Unified Bertotti-Robinson and Melvin spacetimes

We present a solution for the Einstein-Maxwell equations which unifies both the magnetic Bertotti-Robinson and Melvin solutions as a single metric in the axially symmetric coordinates $\{t,\rho ,z,\phi \}$. Depending on the strength of magnetic field the spacetime manifold, unlike the cases of separate Bertotti-Robinson and Melvin spacetime, develops singularity on the symmetry axis ($\rho =0$). Our analysis shows, beside other things, that there are regions inaccessible to all null geodesics.

Figure 1

Plotting of $\rho (z)$ versus $z$ in accordance with the geodesics equation (45), for specific values of $B_0$ (${\lambda }_0=1$). The initial conditions are chosen such that $\rho (0)=1$ and ${\rho }^{\prime }(0)=0$.

Figure 2

Radial fall from $\rho =1$ through $\rho =0$ (in the $z=0$ plane) for a fixed angle as a function of the coordinate time [Fig. 2a]/proper time [Fig. 2b]. The particle crosses $\rho =0$ freely since $\rho =0=z$ is not singular in the chosen interval $\frac{B_0}{{\lambda }_0}=1$.

Figure 3

Stability condition for particles in circular orbits. From Eq. (74), $V^{\prime \prime }({\rho }_c)$ is plotted versus the circular radius ${\rho }_c$. It is observed that for ${\rho }_c<\frac{1}{\sqrt{3}}$ we have stable orbits since it gives $V^{\prime \prime }({\rho }_c)>0$.