Black Hole Formation and Growth by Tiziana Di Matteo & Andrew King & Neil J. Cornish

Author:Tiziana Di Matteo & Andrew King & Neil J. Cornish
Language: eng
Format: epub
ISBN: 9783662597996
Publisher: Springer Berlin Heidelberg

(35)

Here is the hole spin angular momentum vector, and the angular momentum of the disc. Strictly speaking, it is the angular momentum of that part of the disc which feels a significant alignment torque, e.g. where the disc orbits have completed at least one Lense–Thirring precession. The other way of saying this is that is the angular momentum of that part of the disc currently tending to warp under the alignment torque. Over time this radius increases to include more and more of the disc—the warp moves outwards—so in principle the magnitude grows in time. But alignment/counteralignment is faster than this growth (by definition), so in practice we can regard (but not !) as constant.

The degree of misalignment of disc and hole spin is measured by the vector quantity , so all the alignment torques must depend on it. Equation (35), where are functions of the disc structure, is then the most general form possible if the hole feels only precessions. The torques on the rhs are orthogonal to , where the first term on the rhs describes the torque inducing precession, and the second one describes alignment or counteralignment. I will show below that the quantity must be positive if there is dissipation, as we expect, since viscosity is involved in the alignment/counteralignment process.

Taking the scalar product of Eq. (35) with shows that , that is, the magnitude of the spin remains constant during precession and alignment/counteralignment. Thus the tip of the vector moves on a sphere. The total angular momentum is of course a constant vector, representing a fixed direction in space. Using this, and the fact that , we see that

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