物理代写| Linearised Gravity 相对论代考代写 - 代考代写：100%准时可靠您的作业代写专家

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1 物理代写| Linearised Gravity 相对论代考

物理代写| Linearised Gravity 相对论代考

物理代写

8.2 Linearised Gravity
We choose a nearly Cartesian system so that we can write:
$$
g_{\alpha \beta}=\eta_{\alpha \beta}+h_{\alpha \beta},
$$
131
(1) The Author(s), under exclusive license to Springer Nature Switzerland AG 2022
S. Dhurandhar and S. Mitra, General Relativity and Gravitational Waves,
UNITEXT for Physics, https://doi.org/10.1007/978-3-030-92335-8_8
132
8 Gravitational Waves
where $\left|h_{\alpha \beta}\right|<<1$ and $\eta_{\alpha \beta}$ is the Minkowski metric tensor with only nonzero diagonal components $(1,-1,-1,-1) .$ We can think of $h_{\alpha \beta}$ as a “field” over the Minkowski spacetime described by the metric $\eta_{\alpha \beta} .$ This is in the spirit of considering an electromagnetic field described by the electromagnetic vector potential $A_{\alpha}$ over the Minkowski spacetime. From astrophysical considerations, we typically find $h_{\alpha \beta} \sim 10^{-22}$ or less when the waves reach our detectors. Therefore, we are essentially justified in keeping only the first order terms in $h_{\alpha \beta} .$ It is possible that near to the source, $h_{\alpha \beta}$ may not be so small and higher orders are required, but we will not be dealing with such situations in this book.

We start with linearising the Einstein tensor which consists of the Ricci tensor and the scalar curvature which are themselves contractions of the Riemann tensor. Therefore, we first start with the Riemann tensor. Note that,
$$
R_{\alpha \mu \beta v}=\frac{1}{2}\left(g_{\alpha v, \mu \beta}+g_{\mu \beta, \alpha v}-g_{\alpha \beta, \mu v}-g_{\mu v, \alpha \beta}\right)+\Gamma^{2} \text { terms }
$$
Given the weak field limit we find that $\Gamma \sim o(h)$ and so the product of Gamma terms are of the second and higher order in $h_{\alpha \beta}$ and so can be dropped in this approximation. Also the expression in $g_{\alpha \beta}$ between the parenthesis reduces to the expression with $g$ s replaced by $h$ secause the $\eta_{i k}$ are constants whose derivatives are zero. Thus in the weak field limit,
$$
R_{\alpha \mu \beta v} \simeq \frac{1}{2}\left(h_{\alpha v, \mu \beta}+h_{\mu \beta, \alpha v}-h_{\alpha \beta, \mu \nu}-h_{\mu v, \alpha \beta}\right) .
$$
In order to proceed further, that is, compute the Ricci tensor and the scalar curvature, it helps to go over to a new variable,
$$
\bar{h}{\alpha \beta}=h{\alpha \beta}-\frac{1}{2} h \eta_{\alpha \beta},
$$
where $h=h_{\mu}^{\mu}$, is the trace of $h_{\alpha \beta}$. Note that the indices are lowered and raised by the metric tensors $\eta_{\alpha \beta}$ and $\eta^{\alpha \beta}$, because we are keeping terms only to the first order in $h_{\alpha \beta}$. The quantity $\bar{h}{\alpha \beta}$ is called the trace reverse of $h{\alpha \beta}$. By making infinitismal coordinate transformations of the form $x^{\prime \mu}=x^{\mu}+\xi^{\mu}$ where $\xi^{\mu}$ are infinitismal we can choose a coordinate system in which the divergence of $\bar{h}{\alpha \beta}$ vanishes, that is, $$ \bar{h}{, \beta}^{\alpha \beta}=0
$$
This is called the Lorenz gauge in analogy with the Lorenz gauge in electrodynamics (see Eq. (1.7.17)). It can be shown that such a coordinate system can always we found if the relevant functions are well behaved-continuous, differentiable etc. We show this in the next section. Moreover, just as in electrodynamics, this choice of gauge does not exhaust all freedom, because we can still remain in the gauge by making further transformations where the functions $\xi^{i}$ satisfy $\square \xi^{i}=0$. So it is really a family

物理代考

8.2 线性重力
我们选择一个近似笛卡尔的系统，这样我们就可以写出：
$$
g_{\alpha \beta}=\eta_{\alpha \beta}+h_{\alpha \beta}，
$$
131
(1) 作者，获得 Springer Nature Switzerland AG 2022 的独家许可
S. Dhurandhar 和 S. Mitra，广义相对论和引力波，
UNITEXT 物理，https://doi.org/10.1007/978-3-030-92335-8_8
132
8 引力波
其中 $\left|h_{\alpha \beta}\right|<<1$ 和 $\eta_{\alpha \beta}$ 是只有非零对角线分量 $(1,-1,-1, -1) .$ 我们可以将 $h_{\alpha \beta}$ 视为由度量 $\eta_{\alpha \beta} 描述的 Minkowski 时空上的“场”。由 Minkowski 时空中的电磁矢量势 $A_{\alpha}$ 描述的电磁场。从天体物理学的考虑，当波到达我们的探测器时，我们通常会发现 $h_{\alpha \beta} \sim 10^{-22}$ 或更少。因此，我们基本上有理由只保留 $h_{\alpha \beta} 中的一阶项。是必需的，但我们不会在本书中处理这种情况。

我们从线性化爱因斯坦张量开始，它由 Ricci 张量和标量曲率组成，它们本身就是黎曼张量的收缩。因此，我们首先从黎曼张量开始。注意，
$$
R_{\alpha \mu \beta v}=\frac{1}{2}\left(g_{\alpha v, \mu \beta}+g_{\mu \beta, \alpha v}-g_{\alpha \beta, \mu v}-g_{\mu v, \alpha \beta}\right)+\Gamma^{2} \text { terms }
$$
鉴于弱场限制，我们发现 $\Gamma \sim o(h)$ 和 Gamma 项的乘积在 $h_{\alpha \beta}$ 中是二阶和更高阶的，因此可以在这个近似值中删除.同样，括号之间的 $g_{\alpha \beta}$ 中的表达式简化为用 $h$ 替换 $g$ s 的表达式，因为 $\eta_{ik}$ 是导数为零的常量。因此在弱场极限下，
$$
R_{\alpha \mu \beta v} \simeq \frac{1}{2}\left(h_{\alpha v, \mu \beta}+h_{\mu \beta, \alpha v}-h_{\ alpha \beta, \mu \nu}-h_{\mu v, \alpha \beta}\right) 。
$$
为了进一步进行，即计算 Ricci 张量和标量曲率，它有助于转到一个新变量，
$$
\bar{h}{\alpha \beta}=h{\alpha \beta}-\frac{1}{2} h \eta_{\alpha \beta}，
$$
其中$h=h_{\mu}^{\mu}$, 是$h_{\alpha \beta}$ 的轨迹。请注意，指标张量 $\eta_{\alpha \beta}$ 和 $\eta^{\alpha \beta}$ 会降低和提高索引，因为我们只保留 $h_{\ 中的一阶项阿尔法\beta}$。量 $\bar{h}{\alpha \beta}$ 称为 $h{\alpha \beta}$ 的迹逆。通过进行 $x^{\prime \mu}=x^{\mu}+\xi^{\mu}$ 形式的无限坐标变换，其中 $\xi^{\mu}$ 是无限的，我们可以选择一个坐标$\bar{h}{\alpha \beta}$ 的散度消失的系统，即 $$ \bar{h}{, \beta}^{\alpha \beta}=0
$$
这被称为洛伦兹规范，类似于电动力学中的洛伦兹规范（参见方程（1.7.17））。可以证明，如果相关函数表现良好——连续、可微等，我们总能找到这样的坐标系。我们将在下一节中展示这一点。此外，就像在电动力学中一样，这种规范的选择并没有耗尽所有的自由，因为我们仍然可以通过进一步的变换使函数 $\xi^{i}$ 满足 $\square \xi^{i} =0 美元。所以真的是一家人