Quick Notes on Kerr Black Holes and the Innermost Stable Orbit
    ______________________________________________________________


First of all, define the "gravitational radius" Rg == GM/c**2.

Note that Rs = 2Rg, where Rs is the familiar Schwarzchild radius.


Then we have the following (see Frank, King, Raine eq. 7.10):

  Black Hole Rotation        Innermost Stable Orbit Radius
_______________________      ____________________________
zero rotation                          6 Rg
maximum prograde rotation              1 Rg
maximum retrograde rotation            9 Rg


The Schwarzchild solution is for a non-rotating black hole, which
astrophysically is unlikely to exist since the object that forms the BH will
have some non-zero angular momentum. Furthermore, matter accreted by the BH
will have angular momentum too. So in general, BH will be spinning. 
A spinning BH is called a Kerr black hole.


Notice that in the case of a Kerr BH, the accretion disk can extend
all the way down to 1 Rg, six times closer the center than in the
non-rotating Schwarzchild case. This is *much* deeper in the gravitational
potential well, and so much more energy can be liberated by an accretion
disk. While an accretion disk around a Schwarzchild BH can liberate 5.7% of 
the rest mass E of the infalling matter, for a disk in prograde motion around
a maximally spinning Kerr BH a whopping 40% of the rest mass E can be liberated! 
(Compare this with the feeble 0.7 % that comes from thermonuclear fusion.)


Note that mass, spin, and charge are the three *and only three* properties
of a black hole that can be measured from beyond the event horizon.


So can one really observationally tell the difference between a spinning and
non-spinning BH?  

Yes indeedy... see if you can figure out how!





W. Welsh
SDSU
2009 Spring