Hi! I have the following problem I don't really know where to start from:
A bowl with axial symmetry is built in flat Euclidean space ##R^3##, and has a radial profile giveb by ##z(r)##, where ##z## is the axis of symmetry and ##r## is the radial distance from the axis. What radial profile...
I have the feeling the last step, where I integrate ##dt## and get the time the observer takes to fall into the black hole, is not quite correct. But I'm not really sure. Could someone help me out? Thanks, I really appreciate it.
Hi, I have the following problem:
Given the 5-D generalization of the Schwarszschild solution with line element:
ds^2=-\Bigg(1-\frac{r^2_+}{r^2}\Bigg)dt^2+\Bigg(1-\frac{r^2_+}{r^2}\Bigg)^{-1}dr^2+r^2[d\chi^2+\sin^2(\chi)(d\theta^2+\sin^2(\theta)d\phi^2)]
where ##r_+## is a positive constant...
Hi, I'm the given the following line element:
ds^2=\Big(1-\frac{2m}{r}\large)d\tau ^2+\Big(1-\frac{2m}{r}\large)^{-1}dr^2+r^2(d\theta ^2+\sin ^2 (\theta)d\phi ^2)
And I'm asked to calculate the null geodesics.
I know that in order to do that I have to solve the Euler-Lagrange equations. For...
Thanks PeterDonis, I will post my future questions in the homework forum. I've been trying to squeeze my head around it, but I haven't posted my attempt because I literally don't have a decent one.
Yes, the problem says ##r=0##, but you're right. I believe it's a mistake and it should say ##r=2m##.
Hi! I have the following problem I don't really know how to approach. Could someone give me a hand?
The line element of a black hole is given by: ds^2=\Bigg(1-\frac{2m}{r}\Bigg)d\tau ^2+\Bigg(1-\frac{2m}{r}\Bigg)^{-1} dr^2+r^2\Big(d\theta ^2+\sin^2(\theta)d\phi ^2\Big)
It has an apparent...
It was presented to me by its deffinition with Christoffel symbols. I was never explained the physical meaning behind it (geometrical meaning) to help me imagine it.
You mean when I multiply (2) by ##2x/y^2##? I'm repeating the process again, but I can't find the mistake...
Ohhhhh okay I've seen the mistake now. The mistake is on equation (2). The ##2y^3\dot{y}^2## should be ##4y^3\dot{y}^2##.
Oh no. That wasn't the mistake. I still can't see what you're...
Purely mathematical. Two years ago I took a course on differential geometry. It wasn't until this year I started studying General Relativity, but I'm learning it on my own.
Oh, why is my final equation for ##\ddot{y}## not correct?
And what is that change of variables that simplifies the calculation?
Is the solution I have obtained incorrect then?
I need to find all the non-zero components of the Riemann Tensor in a two-dimensional geometry knowing that the only two non-zero components of the Christoffel symbols are:
\Gamma^x_{xx}=\frac{1}{x} and \Gamma^y_{yy}=\frac{2}{y}
knowing that: R^\alpha_{\beta\gamma\delta}=\partial_\gamma...
Hi! I'm asked to find all the non-zero Christoffel symbols given the following line element:
ds^2=2x^2dx^2+y^4dy^2+2xy^2dxdy
The result I have obtained is that the only non-zero component of the Christoffel symbols is:
\Gamma^x_{xx}=\frac{1}{x}
Is this correct?
MY PROCEDURE HAS BEEN:
the...