Melvin metric

The Melvin metric describes the geometry of empty space having cylindrical symmetry containing a magnetic field pointing in the ${\displaystyle z}$-direction. The Melvin metric is given by ^[1]

${\displaystyle ds^{2}=f^{2}(-dt^{2}+dz^{2}+dr^{2})+{\frac {r^{2}}{f^{2}}}d\phi ^{2},\quad f=1+{\frac {B^{2}r^{2}}{4}},}$

with ${\displaystyle t,z\in (-\infty ,+\infty ),r\in [0,\infty ),\phi \in [0,2\pi )}$. The Melvin magnetic solution is used in astrophysical models as a background for more complicated spacetimes.

Einstein tensor

The nonzero components of the Einstein tensor are

${\displaystyle G_{tt}=G_{rr}={\frac {B^{2}}{f^{2}}},\quad G_{zz}=-{\frac {B^{2}}{f^{2}}},G_{\phi \phi }={\frac {B^{2}r^{2}}{f^{6}}}}$

Stress energy tensor

The field strength tensor is

${\displaystyle F=e^{i\psi }B(dz\wedge dt)}$

${\displaystyle \psi }$ is the duality rotation parameter. For ${\displaystyle \psi =\pi /2}$, we have ${\displaystyle F=B(1+B^{2}r^{2}/4)r\;dr\wedge d\phi }$, a magnetic field oriented in the ${\displaystyle z}$-direction. The nonzero components of field strength tensor are

${\displaystyle F_{r\phi }=-F_{\phi r}={\frac {Br}{f^{2}}}}$

Recall the form of the stress energy tensor for a source-free electromagnetic field

${\displaystyle T_{\mu \nu }={\frac {1}{4\pi }}\left(F_{\mu \rho }F_{\nu }^{\;\rho }-{\frac {1}{4}}g_{\mu \nu }F_{\rho \sigma }F^{\rho \sigma }\right)}$

Knowing the form of the Einstein tensor, we know that ${\displaystyle T_{\mu \nu }}$ is nonzero for

${\displaystyle T_{tt}=T_{rr}=-T_{zz}={\frac {B^{2}}{8\pi f^{2}}},\quad T_{\phi \phi }={\frac {B^{2}r^{2}}{8\pi f^{6}}}}$

References

1. Cardoso, Vitor (2024-10-07), An exact solution describing a scalar counterpart to the Schwarzschild-Melvin Universe, arXiv: 2410.02851v1