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:

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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.


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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.

There's a good free paper on this here. In particular, you're looking at equations (15) and (17).

  • $\begingroup$ Minor comment to the post (v1): In the future please link to abstract pages rather than pdf files, e.g., arxiv.org/abs/1001.5429 $\endgroup$ – Qmechanic Nov 7 '17 at 17:38

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