Guilfoyle's Stars: Charged Compact Models

• Guilfoyle's stars are static, spherically symmetric models that generalize the Schwarzschild interior by including electric charge and constant energy density.
• They use a unique Weyl–Guilfoyle ansatz to link gravitational and electric potentials, producing a rich parameter space from regular stars to quasiblack holes.
• Their analysis, which saturates the Buchdahl–Andréasson bound, provides practical insights into the stability and physical characteristics of charged compact objects.

Guilfoyle's stars are a distinguished class of static, spherically symmetric solutions to the Einstein–Maxwell field equations that generalize the classical interior Schwarzschild solution to include electric charge and fluid pressure. These configurations impose a strict relation between the gravitational potential and the electric potential in the stellar interior and require the total energy density—including both matter and electromagnetic contributions—to be constant. When smoothly matched to an exterior Reissner–Nordström spacetime, Guilfoyle’s stars provide explicit models for self-gravitating, electrically charged compact objects and possess a unique parameter space structure encompassing regular charged stars, quasiblack holes, regular charged black holes with diverse cores, naked singularities, and their uncharged analogs. In the limit of infinite central pressure, these stars saturate the Buchdahl–Andréasson bound on compactness and thus play an analogous role for charged matter as the constant-density Schwarzschild interior does for uncharged stars (Lemos et al., 2015, Lemos et al., 2017).

1. Foundational Structure and Ansatz

Guilfoyle’s construction is anchored in two core assumptions. First, the total energy density—comprising both the fluid matter energy density $\rho_m(r)$ and the electromagnetic energy density associated with the interior electric field $Q(r)/r^2$—is held constant:

$$\rho_m(r) + \frac{Q^2(r)}{8\pi r^4} = \frac{3}{R^2},$$

where $Q(r)$ is the enclosed electric charge within radius $r$ and $R$ sets the scale for the effective energy density. This condition is known as the Cooperstock–de la Cruz–Florides ansatz and is the unique way to consistently distribute charge within a uniform-density star (Lemos et al., 2015).

Second, the so-called Weyl–Guilfoyle relation posits a quadratic functional dependence between the time-time component of the metric and the electrostatic potential:

$B(r) = a\left[-\epsilon \phi(r)\right]^2,$

with $a > 0$ the "Guilfoyle parameter" and $\epsilon = \pm1$. The general line element in Schwarzschild-like coordinates is then:

$$ds^2 = -B(r)\,dt^2 + A(r)\,dr^2 + r^2(d\theta^2 + \sin^2\theta\,d\varphi^2),$$

with

$Q(r)/r^2$0

The functional form for $Q(r)/r^2$1 follows from integrating the Einstein–Maxwell equations with the above ansatz (Lemos et al., 2015, Lemos et al., 2017, Banerjee et al., 2017).

2. Solution Families and Parameter Space

Guilfoyle’s model leads to a two-region construction: an interior charged-fluid sphere for $Q(r)/r^2$2, and an exterior Reissner–Nordström vacuum for $Q(r)/r^2$3. The parameter space is conveniently organized via dimensionless variables

$Q(r)/r^2$4

where $Q(r)/r^2$5 is the total charge and $Q(r)/r^2$6 the fluid radius. The metric functions and other physical quantities in the interior are completely specified by $Q(r)/r^2$7, but regularity and boundary conditions at $Q(r)/r^2$8 reduce the number of independent parameters by enforcing:

$Q(r)/r^2$9

$$\rho_m(r) + \frac{Q^2(r)}{8\pi r^4} = \frac{3}{R^2},$$0

The global solution space in $$\rho_m(r) + \frac{Q^2(r)}{8\pi r^4} = \frac{3}{R^2},$$1 encompasses a wide variety of physical and unphysical behaviors, from regular stars to singular configurations, including regular and exotic black holes, depending on the sign and magnitude of $$\rho_m(r) + \frac{Q^2(r)}{8\pi r^4} = \frac{3}{R^2},$$2 (Lemos et al., 2017).

Table: Characteristic Regions in Guilfoyle’s Parameter Space

Designation	$$\rho_m(r) + \frac{Q^2(r)}{8\pi r^4} = \frac{3}{R^2},$$3	Typical Solution Type
Regular under-charged stars (a)	$$\rho_m(r) + \frac{Q^2(r)}{8\pi r^4} = \frac{3}{R^2},$$4	$$\rho_m(r) + \frac{Q^2(r)}{8\pi r^4} = \frac{3}{R^2},$$5, perfect fluid stars
Over-charged tension stars (b)	$$\rho_m(r) + \frac{Q^2(r)}{8\pi r^4} = \frac{3}{R^2},$$6	$$\rho_m(r) + \frac{Q^2(r)}{8\pi r^4} = \frac{3}{R^2},$$7, $$\rho_m(r) + \frac{Q^2(r)}{8\pi r^4} = \frac{3}{R^2},$$8, negative pressure
Quasiblack holes (Q point)	$$\rho_m(r) + \frac{Q^2(r)}{8\pi r^4} = \frac{3}{R^2},$$9, $Q(r)$0	$Q(r)$1
Regular BHs with phantom core (e)	$Q(r)$2, $Q(r)$3	Black holes with $Q(r)$4 near center
Regular “imaginary-charged” stars (i)	$Q(r)$5	Not electric charge; analogue solutions

[See figures in (Lemos et al., 2017) for the full $Q(r)$6 vs.\ $Q(r)$7 classification.]

3. Boundary Matching, Regularity, and Physical Quantities

At the boundary $Q(r)$8, conditions for continuity of the metric and its derivative determine all physical parameters and link the interior configuration smoothly to the exterior Reissner–Nordström metric:

$Q(r)$9

Matching conditions enforce $r$0 and relate the Guilfoyle parameter to the global charges. The full closed-form expressions for matter density, radial pressure, charge function, and field in the interior are given in terms of $r$1, $r$2 and $r$3 (determined by the parameters above), with the notable result:

$r$4

$r$5

Absence of an interior horizon requires $r$6; absence of singularities and infinite pressure restricts the solution to lie outside the “gray” regions in the $r$7–$r$8 diagram abutting singular curves.

The surface redshift, total mass, and total charge are explicitly computable:

$r$9

4. The Buchdahl–Andréasson Compactness Bound

For uncharged stars, the Buchdahl bound asserts $R$0, saturated by the infinite-pressure constant-density Schwarzschild interior. Andréasson generalized this to include electrically charged matter, imposing the condition:

$R$1

with $R$2 the tangential pressure. The resulting Buchdahl–Andréasson bound is:

$R$3

Guilfoyle’s stars precisely saturate this bound in the limit where the central pressure diverges ($R$4), providing the direct charged generalization of the interior Schwarzschild solution. The family interpolates continuously between the classical Buchdahl bound for $R$5 and the extremal quasiblack hole solution at $R$6 (Lemos et al., 2015).

5. Stability Analysis and Limiting Configurations

Linearized radial stability for thin-shell configurations constructed from Guilfoyle’s interiors can be formulated by considering the shell’s effective potential $R$7. The system is in static equilibrium when $R$8, and small radial perturbations are stable if $R$9. Explicit stability regions depend on the detailed algebraic structure involving surface energy density $B(r) = a\left[-\epsilon \phi(r)\right]^2,$0 and pressure $B(r) = a\left[-\epsilon \phi(r)\right]^2,$1, parameterized by $B(r) = a\left[-\epsilon \phi(r)\right]^2,$2 (Banerjee et al., 2017).

As with uncharged Buchdahl stars, stability analysis becomes ill-posed in the $B(r) = a\left[-\epsilon \phi(r)\right]^2,$3 limit, and physically realistic equations of state are expected to prohibit attainment of the limiting bound. The limiting Guilfoyle family thus serves as an idealized theoretical benchmark rather than realistic stellar models.

6. Taxonomy and Physical Interpretation

The unified $B(r) = a\left[-\epsilon \phi(r)\right]^2,$4–$B(r) = a\left[-\epsilon \phi(r)\right]^2,$5 plane demonstrates the rich taxonomy of Guilfoyle’s solutions:

• Regular under-charged stars: Standard perfect fluid charged stars $B(r) = a\left[-\epsilon \phi(r)\right]^2,$6.
• Charged dust (Bonnor) stars: Lie on the $B(r) = a\left[-\epsilon \phi(r)\right]^2,$7 “dust curve.”
• Quasiblack holes: The $B(r) = a\left[-\epsilon \phi(r)\right]^2,$8 extremal point, boundary between star and black hole regimes.
• Regular black holes with de Sitter or phantom cores: Depending on matter content near the center.
• Exotic and singular solutions: Overcharged, negative-mass, or “imaginary charge” regions not corresponding to physical electromagnetic charge.
• Limiting vacua: Schwarzschild, Minkowski, and Kasner limits are recovered for specific parameter choices.

Configurations with $B(r) = a\left[-\epsilon \phi(r)\right]^2,$9 exist mathematically but lack physical interpretation as electric charge (Lemos et al., 2017).

7. Relation to Gravastars and Extensions

Gravastar models avoiding event horizons and singularities can be constructed using Guilfoyle’s electrical charged interiors matched to an external Reissner–Nordström metric across a thin shell. For suitable parameter ranges, these models possess no horizons and admit analysis of stable perturbations. Surface stress–energy (via Lanczos–Israel conditions) and surface redshift are computable in closed form. Ensuring positivity of energy densities and matching requirements yields horizon-free, stable Guilfoyle star solutions, providing alternatives to standard black holes in compact astrophysical configurations (Banerjee et al., 2017).

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For further mathematical details, analytic derivations, and comprehensive parameter-space exploration, see (Lemos et al., 2015, Lemos et al., 2017), and (Banerjee et al., 2017).