高维理想流体与空奇异边界
Higher-dimensional perfect fluids and empty singular boundaries
摘要：
为了探讨在高维引力场中是否会出现空奇异边界，本文研究了（$N+2$）维静态超平面对称理想流体满足状态方程$ρ=η\, p$的Einstein方程的解，其中$ρ$为任意常数，$N\geq2$为常数.我们证明了这个时空有一些奇怪的性质。特别是，在$η>-1$的情况下，它有一个空的（没有物质）排斥奇异边界。我们还研究了测地线的行为和无质量标量场在该时空中传播的柯西问题。对于$η>1$，我们发现，只有垂直零测地线接触的边界和反弹，他们都开始和结束在$z=\infty$ ;而非垂直零以及所有的类时的两个平面之间的初始条件确定的边界。我们得到了无质量标量场传播的柯西问题是适定的，并且波在奇点处完全反射，只要我们要求波具有有限的能量，尽管不需要边界条件.
英文摘要：
In order to find out whether empty singular boundaries can arise in higher dimensional Gravity, we study the solution of Einstein's equations consisting in a ($N+2$)-dimensional static and hyperplane symmetric perfect fluid satisfying the equation of state $ρ=η\, p$, being $ρ$ an arbitrary constant and $N\geq2$. We show that this spacetime has some weird properties. In particular, in the case $η>-1$, it has an empty (without matter) repulsive singular boundary. We also study the behavior of geodesics and the Cauchy problem for the propagation of massless scalar field in this spacetime. For $η>1$, we find that only vertical null geodesics touch the boundary and bounce, and all of them start and finish at $z=\infty$; whereas non-vertical null as well as all time-like ones are bounded between two planes determined by initial conditions. We obtain that the Cauchy problem for the propagation of a massless scalar field is well-posed and waves are completely reflected at the singularity, if we only demand the waves to have finite energy, although no boundary condition is required.
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