Thermodynamics and shadow of Simpson-Visser black hole with phantom global monopoles

We investigate the thermodynamics and shadow of a non-rotating Simpson Visser black hole with a phantom global monopole. The model is governed by three parameters: the coupling constant $ξ$, the energy scale of symmetry breaking $η$, and the bounce parameter $a$, which jointly influence horizon structure and observational signatures. Using specific heat and free-energy analysis, we show that small-horizon configurations are locally thermodynamically stable but never globally favored. Analytical solutions of null geodesics reveal that the photon sphere radius depends on the bounce parameter $a$ and the energy scale of symmetry breaking $η$, while the critical impact parameter is still unaffected by $a$. Moreover, the photon sphere radius and critical impact parameter, showing that increasing $η$ enlarges both quantities for an ordinary global monopole, while reducing them in the phantom case. Our results highlight how the bounce parameter and phantom global monopole significantly alter the black hole’s physical and geometric properties.

💡 Research Summary

This paper investigates the thermodynamic behavior and optical shadow of a non‑rotating Simpson‑Visser (SV) black hole that is endowed with a phantom global monopole (PGM). The spacetime is characterized by three parameters: the bounce (or regularization) length a, the symmetry‑breaking energy scale η, and the coupling constant ξ that distinguishes an ordinary global monopole (OGM, ξ = +1) from a phantom one (PGM, ξ = ‑1). The metric functions become
(f(r)=1-8\pi\xi\eta^{2}-\frac{2M}{\sqrt{r^{2}+a^{2}}}), (h(r)=r^{2}+a^{2}).
The presence of the monopole modifies the lapse function through the term (8\pi\xi\eta^{2}), while the bounce parameter a regularizes the central singularity.

Horizon structure and mass–radius relation
The horizon condition (f(r_{h})=0) yields
(M=\frac{1}{2}\sqrt{a^{2}+r_{h}^{2}},(1-8\pi\xi\eta^{2})).
For OGM a positive mass requires (8\pi\eta^{2}<1); the phantom case imposes no such restriction, allowing any η. Consequently, for the same mass and bounce length, a phantom monopole produces a larger horizon radius.

Thermodynamics
The surface gravity (\kappa=f’(r_{h})/2) gives the Hawking temperature
(T=\frac{1}{4\pi},(1-8\pi\xi\eta^{2})\frac{r_{h}}{\sqrt{a^{2}+r_{h}^{2}}}).
In the phantom case the factor becomes (1+8\pi\eta^{2}), so the temperature is systematically higher than in the ordinary case. The Bekenstein–Hawking entropy, obtained from the area law, reads
(S=\pi\Big