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{equation} \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} \end{equation} 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: \begin{equation} \varpi \equiv \frac{\Sigma}{\rho}\sin\theta, \end{equation} where \begin{equation} \rho^{2} \equiv r^{2} + a^{2}\cos^{2}\theta, \end{equation} \begin{equation} \Sigma^{2} \equiv (r^{2}+a^{2})^{2}- a^{2}\Delta\sin^{2}\theta, \end{equation} and \begin{equation} \Delta \equiv r^{2} + a^{2} - 2Mr. \end{equation} Note also that $\alpha$ is the lapse function defined as \begin{equation} \alpha\equiv \frac{\rho}{\Sigma}\sqrt{\Delta}. \end{equation}

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: \begin{equation} \dot{\Phi} = \dot{\varpi}\frac{\partial\Phi}{\partial \varpi} \end{equation} and \begin{equation} \dot{\Psi} = \dot{\varpi}\frac{\partial\Psi}{\partial \varpi}. \end{equation} 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!

  • $\begingroup$ I am not sure you're going to get a good answer here, but FWIW as far as I can see you're correct. I can't read the paper you've posted, but the spacetime is stationary and that is reflected in the coordinates, so as you say the partial derivatives wrt to t must be zero . $\endgroup$
    – John Davis
    Nov 24 '15 at 11:59
  • $\begingroup$ @JohnDavis I see. Thanks for your comment! $\endgroup$ Nov 24 '15 at 12:53
  • $\begingroup$ I have not read the paper. But in physics it is not unusual to work in, say, a rotating coordinate system. Then the spatial coordinates do change in time. $\endgroup$ Nov 29 '15 at 4:08
  • $\begingroup$ @DavidKetcheson Unless I'm terribly mistaken, the space and time coordinates are independent variables for the PDEs the authors derive in the paper. $\endgroup$ Nov 29 '15 at 4:31

I believe your assumption of the partial derivatives with respect to time being identically zero is incorrect. When you are talking about working in phase space, i.e., Liouville's theorem with functions of positions and velocities, then you are correct that the partial derivatives of one phase space variable with respect to the others is zero. But here (I presume), they're looking at the temporal dynamics of the black hole magnetosphere.

It is a little dense (but if you're working through Equation 4.24 above then you'll have no trouble with this), but you might find the stuff in this paper as filling in a lot of the background equations.

  • $\begingroup$ Hi Dave, thank you for your answer. My understanding is, first, that the authors have derived a partial differential equation satisfied by $\Phi(t,\textbf{r})$ and $\Psi(t,\textbf{r})$ and, second, that $\varpi$ and $\varphi$ are spatial coordinates parametrizing $\textbf{r}$. I thus thought that $\partial\varpi/\partial t$ and $\partial\varphi/\partial t$ are identically zero. Am I mistaken about this? $\endgroup$ Nov 24 '15 at 4:35
  • $\begingroup$ They're treating the stationary and non-stationary cases separately. In Section 3 they handle the stationary case and you'll see there are no time derivatives in the equation (Equation 3.23). In Section 4 they now look to the time-varying effects where the time derivatives of the spatial coordinates now become velocities (e.g., Equation 4.15 has the polodial component of velocity dependent upon $\dot{\varpi}$). In fact, if you set all the time derivatives to zero in Equation 4.24, you recover Equation 3.23. $\endgroup$
    – Dave
    Nov 24 '15 at 12:27
  • $\begingroup$ I have no problem with (partial) time derivatives of scalar or vector fields. In fact, to describe the entire magnetosphere (rather than a trajectory of a particle), I think that the velocity should be a vector field, i.e., a function of $t$ and $\textbf{r}$. In this regard, $\textbf{v}^T = -(\omega-\Omega^F)\textbf{m}/\alpha$ is acceptable, whereas $v^p = -\dot{\varpi}/\alpha$ (Eq 15) isn't. Obviously, $\partial x / \partial t$ should not appear in a PDE with independent variables $x$ and $t$, I believe that $\dot{\varpi} \equiv \partial\varpi/\partial t $ is unacceptable for the same reason. $\endgroup$ Nov 24 '15 at 13:27
  • $\begingroup$ My guess is that it is wrapped up in the FIDO geometry. This goes beyond my expertise, but the geometry is described in this 1986 paper (see, in particular, the second paragraph on page 33). $\endgroup$
    – Dave
    Nov 24 '15 at 16:10
  • $\begingroup$ Thanks for the link to the paper. But the description of what FIDO is in page 916 seems to confirm what I thought. $\endgroup$ Nov 25 '15 at 12:36

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