Gravitomagnetic jets

We present a family of dynamic rotating cylindrically symmetric Ricci-flat gravitational fields whose geodesic motions have the structure of gravitomagnetic jets. These correspond to helical motions of free test particles up and down parallel to the axis of cylindrical symmetry and are reminiscent of the motion of test charges in a magnetic field. The speed of a test particle in a gravitomagnetic jet asymptotically approaches the speed of light. Moreover, numerical evidence suggests that jets are attractors. The possible implications of our results for the role of gravitomagnetism in the formation of astrophysical jets are briefly discussed.

Figure 1

The top panel shows computer-generated plots of admissible solutions $\pm X$ versus $r$ for differential Eq. (2). The thick curve is for the initial data $X(1)=1$ and $X_r(1)=2$; the thin curve is for the data $X(1)=1$ and $X_r(1)=3$. The middle (bottom) panel depicts computer-generated graphs of $F$, $H$, $W$, and $X$ for the thick (thin) solution. The graphs of $F$, $H$, and $X$ vanish at the left end point $r_b$ of the domain of definition of $X$, which is $r_b\approx 0.720$ for the thick solution that allows jets and $r_b\approx 0.787$ for the thin solution that does not allow jets.

Figure 2

Computer-generated phase portrait for system (20). The rest point at $(\delta ,\alpha )=(0,0)$ is a degenerate singularity of system (20), while the other rest point at (1,1) is a simple singularity. This latter isolated singularity is a spiral point that corresponds to the special solution (3).

Figure 3

Plot of $R$ versus $\rho$ corresponding to the spacetime defined by $X$ with $X(1)=1$ and $X_r(1)=2$, as in the middle panel of Fig. 1.

Figure 4

Plots of $(2XQ{)}^{-1}$ versus $\rho =r^{-1/2}$ for the two cases of $X$ given in the top panel of Fig. 1. The upper curve here corresponds to initial data $X(1)=1$ and $X_r(1)=2$; the lower curve corresponds to $X(1)=1$ and $X_r(1)=3$. In general, the maximum of $(2XQ{)}^{-1}$ occurs where $H(r)=rXQ-1$ vanishes; this happens at $r=1$ for the upper curve.

Figure 5

Schematic diagram illustrating the results of our numerical work. Right panel: for a range of parameters close to the jet parameters, the radial coordinate as a function of the particle’s proper time starts from its initial value $r_0$ and reaches $r_J$. It is then fixed at $r_J$ while the test particle executes helical motion up or down parallel to the rotation axis as in jets (see Fig. 6). Left panel: beyond a certain range in the canonical angular momentum $C_{\varphi }$, the test particle cannot be confined; that is, $r_J$ is bypassed and the particle soon reaches the boundary cylinder, thus leaving the spacetime region of interest. In constructing this figure, we have used the solution $X$ given in the middle panel of Fig. 1; therefore, $r_b\approx 0.720$, $r_J\approx 0.774$, and $r_0=1$.

Figure 6

The result of integration of system (55) for timelike geodesics attracted to jets. The initial data at ${\tau }_0=0$ are $t_0=0$, $r_0=1$, $U_0=0$, ${\varphi }_0=0$, $z_0=0$, $X_0=1$, and $Y_0=2$. The parameters are $C_{\varphi }=0.9$ and $C_z=\pm 1$. The left-hand plot (jet going up) is for $C_z=1$ and the right-hand plot (jet going down) is for $C_z=-1$. The coordinates are $(x,y,\widehat{z})$, where $x=\rho \cos \varphi$, $y=\rho \sin \varphi$, and $\rho =r^{-1/2}$; moreover, $\widehat{z}=\ln |\ln z|$ for the left-hand plot and $\widehat{z}=-\ln |\ln |z||$ for the right-hand plot. We use $\widehat{z}$ instead of $z$ for the sake of clarity.

Figure 7

A portion of the $(u,v)$ state space for system (b19) is shown here. We have plotted the line $v=1$, the parabola $v=u^2$, the line $v=(2/3{)}^{1/2}u$ (which is the vertical isocline), and the thick curve $\mathcal{Z}$ that is an approximation of the solution that approaches the rest point at (0,1) tangent to $v=1$. This solution $\mathcal{Z}$ connects the two rest points and crosses the parabola.

Figure 8

A portion of the phase portrait of system (b27) is shown. The shaded region is bounded by the trajectory $\mathcal{J}$ that leaves the source at $(p,q)=(1,0)$ tangent to the line $q=-2p+2$ (which is also depicted) and the trajectory that approaches the sink at (0,1) tangent to the line $q=-2p+1$ (which is also depicted). Solutions that correspond to trajectories that lie in the shaded region are admissible (because their $p$ components remain in the interval $0<p<1$) and allow jets (because they cross the line $q=1$).