Question

general relativity question.

Instructions The first problem is just a test of how well one knows the methods of averaging over directions. This is a standard tool which is used in many areas of classical electrodynamics and general relativity. e.g, in the context of the radiation problem and in the context of particle motion in inhomogeneous electromagnetic fields. The second problem is just a test of how well one understood the content of the present lecture. Solve the problems and prepare a report. The report should contain detailed solutions with justifications. The tasks can be done by hand or with a text(or Tex) editor. You must save the report in pdf format. Review criteria Criteria Solutions of these problems will be evaluated according to the presence of the following items in the report First problem correct calculation of the vector components of the average • the presence of the final answer in the vector form . Second problem correct components of the quadruple tensor correct expression for the radiation intensity correct expression for the velocity of the distance decrease time dependence of the distance .

Answer 1

Part 1:

Consider the spherical polar coordinate system
(r, , છે)
, in which the position vector $\Vec{r}$ and the fixed vector $\Vec{a}$ can be respectively written as

$\Vec{r}= R \hat{e_r}$ and ,

a = aré, + agée + apeo

where $\left ( \hat{e_r}, \hat{e_{\theta}} ,\hat{e_{\phi}} \right )$ are the basis vectors in the
(r, , છે)
system and $\left ( a_r, a_{\theta} ,a_{\phi} \right )$ are the components of $\Vec{a}$ in this system.

Now, $\left (\Vec{r} \cdot \Vec{a} \right )\Vec{r} =(a_r R) R \hat{e_r} = a_r R^2 \hat{e_r}$ .

The above quantity can be averaged over all directions $(\theta,\phi)$ ,

$\left \langle \left (\Vec{r} \cdot \Vec{a} \right )\Vec{r} \right \rangle = \frac{\int a_r R^2 d\Omega}{\int d\Omega} \hat{e_r}$,

where $d\Omega$ is the infinitesimal solid angle and in spherical polar coordinate system, it is given by

$d\Omega= \sin^{2}{\theta} d\theta d\phi$,

with $0< \theta < \pi$ and $0< \phi < 2\pi$ .

Hence

$\left \langle \left (\Vec{r} \cdot \Vec{a} \right )\Vec{r} \right \rangle = \frac{ a_r R^2 \int d\Omega}{\int d\Omega} \hat{e_r} =a_r R^2 \hat{e_r}$.

Part 2:

Let at time , the distance between the two stars of masses $m_1$ and $m_2$ in a double star system be . Since the centripetal force must be equal to the gravitational force between the two stars, hence we can write $\frac{\mu v^2}{r} = \frac{Gm_1 m_2}{r^2}$ ,

where is the gravitational constant, is the relative velocity of the system and $\mu$ is the reduced mass of the system, given by

$\mu =\frac{m_1 m_2}{m_1+m_2}$ .

Hence

$v^2 = \frac{G(m_1+ m_2)}{r}$

or,

$v= \frac{dr}{dt} = \sqrt{\frac{G(m_1+ m_2)}{r}}$

or,

$\sqrt{r} dr = \sqrt{G(m_1+ m_2)} dt$.

Integrating both sides and applying the boundary condition $r(0)=2R$ , we obtain

$-\frac{1}{2} \left (\frac{1}{\sqrt{r}}- \frac{1}{\sqrt{2R}} \right ) = \sqrt{G(m_1+ m_2)} t$

or,

$r(t)=\frac{1}{\left (\frac{1}{\sqrt{2R}}-2 \sqrt{G(m_1+ m_2)} t \right )^2} = \frac{2R}{\left (1-2 \sqrt{2RG(m_1+ m_2)} t \right )^2}$.