On “Hard Stars” in General Relativity
On “Hard Stars” in General Relativity
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论广义相对论中的“硬星”
DOI：
10.1007/s00023-019-00793-4
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
Volker Schlue
中科院分区：
文献类型：
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作者：
G. Fournodavlos;Volker Schlue
我们研究爱因斯坦-欧拉方程的球对称解，该方程模拟了被真空包围的理想化相对论中子星。这些是具有自由边界的正压流体，受状态方程控制，该状态方程将声速设置为等于光速。我们证明了静态解的 1 参数族或“硬星”的存在，并描述了它们的稳定性特性：首先，我们证明小星是在保持粒子总数不变的变化下质量能量泛函的局部最小值。特别是，我们证明了质量能量泛函的第二个变体控制着“质量方面函数”。其次，我们推导了“共动坐标”中围绕小恒星的欧拉-爱因斯坦系统的线性化，并证明了能量的一致有界性陈述，这恰好位于变分论证的水平上。最后，我们展示了线性化系统的时间周期解的存在性，这表明能量有界性对于该问题是最优的。
We study spherically symmetric solutions to the Einstein–Euler equations which model an idealised relativistic neutron star surrounded by vacuum. These are barotropic fluids with a free boundary, governed by an equation of state which sets the speed of sound equal to the speed of light. We demonstrate the existence of a 1-parameter family of static solutions, or “hard stars” and describe their stability properties: First, we show that small stars are a local minimum of the mass energy functional under variations which preserve the total number of particles. In particular, we prove that the second variation of the mass energy functional controls the “mass aspect function”. Second, we derive the linearisation of the Euler–Einstein system around small stars in “comoving coordinates” and prove a uniform boundedness statement for an energy, which is exactly at the level of the variational argument. Finally, we exhibit the existence of time-periodic solutions to the linearised system, which shows that energy boundedness is optimal for this problem.