Reissner-Nordstrom-? metric

[Norman Albers]

The Kerr solution can be derived as a generalization of the Schwarzschild field. Eddington offered a coordinate transform of the time variable: [math]\bar x^0 = x^0 + 2m\log\left| r/2m - 1 \right| [/math]. This yields a form of the metric termed degenerate. From here, substitution of a complex variable where we wrote z gives expression of a system with angular momentum. If I am correct, I have expressed the Minkowski and stress-energy tensors and can apply the Eddington transform to the latter to yield a right-hand side to the Einstein field equations. It looks like it yields first, second, and third orders in the arbitrary m. I would appreciate comments on my methodology.

[Norman Albers]

Part of the challenge here is that the right-hand side of the GR field equations, the stress-energy tensor, is constructed as: [math] T_{ab}=F_{ac}{F^c}_b +1/4g_{ab}F_{cd}F^{cd}[/math]. The form of the metric tensor must by known and expressed in the second term. The LHS of the equation is built from the metric, and the trick in the Kerr solution is generalizing the degenerate form of the Schwarzschild metric to that characterizing the exterior metric of an axially-symmetric mass distribution. I seek an interior solution with the energy distribution of my electron nearfield accounting for the source. . . . It seems to me that if you start with the Kerr metric and drop the terms in m, geometric mass, you are left with the anisotopies depending on a, geometric angular momentum. It seems also that this will describe the vacuum field we recognize in the Ahoronov-Bohm quantum mechanics.

[Norman Albers]

There are RHS terms of order <m> produced whenever indices are raised because the metric form I am assuming (the degenerate form) is : [math]g_{ab}= \eta_{ab} -2m l_a l_b [/math]. The first tensor on the RHS here is the Lorentz diagonal matrix: <1, -1, -1, -1> . I am seeking to combine these terms.

 

For now I must work in Cartesian coordinates even though my source fields are spheric or cylindric. The reason is that I am assuming I can adapt the exterior Kerr solution to an interior one, and the mathematic form here is the degenerate metric. If you work in Cartesian coordinates, the left-hand side of the GR field equations, or Ricci curvature tensor, is transparent to the Lorentz flat-space metric, and this sets the structure of orders of m which is how whole system of equations is parsed. There are more terms produced in my right-hand side expressions, but either way you have to pay your dues in accounting. I wrote out what I think is the whole expression for the <0,0> term, and two lines of it cancelled. This is a good sign.

[Norman Albers]

By gosh and by golly I have some kind of result. By expanding the Kerr to second order in terms of a/r, and without the term in m, I get a new effective potential. The original forms include [math]\mu c^2[/math], for rest mass of the test particle whose geodesic we are examining. Even when this mass has low angular momentum (in the high-AM field) there is a second-order contribution now: [math]V= \mu c^2 \sqrt{1+ a^2/\rho^2}[/math]. [math]\rho[/math] is the real part of the "complexified position" or whatever describes how we wrote 1/r knowing it solves Laplace, then in Cartesian form added to <z> to let it be <z + ia>. This is the trick that pulls Kerr out of the hat. I can certainly expand my SQRT for far-fields where rho converges with r, and say: [math] V(far) = \mu c^2[1 + 1/2(a/r)^2][/math]. This is different from both the Schwarzschild and usual Kerr expansions. Note that it's like coming in through the roof, in that we psyched out the tensor equations through the form of the degenerate metric and found all the info in the Laplace form. Thus I am able to get some real entertainment in the far-field without having solved my interior form yet.

 

[EDIT] In the first line above I should speak in terms of radial coordinate [math]\rho[/math] rather than [math]r[/math]. Only in the far field do they converge. We have expressed a transformed system by allowing: [math] \omega= [x^2 + y^2 + (z-ia)^2]^{1/2} [/math] and we identify the real part of this expression as [math]\rho[/math].

[lakmilis]

Norman, You know from other posts why I say Kerr is better to use than Schw.

 

Schw is a hole with no angualr momentum, any celestial obcject unless has been since the beginning a black hole must of had angular momentum...thus use the kerr metric. Scwz. is just useless...except for when the day comes where we might produce sone ourselves, sure then there is nothign which says we *must* include angular momentum. SQRT? lol that was the abbreviation of the Roman empire was it not? ,)

 

anyway, thats all i can come with with your methodology

 

The question is, are you just trying to work on similar operations as has been before by the people you are emulating or do you wish to find some unique deduction not found before and if so, what are you really looking for?

 

 

lak

[Norman Albers]

When we get the Schwarschild solution we see a characteristic length, or "geometrized mass", of [math]m=GM/c^2[/math]. In the Kerr metric, the rotation gives another characteristic, using this first term. The "geometric angular momentum" is [math]am=GJ/c^3[/math], and if we subsitute for 'm', this shows that [math] a=J/Mc [/math] and has units of length also. In normal stellar physics, stars have Schwarzschild radii of some kilometers' scale, but 'a' is notably smaller. Examine the null hypersurfaces defined by [math]\rho_\pm = m \pm \sqrt{m^2-a^2}[/math] in the Kerr metric. Note that this is distinct from the infinite redshift surface which is oblate outside the outer null surface, and a figure '8' inside the inner one. We can see that if 'a' becomes equal to 'm', the hypersurfaces become degenerate spheres. Now, fast-forward to my considerations of the electron. Here 'm' is of the order E-57m, but on the other hand, angular momentum gives 'a' about E-23m. Thus the electron is "much electromagnetic ado about nothing" as measured by its mass. If we are to investigate what GR might show here, we will make expansions in appropriate places acknowledging relative magnitudes. I am not sure if anyone else has taken things this far. (Newman et. al.) In describing an effective potential, I follow the procedure of determining geodesics of motion, which yields essentially an energy equation for [math] \mu^2\dot{\rho}^2 [/math]. In the Schwarzschild metric this shows as [math](E^2-V^2)/c^4 [/math] and it is clear how we may speak of this potential 'V'. It is not nearly so clear how to interpret the Kerr form, and I am even having an argument with my textbook. However one does yield the statement I made. The expansion I got is distinct from what I read on Schwarzschild in that it does not depend upon the orbit path. One gets a term there of [math] L^2/2\mu r^2[/math]; in the Kerr expansion a "Limit of small 'a'" yields a term of [math] 2macL/\rho^3[/math]. Both of these are dependent on angular momentum [math]L[/math] which is path-dependent. What am I really looking for? Unification, I guess. The same vacuum medium gives polar forms as in E&M, and neutral forms in gravitation. I see frame-dragging outside neutron stars, whose rotating mass I see as a "neutral current", and Ahoronov-Bohm rotation of phase for charged particles, whose source is a polar current of neutral net charge. I want to know how all this connects through the different regimes we are considering in GR.

[Fred56]

Norman: I have found a 1995 paper (you may have seen it) by Cognola & (Italians), and it's titled "One-loop quantum corrections to the entropy for an extremal Reissner-Nordstrom black hole". It's a quantum gravity paper, but maybe the math illustrates something? It does analysis of the scalar fields...?

[Norman Albers]

Fred, I love it and might be able to understand something of it. See, I guess "extremal" means regimes of possible energetics. <<I figure the gold lines in circuitboards are snakes, and the one that appears on the great photo I'm chasing bridged the high-current pins.>> Back to business: It seems pretty clear to me that the solutions we have of both mass and electromagnetics in the interior, sourcing region, are low-field approximations in both senses of characteristic distance. Lense-Thirring portrays a spinning mass of constant "low" density, for instance. I would like to learn if the regime I am suggesting, where to first order angular momentum alone is significant, is used in exterior solutions. Actually it seems to me now that I need to keep terms of [math] <a/\rho, (a/\rho)^2, m/\rho> [/math] to get anything polar out of the equations. No I do not yet know quite what I am doing.

[Norman Albers]

We see many similarities between electromagnetic theory and gravitation, the clearest of which may be inverse square far-field dependence of effects from sources. Yes E&M is a tensor theory of rank one; gravitation is a tensor theory of rank two. I see them both stemming from a common vacuum field, polar and neutral modes respectively, and thus their fundamental difference. I quote Introduction to General Relativity, Adler, Bazin, Schiffer, p.491: "The general theory of relativity succeeded in geometrizing the phenomenon of gravitation by connecting it with the metric of the Riemann space considered. The potential of the gravitational force which occurs in the Newtonian theory was replaced by the metric potentials [math]g_{ab}[/math], the components of the metric tensor. If we wish to obtain an analogous theory for electromagnetic phenomena, we have to establish corresponding relations between the electromagnetic potentials and the metric tensor. However, the components of the metric tensor are already sufficiently determined by the Einstein field equations, and there seems to be no room to imbed also the entire theory of the electromagnetic field into the same differential geometry." The authors go on to describe Weyl's contributions in further geometric possibilities. I feel that in assembling my understanding of the different regimes of interpretability of the Kerr-Newman metric, and combining this with the vision I have developed of polar phenomena, there will become clear a path forward to a more unified expression. The Reimannian metric is a bilinear form and thus a mathematics to deal with squares of things, and also cross-dimension relationships as in rotation tying together [math] d\phi dt[/math]. This is powerful and beautiful mathematics, built on the proposition of smooth changes describable by differential calculus, still in search of complete vacuum physics.

[Norman Albers]

(This letter to my colleague:) SB, I like solstice. I have sat with the mathematics that made me feel stupid half of this year. After casting my spherically (or axially) symmetric electron nearfield and following through the orders of m which are generated whenever you use the metric tensor <<creating the GR stress-energy tensor from the Minkowski tensor>>, it was not clear how to proceed. It is assumed to be of degenerate form, [math]g_{ab} = n_{ab} - 2mk_ak_b[/math] where [math]n_{ab}[/math] is the Lorentz flat-space form, and [math]k_a[/math] is a 4D null vector. In the stationary solution (steady-state) one may say that: [math] k_a=k_0<1, u_1, u_2, u_3>[/math] where the 3-vector is a unit length. Sooooo, in the Schwarzschild solution one sees: [math]k_a =\frac{1}{\sqrt r} <1, x/r, y/r, z/r>[/math] so it is a radial unit vector, no? It's taken a whole year to get here but I finally picture the rotating Kerr forms. It has been shown that the Einstein eqs. yield essentially two relationships which may be bound together in COMPLEX FORM and written as a single Laplacian eq., [math]\nabla^2 \gamma=0[/math]. Brilliantly this is solved by 1/r with the z-variable offset by an imaginary amount. We are solving geometrically the unit vector field, still, though it now has further components. Coordinate systems are flying thick and fast, but we see that the radial component is moderated; there is a z-component going as [math]cos\theta[/math]; there is now a [math]\hat\phi[/math] or azimuthal part also, with [math] sin\theta [/math] dependence. This is nice, isn't it? I did not know what to see in the mess of equations I generated. I stepped back, and wrote them on top of one another, and they are much more simply expressible as only three terms involving this vector field. This is satisfying. I will now investigate whether or not the degenerate form works usefully in spheric coords, or if I should stick to this coord system, actually a mixed Cartesian representation. It was in this form that Kerr wrung the <<exterior>> solution in 1963. ALBERSAWA

[Norman Albers]

I just hit a major roadblock, having realized that there are generated terms with no dependence on m in my RHS source expression, namely the original statement of the source tensor in Lorentz flat-space. All terms on the LHS of the field equations contain orders of m, from <1 to 4>. There are no order-0 terms to balance my source terms, so the assumed form of the metric is not valid for an interior solution. Rather than a catastrophe, this points to the need for a more direct method such as the interior Schwarzschild solution or the Lense-Thirring method.

[Norman Albers]

Analyzing the interior Schwarzschild metric solution I see that it does not conform to metric degeneracy, so I find support for the need to let go of this characteristic in the interior metric solution. This does not change the characterization of the "far-field" as the accomplished Kerr-Newman-schild metric parts. The electrodynamic model I am working with begins to show appreciable vacuum charge response at roughly the classical radius of the electron.

[Norman Albers]

I have arrived at the challenge I asked for: construct a General Relativistic representation of the electron as a strong angular momentum source of circulating electromagnetic energy with negligible total energy, i.e., mass. There certainly are terms remaining in the expression for the Kerr metric, even when the "m"-term is set to zero. I am working to sort through the coordinate transforms.

[Norman Albers]

I have been successfully solving the Kerr metric vacuum solution for the electron as a strong angular momentum source. As distinct from my representations of inhomogeneous possibilities, this is a clean application of General Relativity to an object of spin [math]\hbar/2[/math], and the mass of the electron. The Kerr metric characterizes angular momentum (AM) by the parameter 'a' = J/Mc, which has dimensions of length as does the Schwarzschild geometric mass 'm'. For this value of AM, 'a' is about 2 E-13 meters. On the other hand, 'm' is roughly E-57 meters. My first attempt ignored terms in m since they seem miniscule compared to changes in radius larger than Planck length. However there is a very interesting joker in this deck. In the Schwarzschild solution, the first metric term: [math]g_{00}= 1 - 2m/r [/math] is well-behaved except at the origin of r=0, and in this case the scale is set by m. In the Kerr electron case, however, the coordinates have been put through four or five transforms and usually presented in terms of the final set of variables which present an axially symmetric form. I have tracked the transforms backward to express physics in the original 'lab' Cartesian coordinates. There is a degeneracy in the radial transform, such that [math]\rho[/math] goes strictly to zero in the flat disc region of z=0 and [math] r \le a[/math]. This causes asymptotic behavior where permittivity goes to zero, at least near r=0 and r=a, just as it does at the center singularity in the massive Schwarzschild solution. This behavior characterizes a thin layer, of dimension m, about those regions of the disc. I will be writing up this study and welcome comments from anyone who might have read such an interpretation elsewhere. Other near regions seem generally to have higher permittivity, shading off into the far-field, greater than unity and more like the mass source solution. This might even be the case between r=0 and r=a on the disc of z=0. Thus the singularity itself is a thin torus at a, as well as the small center Schwarzschild singularity. (More later...)

 

When I say "permittivity" of a region, I have solved the null geodesic case and associated lightspeed measured in Cartesian coordinates with what I see as local permittivity, as in my paper on gravitation, and the Polarizable Vacuum interpretation as I am investigating it.

 

This is a mathematician's paradise and possibly a physicist's hell, we'll see. There is a stunning discontinuity between interior behavior approaching the point z=0, r=a, and exterior metrics. I hope to post a paper within a few days.

Edited December 14, 2008 by Norman Albers
multiple post merged

[scalbers]

To put things in a bit of context it might be an interesting challenge (thus strengthening one's paper) to address these skepticisms about whether an electron is a black hole: http://cow.physics.wisc.edu/~ogelman/guide/e/

[Norman Albers]

Electrons are certainly not black holes. Their Schwarzschild dimension is about E-57 meters. All their mass-energy would need to be localized on this scale. I am investigating their nature as singularity. When you solve the massive case, say for a rapidly spinning stellar mass, the angular momentum radius is less than the Scwarzschild dimension. If it could become comparable, the inner and outer singular surfaces merge and there is an opening outward. This implies, however, relativistic velocities of the mass. The investigation I am doing has parameters the other way around. I will offer my paper today or tomorrow.

 

Kerr electron: I have completed an analysis on the first level of GR applied to electrons as angular momentum (AM) sources. By this I mean a mapping of the transformed, axially symmetric coordinates often expressed, back to Cartesian "lab" external coordinates. For now I have covered the expression of null geodesics in the plane z=0 , both inside and outside of r=a. This will be available at the http below in my signature.

Edited December 21, 2008 by Norman Albers
multiple post merged

[MolecularEnergy]

After many months of work I have written the Minkowski tensor in spherical basis representation, for my electron model fields. The radial electric field is as originally worked out in the Reissner-Nordstrom solution. Though I do not use them here, if I am correct the top row of the tensor reads: [math]0, -E_r, -rE_\theta, -rsin\theta E_{\phi } [/math]. Finally also I have the magnetic terms spawned by my assumption of current [math]A_\phi[/math]. Those conversant in tensor manipulation may have wondered why I struggled to transform the Cartesian form of the [math]F_{ab}[/math]. The reason is that there is much differential geometry to master when one starts with the vector potential. You have to go through the curl operation and also the machinery of spherical coordinates. The vector A by itself is a pseudotensor: there is a natural way to see it as contravariant but you cannot transform it by the necessary differential form to covariance. Importantly, when you take the curl you get a true tensor in the Minkowski form, and you get the correct representation in spheric coods. by following coordinate transform multiplication rules for each index. Thus I have constructed the stress-energy tensor for the RHS of the gravitation equations including the magnetic dipole self-field described in the model. I shall be examining the R-N solutions altered by these terms as I have already partly described for the electric components.

 

 

May i see this work...?

 

Electrons are certainly not black holes. Their Schwarzschild dimension is about E-57 meters. All their mass-energy would need to be localized on this scale. I am investigating their nature as singularity. When you solve the massive case, say for a rapidly spinning stellar mass, the angular momentum radius is less than the Scwarzschild dimension. If it could become comparable, the inner and outer singular surfaces merge and there is an opening outward. This implies, however, relativistic velocities of the mass. The investigation I am doing has parameters the other way around. I will offer my paper today or tomorrow.

 

Kerr electron: I have completed an analysis on the first level of GR applied to electrons as angular momentum (AM) sources. By this I mean a mapping of the transformed, axially symmetric coordinates often expressed, back to Cartesian "lab" external coordinates. For now I have covered the expression of null geodesics in the plane z=0 , both inside and outside of r=a. This will be available at the http below in my signature.

 

 

Electrons certainly could be... according to PhD Brian Greene.... Do you have some conclusive proof to put this guy in his place? There are problems, but there are equallly solutions that make electrons as black holes very suitable.

[Norman Albers]

I am sweating through last minute mistakes and proof-reads. Check my cache tomorrow: http://laps.noaa.gov/albers/physics/na . I welcome your observations, MolecularEnergy. I enter this study from the standpoint of someone visualizing the angular momentum alone, at the moment. The Reissner-Nordstrom solution of the charge-induced electric field adds its own physics but at a smaller radius, I think characterized by E-23 meters. Also, as I have indicated, one of my goals is to show how inhomogeneous presentation of circular currents and charge density eliminate the embarrassing orders of infinity. Whatever, a funny thing happened on the way to the theory, and right now I am seeing the interaction of GR with quantum spin to be the current item of business. Angular momentum, and then charge are the essential core of quantum theory, to my reckoning.

 

Look for the paper in my cache called "Kerr Electron", not the earlier "Electron Field Solution ..." I shall add results going up the z-axis, soon. I don't see anything surprising though at first I thought so. You have to analyze carefully where both parts of a fraction go to zero. . . . . . .time passes. . . . . . . . yes there is little change here, and this bespeaks the opening of the lotus when you expand the two inner lobes of the null surfaces beyond a>m. The field folds outward ending up in the extreme as I am trying to describe.

 

Now that I know what I am looking for I see some people do speak about the ring structure I have elicited: http://arxiv.org/abs/hep-th/0507109v1

 

I want to send my paper to A. Burinskii. Can anyone help find an email address for him?

 

Thanks to Severian for supplying an email for Burinskii. I hope he replies; he has done good work on this issue and is dissed by many. The Journal of Mathematical Physics is reading my paper. I don't think I have done anything really new but I have presented a clear discussion of the nature of the most amazing ring structure that previously I did not know anything about. One writeup I think used the term "heterotic string".

 

Burinskii answered and kicks me forward with one of his papers. My joy is extreme. Thank you again Severian.

 

He says yes, understand the coordinates and move on to the Kerr-schild complex solution. I wrote to him 'Yes the argument of the square root becomes negative!', like inside a black hole., where I ask is this not like "collapsed dimensions?" Where we assumed there was propagation of EM energy, here there is absorption as we see in the permittivity becoming imaginary.

Edited December 25, 2008 by Norman Albers
multiple post merged

[Norman Albers]

I am not greatly surprised that the Journal of Mathematical Physics decided after three days they do not need my paper. What I find not acceptable is their statement that this paper is 'not sufficiently mathematical'. I told them this is an offensive statement., but, more to the point, stupid. [The Russian JETP is still reading the paper.]

 

Mama, don't let your babies grow up... to be physicists. They'll... I need help here with lyrics. . . . . . . . . . .There are different realms of mathematics; there is a commonly enjoyed realm of decency, or not.

Edited December 28, 2008 by Norman Albers
multiple post merged

[Norman Albers]

I have never seen a singularity as strange and abrupt as that shown near the ring of z=0, r=a. . Inside, the physics of light propagation is rather "flat" except near the origin of r=0. It is like inside the eye of a hurricane. Examining this "wall" from the outside, light-speeds drop through zero, as near an event horizon (though not the same in details), but on a scale of the Schwarzchild dimension, 2m, near r=a, become imaginary, except that in the final step, become -1. This is the nature of the singular ring approaching from the outside, but there is an essentially discontinuous nature built into the solution. I am just reading of the work of Burinskii and others, who have indeed created a complexification of the mathematics, as is clearly called for! When I have something intelligent to say further, I shall.

 

MolecularEnergy, what do you know?

 

I worked out the details of approaching r=a from the outside and it is weird. The statement above about approaching unity is wrong. The radial form ends up being a small negative real quantity after coming down from an imaginary asymptote. This seems to me characteristic of the superconducting state, a negative reflection. Propagations in z simply blew up asymptotically in the real range. Those in [math]d\phi[/math] went through zero and limit at -i, in the imaginary range.

 

The small negative real residue of [math]dr/dx^0[/math] is equal to 2m/a and is thus roughly E-44. This is dimensionless, as [math]dx^0=cdt[/math]. This must be a fundamental quantity characteristic of the vacuum. Observe also that a superconducting ring is a quantum of magnetic flux. [Did someone say "circular currents"?]

Edited January 2, 2009 by Norman Albers
multiple post merged

[Norman Albers]

The Kerr solution is already "complexified" as I noted on 12/29/07. By applying the Eddington transform to the Schwarzschild metric we get a degenerate metric form. Running this through the GR field equations we see one result constrained by LaPlace's scalar equation: [math]\nabla^2 \phi=0[/math] . We know this has the solution 1/r, (as well as an arbitrary constant) and we usually center circles at the origin. In fact a circle at any center satisfies the equations; look at: [math]r^2=(x-a)^2+(y-b)^2+(z-c)^2[/math] where <a,b,c> are arbitrary constants. By allowing one of these constants to be imaginary, the equations, worked by Kerr in 1963, yield the angular momentum solution. His original form is interesting as it acknowledges [math]\rho[/math] but uses rectilinear coordinates: [math]ds^2=(dx^0)^2-(d\sigma)^2 -\frac {2m\rho} {\rho^4+a^2z^2}\left [dx^0+\frac{\rho}{a^2+\rho^2} (xdx+ydy) +\frac{a}{a^2+\rho^2}(ydx-xdy)+\frac z \rho dz\right]^2 [/math]. Try as you will there is some combination of cylindric and spheric coordinates needed; once I explored cylindric representation but nothing became essentially simpler. The coordinate transforms we often read are the forms necessary to create a fully axially symmetric form of the metric. In my paper I analyze locally the implications for apparent speed-of-light in the "external" Cartesian coordinates.

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Merged post follows:

Consecutive posts merged

After expressing my frustration on 12/27/08 I should have also said that JMP did recommend me to the Journal of Classical and Quantum Gravity. They are reading the paper also. If there is interest I shall take up the offer of my colleague solidspin to let him co-author introductory and other structural elements to a more complete treatise. I could not summarize a field I did not know existed three weeks ago.

Edited January 6, 2009 by Norman Albers
Wrong year. Time warps.

[Norman Albers]

What are we to make of a ring of E-13 (ten to the -13th power) meters, with a "hula hoop" thickess of only E-57 meters? Then too, sliced to eliminate the inner half! I am not used to considering physics below the Planck length. This could be the crunch point between two theoretic representations. Even if we cannot theorize vacuum physics down to the Schwarzschild dimension, we are still left with a radical physical geometry, a crinkle in the vacuum, which we can expect to have quantum properties of current and flux.

Edited January 11, 2009 by Norman Albers

[Norman Albers]

After three weeks the Russian JETP responds that my paper is not sufficiently urgent for them to publish. I certainly understand this! I've done better than re-discovering Saturn; this time I parachuted within the past forty or so years. I am eager to learn what sense people make of these faerie rings, and I shall hopefully produce a vacuum polarizability thesis worthy of all this. I thanked the journal for considering my paper. {{Wouldn't you know it, John Wheeler's name is associated with "geons". Burinskii's early paper uses this term in the context I share. I am trying to find the lay of the 4-space territory. Wheeler speaks of grav. waves, and Burinskii of rings. }}

Edited January 16, 2009 by Norman Albers

[iNow]

Keep at it, Norman, and good luck. Your enthusiasm is contagious. Unfortunately, now I think I need some penicillin.

 

 

Should I be worried if it burns when I pee?

[Norman Albers]

Have you been eating mushrooms? Little circles of them?¿?¿ [bIGTEETH]

Edited January 16, 2009 by Norman Albers