[论文解读] Numerical construction and critical behavior of Kaluza-Klein black holes

Michael Kalisch|arXiv (Cornell University)|Jan 1, 2018

Black Holes and Theoretical Physics参考文献 14被引用 5

一句话总结

本文通过数值方法研究了五维和六维时空中的Kaluza-Klein黑洞，重点关注局域黑洞与黑洞弦之间拓扑结构发生改变的临界点附近的临界行为。采用高精度数值方案，结果证实该临界点附近的物理量由与双锥度规预测一致的普适临界指数所支配，为双锥度规作为该转变解的局部模型提供了有力证据。

ABSTRACT

The idea of extra dimensions provides a promising approach to overcome various problems in modern physics. This includes theoretical as well as phenomenological aspects, such as the unification of the fundamental interactions or the hierarchy problem. Based on the seminal works by Kaluza and Klein that were published nearly 100 years ago, we denote theories with at least one compact periodic dimension as Kaluza-Klein theories. From a gravitational point of view the question arises, what are the fundamental solutions to Einstein's field equations of general relativity under these assumptions. In particular, in this work we are concerned with black hole solutions in Kaluza-Klein theory. Considering only the static case without electric charge, it turns out that there is a much richer phase space than in the usual four-dimensional theory, where only the Schwarzschild solution exists. There are at least two types of solutions with a completely different horizon topology: localized black holes with an ordinary spherical horizon and black strings with a horizon that wraps the compact dimension. Several arguments favor the conjecture that the solution branches of both types are connected via a singular topology changing solution that is controlled by the so-called double-cone metric. We study the regime close to this singular transit solution in five and six spacetime dimensions with the help of a highly accurate numerical scheme that we describe in detail. Consequently, for the first time we are able to show that in this regime the black objects exhibit a critical behavior, indicating that physical quantities are governed by universal critical exponents. Interestingly, such exponents were already derived from the double-cone metric. We show that our data confirms these values extremely well. This provides compelling evidence in favor of the double-cone metric as the local model of the transit solution.

研究动机与目标

• 探索高维中静态、不带电的Kaluza-Klein黑洞的丰富相空间，其中包含局域黑洞和黑洞弦两种类型。
• 研究连接这两种相的奇异拓扑结构转变解的性质，该解被推测由双锥度规描述。
• 数值分析该转变点附近的临界区域，以检验临界指数的普适性。
• 提供证据支持双锥度规作为临界转变解的局部模型。

提出的方法

• 在Kaluza-Klein紧化条件下，采用高精度数值方案求解五维和六维时空中的爱因斯坦场方程。
• 聚焦于静态、不带电的情形，以隔离拓扑转变附近的引力动力学。
• 在接近奇异双锥度规的区域构建数值解，以探测临界行为。
• 分析临界点附近的物理量，如质量、视界面积和曲率不变量。
• 将这些物理量的标度行为与双锥度规的理论预测进行比较。
• 利用观测到的标度指数检验双锥度规作为转变解局部模型的有效性。

实验结果

研究问题

• RQ1 Kaluza-Klein黑洞在局域黑洞与黑洞弦之间发生拓扑结构转变的临界点附近，其临界行为的性质是什么？
• RQ2 在数值模拟中观测到的临界指数是否与双锥度规所预测的普适值一致？
• RQ3 双锥度规是否为连接两种黑洞相的奇异解的有效局部模型？
• RQ4 在五维和六维时空中，质量与视界面积等物理量在临界点附近的标度行为如何？
• RQ5 有哪些数值证据支持在不同视界拓扑之间存在连续相变？

主要发现

• 数值结果证实，临界点附近的物理量表现出由普适临界指数支配的标度行为。
• 测得的临界指数与从双锥度规推导出的数值高度一致，为该度规的有效性提供了强有力的定量支持。
• 本研究首次提供了Kaluza-Klein黑洞在拓扑结构转变附近出现临界行为的数值证据。
• 五维和六维时空中临界区域的标度行为一致，表明其在不同维度间具有普适性。
• 数据强烈支持双锥度规准确描述局域黑洞与黑洞弦之间奇异转变解的猜想。
• 数值方案成功捕捉了临界点附近的精细物理行为，实现了对临界指数的精确测量。