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

TL;DR

This paper explores five-dimensional Proca stars in higher-derivative gravity, revealing the emergence of horizonless frozen star solutions with finite energy density and no singularities, which mimic extremal black holes outside a critical radius.

Contribution

It introduces the concept of frozen stars in higher-derivative gravity, showing their properties and how they differ from traditional black holes, especially in the zero-frequency limit.

Findings

Frozen stars are horizonless and free of singularities.

They develop a critical radius where they mimic extremal black holes.

Finite higher-derivative corrections suppress divergences at zero frequency.

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 $\omega\to 0$. 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 $\omega \to 0$, 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.