D'Inverno Ch 19 - Kerr Solution

[TerryW]

I've tried to work my way through section 19.2 but have run into a problem at the very end.

I've attached a PDF with the Jacobian I used for the transformation and a description of what has gone wrong.

What does D'Inverno mean when he says 'if we require v' and r' to be real' ? you can do this by using a complex value for θ but it doesn't seem to help.

Any thoughts anyone?

TerryW

 

[julian]

I remember having trouble years ago getting the same answer as in D'Inverno. Have you checked the original papers by Newman to make sure there aren't typos in Ray D'InVerno?

I don't like how it isn't explained how the new metric also solves Einstein's equations in D'Inverno. Newman and Janis merely considered the transformation because you get out the Kerr solution from the Schwarzschild solution this way - they then applied it to the Reissner-Nordstrom solution to get out what could be, and as it turned out was, the solution for a charged-rotating black. The proof that the resulting metrics are also solutions of Einstein's equations was only provided a couple of years later I think by someone else.

If you're interested, I've written up the long but easy to follow derivation of the Kerr solution at

http://members.multimania.co.uk/ianbay/chapter 3 of the first set of notes (when you click on pdf file you might get a message saying something corrupted, click on OK then click on pdf again and then it might work. And in actual fact it might not work on your brower, it seems to work in Mozilla firefox though). Anyway if you can't get the file tell me how to post files on this forum and I'll leave it for you.

Got this proof from "Alder" or "Adler" can never remember his name - there were typos in there that anoyed me.julian

 

[julian]

Here's the file for fun View attachment Kerr.pdf if you get stuck on Inverno or for anyone else (there is how to change coordinates in my notes as well if you're interested - not the complex one just the ordinary ones).

Adler or Alder got the sign of $a$ mixed up - I had to correct it to get out the answer that is in all the books...the equation at the top of page 251 of Inverno looks right though.

If I have time might look at Inverno again. Let us know if you sort it out - I'd like to look at it myself if you're inclined to type it out...

julian

 

[julian]

Inverno mentions Adler on page 252...so if you're interested, there you are.

julian

 

[julian]

Here's Newman's papers: View attachment J_Math_Phys_Newman_1_copy.pdf and View attachment J_Math_Phys_Newman_2 (1)_copy.pdf.

 

[julian]

I've tried to work my way through section 19.2 but have run into a problem at the very end.

I've attached a PDF with the Jacobian I used for the transformation and a description of what has gone wrong.

What does D'Inverno mean when he says 'if we require v' and r' to be real' ? you can do this by using a complex value for θ but it doesn't seem to help.

Any thoughts anyone?

TerryW

What he means is that at the stage of eq 19.19 he has ALREADY taken the variable r to be complex in some unspecified way...then when he says 'if we require...' he is saying this particular complex coordinate transformation will turn them into real variables - thereby specifing the way they were complex in the first place. They START out complex, this (complex) transformation is supposed to make them real!

I couldn't get the right answer out either so I skipped to the inverting the contravariant metric - I think that was painful.

 

[TerryW]

Hi Julian,

Thanks for your 5 replies to my query. I'll have a look through what you have sent me to see if it helps. I had already gone past this topic and had used the resulting vectors to solve the next problem, which was to deduce the line element (19.22). This worked out OK provided that I did use the correct version of the complex conjugate of m^a. I then went on holiday for a few weeks and it was only when I got back that I noticed that my original work on the transformation of (19.9) didn't give the correct result. I suppose I could just say that as I get the correct answer for m'^a, then I can just take the complex conjugate and I have reached the right answer, but having found that doing the transformation of the complex conjugate of m^a doesn't work out, I need to know what is going wrong.

If I resolve this, I'll let you know.


Regards


TerryW

 

[TerryW]

Hi Julian,

I've worked my way through the derivation of the Kerr solution. As you say, it is easy to follow but it a very impressive piece of work. I did notice the odd typo still which I could note for you if you want to do any more work on this. I did notice on page 211 that there is a solution (?) to an earlier problem in D'Inverno which PhyPsy posted on 24 Sept 2011 regarding Geodesics under conformal transformations. I added the (?) because I don't think equation (3.89) is correct. I believe ∇_a should be $\partial$_a etc, which appears to invalidate the rest of the proof. Have I missed something?

I've now progressed with D'Inverno and worked through the section on the principle null congruences. This leads to a solution which appears to be independent of θ which surprised me - my expectation was that if the in falling particle is approaching the rotating black hole in the equatorial plane, it would experience the maximum frame dragging effect and that an approach along the axis of rotation would result in no frame dragging. Do you have any pointers to what is actually going on?

Regards


TerryW

 

[julian]

Thanks...I appreciate you telling me if I've got something wrong as I'm writing up a book on this subject of GR. I will look at what you have suggested...

 

[TerryW]

Hi Julian,

Is the document Kerr.pdf a copy of part of your book? Would you like me to send you a document setting out the various typos I spotted, along with a couple of suggestions for improving clarity.


Regards


TerryW

 

[julian]

Hello Terry

Sorry, I only just noticed your message...my alerts for new messages from the physics forum goes to another email account I don't use very much. Yes, yes please do. And yes, it is part of a (incomplete) book - http://www.ianbay.comeze.com/. My email is baynham_ian@hotmail.com.

Ian