A Particular Interior Reissner–Nordström Spacetime

We present an exact interior solution of the coupled Einstein–Maxwell equations describing a static sphere of incompressible charged perfect fluid whose electric charge is distributed in such a way that the charge function grows as the cube of the areal radius, q(r) = Qr³/R³. Under this assumption the isotropy condition reduces to a linear second-order equation which is solved in closed form: the metric function e^Φ is a hyperbolic combination of argument (√11/2)arcsin(·), and the fluid pressure follows algebraically. The solution matches continuously onto the exterior Reissner–Nordström metric at the surface r = R, and reduces to the interior Schwarzschild metric when the charge is switched off. We show that the configuration satisfies the weak, null, strong and dominant energy conditions over a wide parameter range, and that the requirement of a finite central pressure yields a charge-dependent generalisation of the Buchdahl bound, R ≥ 9M/4, which lies remarkably close to the sharp Buchdahl–Andréasson bound. Finally, as an astrophysical application, we evaluate the general relativistic magnetic field generated by an external current loop surrounding such an object and estimate its magnitude. The equipartition field near the horizon scales as B ∝ M⁻¹ᐟ², so that the magnetic field of stellar-mass black holes is several orders of magnitude larger than that of supermassive ones.

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