Evolution of Free Magnetic Fields. II. Including Self-Gravitation

Details Evolution of Free Magnetic Fields. II. Including Self-Gravitation Evolution of Free Magnetic Fields. II. Including Self-Gravitation

Jun 1970

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1970phrvd...1.3239w&link_type=abstract

Physical Review D, vol. 1, Issue 12, pp. 3239-3242

Physics

Scientific paper

This paper extends the work of the preceding paper by including the gravitational (general relativistic) aspects of a magnetic field such as results from an initial uniform current which has been switched off. The solutions to the entire set of Einstein-Maxwell equations are expressed simply in terms of the "pure Maxwell solutions" obtained by neglecting gravitation. The latter solutions, in both graphical and analytical form, were found and discussed in the preceding paper; they are not simple functions of space and time. Appropriate initial conditions for the space outside the wire (ρ>1) are established by setting time derivatives and electric field equal to zero in the Einstein-Maxwell equations, and solving. We then show that the time-dependent solutions are given by B=BMh3h0, and E=EMh3h0, where BM(ρ, τ) and EM(ρ, τ) are the time-dependent Maxwell fields given in the preceding paper, h0, h1, h2, and h3 are the gravitational-field-determining scale factors hα≡|gαα|12 associated with τ≡cta, ρ≡ra, φ, and ζ≡za, respectively (r, φ, and z correspond to standard cylindrical coordinates); a is the wire radius. The hα are expressed in terms of the pure Maxwell field quantities AM and V and a small universal constant K≡(4πGμ0c4)12~=2.91×10-19 m/Wb by the relations h3=1(KAM), h2=aρh3, h1=h0=exp(K2V)h3, AM(ρ, τ)=-(vector potential)M=AM(ρ, 0)+acτEM(ρ, s)ds, AM(ρ, 0)=12b(ρ2- 1) (ρ<=1), AM(ρ, 0)=blnρ (ρ>=1), b≡aBM(1, 0)=μ0I2π, V(ρ, t)=(μ0π) (field energy out to ρ per unit z)- 14b2=a2ρxdxEM2(x, τ)c2+BM2(x, τ)-14b2. The methods and order of investigation presented in these two papers can be a practical modus operandi for finding realistic solutions to the Einstein-Maxwell time-dependent equations.

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