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Harada-Maxwell Static Spherically Symmetric Spacetimes

Alan Barnes
Nov 2023
Very recently Harada has proposed a gravitational theory which is of third order in the derivatives of the metric tensor with the property that any solution of Einstein's field equations (EFEs) possibly with a cosmological constant is necessarily a solution of the new theory. Remarkably he showed that even in a matter-dominated universe with zero cosmological constant, there is a late-time transition from decelerating to accelerating expansion. Harada also derived an exact solution which is generalisation of the Schwarzschild solution. However, this was not the most general static spherically vacuum solution of the theory and the general solution was subsequently obtained by Barnes. Recently Tarciso et al. have considered regular black holes in Harada's theory coupled to non-linear electrodynamics and scalar fields. In particular they exhibit a four-parameter solution with a zero scalar field whose source is a Maxwell electromagnetic field. It is a straightforward generalisation of Harada's vacuum solution analagous to the Reissner-Nordstrom generalistaion of the Schwarzschild solution. However, this solution is not the most general static spherically symmetric solution of the Harada-Maxwell field equations (i.e.\ Harada gravitational fields with a Maxwell electromagnetic source). The most general such solution is obtained in this paper.
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