# Derivation of Kerr Metric

1. Jan 10, 2010

### utku

Hello friends.I study about Kerr metric and black holes.I can deriving Schwarzchild metric basically but i cant derive the Kerr metric.
Anyone know how can i study it with basic concepts?
please suggest to me any lecture note or text.
thanks.

2. Jan 10, 2010

### nicksauce

I don't know of any source that has a simple derivation, and I doubt one exists. After all, it took some 45 years or so for it to be discovered, after GR was invented.

3. Jan 10, 2010

### utku

I want to add an example to here
When ı study Inverno's general relativity book he has used tetrat formalism for deriving the kerr metric but it isnt seems explicitly so i dint understand it.
In chadrasekhar's mathematical theory of blackholes,the derivation of kerr metric so long.
I want to learn ,how can ı derive it too easly?

4. Jan 10, 2010

### alle.fabbri

Try to check on S. Carroll's book...I think there's something...

5. Jan 10, 2010

### nicksauce

Unfortunately, not. To quote from the book "His result, the Kerr metric, is given by the following mess:" So he just states the answer without derivation.

6. Jan 10, 2010

### utku

yes.I know his book.I have studied that book so ı understand many of topics but only except Kerr metric.

7. Jan 10, 2010

### Matterwave

Even the derivation of the Schwarzchild metric is non-trivial (at least, to me it isn't), and that is the simplest solution of the EFEs (apart from the Minkowski metric). I doubt that the Kerr metric can be "easily" derived...

8. Jan 10, 2010

### bcrowell

Staff Emeritus
Have you looked at Kerr's original paper? "Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics," Phys Rev Lett 11 (1963) 237. His derivation is short, but it seems like it depends on knowledge of some special mathematical tricks. If you really want to understand it, one way to go might be to read the references and see if you can dig back far enough to find where the mathematical tricks are developed in the literature.

Another possibility would be to try a series expansion. Here http://www.lightandmatter.com/html_books/genrel/ch06/ch06.html#Section6.2 [Broken] is an example of how to get the Schwarzschild metric by doing a series expansion and then recognizing that the series represents something that can be expressed in closed form. As demonstrated in the link, computer algebra systems can make this kind of thing much less painful than it used to be in the past.

Last edited by a moderator: May 4, 2017
9. Jan 10, 2010

### George Jones

Staff Emeritus
I don't think that there is an easy, straightforward route to Kerr's solution.
For an interesting first-hand account of what Kerr did, see

http://arxiv.org/abs/0706.1109.

10. Jan 10, 2010

### bcrowell

Staff Emeritus
Very cool, George -- thanks for pointing us to that! It's not just historical, it goes into quite a bit of mathematical detail.

11. Jan 11, 2010

### haushofer

Maybe Gron and Hervik have a good treatment on this. Their text is available on the internet and personally I think it's a very nice treatment of GR :)

12. Jan 11, 2010

### CFDFEAGURU

You can also rule out Hartle's book, Gravity, it simply gives the Kerr metric just like Carrol's does.

Thanks
Matt

13. Jan 11, 2010

### Altabeh

I address you to take a look at the following two good and sort of short papers that derive Kerr metric in two different approaches:

2- http://arxiv.org/pdf/gr-qc/0305035v2

It is so interesting that in the second paper, the auther quotes from Landau and Lifgarbage the below saying:

"there is no constructive analytic derivation of the Kerr metric that is adequate in its physical ideas and even a check of this solution of Einstein’s equations involves cumbersome calcualtions",

which may sound unfamiliar to us since we all were talking about a simple derivation of KM while this issue was widely agreed-upon in its non-existence up until the advent of Chandrasekhar's general relativistic approach which, I see, was put aside to be studied due to its massivenes.

Anyways, I think the second paper gives away a very elegant derivation of KM which also calls for some knowledge of gauges introduced by Papapetrou whose beautiful book 'lectures on GR' sadly lacks the proof, but I'm gonna add it to its Farsi version. :tongue:

I don't think you'll catch up with any other derivation, so save your time and start reading these two or be satisfied by the Chandrasekhar's gaudy approach.

Last edited by a moderator: May 4, 2017
14. Jan 11, 2010

### utku

Thanks friends.
I have already read Gron and Hervik's book.I agree with you about that book.I think it is very excellent book.But in that book the writers have used the cartan formalism not directly classical tensor analysis.And when they derive the Kerr metric,They use Ernst equations so i hve no idea about it.But,especially,Introdiction of that book and topics are very large.

I have read some topics of Hartle' book Gravity but not completely.But it is very usefull for beginner to Gr like me.I m going to study Kerr metric in Hartle's book with more attention.

Altabeh and George Jones,i will look to that address.
thanks for help everybody...

15. Jan 11, 2010

### CFDFEAGURU

Good Luck. Let us know how it is going.

Thanks
Matt

16. Sep 8, 2011

### julian

The full derivation is in Adler (2nd edition). I've been trying to go through it but it is difficult.

Last edited: Sep 8, 2011
17. Sep 8, 2011

### Passionflower

Last edited by a moderator: May 5, 2017
18. Sep 12, 2011

### julian

well it wasn't so difficult after all. I'm going to write it up and put it onto the internet

19. Sep 26, 2011

### julian

Just about fininshed writing up the derivation. It is straighforward but lengthy. I'm having trouble uploading it onto the internet. If anybody wants me to email them the file contact me at baynham_ian@hotmail.com.

Last edited: Sep 26, 2011
20. Oct 5, 2011

### julian

If you go to my web page

members.multimania.co.uk/ianbay/

and go to "Chapter 3: Black holes - Event, Isolated and Dynamical Horizons" it looks like the pdf file can now be downloaded. It contains the detailed derivation of the Kerr solution, and how to recast it into Eddington-Finkelstein and Boyer-Lindquist coordinates.