# Quadrupole moment of Kerr spacetime

In this paper this paper, the Kerr black hole is described as having quadrupole moment of $$q=J^2/M$$ (which means $$q=a^2M$$ using $$J=aM$$) whereas in this paper it says in the abstract that the limiting case of Kerr is $$q=0$$. and finally this paper says $$q=-a^2M^3$$ (I think this is due to a different definition of $$a$$ though, as they say $$a=J/M^2$$). Which one is correct? Perhaps in the second paper $$q=0$$ in the approximation they take?

## 1 Answer

The paper of Bini et al. (2009) discusses a metric where the symbol $$q$$ stands for a quadrupole parameter. The origin of $$q$$ is chosen so that it can be understood as parametrizing the deviation from Kerr space-time. On the other hand, the Kerr metric has the magnitude of the quadrupole moment, as precisely defined by Hansen (1974), equal to $$Q = a^2 M$$.

However, you can use different definitions of the multipoles, and in particular, different sign conventions. This is the case of the paper of Bambi & Barrausse (2011) you refer to, where they use the signature convention where a positive $$Q$$ refers to a prolate source/field and a negative $$Q$$ to an oblate source/field. (You have misread their formula which reads $$Q \equiv -J^2/M = -a^2 M$$).

• Thanks, that answers the Bini paper question. Where did you find the formula in the Bambi paper? I was reading the formula just below eqn (10) Apr 4 '19 at 13:39
• @supercoolphysicist In the abstract and, for instance, in equation (1). Note that they use the symbol $a$ in a somewhat different meaning, their $a$ is dimensionless, which is in contrast to the usual definition of the spin parameter that differs by a factor of $M$ and has the dimension of mass (length).
– Void
Apr 4 '19 at 14:13