Bianchi背景下Klein-Gordon方程的统一解
A unified approach to the Klein-Gordon equation on Bianchi backgrounds
摘要：
本文研究了Bianchi背景下Klein-Gordon方程的解。特别是，我们感兴趣的解决方案在沉默的奇异性方向的渐近行为。主要结论是，对于Klein-Gordon方程的给定解$u$，在所考虑的李群上存在光滑函数$u_{i}$，$i=0,1$，使得$u_σ(\cdot,σ)-u_{1}$和$u(\cdot,σ)-u_{1}σ-u_{0}$在奇点方向上渐近收敛到零（其中$σ$是几何定义的时间坐标，使得奇点对应于$σ\rightarrow-\infty$）。这里，$u_{i}$、$i=0,1$应该被认为是关于奇点的数据。有趣的是，我们有可能证明，对于一大类比安奇时空，渐近性是这种形式。此外，该结论适用于物质支配的奇点;真空支配的奇点;甚至当基本的比安奇时空的渐近是振荡的。总而言之，就静奇点方向上的渐近性而言，似乎有一种普遍性。事实上，人们很容易猜想，只要基本的比安奇时空的奇点是沉默的，那么解的渐近性就像上面描述的那样。为了对比上述渐近性与非沉默的设置，我们，通过呼吁已知的结果，提供一个完整的渐近表征的解决方案的Klein-Gordon方程在一个平坦的Kasner背景。在这种情况下，$u_σ$一般不会收敛。
英文摘要：
In this paper, we study solutions to the Klein-Gordon equation on Bianchi backgrounds. In particular, we are interested in the asymptotic behaviour of solutions in the direction of silent singularities. The main conclusion is that, for a given solution $u$ to the Klein-Gordon equation, there are smooth functions $u_{i}$, $i=0,1$, on the Lie group under consideration, such that $u_σ(\cdot,σ)-u_{1}$ and $u(\cdot,σ)-u_{1}σ-u_{0}$ asymptotically converge to zero in the direction of the singularity (where $σ$ is a geometrically defined time coordinate such that the singularity corresponds to $σ\rightarrow-\infty$). Here $u_{i}$, $i=0,1$, should be thought of as data on the singularity. Interestingly, it is possible to prove that the asymptotics are of this form for a large class of Bianchi spacetimes. Moreover, the conclusion applies for singularities that are matter dominated; singularities that are vacuum dominated; and even when the asymptotics of the underlying Bianchi spacetime are oscillatory. To summarise, there seems to be a universality as far as the asymptotics in the direction of silent singularities are concerned. In fact, it is tempting to conjecture that as long as the singularity of the underlying Bianchi spacetime is silent, then the asymptotics of solutions are as described above. In order to contrast the above asymptotics with the non-silent setting, we, by appealing to known results, provide a complete asymptotic characterisation of solutions to the Klein-Gordon equation on a flat Kasner background. In that setting, $u_σ$ does, generically, not converge.
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