# Calculate Komar mass and Komar angular momentum for the Kerr metric [closed]

I have this questions for HW: calculate Komar mass and Komar angular momentum for the Kerr metric.

the quations that I see in the lecture notes are:

in the notes it dosent explain the parameters here and I'm not sure if these are the correct integrals. I saw this other post here: How to calculate angular momentum (J) in the Kerr parameter equation?

can somebody explain what needs to be done here as I'm confused at the moment.

## closed as off-topic by John Rennie, stafusa, Jon Custer, Kyle Kanos, sammy gerbilNov 20 '17 at 11:23

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – John Rennie, stafusa, Jon Custer, Kyle Kanos, sammy gerbil
If this question can be reworded to fit the rules in the help center, please edit the question.

The basic idea behind Komar quantities is that you assume that you're dealing with a stationary spacetime (which includes Kerr), so you have Killing vector fields. Kerr has two Killing fields: one representing time-translation symmetry; the other representing axisymmetry. It looks like your Lecturer's notations for these Killing fields are $k^{(+)}$ and $K^{\phi}$, respectively. The $k^{(+)}$ field is timelike, but has a component in the $\phi$ direction; the $K^{\phi}$ is basically just a vector in the $\phi$ direction.
Now, since these are vector fields, they need vector indices, and it looks like your notes have incorrectly transcribed these indices. Your equations should read \begin{align} M(V) &= - \frac{2}{\chi^2} \oint_{\partial V} \nabla^\mu {k^{(+)}}^{\nu} d\Sigma_{\mu\nu}, \\ J(V) &= \frac{1}{\chi^2} \oint_{\partial V} \nabla^\mu {K^{\phi}}^{\nu} d\Sigma_{\mu\nu}. \end{align} Note that the indices are $\nu$'s instead of $V$'s.
Okay, now $V$ is some (spacelike) volume in your spacetime, and $\partial V$ is its boundary which is assumed to be topologically a two-sphere. Your result should actually be basically independent of your choice of $V$ as long as it encloses all of the sources in your spacetime. However, my guess is that the homework wants you to pick some surface with a nice easy-to-use boundary like constant radius or something. Then, $d\Sigma$ is the volume element of that surface (so it approaches $\sin\theta\, d\theta\, d\phi$). I further guess that you're supposed to figure out what $\chi$ is supposed to be. It should be $\sqrt{16\pi}$, though there's a famous "factor of two anomaly" in Komar's results, so you may encounter that at some point.