# The spin limit of colliding black holes

Geoffrey Lovelace is an Assistant Professor of Physics at California State University, Fullerton. As member of Fullerton’s Gravitational-Wave Physics and Astronomy Center and the Simulating eXtreme Spacetimes collaboration, his research interests focus on using computer simulations to model colliding black holes and neutron stars and the gravitational waves they emit.

A single black hole’s size limits its spin. Do colliding black holes obey this limit?

In our recent paper, published in Classical and Quantum Gravity, we take a first look at how supercomputer simulations can help reveal the answer.

A black hole is an object whose gravity is so strong that nothing, even light, can escape from inside its horizon. An isolated, uncharged black hole can be completely described by just two numbers: its spin and its horizon surface area. All of the black hole’s properties then follow from Kerr’s solution of Einstein’s equations.

Kerr’s solution implies that a single black hole can spin no faster than its horizon area times a constant: spinning any faster would destroy the horizon. Astronomers have found evidence that some black holes spin very close to the limit (but still below it). Mathematical relativists have proven that this spin limit is obeyed not only by Kerr black holes but also by black holes under very general conditions—assuming that their horizons are axially symmetric. (Axial symmetry is used to unambiguously define spin.)

The horizon just after two black holes have merged. At the time shown, the common horizon is near the spin limit. The horizon is colored by its curvature (red = positive, blue = negative).

Do colliding black holes obey this spin limit? The question is challenging because their horizons can be very far from axial symmetry and because there is no exact solution.

In our paper, we look at simulations of colliding black holes that spiral together and merge. To keep things simple, we focus on holes of equal sizes and spins. Defining the spins in the usual way for numerical relativity, in terms of the best available approximate axial symmetries on the horizons, we find that the holes always obey the spin limit, but the limit is almost violated just after the collision.

Horizons from initial data for two merging black holes. The inner horizon by construction violates the spin limit, but it turns out to be enclosed by a larger horizon (cutaway shown) that obeys the spin limit. The horizons are colored by their curvature (red = positive, blue = negative).

Next, we measure an ‘extremality parameter’ closely related to one proposed by Ivan Booth and Stephen Fairhurst. For a single black hole, the extremality is greater than one only when the black hole violates the spin limit. We find that the extremality remains moderately large but less than one.

Finally, we construct initial data for two merging black holes, forcing them to have horizons that exceed the spin limit and have extremalities greater than one. These ‘overspun’ horizons are always surrounded by larger horizons that obey the spin limit and with extremality less than one.

We are eager to explore whether these results will be true in more generic simulations of merging black holes.

Nearly extremal apparent horizons in simulations of merging black holes
Geoffrey Lovelace, Mark A Scheel, Robert Owen, Matthew Giesler, Reza Katebi, Béla Szilágyi, Tony Chu, Nicholas Demos, Daniel A Hemberger, Lawrence E Kidder
Geoffrey Lovelace et al 2015 Class. Quantum Grav. 32 065007

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