The recent detection of gravitational waves from black hole coalescences and the first image of the black hole shadow enhance the possibilities of testing gravitational theories in the strong-field regime. In this paper, we study the physical properties and the shadow image of a class of Kerr-like rotating black holes, whose $\mathbb{Z}_2$ symmetry is generically broken. Such black hole solutions could arise in effective low-energy theories of a fundamental quantum theory of gravity, such as string theory. Within a theory-agnostic framework, we require that the Kerr-like solutions are asymptotically flat, and assume that a Carter-like constant is preserved, enabling the geodesic equations to be fully separable. Subject to these two requirements, we find that the $\mathbb{Z}_2$ asymmetry of the spacetime is characterized by two arbitrary functions of polar angle. The shadow image turns out to be $\mathbb{Z}_2$ symmetric on the celestial coordinates. Furthermore, the shadow is completely blind to one of the arbitrary functions. The other function, although would affect the apparent size of the shadow, it hardly distorts the shadow contour and has merely no degeneracy with the spin parameter. Therefore, the parameters in this function can be constrained with black hole shadows, only when the mass and the distance of the black hole from the earth are measured with great precision.

中文翻译：

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无ℤ2对称性的旋转黑洞及其阴影图像

最近从黑洞合并中探测到的引力波和黑洞阴影的第一张图像增强了在强场区测试引力理论的可能性。在本文中，我们研究了一类类克尔旋转黑洞的物理性质和阴影图像，其 $\mathbb{Z}_2$ 对称性被普遍破坏。这种黑洞解决方案可能出现在基本量子引力理论的有效低能理论中，例如弦理论。在与理论无关的框架内，我们要求类克尔解是渐近平坦的，并假设类卡特常数保持不变，从而使测地线方程完全可分离。满足这两个要求，我们发现时空的 $\mathbb{Z}_2$ 不对称性以极角的两个任意函数为特征。阴影图像在天体坐标上是 $\mathbb{Z}_2$ 对称的。此外，阴影对任意函数之一完全视而不见。另一个函数虽然会影响阴影的表观大小，但它几乎不会扭曲阴影轮廓，并且与自旋参数没有简并性。因此，这个函数中的参数可以用黑洞阴影来约束，只有当黑洞到地球的质量和距离被非常精确地测量时。虽然会影响阴影的表观大小，但它几乎不会扭曲阴影轮廓，并且与自旋参数没有简并性。因此，该函数中的参数可以用黑洞阴影来约束，只有在非常精确地测量黑洞与地球的质量和距离时。虽然会影响阴影的表观大小，但它几乎不会扭曲阴影轮廓，并且与自旋参数没有简并性。因此，这个函数中的参数可以用黑洞阴影来约束，只有当黑洞到地球的质量和距离被非常精确地测量时。