Charged particle motion and electromagnetic field in $\gamma$ spacetime

We consider the electromagnetic field occurring in the background of a static, axially symmetric vacuum solution of Einstein’s field equations immersed in an external magnetic field. The solution, known as the $\gamma$ metric (or Zipoy-Voorhees), is related to the Schwarzschild spacetime through a real positive parameter $\gamma$ that describes its departure from spherical symmetry. We study the motion of charged and uncharged particles in this spacetime and particle collision in the vicinity of the singular surface and compare with the corresponding result for Schwarzschild. We show that there is a sharp contrast with the black hole case; in particular, in the prolate case ($\gamma <1$) particle collision can occur with an arbitrarily high center of mass energy. This mechanism could in principle allow one to distinguish such a source from a black hole.

Figure 1

Radial dependence of the azimuthal component of the magnetic field in $\gamma$ spacetime for the different values of $\gamma$ and the azimuthal angle $\theta$.

Figure 2

Radial dependence of the radial component of the magnetic field in $\gamma$ spacetime for different values of $\gamma$ and the azimuthal angle $\theta$.

Figure 3

Radial dependence of the total magnetic field $B^{\mathrm{tot}}=\sqrt{(B^r{)}^2+(B^{\theta }{)}^2}$ in $\gamma$ spacetime for different values of $\gamma$ and the azimuthal angle $\theta$.

Figure 4

Radial dependence of the effective potential for a charged particle on the equatorial plane for different values of $\gamma$ and magnetic parameter $\beta$.

Figure 5

Radial dependence of the angular momentum of the charged particle moving along circular orbit on equatorial plane for different values of $\gamma$ and magnetic parameter $\beta$.

Figure 6

Value of the ISCO for neutral particles on circular orbits in the equatorial plane for different values of $\gamma$. Vertical dotted lines define the borders of regions with different ISCO structure (see text for details). The corresponding values of $\gamma$ are: $\gamma =1/\sqrt{5}$; $1/2$; and 1. The horizontal dotted line corresponds to the radius of singular surface $r=2m$. The dashed line corresponds to the radius of photon sphere.

Figure 7

Value of the ISCO radius for charged particles on circular orbits in the equatorial plane for different values of $\gamma$ and different values of the magnetic parameter $\beta$. Vertical dotted lines are the borders of the regions, significant in the neutral case, corresponding the following values: $\gamma =1/\sqrt{5}$; $1/2$; and 1. It can be seen that the values of $\gamma$ defining region (ii) change with $\beta$. The horizontal dotted line corresponds to the radius of singular surface $r=2m$. In the case of the motion of charged particles we have two separate curves for the location of the ISCO, depending on the sign of the charge of the particle (solid and dashed lines correspond to positive and negative charges, respectively).

Figure 8

Radial dependence of the center of mass energy of two charged particles moving on circular orbits for the different values of $\gamma$ parameter and magnetic parameter $\beta$. Here we considered both “$+$” and “$-$” type of circular orbits for charged particles corresponding to angular momentum ${\mathcal{L}}_{+}$ and ${\mathcal{L}}_{-}$, respectively.

Figure 9

Radial dependence of the center of mass energy for the collision of a charged particle moving on circular orbit with a neutral particle infalling from infinity for the different values of $\gamma$ and $\beta$ parameters.