General Relativity Problem Questions

[Admetus]

Homework Statement

[1]

(Low quality scan unfortunately: (1) contains Einstein notation for partial differentiation and (2) Einstein notation for the covariant derivative. e(r) and e(θ) are the components of v.)

[2]

[3]
Integral of (1-2m/r)^-1/2 dr, should be recognisable from Schrödinger’s solution

Homework Equations

The Attempt at a Solution

My problem with the first [1] lies in the lack of knowledge of connection coefficients; I can happily deal with these in the 3 dimensional form but not in the unfamiliar 2D Kepler form.
Part (1) was achieved, it is simple partial differentiation. Part (2) I couldn't, I was unable to work backwards due to no sign of partial differentiation in the final answers. Our lecturer only went over a single example of connection coefficients in a 3D case.

[2] is more of "how I should proceed" question, with both x^1 and x^2 are both skewed by an angle theta. I can do the diagram and mark those covariant components, but otherwise am still scratching my head in regard to the ds^2 equations. It may actually be simple vector maths, and this I submit this question gingerly too.

[3] Wondering how to do this integral, just need to know if there are approximations (or series) that I should use, turning the integral into something more palatable!I apologise in advance for any difficulty in answering these questions, which may in fact require some effort in providing the working. Of course, the help would be gratefully appreciated.

 

[George Jones]

My problem with the first [1] lies in the lack of knowledge of connection coefficients; I can happily deal with these in the 3 dimensional form but not in the unfamiliar 2D Kepler form.
Part (1) was achieved, it is simple partial differentiation. Part (2) I couldn't, I was unable to work backwards due to no sign of partial differentiation in the final answers. Our lecturer only went over a single example of connection coefficients in a 3D case.

With respect to polar coordinates, what is the metric for a plane?

 

[Admetus]

I was thinking the same and thought it might be a diagonal metric that I was unaware of. All the information I have available to me is in question [1] is what I have scanned. It may be worth noting that there is a possibility that this was a botched question - the lecturer did mention once that one of the questions in the problem sheets was not doable. It was most likely this one.

 

[George Jones]

The question is doable. I am trying to guide you gently and slowly along the path.

Let's back up a step. With respect to Cartesian coordinates, what is the metric for a plane?

 

[Admetus]

In that case I have written down:

e(r) cos(θ) sin(θ) e(x)
e(θ) -rsin(θ) rcos(θ) e(y)

2x1matrix = (2x2matrix)(2x1matrix)

Not aware of an easier way to represent matrices but hope that is clear enough.

 

[George Jones]

I'm have been asking for expressions for $ds^2$ for a plane, either in Cartesian coordinates, or in polar coordinates.