npc
Feb15-11, 04:12 AM
Have looked on the forum but can't find this query so hope I am not repeating something that might be here already. Also hope this is the right forum for the post.
1. The problem statement, all variables and given/known data
I am working through a paper on particle motion in GR (Cohen PhysRevD 19,8,p2273) but am running into a few hurdles.
2. Relevant equations
We need to work out radial motion for the general metric:
ds^2=g_{tt}dt^2+g_{rr}dr^2
starting from the Lorentz force equation (for the r-component):
\frac{dp_r}{d\tau}+\Gamma^r_{\alpha\beta}p_\alpha\ frac{dx^\beta}{d\tau}\\=-qF^{r\nu}g_{\nu\beta}\frac{dx^\beta}{d\tau}
where
F_{\mu\lambda}=\partial_\lambda A_\mu-\partial_\mu A_\lambda
and
A_{\mu}=(\psi,0,0,0)
The first problem I have is that the paper gives a result:
\frac{d^2r}{d\tau^2}+\left(\frac{\partial_rg_{tt}} {2g_{tt}}+\frac{\partial_rg_{rr}}{2g_{tt}}\right)\ left(\frac{dr}{d\tau}\right)-\frac{\partial_rg_{tt}}{2g_{rr}g_{tt}}
=-\frac{q}{m}\frac{1}{g_{rr}}\frac{d\psi}{dr}\frac{d t}{d\tau}
whereas I get:
\frac{d^2r}{d\tau^2}+\left(\frac{\partial_rg_{tt}} {2g_{tt}}+\frac{\partial_rg_{rr}}{2g_{tt}}\right)\ left(\frac{dr}{d\tau}\right)^2-\frac{\partial_rg_{tt}}{2g_{rr}g_{tt}}
=-\frac{q}{m}\frac{1}{g_{rr}}\frac{d\psi}{dr}\frac{d t}{d\tau}
Workings in attachment.
just wondering if anyone can see where I am wrong.
Also, The next step in the paper is to say that the above equation has first integral:
\left(\frac{dr}{d\tau}\right)^2=\frac{g_{tt}+(K-\frac{q}{m}\psi)^2}{-g_{tt}g_{rr}}
Where K is a constant of integration.
I can't see where this solution is from, any suggestions on getting it? (probably something obivious)
There is a second integral solution given in the paper:
\frac{dt}{d\tau}=\frac{(K-\frac{q}{m}\psi)}{-g_{tt}}
Which I am able to get correctly from the \mu=t component of the lorentz equation.
3. The attempt at a solution
See attachment
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
1. The problem statement, all variables and given/known data
I am working through a paper on particle motion in GR (Cohen PhysRevD 19,8,p2273) but am running into a few hurdles.
2. Relevant equations
We need to work out radial motion for the general metric:
ds^2=g_{tt}dt^2+g_{rr}dr^2
starting from the Lorentz force equation (for the r-component):
\frac{dp_r}{d\tau}+\Gamma^r_{\alpha\beta}p_\alpha\ frac{dx^\beta}{d\tau}\\=-qF^{r\nu}g_{\nu\beta}\frac{dx^\beta}{d\tau}
where
F_{\mu\lambda}=\partial_\lambda A_\mu-\partial_\mu A_\lambda
and
A_{\mu}=(\psi,0,0,0)
The first problem I have is that the paper gives a result:
\frac{d^2r}{d\tau^2}+\left(\frac{\partial_rg_{tt}} {2g_{tt}}+\frac{\partial_rg_{rr}}{2g_{tt}}\right)\ left(\frac{dr}{d\tau}\right)-\frac{\partial_rg_{tt}}{2g_{rr}g_{tt}}
=-\frac{q}{m}\frac{1}{g_{rr}}\frac{d\psi}{dr}\frac{d t}{d\tau}
whereas I get:
\frac{d^2r}{d\tau^2}+\left(\frac{\partial_rg_{tt}} {2g_{tt}}+\frac{\partial_rg_{rr}}{2g_{tt}}\right)\ left(\frac{dr}{d\tau}\right)^2-\frac{\partial_rg_{tt}}{2g_{rr}g_{tt}}
=-\frac{q}{m}\frac{1}{g_{rr}}\frac{d\psi}{dr}\frac{d t}{d\tau}
Workings in attachment.
just wondering if anyone can see where I am wrong.
Also, The next step in the paper is to say that the above equation has first integral:
\left(\frac{dr}{d\tau}\right)^2=\frac{g_{tt}+(K-\frac{q}{m}\psi)^2}{-g_{tt}g_{rr}}
Where K is a constant of integration.
I can't see where this solution is from, any suggestions on getting it? (probably something obivious)
There is a second integral solution given in the paper:
\frac{dt}{d\tau}=\frac{(K-\frac{q}{m}\psi)}{-g_{tt}}
Which I am able to get correctly from the \mu=t component of the lorentz equation.
3. The attempt at a solution
See attachment
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution