A Global Solution of the Einstein-Maxwell Field Equations for Rotating Charged Matter

Affiliation(s)

School of Physics Astronomy and Mathematics, University of Hertfordshire, Hatfield, UK.

School of Physics Astronomy and Mathematics, University of Hertfordshire, Hatfield, UK.

ABSTRACT

A stationary axially symmetric exterior electrovacuum solution of the Einstein-Maxwell field equations was obtained. An interior solution for rotating charged dust with vanishing Lorentz force was also obtained. The two spacetimes are separated by a boundary which is a surface layer with surface stress-energy tensor and surface electric 4-current. The layer is the spherical surface bounding the charged matter. It was further shown, that all the exterior physical quantities vanished at the asymptotic spatial infinity where spacetime was shown to be flat. There are two different sets of junction conditions: the electromagnetic junction conditions, which were expressed in the traditional 3-dimensional form of classical electromagnetic theory; and the considerably more complicated gravitational junction conditions. It was shown that both—the electromagnetic and gravitational junction conditions—were satisfied. The mass, charge and angular momentum were determined from the metric. Exact analytical formulae for the dipole moment and gyromagnetic ratio were also derived. The conditions, under which the latter formulae gave Blackett’s empirical result for rotating stars, were investigated.

A stationary axially symmetric exterior electrovacuum solution of the Einstein-Maxwell field equations was obtained. An interior solution for rotating charged dust with vanishing Lorentz force was also obtained. The two spacetimes are separated by a boundary which is a surface layer with surface stress-energy tensor and surface electric 4-current. The layer is the spherical surface bounding the charged matter. It was further shown, that all the exterior physical quantities vanished at the asymptotic spatial infinity where spacetime was shown to be flat. There are two different sets of junction conditions: the electromagnetic junction conditions, which were expressed in the traditional 3-dimensional form of classical electromagnetic theory; and the considerably more complicated gravitational junction conditions. It was shown that both—the electromagnetic and gravitational junction conditions—were satisfied. The mass, charge and angular momentum were determined from the metric. Exact analytical formulae for the dipole moment and gyromagnetic ratio were also derived. The conditions, under which the latter formulae gave Blackett’s empirical result for rotating stars, were investigated.

Cite this paper

Georgiou, A. (2012) A Global Solution of the Einstein-Maxwell Field Equations for Rotating Charged Matter.*Journal of Modern Physics*, **3**, 1301-1310. doi: 10.4236/jmp.2012.329168.

Georgiou, A. (2012) A Global Solution of the Einstein-Maxwell Field Equations for Rotating Charged Matter.

References

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[2] A. Georgiou, “Rotating Einstein-Maxwell Fields: Smoothly Matched Exterior and Interior Spacetimes with Charged Dust and Surface Layers,” Classical and Quantum Gravity, Vol. 11, No. 1, 1994, pp. 167-186. doi:10.1088/0264-9381/11/1/018

[3] W. Bonnor, “Rotating Charged Dust in General Relativity,” Journal of Physics A: Mathematical and General, Vol. 13, 1980, pp. 3465-3477.

[4] A. Georgiou, “Comments on Maxwell’s Equations,” International Journal of Mathematical Education in Science and Technology, Vol. 13, No. 1, 1982, pp. 43-47. doi:10.180/0020739820130106

[5] C. Moller, “The Theory of Relativity,” 2nd Edition, Oxford University Press, Oxford, 1972.

[6] E. T. Whittaker and G. N. Watson, “A Course of Modern Analysis,” 4th Edition, Cambridge University Press, Cambridge, 1950.

[7] W. Israel, “Singular Hypersurfaces and Thin Shells in General Relativity,” Nuovo Cimento B, Vol. 48, No. 2, 1966, p. 463. doi:10.1007/BF02712210

[8] W. Bonnor, “The Equilibrium of a Charged Sphere,” Monthly Notices of the Royal Astronomical Society, Vol. 129, 1965, p. 433. http://adsabs.harvard.edu/abs/1965MNRAS.129.443B

[9] J. N. Islam, “Rotating Fields in General Relativity,” Cambridge University Press, Cambridge, 1985. doi:10.1017/CB09780511735738

[10] P. M. S. Blackett, “The Magnetic Field of Massive Rotating Bodies,” Nature, Vol. 159, No. 4046, 1947, pp. 658- 666.

[11] D. V. Ahluwalia and T. Y. Wu, “On the Magnetic Field of Cosmological Bodies,” Lettere Al Nuovo Cimento, Vol. 23, No. 11, 1978, pp. 406-408. doi:10.1007/BF02786999

[12] J. A. Nieto, “Gravitational Magnetism and General Relativity,” Revista Mexicana de Fisica, Vol. 34, No. 4, 1988, pp. 571-576.

[13] H. A. Wilson, “An Experiment on the Origin of the Earth’s Magnetic Field,” Proceedings of the Royal Society, Vol. 104A, 1923, p. 451. http://www.jstor.org/stable/94216

[14] Z. Perjes, “Solutions of the Coupled Electromagnetic Equations Representing the Fields of Spinning Sources,” Physical Review Letters, Vol. 27, 1971, p. 1668.

[15] A. Papapetrou, “A Static Solution of the Equations of the Gravitational Field for an Arbitrary Charge-Distribution,” Proceedings of the Royal Irish Academy, Vol. 51, 1945-1948, p. 191.

[16] R. Ruffini and A. Treves, “On a Magnetized Rotating Sphere,” Astrophysical Journal Letters, Vol. 13, 1973, p. 109.

[1] M. A. H. MacCallum, M. Mars and P. Vera, “Stationary Axisymmetric Exteriors for Perturbations of Isolated Bodies in General Relativity, to Second Order,” Physical Review D, Vol. 75, 2007, Article ID: 024017.

[2] A. Georgiou, “Rotating Einstein-Maxwell Fields: Smoothly Matched Exterior and Interior Spacetimes with Charged Dust and Surface Layers,” Classical and Quantum Gravity, Vol. 11, No. 1, 1994, pp. 167-186. doi:10.1088/0264-9381/11/1/018

[3] W. Bonnor, “Rotating Charged Dust in General Relativity,” Journal of Physics A: Mathematical and General, Vol. 13, 1980, pp. 3465-3477.

[4] A. Georgiou, “Comments on Maxwell’s Equations,” International Journal of Mathematical Education in Science and Technology, Vol. 13, No. 1, 1982, pp. 43-47. doi:10.180/0020739820130106

[5] C. Moller, “The Theory of Relativity,” 2nd Edition, Oxford University Press, Oxford, 1972.

[6] E. T. Whittaker and G. N. Watson, “A Course of Modern Analysis,” 4th Edition, Cambridge University Press, Cambridge, 1950.

[7] W. Israel, “Singular Hypersurfaces and Thin Shells in General Relativity,” Nuovo Cimento B, Vol. 48, No. 2, 1966, p. 463. doi:10.1007/BF02712210

[8] W. Bonnor, “The Equilibrium of a Charged Sphere,” Monthly Notices of the Royal Astronomical Society, Vol. 129, 1965, p. 433. http://adsabs.harvard.edu/abs/1965MNRAS.129.443B

[9] J. N. Islam, “Rotating Fields in General Relativity,” Cambridge University Press, Cambridge, 1985. doi:10.1017/CB09780511735738

[10] P. M. S. Blackett, “The Magnetic Field of Massive Rotating Bodies,” Nature, Vol. 159, No. 4046, 1947, pp. 658- 666.

[11] D. V. Ahluwalia and T. Y. Wu, “On the Magnetic Field of Cosmological Bodies,” Lettere Al Nuovo Cimento, Vol. 23, No. 11, 1978, pp. 406-408. doi:10.1007/BF02786999

[12] J. A. Nieto, “Gravitational Magnetism and General Relativity,” Revista Mexicana de Fisica, Vol. 34, No. 4, 1988, pp. 571-576.

[13] H. A. Wilson, “An Experiment on the Origin of the Earth’s Magnetic Field,” Proceedings of the Royal Society, Vol. 104A, 1923, p. 451. http://www.jstor.org/stable/94216

[14] Z. Perjes, “Solutions of the Coupled Electromagnetic Equations Representing the Fields of Spinning Sources,” Physical Review Letters, Vol. 27, 1971, p. 1668.

[15] A. Papapetrou, “A Static Solution of the Equations of the Gravitational Field for an Arbitrary Charge-Distribution,” Proceedings of the Royal Irish Academy, Vol. 51, 1945-1948, p. 191.

[16] R. Ruffini and A. Treves, “On a Magnetized Rotating Sphere,” Astrophysical Journal Letters, Vol. 13, 1973, p. 109.