We study geometrically thick perfect-fluid tori with constant specific angular momentum, so-called ‘Polish doughnuts’, orbiting deformed compact objects with a quadrupole moment. More specifically, we consider two different asymptotically flat, static and axisymmetric vacuum solutions to Einstein’s field equation with a non-zero quadrupole moment, the q-metric and the Erez–Rosen spacetime. It is our main goal to find features of Polish doughnuts in these two spacetimes which qualitatively distinguish them from Polish doughnuts in the Schwarzschild spacetime. As a main result we find that, for both metrics, there is a range of positive (Geroch–Hansen) quadrupole moments which allows for the existence of double tori. If these double tori fill their Roche lobes completely, their meridional cross-section has the shape of a fish, with the body of the fish corresponding to the outer torus and the fish-tail corresponding to the inner torus. Such double tori do not exist in the Schwarzschild spacetime.

我们研究具有恒定特定角动量的几何厚的完美流体环面，即所谓的“波兰甜甜圈”，以四极矩绕变形的紧凑物体运行。更具体地说，我们考虑了爱因斯坦场方程的两种不同的渐近平坦、静态和轴对称真空解，其具有非零四极矩，q-metric 和 Erez-Rosen 时空。我们的主要目标是在这两个时空中找到波兰甜甜圈的特征，从质量上将它们与 Schwarzschild 时空中的波兰甜甜圈区分开来。作为主要结果，我们发现，对于这两个指标，存在一系列正（Geroch-Hansen）四极矩，允许存在双环。如果这些双环完全填满它们的罗氏叶，它们的子午断面呈鱼形，鱼体对应外环，鱼尾对应内环。这种双重圆环在施瓦兹时空中是不存在的。