Talk:Kerr metric

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1 Image Showing Static Limit and Event Horizon
2 Objection to introduction
3 The Boyer/Lindquist coordinate chart metric is wrong
4 To Merge or not to Merge?
5 The spin parameter
6 Will the real monopole please stand up?
7 trying to use Kerr solution for rotating stars is doomed
8 Contravariant components of metric tensor wanted
9 unit error in the Kerr metric
10 For non-physicists,
11 1 November 2007 changes
12 errors
13 Notational consistancy
14 Duplicate figure

[edit] Image Showing Static Limit and Event Horizon

The image showing the static limit and (outer) event horizon appears twice. Moreover, in its second incarnation, the caption incorrectly refers to the outer event horizon as the static limit. I'm not bold enough to decide which copy should go. Certainly one must, and the correct caption used. Warrickball (talk) 23:09, 13 March 2008 (UTC)

[edit] Objection to introduction

Does anyone else object to this section of the text, or is it just me?

"Such black holes have two event horizons where the metric appears to have a singularity. The outer horizon encloses the ergosphere and has an oblate spheroid shape, a flattened sphere similar to a discus. The inner horizon is spherical and marks the "radius of no return"; objects passing through this radius can never again communicate with the world outside that radius."

I would say it's both wrong and misleading. Firstly, the outer surface that encloses the ergosphere is the "stationary limit surface" and not an event horizon (it even says so explicitly in Hawking and Ellis). Secondly, I would say that the metric exists independently of which coordinates you use to describe it and hence whether it appears to have a singularity in a given coordinate system or not isn't really relevant. In Boyer-Lindquist coordinates the metric is not singular at the stationary limit surface/boundary of ergosphere and it is singular at the true singularity (which isn't a horizon). Thirdly, if we are going to call something the "inner horizon" then we really ought to be talking about the inner horizon as the inner Cauchy horizon. I believe that is standard usage - it is where I come from.

--Eujin16 (talk) 05:17, 6 December 2007 (UTC)

[edit] The Boyer/Lindquist coordinate chart metric is wrong

I am certain that the metric as given in the section "Boyer/Lindquist coordinate chart" is wrong. I think that the $r 2 + a 2$ [in $(r 2 + a 2)sin 2 θ d φ 2$ ] needing to be squared and divided by ρ² is the only issue, but I can't be 100% sure without a chance to work it all through.

I admit that I could be wrong, but I am certain enough that there is a problem here that I would rather have the red flag thrown back in my face than let this go unchallenged. --EMS | Talk 17:08, 27 July 2005 (UTC)

Why are you certain? I didn't write this part of the article, but whoever did just copied (5.29) from Hawking & Ellis or (21.1) in Stephani or (8.32) in Ohanian & Ruffini, or the same expression as given in some other standard textbook. Even better, long years ago, I verified that this does give a vacuum solution.

I guess the confusion might have arisen if you have seen the terms in the BL line element collected in a different way (I'll give an example below), but all expressions for the BL line element should agree if you multiply everything out. Another possibility is that you saw the line element in a chart which looks like BL but isn't (for example, de Felice and Clarke introduce a "rotating" BL chart before they get to the standard BL chart).

This "controversy" may be moot since I've been planning to rewrite the article anyway to discuss the Kerr vacuum in much greater detail using various coframes. For example, from the line element as given in (19.27) in D'Inverno or (217) in Chandrasekhar we can read off the following coframe:

$\sigma^0 = -\frac{\delta}{\rho} \, \left( dt + a \sin(\theta)^2 d\phi \right)$

$\sigma^1 = \frac{\rho}{\delta} \, dr$

$\sigma^2 = \rho \, d\theta$

$\sigma^3 = \frac{\sin(\theta)}{\rho} \, \left( -a \, dt + (r^2 + a^2) d\phi \right)$

where

$\delta^2 = r^2 - 2 m r + a^2, \; \; \rho^2 = r^2 + a^2 \cos(\theta)^2$

Then, the metric is

$-\sigma^0 \otimes \sigma^0 + \sigma^1 \otimes \sigma^1 + \sigma^2 \otimes \sigma^2 + \sigma^3 \otimes \sigma^3$

This is one of the simpler coframes, but several others are of great interest. For example, the LNR coframe is given in the textbook by de Felice & Clarke. The Doran coframe is a generalization of the LeMaitre coframe for Schwarzschild. Other charts in common use include Eddington charts, Doran chart (generalization of Painleve chart), Plebanksi chart, Weyl canonical chart, the original Kerr chart, etc., and the coframes can be written in all of these charts. So you might see many different ways of writing down the Kerr solution.---CH (talk) 22:49, 27 July 2005 (UTC)

I don't think that it was copied correctly. I come up with <math>g_{\phi\phi} = \left [ \left (r^2 + a^2 \right ) + 2mra^2/\rho \right] \sin^2 \theta</math> from the metric as given in the article. That is not correct. --EMS | Talk 23:46, 27 July 2005 (UTC)

Ah ha! Now I see what I was missing: The

s i n 2 θ

was squaring also. Now it works out OK. This just is not a form of Kerr that I am used to. The coframe version and the spelled-out versions are ones that I familiar with. --EMS | Talk 23:54, 27 July 2005 (UTC)

[edit] Sorry for the confusion

I was exhausted yesterday, and did not realize just how tired I was. I just cannot do any intellectual "heavy lifting" when I'm like that. I knew that $(r 2 + a 2) 2 / ρ 2$ was involved with $g φφ$ . I had lost track of the fact that it was involved with the coframe version and sorts itself out into a form such as is in the article.

So I was tired and irritated (due to other things) and was not able to respond to my own self-red-flag of "I know I could be wrong". --EMS | Talk 22:03, 28 July 2005 (UTC)

[edit] Roy P. Kerr

Hi, Taxman, don't worry, I am certain the initial is indeed "P."---CH (talk) 20:11, 5 August 2005 (UTC)

[edit] Proposed biographical subcategory

There are many professional biographies on leading figures which have appeared in the journals over the years. I am pretty sure I have seen one on Kerr, for example (can't recall where). Since I have my hands full with writing more technical articles, would someone who lives near a good physics research libraray be interested in systematically searching for such biographies, looking them up, and writing wikibiographies? The biographical articles should cite the published biography as a source. I see someone has written a brief biography of Roy P. Kerr; to avoid cluttering up a category (this one) which is already large (and I plan to enlarge it considerably), I have created the new subcategory "Contributors to general relativity".

[edit] To Merge or not to Merge?

I'm against merging rotating black hole with this article. They are parent subjects, not the same one. The content of these articles are enough different to support so. nihil 09:25, 25 November 2005 (UTC)

Nihil may have a point. The "no hair theorem" is or should be surprising, and Price's theorem is often summarized by saying that any inhomogeneties or anisotropies outside a black hole which can be radiated away, will be radiated away (which restores the Kerr exterior vacuum of a relaxed rotating black hole). So, perhaps the article Rotating black hole can discuss these facts, while referring to Kerr vacuum for the relaxed state.---CH [[User_talk:Hillman|(talk)]] 18:26, 25 November 2005 (UTC)

[edit] The spin parameter

The explanation is incomprehensible as the number a = J/M is not dimensionless. Somebody please clarify. Bo Jacoby 08:56, 26 January 2006 (UTC)

units where c=1. If it troubles you, then use a=J/Mc². -lethe ^talk 09:11, 26 January 2006 (UTC)

Thanks! But the SI-unit of a = J/Mc² is (m²×kg/s)/(kg×m²/s²) = s. Still not dimensionless. Bo Jacoby 10:43, 26 January 2006 (UTC)

Hmm... you're totally right. I do apologize for answering so hastily and incorrectly. It is not dimensionless. And in fact, I guess it shouldn't be dimensionless, it should have the same dimension as r and s. I don't know what the problem is. -lethe ^talk 10:54, 26 January 2006 (UTC)

So that stuff was added by an anonymous editor. I now think it's mistaken, but I won't remove it unless I know what it's supposed to say. I suspect that what is meant is that a can range from 0 to m, rather from 0 to 1, and the anonymous had used a reference with a different notation convention. But I'm not sure, so.... -lethe ^talk 11:12, 26 January 2006 (UTC)

First of all a indeed is not dimensionless at all, nor should it be. In geometrized units (where c=G=1), it is a length, just as M and r are. Secondly, the angular momentum J must be an area so that a=J/M can also be a length. To get these geometrized lengths, you need to multiply a mass by G/c² and an angular momentum by G/c³. --EMS | Talk 15:53, 26 January 2006 (UTC)

P.S. The anon mentioned above was most likely CH, who sometimes forgets to log in. if not, it was someone else who is quite familiar with the Kerr metric. --EMS | Talk 16:01, 26 January 2006 (UTC)

Oops, I mean thanks! -lethe ^talk 18:16, 26 January 2006 (UTC)

FYI - Turns out that I was wrong about the anon being CH. (I should have guessed once I realized what the error was.) My apologies to Chris. Let's just say that this talk page seems to be jinxed for me. :-( --EMS | Talk 04:55, 28 January 2006 (UTC)

I'm the anon. Sorry about the error, I usually use the Kerr metric with M normalized to be 1, and measure everything in R_G 's. In those units a ranges from 0 to 1. By writing the M's in explicitly in the metric, yes, the range changes from 0 to M. What's there now is what I indended, if not what I actually wrote. Sfuerst 22:53, 17 February 2006 (UTC)

The angular momentum J defines a length J/Mc, and the mass M defines another length, the half Schwarzschild radius GM/cc . The ratio Jc/GMM is dimensionless. Is that the spin parameter? Bo Jacoby 12:49, 29 January 2006 (UTC)

Here is an easy way to read off the geometrized units of parameters appearing in metric coefficients (or in one-forms defining a cobasis field). Since metric coefficients are dimensionless in geometric units, a quantity such as m/r must be dimensionless. Since r has units of length, so must m. Angles and trig functions of angles should be dimensionless (as should the argument of any exponential). Thus, $a \, \sin(\theta)^2/\sqrt(r^2+a^2)$ must be dimensionless, so since $\sqrt(r^2+a^2)$ has units of length, so must a. Last but not least, Bo please note that angular momentum has geometrized units of area, just as you would expect from Newtonian theory! See geometrized units. ---CH 23:26, 17 February 2006 (UTC)

[edit] Will the real monopole please stand up?

I have elaborated slightly on this admittedly confusing issue. As I have noted elsewhere, WP requires articles on relativistic moments, moments in weak field gtr (including Weyl moments and the multi-index mass/momomentum/stress moments), and an improved article on multipole moments in Newtonian gravitation. Ultimately, the new section should be linked to this forthcoming articles and can perhaps then be shortened.---CH 22:24, 2 June 2006 (UTC)

[edit] trying to use Kerr solution for rotating stars is doomed

On the article page it says "Open problems

The Kerr vacuum is often used as a model of a black hole, but if we hold the solution to be valid only outside some compact region (subject to certain restrictions), in principle we should be able to use it as an exterior solution to model the gravitational field around a rotating massive object other than a black hole, such as a neutron star--- or the Earth. This works out very nicely for the non-rotating case, where we can match the Schwarschild vacuum exterior to a Schwarzschild fluid interior, and indeed to more general static spherically symmetric perfect fluid solutions. However, the problem of finding a rotating perfect-fluid interior which can be matched to a Kerr exterior, or indeed to any asymptotically flat vacuum exterior solution, has proven very difficult. "

No wonder it is difficult. The Kerr solution has only 2 independent parameters: mass and angular momentum. These determine all the gravitational multipole moments. But a rotating star has a complicated structure and its multipole moments will be controlled by the rotational (angular) velocity within it (which can vary with distance from the center and with latitude or angle from the plane of symmetry), the pressure and density variations with depth and possibly more things like internal magnetic fields. So the external field or various example rotating stars is more than a 2-parameter family. Whoever wrote that stuff should probably fix it. Carrionluggage 06:20, 17 October 2006 (UTC)

[edit] Contravariant components of metric tensor wanted

I think this article is written OK. I checked the metric "ds^2 = ..." and it looks good to me, however, I suggest to include contravariant components of $g i k$ tensor too. greg park avenue 18:33, 4 June 2007 (UTC)

[edit] unit error in the Kerr metric

if i'm not mistaken, there is an error in the Kerr metric. I'm no expert on Relativity, but the last summand in the line element has unit length*time, where it should have length^2 like the other summands. the last summand should probably read (2*r_s*r*alpha/rho^2)*dPhi*(c*dT), since all other uses of time-variables in the line element are scaled by c, too. sorry in advance if i'm talking bullshit, but it puzzled me. i also find the mixed use of "a" and "alpha" in the line element misleading. am i right in assuming they should be the same? Could someone with more knowledge on this subject please verify this? thx Catskineater 14:32, 15 August 2007 (UTC)

Those were little things, but they did need fixing. It is done now. --EMS | Talk 17:30, 15 August 2007 (UTC)

[edit] For non-physicists,

the article is practically unreadable. I came here from the article on Tipler Cylinders, which is readable and easily understandable to a non-physicist. I'm not saying the equations should be taken out, and I also realise this is a fairly specialist article. However, perhaps the introduction should be expanded, in order to further explain to non-physicists like myself what a Kerr metric is, why it is important, and give a general idea of "how it works" as it were. At the moment the introduction is this:

In general relativity, the Kerr metric (or Kerr vacuum) describes the geometry of spacetime around a rotating massive body, such as a rotating black hole. This famous exact solution was discovered in 1963 by the New Zealand born mathematician Roy Kerr.

OK... that's all very well, but it doesn't tell me much. All I get from that is the idea that a Kerr metric is to do with describing how gravity works around a rotating black hole. That's all very well, but why is it important? Does it have application to anything else? I don't know.

WikiReaderer 00:22, 27 September 2007 (UTC)

You certainly have an interesting take on this, not that it is a bad one. To be honest with you, this article is so dense that I even have trouble with it, and I have studied general relativity is all of its glory.

In a nutshell, the Kerr solution is important because it is the simplest case of the spacetime around a rotating massive object. In astrophysics, it helps to predict phenomena in the vicinity of a gravitationally collapsed and rotating massive object. It also has a number of interesting features, some of which are being checked for by Gravity Probe B.

Rewriting this article is something that I would like to do someday to make it more accessible. This cannot and will not be a topic that a non-physicist can get a good grip on, but that is no reason not to have a fairly non-techinical introduction that can give someone like you a sense of what this metric is about before diving into the more "gory" details.

FTI - Some features are:

A frame dragging effect that pulls spacetime around the rotating object. This means that a beam of light traveling with the rotation will be found by distant observer to go around the object faster than a beam of light that travels against the rotation. it also means that an object that appears to a distant observer to be going around the obejct may locally be falling straight into it.
There is a region called the ergosphere close to the object (near the outer event horizon) where an object cannot be at rest with respect to a distant observer but instead must rotate with the object. (Even light must do so in the ergophere, including light that is locally traveling against the rotation.)
There are two event horizons for a "slowly" rotating black hole: An outer one and an inner one. Entry into the outer horizon demands that you pas though the inner one, after which you are stuck there.
The inner region of a Kerr black hole includes a ring singularity and also "closed timelike curves". The latter expression means that you can come back not only to the same place but also the same time! In essense, you can do time travel in the inner region of a Kerr black hole.
For a "rapidly" rotating black hole, the event horizons are not present, and the area where time travel is theoretically possible potentially becomes accessible.

Hopefully this is some help to you. (If it is, an idea may be to put this list into the article.) --EMS | Talk 02:43, 27 September 2007 (UTC)

That was a quick response. Thanks! All the above is useful, and pretty informative. I have already heard of some of these things before (mostly from Stephen Baxter sci-fi!), but hadn't really connected them up with this Kerr metric thing. I would definitely recommend that you insert this list into the article: the only question is where...

WikiReaderer 23:12, 30 September 2007 (UTC)

[edit] 1 November 2007 changes

I've clarified some of the material on this page and on the Kerr-Newman and rotating black hole pages. I believe it is now more accurate. Unfortunately, I've noticed in editing this that there is now some duplication (for instance, the figure on this page). If any frequent wikipedians want to clean these pages further, that would be great. 91.37.241.229 20:44, 1 November 2007 (UTC)

[edit] errors

So, this page has some real problems. For instance, from the introduction(!)

The Kerr metric is often used to describe rotating black holes, which exhibit even more exotic phenomena. Such black holes have two event horizons where the metric appears to have a singularity. The outer horizon encloses the ergosphere and has an oblate spheroid shape, a flattened sphere similar to a discus.

The two event horizons are the outer and inner event horizons, the outer of which cannot be crossed from inside out (I'm not sure about the inner one). In Boyer-Lindquist coordinates, the ergosphere is described by r = M + \sqrt{ M^2 + a^2 \cos^2 \theta }. Not, by the way, an oblate spheroid, as far as I can tell, because as a->1 it looks more like a peanut. Moreover, it is also neither of the horizons, which are located at r_\pm = M \pm \sqrt{ M^2 - a^2 }.

I'm not an expert, I just have a copy of Hartle's Gravity: An introduction to Einstein's general relativity, and have read the chapter on the Kerr metric.

Could someone put up a warning to folks that the info on this page is questionable and that the page needs help from an expert?

Or, of course, correct me if I'm wrong :/

Thanks — gogobera (talk) 05:05, 2 April 2008 (UTC)

PS: Actually, this has been brought up earlier on this talk page!

This was my fault, sorry! I got the math correct, as you can see from the article, but my description of it was incorrect. My basic confusion was describing a "surface where the metric is singular" as "event horizon", which are not the same concept. I also didn't mean oblate spheroid in its exact mathematical sense, but qualitatively; I was trying to describe the shape as a flattened sphere. I'm pretty sure the the outer surface doesn't always have a dimple — consider the limit as the angular momentum gets small, that is, as the ratio of length-scales α/r_s gets small. In that limit, we can expand the square root in a binomial expansion to obtain (if I did the math correctly!)

$r_{outer} = r_{s} \left[ 1 - \frac{\alpha^{2} \cos^{2}\theta}{r_{s}^{2}} + \ldots \right]$

which approximates an oblate spheroid, wouldn't you agree? The dimple appears as the angular momentum gets larger.

Anyway, I'm sorry for my errors and grateful that you pointed them out. I'm keenly conscious that I'm no expert, although I'm trying my best. It would be great if an expert could come here and make the subject more intelligible for lay-people. (hint, hint) ;) Willow (talk) 09:36, 2 April 2008 (UTC)

[edit] Notational consistancy

This page needs to be edited for notational consistency. In the first part in the definition of the metric, the length scales Lambda and alpha are introduced. Then in the rest of the article, the dimensionless parameter a and the parameter Delta are used without definition. Either they need to be defined, or more probably, the Kerr metric should be written using the conventional parameters Delta and a instead of with Lambda and alpha. —Preceding unsigned comment added by 142.103.234.23 (talk) 18:40, 7 April 2008 (UTC)

[edit] Duplicate figure

I commented out the duplicate copy of the ergosphere picture. Caption was:

Two important surfaces around a rotating black hole. The inner sphere is the static limit (the event horizon). It is the inner boundary of a region called the ergosphere. The oblate spheroidal surface, touching the event horizon at the poles, is the outer boundary of the ergosphere. Within the ergosphere a particle is forced (dragging of space and time) to rotate and may gain energy at the cost of the rotational energy of the black hole (Penrose process)

in case someone wants to put some of it back into the other copy. Wwheaton (talk) 16:27, 29 April 2008 (UTC)

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