关于视界的Birkhoff不变量和Thorne Hoop猜想
More about Birkhoff's Invariant and Thorne's Hoop Conjecture for Horizons
摘要：
最近在四维时空中对箍猜想的一个精确表述是，能量为$E$和面积为$A$的视视界的伯克霍夫不变量$β$（任何被圆扫出或叶化的最小最大长度）应该满足$β\le 4 πE$。这个猜想与宇宙审查或等周不等式一起，意味着最短非平凡闭测地线的长度$\ell$满足$\ell^2 \le πA$。我们已经测试了所有四个带电的旋转黑洞的解决方案的未规范的超引力理论的视野，并发现它们总是成立。在负宇宙学常数的存在下，以及在规范超引力下的多电荷旋转解中，它们仍然成立。令人惊讶的是，它们也适用于沉浸在磁场中的恩斯特-怀尔德静态黑洞，这是梅尔文解的渐近解。在5个时空维中，我们把$β$定义为二维环面对视界的所有扫掠的最小最大面积，并发现在所有检验的情况下，我们猜想的$ β(g) \le \frac{16 π}{3} E$一般对视视界保持安静。在偶数维时空$D=2N+2$中，我们发现，对于乘积$S^1 \times S^{D-4}$的清除，$β$从上到下被能量$E$的某个依赖于维数的倍数所限制。我们还发现，$\ell^{D-2}$从上方受视界面积$A$的某个维度相关倍数的限制。最后，我们表明，$\ell^{D-3}$是有界的，从上面的一定的维数依赖的能量的倍数，所有的Kerr-AdS黑洞。
英文摘要：
A recent precise formulation of the hoop conjecture in four spacetime dimensions is that the Birkhoff invariant $β$ (the least maximal length of any sweepout or foliation by circles) of an apparent horizon of energy $E$ and area $A$ should satisfy $β\le 4 πE$. This conjecture together with the Cosmic Censorship or Isoperimetric inequality implies that the length $\ell$ of the shortest non-trivial closed geodesic satisfies $\ell^2 \le πA$. We have tested these conjectures on the horizons of all four-charged rotating black hole solutions of ungauged supergravity theories and find that they always hold. They continue to hold in the the presence of a negative cosmological constant, and for multi-charged rotating solutions in gauged supergravity. Surprisingly, they also hold for the Ernst-Wild static black holes immersed in a magnetic field, which are asymptotic to the Melvin solution. In five spacetime dimensions we define $β$ as the least maximal area of all sweepouts of the horizon by two-dimensional tori, and find in all cases examined that $ β(g) \le \frac{16 π}{3} E$, which we conjecture holds quiet generally for apparent horizons. In even spacetime dimensions $D=2N+2$, we find that for sweepouts by the product $S^1 \times S^{D-4}$, $β$ is bounded from above by a certain dimension-dependent multiple of the energy $E$. We also find that $\ell^{D-2}$ is bounded from above by a certain dimension-dependent multiple of the horizon area $A$. Finally, we show that $\ell^{D-3}$ is bounded from above by a certain dimension-dependent multiple of the energy, for all Kerr-AdS black holes.
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