Conservative form of Boltzmann's equation in general relativity
Conservative form of Boltzmann's equation in general relativity
复制标题
广义相对论中玻尔兹曼方程的保守形式
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
S. Yamada
中科院分区：
文献类型：
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作者：
M. Shibata;H. Nagakura;Y. Sekiguchi;S. Yamada
辐射场及其与物质场的相互作用在一般相对论天体物理现象中往往起着至关重要的作用；例如，中微子的冷却和加热在核坍缩超新星中起着特殊的作用。原因是广义相对论现象通常伴随着中微子强烈相互作用的高密度和高温物质。在数值相对论的物理模拟中，经常需要考虑中微子的辐射传递效应。为此，有必要求解辐射传递方程。为了严格处理辐射传递效应，有必要数值求解Boltzmann方程，考虑吸收、发射和散射源的项。这个方程有一个1维的形式(分别在实空间和动量空间中是3维的，在时间上是1维的)；因此，计算区域必须覆盖六维空间才能进行模拟(除非我们强加任何空间对称性)。除非强加了球对称等高空间对称性(例如，基于SN方案的公式见[1-4]，球对称和广义相对论模拟的结果见[5，6]，基于长特征的替代方法及其通过切线方案应用于全广义相对论质子中子星演化的[7])，否则对该方程进行具有足够网格分辨率的良好数值模拟是一项极具挑战性的任务。到目前为止，还没有人对这个问题提出质疑(但请参阅牛顿引力中的[8])。事实上，即使对于适合于数值模拟的公式，也只报告了几次尝试[9，10]，而且到目前为止还没有确定的公式(见参考文献)。[10]这一领域的审查)。Cardall和Mezzacappa[9]给出了玻尔兹曼方程的保守公式，但不是适用于数值相对论模拟的形式。Cardall，Endeve和Mezzacappa[10]首先在时空坐标下用实验室标架，在动量-空间坐标下用流体静止标架导出了玻尔兹曼方程的3塔1公式。由于采用流体静止标架的动量空间坐标基础，所得到的方程相当复杂。文献[11]尝试用共形平坦近似的谱方法求解Boltzmann方程，但他们的公式不是保守形式。本文给出了广义相对论中玻尔兹曼方程守恒形式的一个更简洁、更一般的表述。这篇论文的组织方式如下。在证券交易委员会。在简要回顾了玻尔兹曼方程的基本知识之后，我们简明扼要地描述了它的守恒形式。给出了几种坐标条件下黑洞时空中的守恒形式，以及在真实时空和动量空间坐标下的守恒型。第三节是小结。在整篇文章中，我们使用普朗克常数h、光速c和引力常数G为一的单位。拉丁文索引a、b、c和d表示抽象索引，而希腊文索引α；β；γ；…拉丁字母i、j、k和L分别表示时空成分和空间成分。
Radiation fields and their interaction with matter fields often play a crucial role in general-relativistic astrophysical phenomena; e.g., the neutrino cooling and heating play a special role in core-collapse supernova. The reason is that general-relativistic phenomena are usually accompanied by the high-density and high-temperature matter with which neutrinos strongly interact. For a physical simulation in numerical relativity, we are often required to take into account the neutrino radiation transfer effects. For this, it is necessary to solve radiation transfer equations. For strictly handling radiation transfer effects, it is necessary to numerically solve Boltzmann’s equation, taking into account the absorption, emission, and scattering source terms. This equation has a 3þ 3þ 1-dimensional form (three dimensions in real and momentum space, respectively, and one dimension in time); hence, the computational domain has to cover six-dimensional space for a simulation (unless we impose any spatial symmetry). It is an extremely challenging task to perform a wellresolved numerical simulation with a sufficient grid resolution for this equation, unless a high spatial symmetry such as spherical symmetry is imposed (e.g., see [1–4] for formulations based on the SN schemes, [5,6] for results in spherically symmetric and general-relativistic simulations, and also [7] for an alternative approach based on long characteristics and its application to protoneutron star evolution in full general relativity by means of a tangentray scheme). To date, no challenge has been made on this issue (but see [8] in Newtonian gravity). Indeed, even for formulations suitable for a numerical simulation, only a few attempts [9,10] have been reported, and there is no established formulation to date (see Ref. [10] for a review in this field). Cardall andMezzacappa [9] gave a conservative formulation of Boltzmann’s equation but not in a 3þ 1 form applicable to numerical-relativity simulations. A 3þ 1 formulation of Boltzmann’s equation was first derived by Cardall, Endeve and Mezzacappa [10] with laboratory frame in spacetime coordinates and fluid rest frame in momentum-space coordinates. As a consequence of adopting a fluid-rest-frame momentum-space coordinate basis, the resulting equations are rather complicated. An attempt to solve Boltzmann’s equation using spectral methods with the conformally flat approximation was done by [11], but their formulation was not in a conservative form. In this paper, we derive a more concise and general formulation for the conservative form of Boltzmann’s equation in general relativity. The paper is organized as follows. In Sec. II, after we briefly review the basics of Boltzmann’s equation, we describe its conservative form in a concise way. We also give conservative forms in black-hole spacetime with several coordinate conditions in real-spacetime and momentumspace coordinates. Section III is devoted to a summary. Throughout this paper, we employ the units in which the Planck constant, h, speed of light, c, and gravitational constant, G, are unity. Latin indices a, b, c, and d denote the abstract index while greek ones α; β; γ;... and latin ones i, j, k, and l denote the spacetime and spatial components, respectively.