Norman Albers

Very cool question as I am just at this looking into General Relativity. What do we do with the concept of the vector "norm" when [math]ds^2[/math] is negative? I'm trying to make sense of |ds|= i . Somebody brilliantly defined the square root of minus one. Can we define some sort of fantasy numbers? They'd need to be orthogonal to the complex plane...

It may be that if we know how to work with a given coordinate choice in GR, it should yield similar physics as any other. It seems though that choosing a possible form of the solution at the start is what can mathematically limit, or perhaps, illuminate our understanding. If you just look at the axially symmetric form of the Kerr solution, you can feel some satisfaction of geometric understanding, but in what coordinate space? I worked this back into external coordinates and am marvelling at the "faerie ring" singularity. On the other hand if you assume isotropic coordinates are appropriate in massive singularities, you may be missing much. They cannot map inside the event horizon.

B-B-Bingo! Good answers. [bIG TEETH]

Indeed the expression of [math] ds^2 = g_{ab} dx^a dx^b [/math] produces negative answers if we consider spacelike intervals. Thus a complex field is clearly implied. I am investigating [math] ds^a={T^a}_b dx^b.[/math] I can show that this tensor, [math]T_{ab}[/math], is Hermitian. <<Who was Herman, or Hermit?>> Merged post follows: Consecutive posts mergedIf anyone is aware of current work in this area I'd appreciate hearing. I am feeling my way in constructing this linear form, as opposed to GR usually speaking only of [math]ds^2[/math]. Right now I am using real Cartesian coordinates, [math]dx^a[/math], and allowing a complex [math]T_{ab}[/math] and thus, also ds as a complex 4-vector. Merged post follows: Consecutive posts mergedIn addition to having to be Hermitian, the matrix T must be unitary. Is this like saying it is a rotation in complex vector space? Merged post follows: Consecutive posts mergedCome to think of it my coordinates may or may not be Cartesian but they are those of an external, essentially flat Minkowski space. Merged post follows: Consecutive posts mergedSince I allow a complex vector ds, now [math]ds^2=g_{ab}ds^{*a} ds^b[/math] ((I hope.)) Merged post follows: Consecutive posts mergedNot a simple square root, I can say that: [math] g_{ab}= T_{ac} {T^c}_b [/math] Merged post follows: Consecutive posts mergedIt is not clear to me that this last statement adds information, since we must use the metric tensor to raise the index of the second T. Merged post follows: Consecutive posts mergedWhoa, I read of mathematician Charles Hermite. Cool.

Can you find a complex number which, when multiplied by its complex conjugate, is -1? Today, I tried to find one.

Thanks, ajb. I work in the zone between geometry and algebra, between knowledge and imagination, light and dark. Do not try to adjust your monitor...

Perhaps more to the point, which symbol is used more these days for c.c., the asterisk or a bar-over?

Sounds a little like quantum foam, not that I know much of theory.

In post #36 I made a statement about the implications of the Kerr low-mass AM solution that I now see as incorrect. The GR metric terms do not become imaginary. They do become negative; furthermore, if you solve for null geodesics, they have terms stirred by the AM to complex nullspeeds, but is not a property of the original metric tensor. Merged post follows: Consecutive posts mergedThe singular ring analyzed in the original coordinates introduces a quadratic equation when you look at radial displacements. Thus, when we set ds=0, there is not yet a direct relation between cdt and dr.

solidspin is mentioning hyperbolic and elliptic coordinates so you are hot, kleinwolf. He says the former manifests the latter in 3-D... . Merged post follows: Consecutive posts mergedI think my adventurous friend solidspin has folded a spatial dimension into the hyperbolic tangent, which is a coordinate mapping that sort of brings far things near... in the far field it approaches unity.

I seek it once in a while. I shall try further to investigate the farm feeding of these creatures.

Yah, no, and all the above. I am not analyzing massive things. Yes you speak well of the "outer lotus" . Mass is not much in my considerations. Let the inner lotus open as you allow a system with more Angular Momentum than total energy. I guess life is a polka. So did physicists in the early 1900's. We dance the Planck Polka.

I have gotten clear on the analytic properties of the coordinates in which the Kerr GR solution is expressed, vis-a-vis my near-field nullspeed statements. We already knew that [math]ds^2[/math] could be negative on spacelike intervals. When you start looking at nullspeeds, or [math]dx^a/dx^o[/math] the range goes into the complex-imaginary. This does not mean that the metric tensor does this, and indeed it is real-valued regardless of the coordinate transforms discussed here. It is the multiple, mixed-term entries which begin from the Eddington transform for the time variable, that produce complex solutions in the Kerr. Approaching the singular ring from the outside r=a+, z=0, the value of [math]g_{oo}[/math] goes negative, which by itself would demand an imaginary nullspeed. However there are also mixed tensor terms in [math]g_{01}[/math] and this makes necessary a quadratic solution and complex expression for radial nullspeed. It might seem at first that this messes up our need for real [math]ds^2[/math] but it does not. Following the rules for whichever coordinate system you are in, and adding the complete bilinear sum, consistency follows.

I am working in OpenOffice and this didn't cut it. It's looking for a multiplicand. Merged post follows: Consecutive posts mergedI got into the OpenOffice forum and the answer is: "*" in the superscript.

I very much appreciate this discussion. I may not agree with, say, Moontanman's attitudes but I understand and respect the thinking behind them. It is important to think on these things to establish my own center from which to react, or not.

I am not quite sure where the discussion is, but one of the most surprising things I ever read was in Dirac's 'little black book' on quantum theory. He said, we measure electrons always with the same charge and the same mass. We have no idea why these quantities are thus, but we accept these as eigenvalues.

Hindsight is not allowed, though it also seems it may be necessary for wave-fields with non-locality.

I'm sorry. I shall promote something better.

Thank you sincerely, Moontanman. Do you have much of a moontan??? OK, I have lived through an era of abuse of government power. I think that an armed populace will pose an unacceptable opposing conflict, yes.

Moontanman, I do not know what to say entirely, but I have deep distrust of power in the large, and respect the need for individuals to be strong when things get extreme. I feel your discussion is not facing this.

Why was it possible that 6,000 men were massacred in Sc`rebrenika, in the Croatian Wars in the '90's? I was recently at a neighbor's barbecue where two guys were talking of the assault rifles they owned. I asked if they had shot in the military and they said no.

Fools do not know that they do not know what they do not know.

What's the code for complex conjugate, you know, something "star"?

I had a neighbor, a tall hardworking good fellow. They restored an old house at the bottom of the road. One day last year he caught a thief in the dining room!!! Drawers were open, etc., so Tom yelled "Get out!" The guy leaped up, crashing his head on an open drawer and drawing considerable blood. He bolted for the back door which was locked. Tom did not attack him and he ran out the front door, got in his car to leave. Tom shouldered his rifle and shot right through both rear doors of his car. Goddamned nice response I thought.

This is, for this old dog, new tricks. I've not before encountered these representational possibilities. . . . . . . .Looking in Wikipedia, 'tis a cool way to map the plane. I'll hang out with it to get some feel.