Solving D'Inverno Ch 19 Problem 19.5

[TerryW]

Has anyone worked all the way through D'Inverno Chapter 19 Problem 19.5?

I've had a good thrash at it - it is quite a bit of work, but I haven't been able to get to a satisfactory conclusion. I've worked out all the components of g_ab and J_ab (for the cartesian frame) but when when I work through the transformations into the spherical co-ordinates, I get the correct answers for g'₀₀, g'₀₂, g'₀₃, g'₁₂, g'₂₂, g'₂₃ and g'₃₃. The results for g'₁₀, g'₁₁, g'₁₃ just will not come out with results that let me work through to the end of the problem successfully. The result for g'₁₂ is fine and for this I used all the components of g'_ab and J_ab (except for g₀₀) which is trivial.

In the case of g'₁₀, my result has a divisor of (r²+a²), but I've worked through two different ways to find that the answer probably ought to have a divisor of Δ. I just cannot see where this can come from given the values for g_ab and J_ab.

Any ideas anyone?


Regards

TerryW

 

[elfmotat]

Would you mind posting the problem?

 

[TerryW]

Hi Elfmotat,

I've a pdf with the relevant information from pages 251 and 252 of D'Inverno. The problem is as follows:

Apply the transformation (19.29) to (19.28) and then the transformation (19.24) to the result, to obtain the form (19.27).

Clearly the answer to part 1 is going to be something like (19.22). I used this idea to 'reverse engineer' g'₀₁ leaving me with two expressions which were the same apart from a term which is r² + a² when I do the transformation and Δ (as defined by(19.26) when I do the 'reverse engineer'.

I'm going to see what happens if I replace r² + a² with Δ in (19.28), Maybe you can spot what is going wrong.

Regards


TerryW