Kerr solution as complex transformation

[dpidt]

I've been reading Ray D'Inverno's book about general relativity, and when he derives the Kerr metric he uses a trick with a complex transformation. I can follow the derivation, but I have no clue why it works. Anyone care to explain?



The derivation is:

They start with schwarzchild in Eddington-finklestein coordinates and use a null basis:

[tex]g^{a b} = l^a n^b + n^a l^b - m^a \overline{m}^b - \overline{m}^a m^b[/tex]

where

[tex]l^a = \delta^a_1 [/tex]
[tex]n^a = -\delta^a_0 - \frac 1 2 [1-m(r^{-1}+\overline{r}^{-1})]\delta^a_1 [/tex]
[tex]m^a = \frac 1 {\sqrt{2} \overline{r}} (\delta^a_2 + \frac i {sin\theta} \delta^a_3) [/tex]
[tex]\delta^a_0 = \partial_v[/tex]
[tex]\delta^a_1 = \partial_r[/tex]
[tex]\delta^a_2 = \partial_{theta}[/tex]
[tex]\delta^a_3 = \partial_{phi}[/tex]

Then they transform
[tex] v' = v + i a cos\theta [/tex]
[tex] r' = r + i a cos\theta [/tex]

and end up with:

[tex]l^a = \delta^a_1 [/tex]
[tex]n^a = -\delta^a_0 - \frac 1 2 (1 - \frac {2 m r'} {r'^2 + a^2 cos^2 \theta})\delta^a_1 [/tex]
[tex]m^a = \frac 1 {\sqrt{2} (r'+i a cos \theta)} (-i a sin \theta (\delta^a_0 + \delta^a_1) + \delta^a_2 + \frac i {sin\theta} \delta^a_3) [/tex]

Then they use

[tex]g^{a b} = l^a n^b + n^a l^b - m^a \overline{m}^b - \overline{m}^a m^b[/tex]

to get the Kerr metric.

 

[George Jones]

Chris Hillman referred to this derivation as "rather mysterious," and he gives a few more details. See posts #4 and #5 in https://www.physicsforums.com/showthread.php?p=182120#post182120".

 

[Entropia]

The complex transformation used in the derivation of the Kerr metric is known as the Kerr solution as a complex transformation. This technique is a mathematical tool that is commonly used in physics, particularly in the field of general relativity, to simplify and solve complex equations and problems. In this case, the complex transformation is used to transform the Schwarzchild metric into the Kerr metric, which describes the geometry of a rotating black hole.

The reason why this technique works is because it allows us to manipulate the equations in a way that makes them more manageable and easier to solve. In the Kerr solution, the complex transformation is used to simplify the null basis vectors, making it easier to express the metric in terms of these vectors. This simplification allows for a more straightforward derivation of the Kerr metric.

Furthermore, the use of complex numbers in general relativity is not uncommon. In fact, complex numbers are often used to represent physical quantities in a more elegant and compact way. In the case of the Kerr solution, the use of complex numbers helps to describe the rotation of a black hole in a more concise and efficient manner.

In conclusion, the Kerr solution as a complex transformation is a powerful mathematical tool used in general relativity to solve complex problems. It allows for a simpler derivation of the Kerr metric and provides a more elegant representation of the rotation of a black hole.