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Title: Does the quadrupole moment of supermassive black holes equal $-J^{2}/(M c^{2})$?
Plain: A Kerr black hole's shape is fixed by its mass and spin, so its flattening, measured by the quadrupole moment, must take one exact value. Small bodies spiraling into massive black holes, to be observed by the LISA space detector, can measure it.
Precise: For Kerr the mass quadrupole moment is $Q=-J^{2}/(M c^{2})$ (in g cm^2). Using extreme mass-ratio inspirals (a stellar-mass body orbiting a supermassive black hole) observed by LISA over about $10^{5}$ orbital cycles, measure M, J and Q independently and test this relation; Barack and Cutler (2007) estimated fractional precision of about $10^{-4}\ \text{to}\ 10^{-2}$. An answer is the measured deviation of $Q$ from $-J^{2}/(M c^{2})$ with its uncertainty.
Settled by: LISA extreme mass-ratio inspiral observations with an independent fit of the central quadrupole moment.
Kind: well-posed
Answer type: value
References: (empty)
Literature status: open
Status note: LISA is planned for launch in the mid-2030s.
Posed since: (empty)