How can I easily study the derivation of Kerr Metric?

[utku]

Hello friends.I study about Kerr metric and black holes.I can deriving Schwarzschild metric basically but i can't derive the Kerr metric.
Anyone know how can i study it with basic concepts?
please suggest to me any lecture note or text.
thanks.

 

[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.

 

[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 isn't seems explicitly so i dint understand it.
In chadrasekhar's mathematical theory of black holes,the derivation of kerr metric so long.
I want to learn ,how can ı derive it too easly?

 

[alle.fabbri]

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

 

[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.

 

[utku]

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.

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

 

[Matterwave]

Even the derivation of the Schwarzschild 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...

 

[bcrowell]

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 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.

 

[George Jones]

I don't think that there is an easy, straightforward route to Kerr's solution.

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.

For an interesting first-hand account of what Kerr did, see

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

 

[bcrowell]

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

 

[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 :)

 

[CFDFEAGURU]

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

Thanks
Matt

 

[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:

1- http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_2/20005a7e_1322.pdf
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 going to add it to its Farsi version.

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.

 

[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...

 

[CFDFEAGURU]

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.

Good Luck. Let us know how it is going.

Thanks
Matt

 

[julian]

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

 

[Passionflower]

See for instance "The Kerr Spacetime" by Wiltshire and Visser (Cambridge 2009)
https://www.amazon.com/dp/0521885124/?tag=pfamazon01-20

 

[julian]

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

 

[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.

 

[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.

 

[phyzguy]

If you're willing to learn some new mathematical techniques, Doran's book "Geometric Algebra for Physicists" has a very simple derivation of the Kerr metric using the techniques of Geometric Algebra. Using these techniques, the Riemann tensor for the Kerr metric takes this remarkably simple form (p.558 from above text):
R(B) = \frac{M}{2(r-Iacos(\theta))^3} (B+3\sigma_r B \sigma_r)

 

[julian]

I'll have a look at Doran, I've already done a fair amount of differential geometry. - Can you download it for free?

By the way are the links on my web page working for chapter 3? It probably won't work if you click and do "save link as", but if you just click on pdf or ps they may work, you might have to try more than once. My internet provider is being a pain at the moment.

 

[julian]

If you click on pdf you might get a message saying the file is corrupted or something...click on "ok" then try clicking on pdf again and it will work (probably). It's a pain - sorry.

 

[phyzguy]

I'll have a look at Doran, I've already done a fair amount of differential geometry. - Can you download it for free?

I don't think you can download the whole book for free, but most of the content is in the publications list at the group's website here:

http://www.mrao.cam.ac.uk/~clifford/index.html