Black holes in higher dimensions

广义相对论是描述引力的最佳理论。其中最引人注目的预言之一是黑洞的存在。后者是任何东西，甚至光，都无法逃离的时空区域。在四个时空维度中，平衡黑洞的特性被很好地理解了：它们是由它们的质量和自旋唯一指定的，它们是球形的，它们是动态稳定的。最后一个性质非常重要，因为它使它们成为恒星引力坍缩的终点，在天体物理学上具有相关性。事实上，根据我们目前的理解，在一颗足够大的恒星耗尽其核燃料后，它别无选择，只能坍缩并形成一个黑洞。然而，黑洞内部有奇点，在那里引力变得如此极端，以至于广义相对论失效。在这种情况下，需要一种结合了量子力学定律和广义相对论的新理论，即量子引力理论。近年来，由于各种原因，对大于4维的广义相对论的研究受到了很多关注。首先，广义相对论包含在弦理论中，而弦理论是量子引力理论的最佳候选。然而，它的一致性需要超过四个时空维度。其次，根据最近的发展，弦理论（以及引力）在特定时空中相当于普通的粒子理论，没有引力，在少一个时空维度。典型地，在弦理论简化为广义相对论的状态下，相应的粒子物理理论不能用粒子物理的标准方法来描述。因此，五维引力，特别是黑洞，为解决标准四维粒子理论中的问题提供了一种新的工具。这个新工具已被证明在各种上下文中非常有用。例如，已经有可能了解加速器（如大型强子对撞机）中产生的夸克-胶子等离子体的一些特性。最后，一些理论预测在大型强子对撞机中会产生更高维度的黑洞。我研究的主要目标是更好地理解高维黑洞。为此，我提出：研究黑洞的动力学稳定性，寻找新的高维黑洞，研究黑洞的形成。高维黑洞的一个新特性是，如果它们旋转得足够快，它们就会变得不稳定。在四维空间中，这是不可能发生的，因为黑洞的自旋总是有上界的。然而，这些不稳定性的细节尚不完全清楚。在我的研究中，我将对这些不稳定性进行分析和数值研究。这些不稳定性通常是新黑洞存在的信号。因此，他们的研究应该为我们提供有关黑洞种类以及它们之间关系的线索。其次，高维黑洞的一般性质还没有被完全理解。特别是，我们甚至不知道会有什么样的黑洞。例如，已经在五个维度上发现了甜甜圈形状的黑洞，它们明确地表明，除了质量和自旋之外，还需要其他参数来完全描述黑洞。在我的项目中，我将开发系统地分析发现新黑洞的方法。最后，我将使用五维重力来研究相应的粒子平衡理论。这个过程不能用标准方法进行分析研究，但从引力的角度来看，它对应于黑洞的形成。此外，我的结果可能会对广义相对论中的奇点形成问题产生影响，这本身就是一个有趣的数学问题。

General relativity is our best theory to describe the gravitational force. Among one of its most striking predictions is the existence of black holes. The latter are regions of the spacetime from which nothing, even light, can escape. In four spacetime dimensions, the properties of equilibrium black holes are well-understood: they are uniquely specified by their mass and spin, they are spherically shaped, and they are dynamically stable. This last property is very important since it makes them astrophysically relevant as the endpoint of the gravitational collapse of a star. Indeed, according to our current understanding, after a sufficiently massive star has exhausted its nuclear fuel, it has no other option but to collapse and form a black hole. However, black holes contain in their interior singularities where the gravitational force becomes so extreme that general relativity breaks down. It is expected that in these situations a new theory that combines the laws of quantum mechanics with general relativity, namely a theory of quantum gravity, is required.In recent years the study of general relativity in dimensions greater than four has received a lot of attention for various reasons. First, general relativity is contained in string theory and the latter is our best candidate for a theory of quantum gravity. However, its consistency requires more than four spacetime dimensions. Second, according to recent developments, string theory (and hence gravity) in certain spacetimes is equivalent to an ordinary theory of particles, without gravity, in one spacetime dimension less. Typically, in the regime in which string theory reduces to general relativity the corresponding particle physics theory cannot be described by the standard methods in particle physics. Therefore, five-dimensional gravity, and in particular black holes, provide a novel tool to address problems in standard four-dimensional theories of particles. This new tool has proven to be very useful in a variety of contexts. For example, it has been possible to understand some properties of the quark-gluon plasma that is being produced in accelerators (e.g., the LHC). Finally, some theories predict the production of higher dimensional black holes at the LHC. The main goal of my research is to get a better understanding of higher dimensional black holes. To achieve this I propose to: study the dynamical stability of black holes, find new higher dimensional black holes and study the formation of black holes. One novel property that higher dimensional black holes have is that they can become unstable if they spin sufficiently fast. In four dimensions this cannot happen since the spin of a black hole always has an upper bound. However, the details of these instabilities are not fully understood. In my research I will study these instabilities both analytically and numerically. These instabilities typically signal the existence of new black holes. Hence, their study should give us clues about the kinds of black holes that there are and how they are related. Secondly, the general properties of higher dimensional black holes are not fully understood. In particular, it is not even known what kinds of black holes there can be. For instance, doughnut-shaped black holes have been discovered in five dimensions and they explicitly demonstrate that parameters other than the mass and the spin are needed to fully specify the black hole. In my project I will develop methods to systematically find new black holes analytically.Finally, I will use five-dimensional gravity to study the equilibration of the corresponding particle theory. This process cannot be studied analytically by standard methods, but from the gravity side it corresponds to the formation of a black hole. In addition, my results may have implications for the issue of singularity formation in general relativity, which is an interesting mathematical problem in its own right.