# What's the "size" of the charged rotating object described by the Kerr-Newman solution?

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The Kerr-Newman metric describes a rotating charged black hole when

$q^2 + a^2 < m^2$

where q is the charge, m is the mass and a = J/m is the angular momentum per mass, in units where G = c = 1.  There are various imaginable notions of the "size" of this black hole, since there is a "stationary limit surface" within which a particle cannot stay at rest relative to the fixed stars, and then a region called the "ergosphere", then an event horizon called the "outer horizon", then an "inner horizon", and finally a "ring singularity" inside all those.  From the classic book by Hawking and Ellis:

Defining the "size" of these things is a challenge, since the radius of some structure in some arbitrary coordinate system may not have much physical meaning: it's a curved spacetime and the Killing vector field describing "time" is not orthogonal to a preferred foliation by surfaces describing "space".   But still, someone should have tried to define and compute the "size" of these various things.  What answers did they get?

I'm actually even more interested in the case where

$q^2 + a^2 > m^2$

In this case there is a naked ring singularity.  Still there should be some ways to define a "size".

recategorized Sep 5, 2016

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As every relativist knows, notions such as "size" and "duration" have to be attached to precise operational definitions, otherwise they are non-unique. But even if you manage to pin down some unique operation defining the "size" of a black hole, the more practical question for observational purposes is "what is it good for?"

The "size" I find most directly linked to observation is the size of the shadow of the black hole, you can find the prescription for the whole Plebanski-Demianski black hole class in the works of Arne Grenzebach, e.g. here.

Naked singularities ($q^2+a^2>m^2$), on the other hand, do not have a shadow and the question is what would we see emerging from them which would define their observational image. I believe that there is simply no meaningful way to define the size of a point-like or a ring-like naked singularity, because in many senses of the word they are "infinitely far away" from many "nearby" points, points on the line-like singularity are also "infinitely far away" from each other, and a whole new infinite universe fits inside the central Kerr-Newman loop. One can force some kind of definition of size but this would be just a smoke-screen which is put up to hide the naked horror.

answered Sep 8, 2016 by (1,645 points)

As every relativist knows, notions such as "size" and "duration" have to be attached to precise operational definitions, otherwise they are non-unique.

Yeah - since I'm a relativist, and every relativist knows this, I know this.  I explained it in my question.   I invited the reader to supply their own operational definition, to increase the chance of getting an interesting answer.

Given an angular momentum $J$ and mass $m$ we can define a length scale $J/mc$.  This length, possibly times a dimensionless constant, should be the answer to some operational question about the Kerr metric, even when that metric has a naked singularity.  I feel someone should have thought about this.

@JohnBaez: The length scale is not obtained with combining dimensional constants by hand. It should follow naturally from the equation solution. It is the only meaningful way of getting scales and other things - from the solutions. For example, in a Coulomb field a projectile with the energy $E$, etc., has a natural combination of the problem constants resulting into "the closest distance" between the projectile and the Coulomb center (in a scattering problem).

@JohnBaez Well, but $J/m$ is simply the parameter known as $a$. I.e., the Kerr-Newman solution is defined by three length scales (in geometrized units), $m,q,a$. This seems to be a general feature of all naked singularities including naked Kerr-Newman or naked Reissner-Nordström, but also e.g. the Bach-Weyl or Appel ring; they seem to require at least two length-scales to meet in one object.

I recall one single special case where you can get rid of the second scale and that is a plane wave. I.e., you take the Aichelburg-Sexl limit of the objects and what you get is a single-scale object moving at the speed of light. For a Kerr black hole you eliminate the mass $M$, and $a$ is then some kind of radius-scale of the ring of particles moving at the speed of light; this has been done by Valeria Ferrari and Paolo Pendenza in 1990.

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