Foliations of null hypersurfaces and the Penrose inequality
Foliations of null hypersurfaces and the Penrose inequality
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DOI：
10.3929/ethz-a-005713669
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发表时间：
2008
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影响因子：
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通讯作者：
J. Sauter
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作者：
J. Sauter
本文的出发点和动机在于研究渐近平坦真空时空(M，g)的零情形下的Penrose不等式，假设给定的球拓扑类空边缘外陷面S0的过去光锥C−S0正则延伸到过去的零无穷远。然后，这个不等式读作MB，≥√，A0，16，π，MB，C，−，S0，S0的面积。我们描述了一种策略，该策略被认为可能导致通过使用沿零超曲面C−S0的曲面流并从S0开始，其中曲面的运动定律必须以这样的方式来选择，即某个相关的质量泛函-霍金质量-是单调的。我们给出了两个具有所需单调性的运动定律，其中一个早已为人所知。然而，所产生的叶理的存在或收敛性质从未被研究过。因此，我们自然地引出了C−S0上这两个叶的局部和整体存在性的问题。更一般地，我们证明了一类可以用逆衰减定律描述的叶理的局部存在性，前面提到的两个叶理只是它的特例。然后，我们证明了两个感兴趣的叶的整体存在性，只要零锥C−S0以精确定义的方式接近于Schwarzschild时空的球对称过去光锥。这特别允许分析所得到的叶的渐近行为，这表明所提出的策略总体上不会导致彭罗斯不等式的证明。在分析几个明确的例子以检测这些问题的原因时，我们得到了一类无剪圆锥，它具有特别简单的结构，并且还具有至少一个上述叶层全局存在于C−S0上的性质。因此，这些锥体非常适合于说明与所提出的策略相关的问题，更是因为几何是显式可积的。然而，这一事实允许以某种方式直接分析彭罗斯不等式，导致在无剪切情况下对后者的基本但仍有启发的证明。
Starting point and motivation for this thesis rest upon a study of the Penrose inequality in the null case for asymptotically flat vacuum spacetimes (M, g) under the assumption that the past light cone C− S0 of a given spacelike marginally outer trapped surface S0 of spherical topology extends regularly to past null infinity. The inequality then reads mB ≥ √ A0 16π with mB the Bondi mass of C − S0 and A0 the area of S0. We describe a strategy which has been thought to possibly lead to a proof of this inequality by using a flow of surfaces along the null hypersurface C− S0 and starting from S0, where the law of motion of the surfaces has to be chosen in such a way that a certain associated mass functional – the Hawking mass – is monotone. We present two laws of motion with the desired monotonicity property one of which has been known for a long time. However, neither existence nor convergence properties of the resulting foliation have ever been investigated. We are thus naturally led to the question of local as well as global existence of these two foliations on C− S0. More generally we prove local existence for a whole class of foliations which may be described by an inverse lapse law, the two foliations mentioned before just being particular examples thereof. We then prove global existence for the two foliations of interest provided that the null cone C− S0 is close – in a precisely defined manner – to the spherically symmetric past light cone of Schwarzschild spacetime. This in particular allows to analyze the asymptotic behavior of the resulting foliations which indicates that the proposed strategy does not lead to a proof of the Penrose inequality, in general. In analyzing several explicit examples in order to detect the reason for these problems we are led to the class of shearfree cones which are of particularly simple structure and also have the property that at least one of the aforementioned foliations exists globally on C− S0 . Hence these cones are well suited to illustrate the problems associated with the proposed strategy, all the more because the geometry is explicitly integrable. This fact however allows to analyze the Penrose inequality in a somehow direct way, leading to an elementary but nevertheless instructive proof of the latter in the shearfree case.