- #1
VuNguyen
- 1
- 0
How can I obtain a Kerr metric by using the Einstein equation?
VuNguyen said:How can I obtain a Kerr metric by using the Einstein equation?
Stingray said:Its not simple. The most elegant way is by using the Kerr-Schild construction. Let the metric be of the form g=eta+k.k, where eta is the minkowski metric, and k is a null vector. There are various nice theorems that can be proven with this (even though it looks perturbative, the results are exact) that eventually lead to Kerr. I believe Chandrasekhar's black hole book does this, although it may not be the best place to start.
VuNguyen said:How can I obtain a Kerr metric by using the Einstein equation?
Stingray said:Its not simple. The most elegant way is by using the Kerr-Schild construction. Let the metric be of the form g=eta+k.k, where eta is the minkowski metric, and k is a null vector. .
Probably not; the discussion is perhaps a bit too sketchy.maddy said:Was trying to follow the construction by way of complex transformations of the Schwartzschild metric (in advanced Eddington-Finkelstein coordinates) through the null tetrad formalism in Ray D'Inverno's Introducing Einstein's Relativity. The result is supposed to be the Boyer-Lindquist form of the Kerr metric. But there are gaps in the complex transformation process that I don't understand. For example:- it suddenly comes up with a choice of the null tetrad for the Schwartzschild metric (in advanced Eddington-Finkelstein coordinates) and asks the reader to check that [itex]g^{ab}=l^a n^b + l^b n^a - m^a \bar{m}^b - m^b \bar{m}^a[/itex], and I'm wondering whether there is a typo inside that choice [itex]n^a=(-1,-\frac{1}{2} (1-\frac{2m}{r}),0,0)[/itex] (shouldn't it be [itex]n^a=(-1,-\frac{1}{2} (1-\frac{2m}{r})^{-1},0,0)[/itex]?), etc. Has anyone out there followed the construction? Am I missing some more fundamental math that's able to generate that choice of the null tetrad?
maddy said:Are such algebraic stuff (construction of the Kerr metric) more suited for computer programs to work out? Is it really necessary to learn how to do it by hand?
A Kerr metric is a mathematical representation of the spacetime around a rotating black hole. It was first described by the physicist Roy Kerr in 1963 as a solution to Einstein's equation of general relativity.
A Kerr metric is calculated using the Einstein equation, which is a set of differential equations that describe the relationship between the curvature of spacetime and the distribution of matter and energy within it. To solve the equation for a Kerr metric, specific assumptions are made about the properties of the black hole, such as its mass and angular momentum.
The Kerr metric is significant because it provides a theoretical understanding of the properties of rotating black holes. It also has practical applications in astrophysics, as it can be used to calculate the effects of a rotating black hole on the surrounding space and matter.
Yes, the Kerr metric can also be used to describe the spacetime around other rotating objects, such as neutron stars. However, it is most commonly associated with black holes due to their extreme gravitational pull and rotation.
While the Kerr metric is a useful tool for understanding the properties of rotating black holes, it does have some limitations. For example, it does not take into account the effects of quantum mechanics or other forces besides gravity. Additionally, the Kerr metric is based on certain assumptions and may not accurately describe all real-world scenarios.