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

Understand

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.