# 物理代写|广义相对论代写General relativity代考|МАТH7105

## 物理代写|广义相对论代写General relativity代考|Three Types of Mass

1. Inertial mass: This mass is a measure of the body’s resistance to change its state, i.e., either in a rest position or in motion. The mass appears in Newton’s second law $F=m^I f$, i.e., here $m^I$ is known as inertial mass.
2. Passive gravitational mass: Passive gravitational mass $m^p$ deals with a body’s response when it is placed in a gravitational field. Suppose the gravitational potential of some source at some point is $\varphi$, then if we place the body of mass $m^p$ at this point, it will experience a force, which is given by $F=-m^p \operatorname{grad} \varphi$.
3. Active gravitational mass: Active gravitational mass $m^A$ determines the strength of the gravitational field what the body produces. Suppose we place a body of mass $m^A$ at a particular point, then the gravitational potential it produces at any point of distance $r$ is given by $\varphi=-\frac{G m^4}{r}$.

In Newtonian physics, these concepts are identical. These identification are valid in general relativity, too.

$$G_{\mu v}=R_{\mu v}-\frac{1}{2} g_{\mu v} R .$$
Einstein tensor (symmetric tensor of rank two) specifies the geometry of the spacetime. We have already shown in Section $1.17$ that $G_{\mu ; \vartheta}^v=0$.

The energy-momentum tensor is represented as a bulk properties of matter. This is a symmetric tensor of rank two. Also, conservation law for the energy-momentum tensor $T^{\mu \nu}$ implies that $T_{\mu ; v}^\nu=0$. Again, one of the forms of Mach’s principle is that the matter distribution in the universe is responsible for inertial effects. It seems, Einstein was guided by these two results and considering equivalence principle, principle of covariance along with Mach’s principle to give the final form of his gravitational field equations, which should be tensorial equations, i.e.,
$$G^{a b} \alpha T^{a b} \Rightarrow G^{a b}=-k T^{a b},$$
where $k=\frac{8 \pi G}{c^4}=$ coupling constant.
Einstein gave the above equation by applying his own intuition. Later there have been many methods developed to construct the Einstein field equation.

## 物理代写|广义相对论代写General relativity代考|Some Useful Variations

1. $\delta \sqrt{-g}=\frac{1}{2} \sqrt{-g} g^{i k} \delta g^{i k}$.
2. $\delta g_{i k}=-g_{i l} g_{k m} \delta g^{I m}$.
3. $\delta L=\frac{\partial L}{\partial g_{i k}} \delta g_{i k}+\frac{\partial L}{\partial g_{i L}} \delta\left(g_{i k, l}\right)$.
4. $\delta\left(g_{i k, l}\right)=\left(\delta g_{i k}\right)_r$.
5. $(\sqrt{-g}){, l}=\frac{1}{2} \sqrt{-g} g^{i k} g{i k, l}$.
6. $\delta R_{\alpha \beta}=\left[\delta \Gamma_{\alpha v}^v\right]{; \beta}-\left[\delta \Gamma{\alpha \beta}^v\right]{; v}$. Hint: $\delta R{\mu v}=\delta\left[\frac{\partial \Gamma_{\mu v}^\rho}{\partial x^\rho}-\frac{\partial \Gamma_{\mu \rho}^\rho}{\partial x^v}+\Gamma_{\mu v}^\sigma \Gamma_{\rho \sigma}^\rho-\Gamma_{\mu \rho}^\sigma \Gamma_{v \sigma}^\rho\right]$.
In a geodesic coordinate system, we have
\begin{aligned} \delta R_{\mu v} &=\delta\left[\frac{\partial \Gamma_{\mu v}^\rho}{\partial x^\rho}-\frac{\partial \Gamma_{\mu \rho}^\rho}{\partial x^v}\right] \ &=\frac{\partial\left(\delta \Gamma_{\mu v}^\rho\right)}{\partial x^\rho}-\frac{\partial\left(\delta \Gamma_{\mu \rho}^\rho\right)}{\partial x^v} \ &=\left(\delta \Gamma_{\mu v}^\rho\right){; \rho}-\left(\delta \Gamma{\mu \rho}^\rho\right)_{; v} \end{aligned}
Since this is a tensorial equation, therefore, it is true for all coordinate systems and at all points of space and time.
7. $\delta \Gamma_{\mu v}^\lambda=\frac{1}{2} g^{\lambda \rho}\left[D_v \delta g_{\rho \mu}+D_\mu \delta g_{v \rho}-D_\rho \delta g_{\mu v}\right]$.
$D \equiv$ covariant derivative.
8. $V_{; \alpha}^x=\frac{1}{\sqrt{-g}} \frac{\partial}{\partial x^x}\left(\sqrt{-g} V^x\right)$.

## 物理代写|广义相对论代写General relativity代考|Three Types of Mass

1. 惯性质量: 该质量是衡量身体改变其状态 (即处于静止位置或运动中) 的阻力的量度。质量出现在牛顿 第二定律中 $F=m^I f$, 即这里 $m^I$ 被称为惯性质量。
2. 被动重力质量: 被动重力质量 $m^p$ 处理物体放置在引力场中时的反应。假设某个源在某个点的引力势为 $\varphi$, 那么如果我们把质量体 $m^p$ 在这一点上，它将经历一个力，由 $F=-m^p \operatorname{grad} \varphi$.
3. 主动重力质量: 主动重力质量 $m^A$ 决定了身体产生的引力场的强度。假设我们放置一个质量体 $m^A$ 在特 定点，然后它在任何距离点产生的引力势 $r$ 是 (谁) 给的 $\varphi=-\frac{G m^4}{r}$.
在牛顿物理学中，这些概念是相同的。这些识别在广义相对论中也是有效的。
$$G_{\mu v}=R_{\mu v}-\frac{1}{2} g_{\mu v} R .$$
喛因斯坦张量 (二阶对称张量) 指定时空的几何形状。我们已经在章节中展示了 $1.17$ 那 $G_{\mu ; \vartheta}^v=0$.
能量-动量张量表示为物质的整体性质。这是一个二阶对称张量。此外，能量-动量张量的守恒定律 $T^{\mu \nu}$ 暗 示 $T_{\mu ; v}^\nu=0$. 同样，马赫原理的一种形式是宇宙中的物质分布是造成惯性效应的原因。看来，爱因斯坦是 以这两个结果为指导，考虑等价原理、协方差原理以及马赫原理，给出了他的引力场方程的最終形式，应 该是张量方程，即，
$$G^{a b} \alpha T^{a b} \Rightarrow G^{a b}=-k T^{a b},$$
在哪里 $k=\frac{8 \pi G}{c^4}=$ 耦合常数。
爱因斯坦应用他自己的直觉给出了上述方程。后来发展了许多方法来构造爱因斯坦场方程。

## 物理代写|广义相对论代写General relativity代考|Some Useful Variations

1. $\delta \sqrt{-g}=\frac{1}{2} \sqrt{-g} g^{i k} \delta g^{i k}$.
2. $\delta g_{i k}=-g_{i l} g_{k m} \delta g^{I m}$.
3. $\delta L=\frac{\partial L}{\partial g_{i k}} \delta g_{i k}+\frac{\partial L}{\partial g_{i L}} \delta\left(g_{i k, l}\right)$.
4. $\delta\left(g_{i k, l}\right)=\left(\delta g_{i k}\right)_r$.
5. $\$(\backslash \operatorname{sqrt}{-\mathrm{g}}){, l}=\left\lfloor\right.$frac${1}{2} \mid \operatorname{sqrt}{-g} g^{\wedge}{i k} g{\mathrm{ik}, 1} \。 \backslash beta }^{\wedge} \vee \backslash right] {; v}. Hint : \delta R{\backslash \mathrm{mu} v}=\backslash delta \backslash left \backslash frac {\backslash \text { partial } \backslash \text { Gamma_{\mu v}}^{\wedge} \backslash rho } \backslash \backslash partial . Inageodesiccoordinatesystem, wehave\ 6. \backslash begin{aligned } lend{aligned } 7. \ \
8. 因为这是一个张量方程，所以它适用于所有坐标系和所有空间和时间点。
9. $\delta \Gamma_{\mu v}^\lambda=\frac{1}{2} g^{\lambda \rho}\left[D_v \delta g_{\rho \mu}+D_\mu \delta g_{v \rho}-D_\rho \delta g_{\mu v}\right]$.
10. $D \equiv$ 协变导数。
11. $V_{; \alpha}^x=\frac{1}{\sqrt{-g}} \frac{\partial}{\partial x^2}\left(\sqrt{-g} V^x\right)$.

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