All Questions
67 questions
3votes
2answers
285views
The inverse of a specific metric tensor [closed]
I am studying general relativity and here is a problem I encountered: Suppose $$ \mathrm{d}s^2=-M^2(\mathrm{d}t-M_i\mathrm{d}x^i)(\mathrm{d}t-M_j\mathrm{d}x^j)+g_{ij}\mathrm{d}x^i\mathrm{d}x^j $$ or ...
- 129
4votes
1answer
95views
Circumference of ellipse in post-Newtonian metric
The post-Newtonian metric, in harmonic coordinates, is: $$\tag{1} \mathrm{d}s^2=-\left(1+\dfrac{2\phi}{c^2}\right)c^2\mathrm{d}t^2 + \left(1-\dfrac{2\phi}{c^2}\right)\mathrm{d}\mathbf{x}^2$$ where $\...
- 369
1vote
0answers
97views
Robertson-Walker metric exercise [closed]
I'm trying to solve an exercise from my astrophysics and cosmology class, the request is the following, starting from the RW metric expression: $$ \begin{equation*} ds^2=c^2 dt^2 - a^2 \left ( \frac{...
- 73
2votes
0answers
74views
Do two coordinate systems cover the same patch of the de Sitter manifold
I am self studying general relativity and there is some especially hard problem (it is called bonus problem in book) I am currently working on it, but I am trully stuck, so I would appreaciate all the ...
2votes
0answers
88views
Proving that the Christoffel connection transforms like a connection
In Sean Carroll's intro to GR, he shows that a connection transforms as follows: $$\Gamma^{\upsilon'}_{\mu'\lambda'}=\frac{\partial x^\mu}{\partial x^{\mu'}}\frac{\partial x^\lambda}{\partial x^{\...
- 187
2votes
1answer
329views
How to obtain orthonormal tetrad basis for an infalling observer?
An eternal Schwarzschild spacetime in Painlevé-Gullstrand coordinates reads as $ds^2 = -\left(1-\dfrac{2m}{r}\right)c^2~dT^2 + 2\sqrt{\dfrac{2m}{r}}c~dTdr + dr^2 + r^2\left(d\theta^2+\sin{^2\theta}~d\...
- 1,177
0votes
4answers
397views
Why is $dt/d\tau=\gamma$? What is $dt/d\tau$ supposed to mean exactly?
I'm a math student trying to learn some physics by reading Susskind's The Theoretical Minimum. In the volume on special relativity he derives that $\frac{dt}{d\tau}=\gamma=1/\sqrt{1-v^2}$ and uses it ...
- 121
1vote
2answers
430views
Metric tensor determinant under coordinate transformation
I've been studying GR through Wald's and Carroll's books, and I've been trying to derive one expression. $$g(x^{\mu^\prime}) = \left|\dfrac{\partial x^{\mu^\prime}}{\partial x^{\mu}}\right|^{-2} g(x^\...
- 378
4votes
1answer
939views
What is the Schwarzschild metric in cylindrical coordinates?
I was researching online for different metrics of spacetime out of curiosity, and I found one that was said to be Schwarzschild metric in cylindrical coordinates: $$ds^2 = -\left(1-\frac{r_s}{r}\right)...
- 2,118
0votes
0answers
58views
Tensor index of expression
When I do photon path integral quantization, I need to change variables like: $$A^{\mu \prime}(x) \equiv A^{\mu }(x)-(\partial^2 g^{\mu \nu}-(1-\frac{1}{\xi}) \partial^{\mu}\partial^{\nu})^{-1} J_{\nu}...
- 1,505
1vote
1answer
117views
String action in light-cone coordinates
I am going through textbook Einstein Gravity in a Nutshell by A. Zee and I got mathematically stuck at page 147 where he is talking about the classical string action using light cone coordinates. ...
2votes
1answer
295views
Finding Locally flat coordinates on a unit sphere
I know this is more of a math question, but no one in the Mathematics community was able to give me an answer, and since physicists are familiar with General Relativity, I thought I might get an ...
- 1,376
0votes
1answer
161views
Change of Metric Under Coordinate Transformation
Under a local change of coordinates $x\to x'=x+\delta x$, the metric transforms as $$g_{\mu \nu}^{\prime}\left(x^{\prime}\right)=g_{\lambda \rho}(x) \frac{\partial x^{\lambda}}{\partial x^{\prime \mu}}...
- 381
0votes
1answer
237views
Finding a coordinate transformation to diagonalize the metric
I'm reading a solved problem which states that we have a bidimensional metric space whose metric is $$ ds^2 = dv^2 - v^2 du^2 $$ and we want to find a coordinate transformation such that we get the ...
- 611
1vote
2answers
138views
How do you identify a world line is in proper time?
I’m attempting to answer the question: Show that the world line: $$ x(\lambda) = \begin{pmatrix}c\lambda \\ 0 \\ 0 \\ c\sin(\omega \lambda)\end{pmatrix}$$ where $ (\lambda,\omega>0) $ isn’t ...
