Proca stars and their frozen states in an infinite tower of higher-derivative gravity

TL;DR

This work analyzes static, spherically symmetric Proca stars in five-dimensional gravity with an infinite tower of higher-curvature corrections. Using a Proca field with an ansatz $\mathcal{A}=[F(r) dt+i H(r) dr] e^{-i ω t}$ and a metric $ds^2=-\sigma(r)^2 N(r) dt^2+dr^2/N(r)+r^2 d\Omega_3^2$, the authors derive the coupled field equations and explore the impact of curvature order $n$ on solution space. They find that for $n=1$ and $n=2$ the mass–charge relations exhibit spirals and, in the $n=2$ case, central divergences as $ω\to 0$, while for $n\ge 3$ the fields remain finite and a new class of horizonless frozen-star solutions appears, characterized by a critical radius $r_c$ where $-g_{tt}$ and $1/g_{rr}$ vanish and outside of which the exterior spacetime matches that of an extremal black hole. Frozen stars are regular and horizonless, yet their exterior geometry is observationally indistinguishable from extremal BHs, offering a geometric mechanism to address singularities and information-loss without invoking event horizons. The results highlight that horizonless, highly compact objects can arise at finite curvature order and motivate future work on their stability and tidal properties.

Abstract

In this work, we investigate the five-dimensional Proca star under gravity with the infinite tower of higher curvature corrections. We find that when the coupling constant exceeds a critical value, solutions with a frequency approaching zero appear. In the finite-order corrections case $n=2$ (Gauss-Bonnet gravity), the matter field and energy density diverge near the origin as $ω\to 0$. In contrast, for $n\geq 3$, the divergence is efficiently suppressed, both the field and the energy density remain finite everywhere, and both the matter field and energy density remain finite everywhere. In the limit $ω\to 0$, a class of horizonless frozen star solutions emerges, which are referred to ``frozen stars". Importantly, frozen stars contain neither curvature singularities nor event horizons. These frozen stars develop a critical horizon at a finite radius $r_c$, where $-g_{tt}$ and $1/g_{rr}$ approach zero. The frozen star is indistinguishable from that of an extremal black hole outside $r_c$, and its compactness can reach the extremal black hole value.

Figures (10)

• Figure 1: The ADM mass $M$ (left panel) and Noether charge $Q$ (right panel) as functions of the frequency $\omega$ for $n=1$ (dashed) and $n=2$ (solid) with different values of the coupling $\alpha$. The insets show a magnified view of the high-frequency region near $\omega\to 1$.
• Figure 2: The metric components $-g_{tt}$ (top left) and $1/g_{rr}$ (top right), and the field functions $F(x)$ (bottom left), $H(x)$ (bottom right) as functions of the radial coordinate for $n=1$ (dashed line) and $n=2$ (solid lines) with different coupling constants $\alpha$. All solutions correspond to the second branch with a frequency $\omega=0.965$.
• Figure 3: The metric components $-g_{tt}$ (top left) and $1/g_{rr}$ (top right), and the field functions $F(x)$ (bottom left), $H(x)$ (bottom right) as functions of the radial coordinate for $n=2$ with different frequencies $\omega$. All solutions have $\alpha=5$.
• Figure 4: The ADM mass $M$ (left panel) and Noether charge $Q$ (right panel) as functions of the frequency $\omega$ for $n=2,3,4,\infty$. Dashed curves denote $n=2$, while solid curves correspond to higher-order corrections. All solutions have $\alpha=5$.
• Figure 5: The metric components $-g_{tt}$ (top left) and $1/g_{rr}$ (top right), and the field functions $F(x)$ (bottom left), $H(x)$ (bottom right) as functions of the radial coordinate for $n=3,4,\infty$. Dashed curves denote $n=2$. All solutions have $\omega=0.0002$ and $\alpha=5$.