Characterisation of the energy of Gaussian beams on Lorentzian manifolds: with applications to black hole spacetimes
Characterisation of the energy of Gaussian beams on Lorentzian manifolds: with applications to black hole spacetimes
复制标题
洛伦兹流形上高斯光束能量的表征：及其在黑洞时空中的应用
DOI：
--
复制
发表时间：
2013
期刊：
影响因子：
--
通讯作者：
Jan Sbierski
中科院分区：
文献类型：
--
作者：
Jan Sbierski
我们知道，利用高斯束近似可以证明，在一个一般的全局双曲洛伦兹流形上存在波动方程的解，该流形的能量在有限但任意长的时间内沿给定的零测地线局部化。在本文中，我们证明了这种局域解的能量是由底层零测地线的能量决定的。这一结果为高斯光束在不允许全局类时杀戮向量场的洛伦兹流形上的各种应用打开了大门。特别地，我们证明了在Kerr黑洞的外部或在极端Reissner-Nordstr\ ' m黑洞的视界中捕获必然导致局部能量衰减陈述中的“导数损失”。我们还展示了史瓦西黑洞视界上的红移效应对未来散射结构（红移变成蓝移）形成的阻碍：我们构造了能量在有限但任意长时间内呈指数增长的逆向问题的解。最后，我们给出了亚极值和极值黑洞柯西视界附近蓝移效应的一个简单的数学实现：我们构造了波方程的一系列解，其初始能量是均匀有界的，而柯西视界附近的能量趋于无穷大。
It is known that using the Gaussian beam approximation one can show that there exist solutions of the wave equation on a general globally hyperbolic Lorentzian manifold whose energy is localised along a given null geodesic for a finite, but arbitrarily long time. In this paper, we show that the energy of such a localised solution is determined by the energy of the underlying null geodesic. This result opens the door to various applications of Gaussian beams on Lorentzian manifolds that do not admit a globally timelike Killing vector field. In particular we show that trapping in the exterior of Kerr or at the horizon of an extremal Reissner-Nordstr\"om black hole necessarily leads to a `loss of derivative' in a local energy decay statement. We also demonstrate the obstruction formed by the red-shift effect at the event horizon of a Schwarzschild black hole to scattering constructions from the future (where the red-shift turns into a blue-shift): we construct solutions to the backwards problem whose energies grow exponentially for a finite, but arbitrarily long time. Finally, we give a simple mathematical realisation of the heuristics for the blue-shift effect near the Cauchy horizon of sub-extremal and extremal black holes: we construct a sequence of solutions to the wave equation whose initial energies are uniformly bounded, whereas the energy near the Cauchy horizon goes to infinity.
影响因子：
4.9
作者：
Andersson, Lars;Blue, Pieter
通讯作者：
Blue, Pieter