Will Clifford M
GReCO, Institut d'Astrophysique de Paris, CNRS, Université Pierre et Marie Curie, 98 bis Bd. Arago, 75014 Paris, France.
Phys Rev Lett. 2009 Feb 13;102(6):061101. doi: 10.1103/PhysRevLett.102.061101. Epub 2009 Feb 12.
For a test body orbiting an axisymmetric body in Newtonian gravitational theory with mass m and multiple moments Q_{l} (and for a charge in orbit about a charge distribution with the same multipole moments) we show that there exists, in addition to the energy and angular momentum component along the symmetry axis, a conserved quantity analogous to the Carter constant of Kerr spacetimes for rotating black holes in general relativity, if the odd-l moments vanish, and the even-l moments satisfy Q_{2l}=m(Q_{2}/m);{l}. Strangely, this is precisely the relation among mass moments enforced by the no-hair theorems of rotating black holes. By contrast, if Newtonian gravity is supplemented by a multipolar gravitomagnetic field, whose leading term represents frame dragging, we are unable to find an analogous Carter-like constant. This further highlights the special nature of the Kerr geometry.
对于在牛顿引力理论中围绕具有质量(m)和多个矩(Q_{l})的轴对称体做轨道运动的测试体（以及对于围绕具有相同多极矩的电荷分布做轨道运动的电荷），我们表明，若奇数(l)阶矩为零，且偶数(l)阶矩满足(Q_{2l}=m(Q_{2}/m)^{l})，那么除了沿对称轴的能量和角动量分量外，还存在一个守恒量，它类似于广义相对论中旋转黑洞的克尔时空的卡特常数。奇怪的是，这恰好是旋转黑洞无毛定理所强制的质量矩之间的关系。相比之下，如果牛顿引力由一个多极引力磁场补充，其首项表示参考系拖拽，我们就无法找到类似卡特常数的量。这进一步凸显了克尔几何的特殊性质。
