The so-called hybrid metric-Palatini theory of gravity (HMPG), proposed in 2012 by T. Harko et al., is known to successfully describe both local (solar-system) and cosmological observations. This paper gives a complete description of static, spherically symmetric vacuum solutions of HMPG in the simplest case where its scalar-tensor representation has a zero scalar field potential V(ϕ), and both Riemannian (R) and Palatini \((\mathcal{R})\) Ricci scalars are zero. Such a scalar-tensor theory coincides with general relativity with a phantom conformally coupled scalar field as a source of gravity. Generic asymptotically flat solutions either contain naked central singularities or describe traversable wormholes, and there is a special two-parameter family of globally regular black hole solutions with extremal horizons. In addition, there is a one-parameter family of solutions with an infinite number of extremal horizons between static regions and a spherical radius monotonically changing from region to region. It is argued that the obtained black hole and wormhole solutions are unstable under monopole perturbations. As a by-product, it is shown that a scalar-tensor theory with V(ϕ) = 0, in which there is at least one nontrivial (ϕ ≠ const) vacuum solution with R ≡ 0, necessarily reduces to a theory with a conformal scalar field (the latter may be usual or phantom).

中文翻译：

---

混合公制-帕拉蒂尼引力中的球对称黑洞和虫洞

T. Harko等人在2012年提出的所谓混合度量-帕拉蒂尼重力理论（HMPG），可以成功地描述局部（太阳系）和宇宙学观测。本文给出了静态的完整描述，在最简单的情况HMPG的球对称真空解决方案，其中其标量张量表示具有零标量场电位V（φ），并且两个黎曼（[R ）和膀\（（\ mathcal { R}）\）里奇标量为零。这种标量-张量理论与广义相对论相吻合，其中幻像共形耦合标量场为引力源。一般渐近平坦的解决方案要么包含裸露的中心奇异点，要么描述可遍历的虫洞，并且存在一个特殊的两参数类全局极黑洞的全局规则黑洞解决方案。此外，还有一个单参数解决方案系列，其中静态区域之间的极值层数无限，并且球半径在各个区域之间单调变化。有人认为，在单极扰动下，获得的黑洞和虫洞解是不稳定的。作为副产品，表明具有V（ϕ的标量张量理论）= 0，其中至少有一个R = 0的非平凡（ϕ ≠const）真空解，必然会简化为具有保形标量场的理论（后者可能是普通的或幻像的）。