Norman Albers

"Boiling the vacuum" he called it. He retired as a QM professor. . . . . . . . On a different note I am looking at a paper of the Astronomical Society of Australia by Qinghuan Luo and R.J.Protheroe , Vol.15, #2, called "Pair production by photons in the field of the nucleus" and this sounds like what Swansont said. These authors are looking at polar caps of pulsars: "Resonant inverse Compton scattering above polar caps... for young pulsars". I quote the text: "Electron positron pair cascades can be started by accelerated ions in thermal photon fields. A photon can decay into an electron positron pair in the Coulomb field of a nucleus when the center of momentum frame energy exceeds the rest mass of the numcleus plus two electrons... This condition can be satisfied for soft photons from the polar cap in the presence of accelerated ions..." It seems to me they include the strong magnetic field lines from the pole and these are confining the ions. Check out the specs in the discussion: star surface temps are 5 million Kelvins and 100 million for the caps. The B-field is estimated at 3 x 10^9 Teslas, whoa Betty. solidspin could really do some analysis with such a magnet! Cameron, wherever there are such large amounts of energy sloshing around, different forms have probabilities of manifesting. First we know we have to satisfy conservation of momentum and of energy, as well of charge, in the possible outcomes.

Yah, that's one of the important points I have gained from these discussions. I may call them on their toll-free line and ask about corn. Are there other questions I might ask them?

I have not yet raised an email reply from Carmen Creek bison meat producers up in Canada. Go to their website and click on 'producers'. What are they saying as feeding rules: "...feed a grain-based diet for at least 100 days prior to slaughter"? Is corn a grain?

Yes but at Stanford a handful of years ago Seb Doniach told me they were cross-aiming duelling x-ray lasers from the two storage rings, to actually "cook the vacuum" and create particle pairs. I've not kept tabs on this.

The only previous encounter I've had with quaternions was reading Doug Sweetser on another distant forum in a faraway galaxy. If I recall correctly he rejected their usefulness, but my exchanges there were truncated.

Very cool, scalbers. The first ref. refers to 'gauge phases' ; the second and third are telling me I'm on the right mathematic track. I have been steadily and slowly advancing on my thesis described more down in the Mathematics Linear Algebra section, of a complete first-order complexification connecting GR and its representation of angular momentum, with quantum theory. http://www.scienceforums.net/forum/showthread.php?p=489040#post489040

Here is the system of equations I'm looking at. <A,B,C> are the <T_00,T_11,T_01> of the 2x2 essence of the Schwarzschild solution, taken through the Eddington coordinate transform. I have defined [math]S=1-2m/r[/math] and wish to solve for <A,B,C> in terms of S:.................................................... <0,0>: [math]-A^2(S-2)~+~2AC(S-1)~-~C^2S~=~S[/math] ............................................................. <0,1>:[math]~AB(S-1)~-~AC(S-2)~+~C^2(S-1)~-~SBC~=~S-1[/math] ......................................<1,1>:[math]~-C^2(S-2)~+~2BC(S-1)~-~SB^2~=~S-2[/math] . . . . . . . . . . . . . I would appreciate analytic clues. Merged post follows: Consecutive posts mergedThere is another equation generated by knowing the determinant of the system. I know I can start with Cartesian coordinates of the Schw. solution and generate my T matrix. Then I can apply the Eddington coordinate transform to get the answer needed. The determinant stays at +1 for these (as opposed to the determinant of either metric). Thus I may say that: [math]AB~-~C^2~=~1[/math]. It seems this does not add new information, though, since I can substitute it into one of the first three equations and produce one of the others. It seems to me this is a rank-3 system. Three matrices are needed. When I solved the simpler Cartesian case there did appear a degeneracy expressed as a floating ratio of A, B, meaning that their ratio is constrained.

I am working in the Eddington off-diagonal form of the Scwarzschild solution and trying the generate the tensor T. I actually have the answer but I cannot yet generate it "raw from scratch". I got it by tensor coordinate transform of my answer in the original Schwarschild coordinates. I should be able to generate it by just hardballing and solving three equations in three variables. We need look only at the 2x2 matrices of the time and radial variables, so just say T_00 = A, T_11 = B, and T_01 = C. I invert the Eddington metric matrix to raise an index, and then make the tensor product I mentioned above. This should equal the metric tensor of the system. The problem is I generate three equations which are quadratic in the three variables, <A,B,C>. What is the analytic nature of such a system; I don't think I've looked at such before. The solution from first approach does indeed satisfy these equation constraints, but there is perhaps more to understand.

solidspin, as usual some of what you said is now sinking in, a week later. I am succeeding in working out the form of the proposed tensor T in the Schwarzschild case. It seems to me now that I was not quite understanding unitarity, and welcome advice. The Kronecker delta is the form of the metric tensor of mixed indices, so you lower the upper index and get the statement I am working with: [math] g_{ab}= T_{ac}{T^c}_b [/math]. It seems I'm working toward a paper on this, stay tuned. I tried starting with the Eddington transformed coordinates and the algebra got nasty. I got a good result in the Schwarzschild coordinates and it is not difficult to run that tensor through this coord. t-form. It is quite cool that starting with a diagonal Schwarzschild metric tensor, I get <0,1> off-diagonals in T, without any implication of the Eddington time transform. All this without angular momentum, seemingly. There is a strong hint when we allow the time coordinate to shift with radius in the Eddington transform. This is the mathematic step we take on the way to the degenerate metric and the Kerr AM solution... . . This seems to be a two-step polka... . . Merged post follows: Consecutive posts mergedThis morning as I got to work it seemed that the exciting result of the off-diagonal terms from simply the Schwarzschild form had evaporated. My, my, that was disappointing, but hey, the job is to find truth. After another long day's work, now I can see that there is degeneracy in that the product of the <0,0> and <1,1>. If we define: [math]S=1-2m/r[/math], the quotient must equal [math]T_{00}/T_{11}=S^2[/math] but they are not otherwise determined. First of all observe that always here the signs of the two terms are similar. If you choose the diagonal to be <S, 1/S>, then, in this particular case, the off-diagonal term [math] T_{01}[/math] vanishes. There is nothing new here compared with the metric tensor itself, except the signature of the diagonals. However, we can choose, say, [math] <S^2, 1>[/math], and get a much more entertaining result with off-diagonals. This is not so exciting in the mass-only Schwarzschild form. Once this level is analytically clear, it will be easier to move to expressions of AM from the Kerr metric.

Yes we all experience 'now' but we are not the same being as when we experienced a past time. The wag may say, thank goodness.

One might well ask why should I care about short-lived energy/particle states. They fall apart so why the big deal. Well, there are indeed physical regimes where these are the relevant alphabits, the constituents of the soup of possibilities.

What's the big deal starting at rest, D H ? Yes we work in a center of mass system so momenta are equal and opposite. Shadow, I rebuild acoustic pianos. I try to not need to move them.

This sounds like a "retarded time" issue as we do in electrodynamics. As long as you are not moving at a huge relativistic velocity, say [math] \beta[/math] greater than one-tenth, you'll get a close answer.

Do you know the difference between a pigeon and an investment banker? The pigeon can still put down a deposit on a BMW.

Tree, the question I am looking at is the relevance of quaternions to angular momentum formulation in QM.

My brother wrote this to me yesterday: "While we are thinking orthogonally, how about something like a second time dimension? Does that fit off of your complex plane? I saw Brian Greene give a talk the other day and he mentioned (often humorously) the case for string theory and how it could resolve incompatibilities to help GR work on quantum spatial scales. In response to a question, he did acknowledge that string theory has so far been applied more to concepts of space rather than of time. If you think about it if you want to supersede GR you may need to have an improved understanding of time. So he reminded me that in considering multiple dimensions, one could possibly have multiple time dimensions. In general though he suggested that future progress in string theory may well be in the direction of time."

Life's a bitch and then we die.

In allowing a complex addition to a spatial dimension, like [math] i\hbar \kappa /c^3[/math], we let GR cook up angular momentum, in this case on the quantum scale. What might we get doing this to [math]dx^0[/math], the time dimension? . . . . .[i use [i]kappa[/i] for grav. const.] Merged post follows: Consecutive posts mergedI made a misstatement and actually wrote the form of the geometric angular momentum, which is the constant a times the geometric mass, or Schwarzschild term: [math] ma= \kappa J/c^3[/math] and [math]m=\kappa M/c^2[/math]. Thus [math]a=J/Mc[/math] and the imaginary term is: [math]i\hbar/Mc [/math].

Yowsa, that is a far-reaching question and I have stopped short of this for now. In the Kerr metric we get twisting of spacetime but real-valued metric terms. To reach for an answer, the constraint of the tensor T, where I posit physics, is that it is unitary, and also Hermitian. This makes me look for expression of polar theoretics as we need for particles and electrodynamics. There must be a relation to the 3-space unit vector defined as part of the degenerate metric: [math]g_{ab}= \eta_{ab} + 2mk_a k_b [/math] where [math]\eta[/math] is the Lorentz flatspace: <1, -1, -1, -1> and there is a null 4-vector k such that: [math] k_a = k_0 <1, u_1, u_2, u_3> [/math]. For now I figure that the metric tensor is real-valued, as gravitation is a neutral field level, clear to us when large aggregates of neutral assemblies of charged particles gather. This still gets us into plenty of existential hot water, as with my interpretation of sideways nullspeed inside a black hole having an imaginary value. Yes I am deeply, and hopefully, productively confused and working.

Remember, in the dielectric model an increase of the polarizability to a value of 3 creates "dielectric runaway", or maybe an event horizon.

We and our detectors are atomic systems, interacting with light and mass. We find these experiences to be relative in different Lorentz frames, which is to say locally particles are as we know, but observing a distant locale, or one of high gravitational field, demands acknowledging metric transforms.

Time is oscillation.

Over in Physics threads I added to my 'Reissner-Nordstrom...' thread about the necessary complexification of variables introduced by angular momentum terms into GR. http://www.scienceforums.net/forum/showthread.php?p=484892#post484892 Again I thank D H for gifts which I will come to know (soon).

I figure you mathheads can characterize such things well in terms of symmetries and such. Given i, successive multiplications give us a rotation or toggling between <1, i, -1, -i>, and this is hugely useful as complex analysis. 'twould be cool if this could be extended to the fantasy numbers. Good one, Tree, this makes me want to learn what quaternions are. . . . .Having started reading Wiki on them, this is quite exciting, thank you!!! Is this Hamilton the Hamilton of mechanics and such? I see there was a Quaternion Society, I shall look it up, along with Phlogiston. Atheist, no problem, I've been called worse. Seriousness aside, I don't buy it when people say, here timelike intervals are spacelike and vice versa. This seems like a copout to me and so I have suggested we acknowledge that, say inside a black hole, radial propagation has a positive [math] [\frac {dr}{cdt}]^2 [/math], but not tangential, like [math] [\frac {rd\theta}{cdt}]^2. [/math] I am getting into complexifying GR here: http://www.scienceforums.net/forum/showthread.php?p=484892#post484892 .

I am pursuing the idea of a more fully complexified GR. I do not know yet what to say about the concept of a vector norm whose square is negative. At root, when Kerr solved the angular momentum (AM) problem, he took the simple 1/r solution of Laplace's equation and let z be offset by an imaginary constant. This brilliantly clever move might seem innocuous but it is more than just adding an imaginary constant. Now the expression for radius is: [math] r= \sqrt{x^2+y^2+(z+ia)^2} [/math]. The kicker comes when you differentiate by z: [math] \partial r =(z+ia)\partial z/r [/math]. Aha, the spatial differential has a complex coefficient now! This is the sort of system I am trying to formulate.