Gravitational Self-force in a Radiation Gauge
Gravitational Self-force in a Radiation Gauge
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辐射计中的重力自力
DOI：
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发表时间：
2010
期刊：
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通讯作者：
L. Price
中科院分区：
文献类型：
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作者：
T. Keidl;Abhay Shah;J. Friedman;Dong;L. Price
在这两篇论文的第一篇中，我们提出了一种在修正的辐射规中求出粒子在Schwarzschild或Kerr时空中在测地线上运动的引力自力的方法。Wald的一个早期结果的扩展被用来证明自旋重量±2微扰的Weyl标量(ψ0或ψ4)决定了粒子外部的度规微扰，直到规范变换和质量和角动量的无限小变化。赫兹势被用来构造延迟度规微扰的一部分，该部分不涉及从辐射规中的ψ0开始的质量或角动量变化。度规微扰是通过在任何方便的规范中将背景时空的质量和角动量的变化添加到粒子的径向坐标r0之外来完成的。由此产生的度规摄动只在粒子的轨迹上是奇异的。然后使用模式和方法来重新规格化自力。Gralla证明了即使在与粒子轨道垂直的宇称变换下，重整化自力也可以用来求出对规范中的测地线轨道的修正，对于该规范，即使在与粒子轨道垂直的宇称变换下，度规扰动中的领先项O(ρ-1)也具有空间分量，并且我们验证了辐射规范中的度规扰动满足这一条件。我们证明了在L=L+1/2规范中，度规微扰的奇异行为和裸自力的表达式与洛伦兹规范(具有不同的系数)具有相同的幂定律行为。我们显式计算了Schwarzschild几何中圆轨道粒子的奇异Weyl标量及其模和分解到L中的子载荷级，并得到了重整化场。由于奇异场可以定义为这个模和，所以这个和中每个角谐的系数必须与延迟场的相应系数的L大极限一致。因此，人们可以通过将延迟场与L中的幂函数进行数值匹配来计算奇异场。为了验证数值方法的准确性，我们解析地计算了ψ0奇异展开式中的前导项和次引项，并将重整化常数的数值和解析值进行了比较，发现两者的精度很高。文中给出了这一测试用例的扰动度规、自力和量hαβu a uβ(螺旋对称规范变换下的规范不变量)的详细数值计算。
In this, the first of two companion papers, we present a method for finding the gravitational self-force in a modified radiation gauge for a particle moving on a geodesic in a Schwarzschild or Kerr spacetime. An extension of an earlier result by Wald is used to show the spin weight ±2 perturbed Weyl scalar (ψ 0 or ψ 4 ) determines the metric perturbation outside the particle up to a gauge transformation and an infinitesimal change in mass and angular momentum. A Hertz potential is used to construct the part of the retarded metric perturbation that involves no change in mass or angular momentum from ψ 0 in a radiation gauge. The metric perturbation is completed by adding changes in the mass and angular momentum of the background spacetime outside the radial coordinate r 0 of the particle in any convenient gauge. The resulting metric perturbation is singular only on the trajectory of the particle. A mode-sum method is then used to renormalize the self-force. Gralla shows that the renormalized self-force can be used to find the correction to a geodesic orbit in a gauge for which the leading, O(ρ -1 ), term in the metric perturbation has spatial components even under a parity transformation orthogonal to the particle trajectory, and we verify that the metric perturbation in a radiation gauge satisfies that condition. We show that the singular behavior of the metric perturbation and the expression for the bare self-force have the same power-law behavior in L = l + 1/2 as in a Lorenz gauge (with different coefficients). We explicitly compute the singular Weyl scalar and its mode-sum decomposition to subleading order in L for a particle in circular orbit in a Schwarzschild geometry and obtain the renormalized field. Because the singular field can be defined as this mode sum, the coefficients of each angular harmonic in the sum must agree with the large L limit of the corresponding coefficients of the retarded field. One may therefore compute the singular field by numerically matching the retarded field to a power series in L. To check the accuracy of the numerical method, we analytically compute leading and subleading terms in the singular expansion of ψ 0 and compare the numerical and analytic values of the renormalization constants, finding agreement to high precision. Details of the numerical computation of the perturbed metric, the self-force, and the quantity h αβ u a u β (gauge invariant under helically symmetric gauge transformations) are presented for this test case in the companion paper.