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In general relativity, the Melvin metric describes the geometry of spacetime containing a bundle of cylindrical symmetric magnetic or electric field pointing in the $z$ -direction, held together by its own gravity.^[1]^: 103 It was first obtained by William Bonnor in 1954,^[2] although it is named after Mael Melvin who rediscovered it in 1964.^[3]^[1]^: 102 The Melvin magnetic solution is used in astrophysical models as a background for more complicated spacetimes. The concept is also referred to as “Melvin’s magnetic universe”.^[4]

Studying the Melvin universe is useful because the equations of general relativity are complex and nonlinear. Exact solutions even in unrealistically symmetric and limited cases is part of developing new ideas for realistic cases which may require numerical computation.^[5] The Melvin solution can be viewed as a spacetime with a uniform magnetic field whose own stress-energy concentrates space close to the symmetry axis.^[6]

Qualitatively, the Melvin metric relates to the idea that an intense gravitational field such as in a black hole would hold magnetic field lines together similar to the way an iron bar does. The magnetic field energy reacts to gravity — gravitates — because energy is equivalent to mass; conversely the field creates gravitiation force.^[7]^: 263 However, the two dimensional cylindrical geometry of Melvin’s metric fails the hoop conjecture: the magnetic field line repulsion cannot be overcome by gravitational forces.

Nevertheless the interaction between magnetic fields and intense gravity of black holes is studied by astrophysicists^[8] leading to analysis of “Schwarzschild–Melvin black holes”^[1]^: 149 and, when rotation is added, “Kerr–Melvin black holes”.^[9] Models of Kerr–Melvin black holes have been compared to the M87* images by the Event Horizon Telescope to infer the strength of the star’s magnetic field. Gray regions in the image are believed to be caused by photons trapped in stable orbits by the Melvin magnetic field effects.^[10]

Description

The Melvin metric is given in axisymmetric coordinates by^[11]

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 $t,z\in (-\infty ,+\infty ),r\in [0,\infty ),\phi \in [0,2\pi )$ , and $B$ parametrizing the strength of the electromagnetic field. The nonzero components of the Einstein tensor are^[12]^: 3

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}}}

The metric follows from an electromagnetic field configuration described by a complex self-dual field strength tensor^[1]^: 102

F+i{\tilde {F}}=e^{-i\psi }B(dz\wedge dt+f^{-2}rdr\wedge d\phi ),

where $\psi$ is the duality rotation parameter that parametrizes a family of solutions and ${\tilde {F}}$ is the Hodge dual field strength tensor. For $\psi =\pi /2$ , we have $F=Bf^{-2}r\;dr\wedge d\phi$ , a magnetic field oriented in the $z$ -direction while for $\psi =0$ we have $F=Bdz\wedge dt$ , which describes an electric field pointing along the z-direction. For the magnetic solution, the nonzero components of field strength tensor are

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

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

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),

the non-zero components of the stress-energy tensor are found to be^[12]^: 3

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}}},

which matches the result acquired from the Einstein tensor.

References

^ ^a ^b ^c ^d Griffiths, Jerry B.; Podolský, Jiří, eds. (2009). Exact Space-Times in Einstein’s General Relativity. Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511635397.010. ISBN 978-0-521-88927-8.
^ Bonnor, W. B. (1954). “Static Magnetic Fields in General Relativity”. Proc. Roy. Soc. Lond. A. 67 (3): 225–232. Bibcode:1954PPSA…67..225B. doi:10.1088/0370-1298/67/3/305.
^ Melvin, M.A. (1964). “Pure magnetic and electric geons”. Phys. Lett. 8 (1): 65–70. Bibcode:1964PhL…..8…65M. doi:10.1016/0031-9163(64)90801-7.
^ Thorne, Kip S. (July 1965). “Absolute Stability of Melvin’s Magnetic Universe”. Physical Review. 139 (1B): B244–B254. Bibcode:1965PhRv..139..244T. doi:10.1103/PhysRev.139.B244. ISSN 0031-899X.
^ Cite error: The named reference Cardoso-2025 was invoked but never defined (see the help page).
^ Thorne, Kip S. (July 12, 1965). “Absolute Stability of Melvin’s Magnetic Universe”. Physical Review. 139 (1B): B244–B254. Bibcode:1965PhRv..139..244T. doi:10.1103/PhysRev.139.B244. ISSN 0031-899X.
^ Thorne, Kip S.; Hawking, Stephen (1994). Agrawal, Milan (ed.). Black Holes and Time Warps: Einstein’s Outrageous Legacy (1st ed.). W. W. Norton & Company. ISBN 978-0-393-31276-8. Retrieved 12 April 2019.
^ Ernst, Frederick J. (January 1976). “Black holes in a magnetic universe”. Journal of Mathematical Physics. 17 (1): 54–56. Bibcode:1976JMP….17…54E. doi:10.1063/1.522781. ISSN 0022-2488.
^ Hou, Yehui; Zhang, Zhenyu; Yan, Haopeng; Guo, Minyong; Chen, Bin (September 30, 2022). “Image of a Kerr-Melvin black hole with a thin accretion disk”. Physical Review D. 106 (6) 064058. arXiv:2206.13744. Bibcode:2022PhRvD.106f4058H. doi:10.1103/PhysRevD.106.064058. ISSN 2470-0010.
^ Chen, Songbai; Jing, Jiliang; Qian, Wei-Liang; Wang, Bin (June 2023). “Black hole images: A review”. Science China Physics, Mechanics & Astronomy. 66 (6) 260401. arXiv:2301.00113. Bibcode:2023SCPMA..6660401C. doi:10.1007/s11433-022-2059-5. ISSN 1674-7348.
^ Cardoso, Vitor; Natário, José (October 2025). “An exact solution describing a scalar counterpart to the Schwarzschild-Melvin Universe”. General Relativity and Gravitation. 57 (10) 138. arXiv:2410.02851. Bibcode:2025GReGr..57..138C. doi:10.1007/s10714-025-03476-0. ISSN 0001-7701.
^ ^a ^b Barbosa, L.G.; Santos, L.C.N.; Zamperlini, J.V.; da Silva, F.M.; Barros, C.C. (2025). “Charged Scalar Boson in Melvin Universe”. Universe. 11 (6): 193. arXiv:2506.07329. Bibcode:2025Univ…11..193B. doi:10.3390/universe11060193.

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