fjebaker
commented
1 year ago I'm trying to solve the circular orbits of charged particles in the Kerr-Newman spacetime analytically, and have made some progress, but am now concerned this method is not possible without using a non-linear solver.
In short, stemming from the method for calculating the Keplerian Frequency, writing the acceleration equation with the Lorentz force ansatz:
$$ \frac{\partial^2 x^\mu}{\partial \lambda^2} = - \Gamma^\mu{\phantom{\mu}\alpha \beta} \frac{\partial x^\alpha}{\partial \lambda} \frac{\partial x^\beta}{\partial \lambda} + e F^{\mu}{\phantom{\mu}\nu} \frac{\partial x^\nu}{\partial \lambda}, $$
which, by the same expansions in the blog post, with
$$ \frac{\partial x^r}{\partial \lambda} = \frac{\partial x^\theta}{\partial \lambda} = 0, $$
can be whittled down to
$$ 0 = \frac{1}{2} \frac{\partial x^\alpha}{\partial \lambda} \frac{\partial x^\beta}{\partial \lambda} \partialr g{\alpha \beta} + g{\sigma r} e F^{\sigma}{\phantom{\sigma}\mu} \frac{\partial x^\mu}{\partial \lambda}. $$
The method I'd been using is then to expand in terms of $t$ and $\phi$ components, and define the Keplerian velocity via
$$ \Omega := \frac{\dot{x}^\phi}{\dot{x}^t}, $$
where the dot denotes differentiation with respect to $\lambda$, and to solve for the Keplerian velocity purely in terms of metric factors -- and now also $F^r{\phantom{r}t}$ and $F^r{\phantom{r}\phi}$. The issue with this is that I made a mistake in thinking dividing by $(\dot{u}^t)^2$ would eliminate all velocity terms except for $\Omega$, however this is not the case, and one ends up with a function in the form
$$ 0 = \mathcal{P}(\Omega) + g{rr} \frac{e}{\dot{x}^t} \left(F^r{\phantom{r}t} + F^r_{\phantom{r}\phi}\Omega \right), $$
where $\mathcal{P}$ is a polynomial in $\Omega$ with coefficients derived from the metric.
One can use the constraint on velocity to find a formulation of $\dot{x}^t$ that depends on $\sqrt{\mathcal{\tilde{P}}(\Omega)}$, but the resulting expression I am unable to rearrange to solve for $\Omega$, once substituting the square root terms.
This may be accomplished numerically, with the full complex form written out and then using a root finder to determine $\Omega$, with some a priori condition to ensure either the prograde or retrograde solution is chosen.
The version implemented in the PR forgot that the $1/\dot{x}^t$ terms appear in the constraining equation, and solved the polynomial with coefficients modified by the Maxwell tensor terms. This, as one might expect, consistently calculates $\Omega$ too small compared to $\Omega$ derived through numerical solving methods (the setup here is arbitrary with $eQ \neq 0$):
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