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I am trying to calculate the Komar conserved quantities for black hole having Killing vectors.

I followed the standard procedure and calculated the Komar mass for Reissner-Nordstrom black hole (plz see attached file).

But failed to calculate the same for Kerr-Newman black hole. Although in this paper calculation is done through dual form method. Please provide your suggestion.

$$f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}$$ Komar Mass $M_K$ is given by:$$M_K=\frac{1}{8\pi}\oint\nabla_{\mu} k_{\nu} ds^{\mu\nu}$$ $$k^{\mu}=[1, 0, 0, 0]$$ $$k_{\nu}=[f(r), 0, 0, 0]$$ Because at large $r$, $T$ is Minkowski flat, $$\nabla_\mu k_{\nu}\rightarrow\partial_\mu k_\nu$$ \begin{align} \\\partial_\mu k_\nu&=\partial_rf(r) \\&=\bigg(\frac{\partial M}{r^2}-\frac{2Q^2}{r^3}\bigg) \end{align} \begin{align} \\M_K&=\frac{1}{8\pi}\int\bigg(\frac{\partial M}{r^2}-\frac{2Q^2}{r^3}\bigg)\sqrt{g_{\theta\theta}g_{\phi\phi}}\ d\theta\ d\phi \\&=\frac{1}{8\pi}\int\bigg(\frac{\partial M}{r^2}-\frac{2Q^2}{r^3}\bigg)r^2\sin\theta\ d\theta\ d\phi \\&=\frac{1}{8\pi}\int^\pi_0\bigg(\partial M-\frac{2Q^2}{r}\bigg)2\pi\sin\theta\ d\theta \\&=M-\frac{Q^2}{r} \end{align}

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  • $\begingroup$ Can you explain where you get stuck with Kerr-Newman? Have you tried the plain Kerr case? $\endgroup$ – mmeent Jul 27 '18 at 6:42
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I would like to split my answer into 3 parts. Even though I'm not going to present explicit calculation of Komar's mass for the case of interest (due to its considerable length) I will present (what I hope to be) useful hints and leads.


Part I: Another def of Komar's conserved quantities

The article you've presented defines Komar's conserved quantities in terms of differential forms, which I will discuss in Parts II and III of this answer.

The same things may also be written by means of index notation. This definition may be found in paragraph 6.4 of Sean Carroll's book "Spacetime and Geometry". The good thing about this paragraph is that one can also find explicit utilization of this approach for Schwarzschild black hole.

Application of this approach for Kerr-Newman black hole seems to follow exactly the same steps. I must admit that I didn't have enough time to arrive to final answer myself due to the length of calculation. Still calculations themselves are not that difficult. Just a lot of derivatives.

$$ E_{R} = \frac{1}{4\pi G} \int_{ \partial S} d^2 x \sqrt{\gamma^{(2)}}\cdot n_{\mu} \sigma_{\nu} \nabla^{\mu} K^{\nu} \tag{1} \,,$$ where:

  • $S$ is a 3-dimensional constant time slice;

  • $n_{\mu}$ is a unit vector orthogonal to it;

  • $\partial S$ is boundary of $S$;

  • $\sigma_{\mu}$ is unit vector orthogonal to this boundary (in our case it is space-like vector looking in radial direction);

  • $\gamma^{(2)}$ is induced metric on $\partial S .$


Part II: Equivalence of two definitions

Note that different sources define Komar's conserved quantities with different numerical coefficients in front of integral. I will talk about equivalence of definitions up to this coefficient. There are also slight differences in notation for different formulas. Yet I wanted to keep them the same way they were presented in the sources.

Aforementioned article gives following def of conserved Komar's quantities: $$ K_{\xi^{\mu}_{(t)}} = -\frac{1}{8 \pi} \int_{\partial S} * \mathrm{d}\sigma \tag{2} \,,$$ where:

  • $\sigma = \xi_{ (t) \mu} \mathrm{d}x^{\mu}$ is a time-Killing one form;

  • $\xi_{ (t)}^{ \mu}$ is Killing vector corresponding shifts in time direction.

Before diving into details of calculation Komar's mass by means of this formula, I would strongly recommend to prove equivalence of (1) and (2). Here are some useful tips.

First you might like to go through Appendix E of Sean Carroll's book "Spacetime and Geometry". Stock's theorem derived there allows to show equivalence of (1) and the following expression: $$ Q_{S} = -\int_{S} * J \tag{3} \,$$ where $J$ is divergentless current corresponding to Killing vectors $$ J^{\mu}_R = K_{\nu} R^{\mu \nu} = \nabla_{\nu} \nabla^{\mu} K^{\nu} $$ second equality comes from Killing equation, and $R^{\mu \nu}$ is Ricci tensor.

Now (3) and (2) are equivalent (up to constant) as long as following expression holds (use Stock's theorem to obtain it): $$ \frac{1}{2} \mathrm{d}(*\mathrm{d}\sigma) = *J \,.$$

The easiest way (at least for me) to show that this is indeed the case was to write both sides explicitly in coordinate notation and carefully compare them. Along the way you might find useful this site, page 21 of this guide and "Exterior derivative", Wikipedia.


Part III: Notes about approach from the article

If, once you managed to go through calculations from Part II, it will be much easier to follow the approach from the article. It is true that they only present explicit calculation of $J_{\text{eff}} .$ Yet if you have access to the article J. M. Cohen, F. De Felice, J. Math. Phys. 25, 992 (1984), you can see that exactly the same approach is used.

Since I don't know which steps are needed to be elaborated, I will take free rein. If something is still missing I could edit this post later.

From my point of view, one unexplained thing was change of basis of 1-forms from $\mathrm{d}t, \, \mathrm{d}r, \, \mathrm{d}\theta, \, \mathrm{d}\phi$ to $\mathrm{d}\hat{\chi}_0 \dots \mathrm{d}\hat{\chi}_3 .$ The reason is the following one. Original basis of 1-forms is not orthogonal(metric has off-diagonal terms). Once we go to orthonormal basis it becomes much easier to write explicit expression for Hodge star operator.

Everything else seems to be more or less clear, at least after all the training from Part II.

Hope this information helps. If something is unclear or missing, please let me know.

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  • $\begingroup$ By the way your approach for Reissner-Nordstrom black hole seems to be equivalent to (1) def from my answer. I don't know what went wrong for you with it. Hopefully chapter from Carroll can clarify some things. $\endgroup$ – Yaroslav Shustrov Aug 11 '18 at 18:11

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