Quasiblack holes with pressure: Relativistic charged spheres as the frozen stars

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In general relativity coupled to Maxwell's electromagnetism and charged matter, when the gravitational potential $W^2$ and the electric potential field $\phi$ obey a relation of the form $W^{2}= a\left(-\epsilon\, \phi+ b\right)^2 +c$, where $a$, $b$ and $c$ are arbitrary constants, and $\epsilon=\pm1$ (the speed of light $c$ and Newton's constant $G$ are put to one), a class of very interesting electrically charged systems with pressure arises. We call the relation above between $W$ and $\phi$, the Weyl-Guilfoyle relation, and it generalizes the usual Weyl relation, for which $a=1$. For both, Weyl and Weyl-Guilfoyle relations, the electrically charged fluid, if present, may have nonzero pressure. Fluids obeying the Weyl-Guilfoyle relation are called Weyl-Guilfoyle fluids. These fluids, under the assumption of spherical symmetry, exhibit solutions which can be matched to the electrovacuum Reissner-Nordstr\"om spacetime to yield global asymptotically flat cold charged stars. We show that a particular spherically symmetric class of stars found by Guilfoyle has a well-behaved limit which corresponds to an extremal Reissner-Nordstr\"om quasiblack hole with pressure, i.e., in which the fluid inside the quasihorizon has electric charge and pressure, and the geometry outside the quasihorizon is given by the extremal Reissner-Nordstr\"om metric. The main physical properties of such charged stars and quasiblack holes with pressure are analyzed. An important development provided by these stars and quasiblack holes is that without pressure the solutions, Majumdar-Papapetrou solutions, are unstable to kinetic perturbations. Solutions with pressure may avoid this instability. If stable, these cold quasiblack holes with pressure, i.e., these compact relativistic charged spheres, are really frozen stars.