# 物理代写|广义相对论代写General relativity代考|MATH7105

## 物理代写|广义相对论代写General relativity代考|The Tolman–Oppenheimer–Volkoff Equation

The Tolman-Oppenheimer-Volkoff (TOV) equation is a constraint equation for constructing a spherically symmetric body of isotropic material, which is in static gravitational equilibrium.
After integrating, Eq. (6.45) yields
$$e^{-\lambda}=1-\frac{2 m(r)}{r}$$
where
$$m(r)=4 \pi G \int_0^r \rho r^2 d r,$$
i.e.,
$$\frac{d m}{d r}=4 \pi G r^2 \rho .$$
Eqs. (6.45) and $(6.46) \Longrightarrow$
$$8 \pi G(p+\rho)=\frac{e^{-\lambda}}{r}\left(\lambda^{\prime}+v^{\prime}\right) .$$
Eliminating $\lambda$ from Eq. (6.52) and using Eq. (6.49), we get
$$8 \pi G(p+\rho)=\left(1-\frac{2 m}{r}\right) \frac{v^{\prime}}{r}+\frac{1}{r}\left(8 \pi G \rho r-\frac{2 m}{r^2}\right) .$$
Again, putting the value of $v^{\prime}$ from Eq. (6.48), we finally get TOV equation as
$$\frac{d p}{d r}=-\frac{(p+\rho)\left(4 \pi G p r^3+m\right)}{r(r-2 m)} .$$ We have to integrate Eq. (6.53) to find the interior solution of the spherically symmetric object. To find the exact solution, we will have to consider an additional equation connecting $p$ and $\rho$. This equation helps us construct the stellar model.

## 物理代写|广义相对论代写General relativity代考|The Structure of Newtonian Star

Now, we try to find Newtonian limit of the TOV equation. In Newtonian circumstances, we take $p \ll \rho$, therefore $4 \pi G r^3 p \ll m$. Moreover, in Newtonian limit the metric is nearly Minkowskian, therefore, in Eq. (6.49), we have $m<<r$. These inequalities help to simplify Eq. (6.53) as
$$\frac{d p}{d r}=-\frac{\rho m}{r^2} .$$
This equation coincides with the equation of hydrostatic equilibrium for Newtonian stars.

We can relate the matter energy density $\rho$, pressure $p$, temperature $T$, and entropy $S$ in volume $V$ for a relativistic fluid through the first law of thermodynamics as
$$d(\rho V)=-p d V+T d S .$$
It is known that Baryon number is conserved, so the above equation is expressed in terms of baryon number density $n$ and entropy per baryon $s$ as
\begin{aligned} &d\left(\frac{\rho}{n}\right)=-p d\left(\frac{1}{n}\right)+T d s \ &{\left[V \equiv \frac{\text { Number of baryons }}{n}\right],} \end{aligned} or
$$d \rho=(p+\rho)\left(\frac{d n}{n}\right)+n T d s .$$
If the fluid flow is isentropic, then $\frac{d s}{d t}=0$, i.e., $s=$ constant. Hence, the first law of thermodynamics yields for a perfect fluid having equation of state $\rho=\rho(n)$ as
$$\frac{d \rho}{\rho}=\frac{p+\rho}{\rho} \frac{d n}{n} .$$
Let us consider the acoustic wave is developed from a perturbation (isentropic) in a uniform static fluid comprising the parameters $\rho_0, p_0$, and $n_0$ and perturbations are $\rho_1, p_1$, and $n_1$ (i.e., $p=p_0+p_1$ and $\left.\rho=\rho_0+\rho_1\right)$. Let us also consider the fluid velocity to be $\left(1, \overrightarrow{v_1}\right)$ in the rest frame of unperturbed fluid. Now, we calculate the first-order perturbation terms in conservation equation, $T_{; v}^{\mu v}=0$ as
$\nabla \cdot \overrightarrow{v_1}=-\frac{\partial \rho}{\partial t} \frac{1}{\rho_0+p_0}, \quad$ for $\mu=0$,
$\frac{\partial \overrightarrow{v_1}}{\partial t}=-\nabla p_1 \frac{1}{\rho_0+p_0}, \quad$ for $\mu=1,2,3 .$

## 物理代写|广义相对论代写广义相对论代考|托尔曼-奥本海默-沃尔科夫方程

Tolman-Oppenheimer-Volkoff (TOV)方程是构造处于静态重力平衡的各向同性球对称体的约束方程。积分后，Eq.(6.45)得到
$$e^{-\lambda}=1-\frac{2 m(r)}{r}$$
where
$$m(r)=4 \pi G \int_0^r \rho r^2 d r,$$

$$\frac{d m}{d r}=4 \pi G r^2 \rho .$$
(6.45)和$(6.46) \Longrightarrow$
$$8 \pi G(p+\rho)=\frac{e^{-\lambda}}{r}\left(\lambda^{\prime}+v^{\prime}\right) .$$

$$8 \pi G(p+\rho)=\left(1-\frac{2 m}{r}\right) \frac{v^{\prime}}{r}+\frac{1}{r}\left(8 \pi G \rho r-\frac{2 m}{r^2}\right) .$$

$$\frac{d p}{d r}=-\frac{(p+\rho)\left(4 \pi G p r^3+m\right)}{r(r-2 m)} .$$我们必须对Eq.(6.53)进行积分，以找到球对称物体的内部解。为了找到精确的解，我们必须考虑一个连接$p$和$\rho$的附加方程。这个方程帮助我们构建了恒星模型

## 物理代写|广义相对论代写广义相对论代考|牛顿星的结构

$$\frac{d p}{d r}=-\frac{\rho m}{r^2} .$$

$$d(\rho V)=-p d V+T d S .$$

\begin{aligned} &d\left(\frac{\rho}{n}\right)=-p d\left(\frac{1}{n}\right)+T d s \ &{\left[V \equiv \frac{\text { Number of baryons }}{n}\right],} \end{aligned}或
$$d \rho=(p+\rho)\left(\frac{d n}{n}\right)+n T d s .$$

$$\frac{d \rho}{\rho}=\frac{p+\rho}{\rho} \frac{d n}{n} .$$

$\nabla \cdot \overrightarrow{v_1}=-\frac{\partial \rho}{\partial t} \frac{1}{\rho_0+p_0}, \quad$为$\mu=0$，
$\frac{\partial \overrightarrow{v_1}}{\partial t}=-\nabla p_1 \frac{1}{\rho_0+p_0}, \quad$为$\mu=1,2,3 .$

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