Black holes in higher dimensions

爱因斯坦的广义相对论是一个数学理论，目前提供了在我们的宇宙中观察到的引力的最准确的描述。它预言存在质量如此之大的物体，以至于它们扭曲了空间（和时间），从而创造了一个任何东西都无法逃脱的区域-这样的区域被称为黑洞。黑洞为引力的研究提供了如此极端的环境，以至于在量子力学（基本粒子理论）中研究的效应变得非常重要。因此，黑洞的研究是一个理想的竞技场，在这个竞技场中，我们可以进一步理解引力和量子力学是如何统一起来的，从而形成一个长期以来一直在追求的量子引力理论。从表面上看，我们的宇宙似乎有三个空间维度和一个时间维度：这些维度合在一起被称为四维时空。一个基本的问题是，事实上是否有更多的维度我们看不见，要么是因为它们太小，要么是由于其他机制。事实上，弦理论，目前我们在统一引力理论和其他力方面最好的尝试之一，预言了额外维度的存在。还有一些更抽象的原因，说明为什么在四维以上的时空中研究引力和黑洞这样的物体是重要的。最近一个重要的理论发展是认识到某些五维量子引力理论实际上等价于普通四维时空中的粒子物理学理论。这导致了令人兴奋的可能性，我们可以了解粒子物理理论的制度，无法使用标准的技术，通过研究五维理论的引力。开放的问题，在高维广义相对论我建议调查：（1）分类所有可能的平衡（时间无关）黑洞和（2）这些黑洞的动力学稳定性。在四维时空中，这两个数学问题都得到了解决。事实证明，一旦指定了它的质量，自旋和电荷，就会有一个独特的黑洞，并且它是稳定的。此外，它的事件视界（即黑洞的边界）不可能有任何洞-就像（被压扁的）球的表面一样。这些结果显然具有天体物理学的重要性。在高维数中，这些问题要复杂得多。其中一个原因是黑环的存在：一个具有甜甜圈状事件视界的五维黑洞解。它表明事件视界可以有洞，因此不需要是球形的。此外，与四维不同，质量，自旋和电荷不足以指定黑洞，因为人们可以同时具有球形和环状视界。我打算在问题（1）和（2）上取得进展，方法是集中研究黑洞的某些子集，这些子集虽然在物理上仍然令人感兴趣，但在数学上更易于分析。我已经在发展针对问题（1）的新方法，把重点放在黑洞的某些子集上，并成功地用这些方法回答了某些悬而未决的问题。我也已经对问题（2）进行了研究，对高度对称黑洞的某个子集进行了第一次稳定性分析。这给了我在（1）和（2）方面的丰富经验，我打算用它来解决更一般的黑洞子集的这些问题，这些高维广义相对论问题的结果可以直接应用于弦理论和量子引力，我希望研究这些问题。他们将有助于解决重要的开放问题，如弦理论中的黑环的量子描述和普通四维粒子物理理论中某些黑洞的量子描述。解决这些问题将加深我们对量子引力的理解。

Einstein's theory of General Relativity is a mathematical theory which currently provides the most accurate description of the force of gravity as observed in our universe. It predicts the existence of objects so massive that they distort space (and time) in such a way as to create a region from which nothing can escape -- such a region is called a black hole. Black holes provide such extreme settings for the study of gravitation that effects studied in quantum mechanics, the theory of the fundamental particles, become important. Therefore the study of black holes is an ideal arena within which to further our understanding of how gravity and quantum mechanics might be unified leading to a long sought after theory of quantum gravity.Superficially it appears that our universe has three spatial dimensions and one time dimension: together these are referred to as four dimensional space-time. A fundamental question is whether there are in fact more dimensions which we cannot see, either because they are too small or due to some other mechanism. In fact, String Theory, currently one of our best attempts at unifying the theory of gravity with the other forces, predicts the existence of extra dimensions. There are also more abstract reasons why it is important to investigate gravitation and objects such as black holes in more than four space-time dimensions. An important recent theoretical development is the realization that certain five dimensional theories of quantum gravity are in fact equivalent to particle physics theories in ordinary four dimensional space-time. This leads to the exciting possibility that we can learn about particle physics theories in regimes which are inaccessible using standard techniques, by studying five dimensional theories of gravity.The open problems within higher dimensional general relativity I propose to investigate are: (1) classification of all possible equilibrium (time independent) black holes and (2) dynamical stability of these black holes. In four space-time dimensions, both of these mathematical problems have been solved. It turns out that there is a unique black hole once one specifies its mass, spin and electric charge and it is stable. Furthermore, its event horizon (i.e. the boundary of the black hole) cannot have any holes - just like the surface of a (squashed) ball. These results are of clear astrophysical importance.In higher dimensions these problems are much more complicated. One reason for this is the existence of the black ring : a five dimensional black hole solution with a doughnut-like event horizon. It shows that an event horizon can have holes and thus need not be spherical. Furthermore, unlike in four dimensions, the mass, spin and electric charge are insufficient to specify a black hole as one can have both spherical and ring-like horizons. I intend to make progress on problems (1) and (2) by focusing on certain subsets of black holes which, while still physically interesting, are mathematically more amenable to analysis. I have already been developing new methods aimed at problem (1) by focusing on certain subsets of black holes and have used these successfully to answer certain open problems. I have also already worked on problem (2) by providing the first stability analysis for a certain subset of highly symmetric black holes. This has given me extensive experience with both (1) and (2) and I intend to use this to address these problems for more generic subsets of black holes.The results of these problems of higher dimensional general relativity have direct applications to string theory and quantum gravity, which I hope to study. They will contribute to the solution of important open problems such as the quantum description of black rings within string theory and the quantum description of certain black holes in terms of ordinary four dimensional particle physics theories. Solving such problems will deepen our understanding of quantum gravity.