Black holes in higher dimensions

爱因斯坦的广义相对论是一种数学理论，它目前对我们在宇宙中观察到的引力提供了最准确的描述。它预言存在一些质量巨大的物体，它们会扭曲空间（和时间），从而形成一个任何东西都无法逃脱的区域——这样的区域被称为黑洞。黑洞为研究引力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，对黑洞的研究是一个理想的领域，在这里我们可以进一步了解如何将引力和量子力学统一起来，从而得出一个长期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：这两个维度合在一起被称为四维时空。一个基本的问题是，是否真的有更多的维度是我们看不到的，或者是因为它们太小，或者是由于一些其他的机制。事实上，弦理论预测了额外维度的存在，弦理论是目前我们将引力理论与其他力统一起来的最佳尝试之一。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等物体很重要。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同于普通四维时空中的粒子物理理论。这导致了一种令人兴奋的可能性，即我们可以通过研究五维引力理论，在标准技术无法达到的状态下了解粒子物理理论。我建议研究高维广义相对论中的开放问题是：(1)所有可能的平衡（时间无关）黑洞的分类和(2)这些黑洞的动力稳定性。在四维时空中，这两个数学问题都得到了解决。事实证明，一旦确定了黑洞的质量、自旋和电荷，就会有一个独特的黑洞，它是稳定的。此外，它的视界（即黑洞的边界）不能有任何洞——就像一个（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。在更高的维度中，这些问题要复杂得多。其中一个原因是黑环的存在：一个具有甜甜圈状视界的五维黑洞解决方案。它表明事件视界可以有洞，因此不必是球形的。此外，与四维空间不同，质量、自旋和电荷不足以确定黑洞，因为黑洞既可以有球形视界，也可以有环状视界。我打算在问题(1)和(2)上取得进展，重点关注黑洞的某些子集，这些子集虽然在物理上仍然很有趣，但在数学上更易于分析。我已经通过关注黑洞的某些子集开发了针对问题(1)的新方法，并成功地使用这些方法来回答某些开放问题。我也已经对问题(2)进行了研究，为高度对称黑洞的某个子集提供了第一个稳定性分析。这给了我关于(1)和(2)的丰富经验，我打算用它来解决黑洞更一般子集的这些问题。这些高维广义相对论问题的结果对弦理论和量子引力有直接的应用，这是我希望研究的。他们将有助于解决重要的开放问题，如弦理论中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.