Dynamic and thermodynamic stability of relativistic, perfect fluid stars
Dynamic and thermodynamic stability of relativistic, perfect fluid stars
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DOI：
10.1088/0264-9381/31/3/035023
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发表时间：
2013-09
影响因子：
3.5
通讯作者：
Stephen R. Green;Joshua S. Schiffrin;R. Wald
中科院分区：
文献类型：
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作者：
Stephen R. Green;Joshua S. Schiffrin;R. Wald
我们考虑广义相对论中的理想流体体(恒星)，其流体的局域状态由它的4-速度u_A、它的‘粒子数密度’n和它的‘每粒子熵’S来表示。如果一颗恒星是具有环流的爱因斯坦流体方程的定常轴对称解，则它被称为处于动态平衡状态。如果处于动态平衡状态的恒星称为热力学平衡恒星，且其总熵S是满足爱因斯坦约束方程且总质量、M、粒子数、N和角动量J固定的初始数据的所有变量的极值。证明了对于处于动态平衡状态的恒星，热力学平衡的充要条件是角速度Ω、红移温度T˜和红移化学势μ˜不变。如果与恒星的线性扰动有关的所有物理规范不变量在时间上保持有界，则处于动态平衡状态的恒星称为线性动态稳定的；如果没有非纯规范的指数增长解，则称为模式稳定。处于热力学平衡状态的恒星称为线性热力学稳定的，如果对所有一阶变分，δ2Sδ2M−T˜δ2S−μ˜δ2N−Ωδ2J>；0？gt；满足δM=δN=δJ=0(因此，δS=0)。弗里德曼之前曾将正则能量E？>；作为动力学稳定性的一个判据，并认为所有自转恒星在非轴对称扰动下都是动力学不稳定的(CFS不稳定性)，所以我们的主要关注点是轴对称稳定性(尽管我们发展了我们的公式，并证明了许多非轴对称扰动的结果)。我们证明了，对于处于动态平衡状态的恒星，如果E？>；在轴对称拉格朗日扰动的某个子空间V？>；上为正，则模稳定性对于所有轴对称扰动都成立，特别是在角动量密度具有零拉格朗日变化的情况下。相反，如果E？>；在V？>；上不是正的，则存在不能在后期变得渐近平稳的扰动。我们进一步证明，对于处于热力学平衡的恒星，对于所有的拉格朗日扰动，我们有Er=δ2M−Ωδ2J？>；，其中Er？>；表示“旋转标架中的正则能”，因此，对于δJ=0的微扰，Er？>；的正性是热力学稳定性的必要条件。对于轴对称摄动，我们有E=Er？>；，因此，对于轴对称摄动，热力学稳定性的一个必要条件是E？>；对于所有δJ=0的摄动都是正性的，而不仅仅是关于V？>；中的摄动。我们的许多结果与霍兰兹和沃尔德关于黑洞理论的结果非常相似。
We consider perfect fluid bodies (‘stars’) in general relativity, with the local state of the fluid specified by its 4-velocity, ua, its ‘particle number density’, n, and its ‘entropy per particle’, s. A star is said to be in dynamic equilibrium if it is a stationary, axisymmetric solution to the Einstein-fluid equations with circular flow. A star is said to be in thermodynamic equilibrium if it is in dynamic equilibrium and its total entropy, S, is an extremum for all variations of initial data that satisfy the Einstein constraint equations and have fixed total mass, M, particle number, N, and angular momentum, J. We prove that for a star in dynamic equilibrium, the necessary and sufficient condition for thermodynamic equilibrium is constancy of angular velocity, Ω, redshifted temperature, T˜?>, and redshifted chemical potential, μ˜?>. A star in dynamic equilibrium is said to be linearly dynamically stable if all physical, gauge invariant quantities associated with linear perturbations of the star remain bounded in time; it is said to be mode stable if there are no exponentially growing solutions that are not pure gauge. A star in thermodynamic equilibrium is said to be linearly thermodynamically stable if δ2S δ2M−T˜δ2S−μ˜δ2N−Ωδ2J>0?> for all variations that, to first order, satisfy δM = δN = δJ = 0 (and, hence, δS = 0). Friedman previously identified positivity of canonical energy, E?>, as a criterion for dynamic stability and argued that all rotating stars are dynamically unstable to sufficiently non-axisymmetric perturbations (the CFS instability), so our main focus is on axisymmetric stability (although we develop our formalism and prove many results for non-axisymmetric perturbations as well). We show that for a star in dynamic equilibrium, mode stability holds with respect to all axisymmetric perturbations if E?> is positive on a certain subspace, V?>, of axisymmetric Lagrangian perturbations that, in particular, have vanishing Lagrangian change in angular momentum density. Conversely, if E?> fails to be positive on V?>, then there exist perturbations that cannot become asymptotically stationary at late times. We further show that for a star in thermodynamic equilibrium, for all Lagrangian perturbations, we have Er=δ2M−Ωδ2J?>, where Er?> denotes the ‘canonical energy in the rotating frame’, so positivity of Er?> for perturbations with δJ = 0 is a necessary condition for thermodynamic stability. For axisymmetric perturbations, we have E=Er?>, so a necessary condition for thermodynamic stability with respect to axisymmetric perturbations is positivity of E?> on all perturbations with δJ = 0, not merely on the perturbations in V?>. Many of our results are in close parallel with the results of Hollands and Wald for the theory of black holes.