurface moves in from and has constant energy density and pressure before the collision. After the collision, it collapses to a singularity at .

### ii.3

In this subcase, from Eq.(II) we can see that both of the two branes violate all the three energy conditions HE73 . Dividing the spacetime into the following four regions,

I: | (2.25) | ||||

II: | |||||

III: | |||||

IV: |

as shown in Fig. 4, we find that

(2.26) |

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 . Before the collision the two branes move in from all with constant energy density and pressure. After the collision, their energy densities and pressures all decrease like , while the spacetime on these two branes is expanding like , where is the proper time on each of the two branes, and 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 five-dimensional 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 MT04 . When only one of the two branes satisfies the energy conditions, the other brane either starts to expands from a singular point [cf. Fig. 2], or comes from and then focuses to a singular point after the collision [cf. Fig. 3]. However, if both of the two colliding 3-branes 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 , they start to expand as , where denotes their expansion factor, and their proper time. As the branes are expanding, their energy densities and pressures decrease as , in contrast to that of in the four-dimensional FRW model.

As argued in LFT06 , 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.

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