Multipole moments for black objects in five dimensions

In higher dimensions than four, conventional uniqueness theorem in asymptotically flat space-times does not hold, i.e., black objects cannot be classified only by the mass, angular momentum, and charge. In this paper, we define multipole moments for black objects and show that Myers-Perry black hole and black ring can be distinguished by quadrupole moments. This consideration gives us a new insight for the uniqueness theorem for black objects in higher dimensions.

Figure 1

Rod structure of Schwarzschild black hole.

Figure 2

Rod structure of static black ring (BR).

Figure 3

Normalized horizon area $a_H=\sqrt{2\nu (1-\nu )}$ vs spin. Even if given the spin $j\sim \nu$, the fat ring ($\nu <1/2$) and thin ring ($\nu >1/2$) are not distinguished.

Figure 4

Normalized horizon area vs quadrupole moment. If given the quadrupole $q$, $\nu$ is completely determined, that is, thin or fat is specified.

Figure 5

Rod structure of static black Saturn (BS).

Figure 6

Rod structure of static orthogonal black di-ring (OBD).

Figure 7

Rod structure of black Saturn.

Figure 8

Rod structure of orthogonal black di-ring.