广义波坐标下的Burnett猜想
Burnett's conjecture in generalized wave coordinates
AI总结：
本研究证明广义相对论中满足广义波坐标条件的洛伦兹度量序列高频弱收敛时，极限度量满足Einstein--无质量Vlasov系统，其Vlasov场为对应微局部缺陷测度，证明借助Einstein方程的补偿现象。
摘要：
我们证明了广义相对论中满足广义波坐标条件时的Burnett猜想：即假设$\{g_n\\_\{n=1\}^\infty\}$是一列满足广义波坐标条件的洛伦兹度量（维度$d \geq 3$任意），且$g_n$以适当的弱“高频”方式收敛于$g$，则极限度量$g$满足Einstein--无质量Vlasov系统。此外，我们证明极限度量的Vlasov场可取为对应于该收敛的合适微局部缺陷测度。证明利用了基于Einstein方程线性与非线性结构的补偿现象。
英文摘要：
We prove Burnett's conjecture in general relativity when the metrics satisfy a generalized wave coordinate condition, i.e., suppose $\{g_n\}_{n=1}^\infty$ is a sequence of Lorentzian metrics (in arbitrary dimensions $d \geq 3$) satisfying a generalized wave coordinate condition and such that $g_n\to g$ in a suitably weak and "high-frequency" manner, then the limit metric $g$ satisfies the Einstein--massless Vlasov system. Moreover, we show that the Vlasov field for the limiting metric can be taken to be a suitable microlocal defect measure corresponding to the convergence. The proof uses a compensation phenomenon based on the linear and nonlinear structure of the Einstein equations.
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