# Does this black hole magnetohydrodynamics equation even superficially make sense?

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My question is about the journal paper mentioned in an Academia Stack Exchange post. Please understand that this paper has never been posted on arXiv, and I can provide only a link whose content is behind a paywall.

Summary of my question: it boils down to "whether a spatial coordinate of a fiducial observer can have a nonzero partial derivative with respect to the coordinate time."

I am interested in the validity of the central result of this paper. It is Eq. 4.24, which reads

\begin{split}
&\nabla\cdot\left[\frac{\alpha}{\varpi^{2}}\left\{1-\left(\frac{\omega-\Omega^{F}}{\alpha}\varpi\right)^{2}\right\}\nabla\Psi\right]
- \frac{\omega-\Omega_{F}}{\alpha}\nabla\Omega^{F}\cdot\nabla\Psi\\
&+ \frac{4\pi\dot{\varpi}}{\alpha^{2}\varpi}\left(1-\frac{\dot{\Phi}}{4\pi}\right)\frac{\partial \Omega^{F}}{\partial z} + 4\pi \frac{\partial}{\partial z}\left[\frac{\dot{\varpi}}{\alpha\varpi}\frac{\omega-\Omega^{F}}{\alpha}\left(1-\frac{\dot{\Phi}}{4\pi}\right)\right]\\
&+\frac{1}{\alpha\varpi^{2}}\left[\left(\frac{\dot{\alpha}}{\alpha}+\frac{\dot{\varpi}}{\varpi}\right)\dot{\Psi}-\ddot{\Psi}\right] + \frac{\dot{\varphi}}{\varpi}\frac{\omega-\Omega^{F}}{\alpha}\frac{\partial\Psi}{\partial\varpi}\\
&-\frac{16\pi^{2}\xi}{\varpi^{2}}\left(1-\frac{\dot{\Phi}}{4\pi}\right) = 0,
\end{split}

where a dot on top of a symbol denotes a partial derivative with respect to the coordinate time $t$.

The above partial differential equation is supposed to describe the magnetosphere of a Kerr black hole. The authors use the spherical coordinates $(r,\theta,\varphi)$ and define $\varpi$ as follows:

\varpi \equiv \frac{\Sigma}{\rho}\sin\theta,

where

\rho^{2} \equiv r^{2} + a^{2}\cos^{2}\theta,

\Sigma^{2} \equiv (r^{2}+a^{2})^{2}- a^{2}\Delta\sin^{2}\theta,

and

\Delta \equiv r^{2} + a^{2} - 2Mr.

Note also that $\alpha$ is the lapse function defined as

\alpha\equiv \frac{\rho}{\Sigma}\sqrt{\Delta}.

The functions $\Psi(t,\textbf{r})$ and $\Phi(t,\textbf{r})$ denote the magnetic and electric fluxes through an $\textbf{m}$-loop passing through $\textbf{r}$, where $\textbf{m} \equiv \varpi\hat{e}_{\varphi}$ is the Killing vector associated with axisymmetry.

What confuses me is the following: $\varpi$, $\varphi$, and $\alpha$ are simply spatial coordinates or combinations thereof, and their (partial) time derivatives should all be identically equal to zero because space and time coordinates are independent variables. This would render many parts of Eq. 4.24 nothing but convoluted ways to express the number zero.

I have also tried to follow the derivation of Eq. 4.24, and figured out that the authors implicitly assumed the following relations:

\dot{\Phi} = \dot{\varpi}\frac{\partial\Phi}{\partial \varpi}

and

\dot{\Psi} = \dot{\varpi}\frac{\partial\Psi}{\partial \varpi}.

Recall that a dot means a partial derivative with respect to time. As $\dot{\varpi}$ is identically zero, the above relations seem to be wrong.

However, what makes me somewhat unsure about my conclusion is that this paper is published in The Astrophysical Journal, a renowned peer-reviewed journal in astrophysics. (I have little expertise in astrophysics.)

Could someone verify whether my suspicion is well founded or correct me where I am wrong? Thanks in advance!

edited Nov 25, 2015

In fact, I posted exactly the same question on Physics Stack Exchange, but it hasn't attracted much attention. I interpreted the PhysicOverflow policy that one can make one's own PhysSE post native here as that I can copy and paste my post from PhysSE. If I misunderstood the policy, please let me know how I can make this post in compliance with the rules here.

Added Note: I later deleted the PhysSE post and reposted it on Astronomy SE.

Hi arendellean, welcome to PhysicsOverflow.

It is correct that you are allowed to make your own (imported) posts native; there is also nothing wrong with copy-pasting your own posts from elsewhere to here.

It seems you have studied the paper in some detail, so maybe you would also be interested in submitting it to our reviews section and write a review for it later?

@Dilaton Hi Dilaton, thank you for your welcome and also for comfirming the policy. As for a review, I should admit that I'm not capable of writing a worhwhile one. Of course, if my suspicion turned out to be true, this paper would be so obviously faulty that it wouldn't deserve an additoinal review. In case my suspicion is wrong, the problem is that I don't have enough knowledge of this field to discuss the physics described by the paper (as opposed to rederivation of equations)..

I cannot access the paper now, but it all depends on whether your fiducial observer is moving or not, the meaning of the coordinates $r,\theta$ is then in fact $r(\tau),\theta(\tau)$ as given by the trajectories (velocity fields) of the observers. Also, I advise caution, the publication seems to be at least in the ethical gray zone, there is a non-zero possibility that it is simply not good science.

@Void Thank you for your comment. My understanding is that for a given coordinate system, a fiducial observer has spatial coordinates that do not change with time. Am I mistaken here? Just in case you missed it, I would like to mention that all time derivatives here are with respect to the coordinate time.

I'm well aware of the potential scientific misconduct associated with this work. The journal paper is almost a duplication of winter school lecture notes published 13 years ago. More surprising is that while the notes are single-authored by Seok Jae Park, another person is added as the first author to the journal paper. The new guy is Yoo Geun Song, who is a Ph.D. student of Park and also a child prodigy well known among Koreans. According to Park, there is nothing wrong with Song being the first author because he derived Eq. 4.24, which (again, according to Park) constituties a major improvement over the earlier lecture notes. This made me curious about the pure scientific value of this equation provided that we set aside the ethics issue.

What I have found is that Eq. 4.24 is really a restatement of Eq. 61 in the winter school notes, modulo the two implicit assumptions $\dot{\Phi} = \dot{\varpi}\,\partial\Phi/\partial\varpi$ and $\dot{\Psi} = \dot{\varpi}\,\partial\Psi/\partial\varpi$ (note that a dot means a partial derivative with respect to $t$). I thought that for a given coordinate system, the partial derivative of a spatial coordinate with respect to the time coordinate is trivially zero, but $\dot{\varpi}$ and $\dot{\varphi}$ appear not only in these implicit assumptions but also in Eq. 4.24 itself.  This seems to make the validity of Eq. 4.24 questionable. Then, on the other hand, this paper was reviewed by experts in the field in which I have little knowledge, and I became somewhat unsure.

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