Diameter of Kerr-Newman Ring Singularity?
Robert L. Oldershaw
There must be a distance across the center of the ring singularity,
i.e., a diameter for the ring singularity.
Theoretically, one would expect that this diameter would be related to
the mass of the black hole, or its angular momentum, or both.
But no discussions of K-N BHs that I have read ever mention this size
issue, although they all mention the ring singularity.
What gives here? Is there some physical or mathematical reason this
subject is not addressed?
Better yet, is there a comprehensible relation that yields the value
of this diameter, or at least allows one to get an order of magnitude
estimate?
Thanks,
RLO
http://www3.amherst.edu/~rloldershaw
[[Mod. note -- There are several points here:
* The ring singularity is a *singularity*, and any reasonable notion
of "distance" in a curved spacetime requires integrating metric
components (which generally blow up at the singularity). I don't
know if a reasonable notion of "distance across the center of the
ring singularity" in Kerr (or Kerr-Newmann) spacetime is finite or not.
* On dimensional-analysis grounds, if the result is finite, then it's
is surely going to be proportional to the mass for a Kerr BH; I don't
know how it would differ for Kerr-Newmann BH. So -- *if* it's finite
-- you can just use the mass (i.e., the Schwarzschild radius) as an
order-of-magnitude estimate.
* The likely reason you've never read about it is that researchers
in general relativity don't find it particularly interesting.
-- jt]]
eric gisse
e9ac-40e1-89a...@b6g2000vbz.googlegroups.com:
> Kerr-Newman black holes have ring singularities.
>
> There must be a distance across the center of the ring singularity,
> i.e., a diameter for the ring singularity.
Looks like the mod note already covered the problem with using the word
"diameter".
>
> Theoretically, one would expect that this diameter would be related to
> the mass of the black hole, or its angular momentum, or both.
>
> But no discussions of K-N BHs that I have read ever mention this size
> issue, although they all mention the ring singularity.
Really? I seem to recall both MTW and Carroll discuss this.
You have the coordinate value of the singularity rho^2 = r^2 + a^2 cos^2
(theta). That's not a point, its' an annulus. The rotation blurs the
singularity, so to speak.
>
> What gives here? Is there some physical or mathematical reason this
> subject is not addressed?
No, as it is discussed at some level in every textbook on the subject I
have ever seen.
[...]
carlip...@physics.ucdavis.edu
> Kerr-Newman black holes have ring singularities.
> There must be a distance across the center of the ring singularity,
> i.e., a diameter for the ring singularity.
As others have said, the diameter is the wrong thing to think
about -- the space "inside" the ring singularity is highly curved,
and it's not even obvious on what slice of "constant time" one
should do the calculation.
The circumference, on the other hand, is easy. It's equal to
2\pi a, where a is the angular velocity parameter. You can find
this in, for example, d'Inverno, _Introducing Einstein's Relativity_,
section 19.5, or Poisson, _A relativist's Toolkit_, section 5.3.8, or
Hobson et al. _General Relativity_, section 13.8. It's also quite
easy to calculate directly in the Kerr-Schild form of the metric.
> Theoretically, one would expect that this diameter would be related
> to the mass of the black hole, or its angular momentum, or both.
Angular momentum, since as the angular momentum goes to zero,
the metric becomes Schwarzschild, with no ring singularity.
> But no discussions of K-N BHs that I have read ever mention this size
> issue, although they all mention the ring singularity.
> What gives here? Is there some physical or mathematical reason this
> subject is not addressed?
I suspect the reason is just that you haven't looked very hard. As I
said, the circumference is very easy to compute in Kerr-Schild
coordinates.
Steve Carlip
Robert L. Oldershaw
>
> The circumference, on the other hand, is easy. It's equal to
> 2\pi a, where a is the angular velocity parameter. You can find
> this in, for example, d'Inverno, _Introducing Einstein's Relativity_,
> section 19.5, or Poisson, _A relativist's Toolkit_, section 5.3.8, or
> Hobson et al. _General Relativity_, section 13.8. It's also quite
> easy to calculate directly in the Kerr-Schild form of the metric.
Sincere thanks for the help.
Sean M. Carroll gives the following ang. vel. param.:
a.v.p. = (specific ang. momentum)/(r+)^2 + (spec. ang. momen.)^2 .
Is this the same parameter found in the books you list (which I do not
have at hand)?
If so, then for the particular Kerr-Newman object that I am interested
in I get "R"(sing)/R(bh) of about 0.428, ( = 3/7).
This result is most interesting, but first I need to verify that the
a.v.p. I used is the same one that you cite for:
circum. = 2/(pi)(a.v.p.)
RLO
http://www3.amherst.edu/~rloldershaw
eric gisse
bbf1-4c1e-bd9...@e37g2000yqa.googlegroups.com:
> On Oct 14, 3:42 am, carlip-nos...@physics.ucdavis.edu wrote:
>>
>> The circumference, on the other hand, is easy. It's equal to
>> 2\pi a, where a is the angular velocity parameter. You can find
>> this in, for example, d'Inverno, _Introducing Einstein's Relativity_,
>> section 19.5, or Poisson, _A relativist's Toolkit_, section 5.3.8, or
>> Hobson et al. _General Relativity_, section 13.8. It's also quite
>> easy to calculate directly in the Kerr-Schild form of the metric.
> -----------------------------------------------------------
>
> Sincere thanks for the help.
>
> Sean M. Carroll gives the following ang. vel. param.:
>
> a.v.p. = (specific ang. momentum)/(r+)^2 + (spec. ang. momen.)^2 .
>
> Is this the same parameter found in the books you list (which I do not
> have at hand)?
Angular momentum is a conserved quantity.
Evaluate the Komar angular momentum equation (google it, or open MTW,
Carroll, D'Inverno, etc) for the metric and you'll be able to get an
answer for it.
What it is I have no idea but the math is straight forward. Unless you
know the meaning of the answer you are getting, the result is
meaningless. Unless all you are interested in really is something
semiplausible that'll get you a number you want...
>
> If so, then for the particular Kerr-Newman object that I am interested
> in I get "R"(sing)/R(bh) of about 0.428, ( = 3/7).
>
> This result is most interesting, but first I need to verify that the
> a.v.p. I used is the same one that you cite for:
>
> circum. = 2/(pi)(a.v.p.)
This ought to be a trivial exercise for someone who claims to have a
theory built around Kerr-Newman black holes.
For t fixed, r fixed, \theta free, and \phi= pi/2, the Kerr-Newman
metric reduces to the following:
ds^2 = (r^2 + a^2 cos^2(theta)) d\theta^2
I'll leave the integration to you. I think you can handle it.
>
> RLO
> http://www3.amherst.edu/~rloldershaw
>
carlip...@physics.ucdavis.edu
> > The circumference, on the other hand, is easy. It's equal to
> > 2\pi a, where a is the angular velocity parameter. You can find
> > this in, for example, d'Inverno, _Introducing Einstein's Relativity_,
> > section 19.5, or Poisson, _A relativist's Toolkit_, section 5.3.8, or
> > Hobson et al. _General Relativity_, section 13.8. It's also quite
> > easy to calculate directly in the Kerr-Schild form of the metric.
invent a flat "background" geometry in which the singularity has
this circumference. But in terms of the real, intrinsic geometry
of the Kerr metric, the circumference of the singularity is infinite.
This is also not surprising -- it is, after all, a singularity.
More precisely, in Boyer-Lindquist coordinates, the line element
along the equator at constant t and r is
ds^2 = [r^2 + (1 + 2m/r)a^2] d\pi^2
so the physical circumference of a circle at constant r is
2\pi \sqrt{r^2 + (1 + 2m/r)a^2}
This circumference approaches 2\pi r for very large r, and decreases
as r decreases until one reaches r^3 = 2ma^2. After that, further
decrease of r corresponds to an *increase* in the circumference,
which goes to infinity at the singularity r=0.
Note that this is a particular limit, in which one first restricts to
the equatorial plane and then takes r->0. One can alternatively
start with the surface r=0 (which is not singular except at the
equator) and then take the limit \theta -> \pi/2. It turns out that
the two limits are not equal -- yet another illustration of the fact
that the singularity really is singular.
[...]
> a.v.p. = (specific ang. momentum)/(r+)^2 + (spec. ang. momen.)^2 .
that's something -- it's the angular velocity of the event horizon.
> Is this the same parameter found in the books you list (which I do not
> have at hand)?
momentum (angular momentum as measured by a very distant
observer) and M is the ADM mass (again, mass as measured by a very
distant observer).
> If so, then for the particular Kerr-Newman object that I am interested
> in I get "R"(sing)/R(bh) of about 0.428, ( = 3/7).
not spherical. What cross-section are you looking at?
Steve Carlip