论文标题
Minkowski时空的全球稳定性用于自旋1/2场
Global stability of Minkowski spacetime for a spin-1/2 field
论文作者
论文摘要
我们研究了爱因斯坦 - 迪拉克系统的初始值问题，并通过使用广义波坐标在无质量的情况下显示了Minkowski解决方案的稳定性。这需要在弯曲时空中理解狄拉克方程，我们为此建立了各种估计。该证明基于矢量场方法，该方法广泛用于其他爱因斯坦耦合系统的Minkowski问题稳定性的作品。在四局的特定选择下，我们表明，通过解决衍生物问题的潜在损失，Dirac场的组成部分满足了准电波方程。我们还表明，该方程的半线性非线性的表现就像无效形式。这样，我们沿光锥获得了田野的尖锐衰变。能量量张量的结构会导致公制的某些组成部分的行为差，而不是vaccum情况，但是额外的结构表明，对全球存在没有损害。此外，我们对dirac方程本身的估计值估计为指标的衰减，因为与二阶方程的估计值相比，这提供了更好的内部估计值。这些估计的组合导致对狄拉克场的良好控制有助于关闭证明。我们还将在这里看到我们的论点如何证明系统在大规模案例中的良好性。
We study the initial value problem of the Einstein-Dirac system, and show the stability of the Minkowski solution in the massless case with the use of generalized wave coordinates. This requires the understanding of the Dirac equation in curved spacetime, for which we establish various estimates. The proof is based on the vector-field method which is widely used in works on the stability of Minkowski problems for other Einstein-coupled systems. Under a specific choice of the tetrad, we show that components of the Dirac field satisfy a quasilinear wave equation, by resolving a potential loss of derivative problem. We also show that the semilinear nonlinearity of this equation behaves like a null form. In this way, we obtain the sharp decay of the field along the light cone. The structure of the energy-momentum tensor causes worse behavior of some components of the metric than the vaccum case, but an additional structure shows that there is no harm to the global existence result. In addition, we develop an estimate of the Dirac equation itself adpated to the decay of the metric, as this provides better estimates in the interior compared with the estimates from the second order equation. The combination of these estimates leads to a good control of the Dirac field that helps close the proof. We shall also see how our argument here gives a proof of the wellposedness of the system in the massive case.
