Norman Albers

Thanks, D H, I would not have figured that out. Also, I didn't know you need not give brackets after \frac , as the space delineates the argument change. Correct?

How can I fly with eagles when I'm penned with turkeys? No offense to contributing friends.

Help! How do you compose the vertical line evaluating an integral at limits?

I was told CRT's wear out, but don't know if it's phosphors, or maybe gas?

WHAT, ME, WORRY?? I am a bit confused about units so here's how I proceed: [math] L=\int_{2m}^r dr/\sqrt{(1-2m/r) } =\int_{2m}^r dr \sqrt{r \over {r-2m} }[/math], where the upper limit is "far away" . Rescale according to: R=r/2m, and write: [math]L=2m\int_1^R dR \sqrt{R\over {R-1}} = 2m \int_1^R dR \frac{R}{ \sqrt{R^2-R}} [/math]. Add and subtract to make a perfect differential: [math] L=m \int_1^R dR \left( \frac {2R-1}{\sqrt{R^2-R}}+\frac{1} {\sqrt{R^2-R}}\right) = 2m\sqrt{R^2-R}|_1^R~~ +m\int_1^R {dR \over{\sqrt{R^2-R}} } [/math]. The latter integral from CRC tables, is:[math] m log(2\sqrt{R^2-R} +2R-1)|_1^R [/math]. We can see that at large R the expression tends to 2mR; at R=1, the event horizon, the first term is zero, and the argument of the logarithm becomes 1 and so the log is zero. This says the total distance covered by a string will be that extra 2m, compared with the external coordinate measure by which we are observing. This is a pretty fun result. I get messed up if I don't do the rescaling first, as I don't understand what goes into the log argument, but I trust this solution, as unitized above, better. Satisfyingly, the answer to the scaling problem here came to me at 3am. It made me nervous that you come up with [math]log 2m [/math] if you don't unitize things by rescaling. HOWEVER, this only amounts to a constant of integration, so the problem goes away. Hand-waving and cheering. We'll see if BenTheMan buys this as a valid case (of hand-waving).

Limits of evaluation: [math] I®\vline _1^R [/math]. Larger, [math] \frac{F}{G}|_1^R[/math]. Nope, how do we do "evaluated at"?

SHUT UP.

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.

Losfomot I appreciate you hazarding an answer. I get a finite result integrating from afar, into and even including r=2m. This says that locally a measurement of small distance change is [math]\sqrt{-g_{11}} dr [/math], "proper distance". This is, in the Schwarzschild metric, integrated as: [math] \int_{2m}^{r_{far}} (1-2m/r)^{-1/2} dr [/math]. Also a further note to Spyman, that I think I can transform the result I described, where my expression for gravitational acceleration, [math] d^2r/dt^2[/math] went to zero. There is one order of differential in distance to be stated as proper distance, or multiplied by the square root above, and then also the denominator has two orders of time differential, which both transform to proper time with multiplication by [math]\sqrt{g_{00}}[/math]. So you gain three factors of the square root and that changes the initial expression in flat-space coordinates, which goes to zero, into an expression in proper coordinates which blows up with the square root of the zero in the denominator. This is the result in your reference. P.S. I did need integral tables to analyze the seemingly embarrassing infinity.

I have posed a valid and good question and await answers.

Disappointing? Sounds like great material, to me. Give the wheel some sort of Maxwell's demon so it keeps pulling energy from space... Seriously, I am not sure of the implications and want other people's comments. Then again, check out neutron stars, or even the electron as I model it. You really have to come up with some attitude to create strong circulations.

OK I'll bite on talking about distance of circumference. It will be measured as longer than [math]2\pi r[/math] by someone on a relativistic wheel. This seems to say to me that tidal stress has torn things apart, unless we built with elastic materials...Thwwppp.

Fast enough so [math]\omega r [/math] approaches c.

I argue that acceleration is not what is being figured here. Consider that the rim velocity is [math]\omega r[/math] but that acceleration is [math]\omega^2 r[/math]. I may keep the first product constant but work with a larger and larger wheel to reduce the acceleration to an arbitrarily small value. You'd like to think that acceleration is equivalent to gravitation and so we are doing general relativity, but this is not the case. I would like to know the answer to characterizing a smaller and smaller wheel... Running some ballpark numbers on the Wheel of Doom, what sort of rim velocity, or [math]\gamma[/math] would it take to get near creating a black hole from the increasing relativistic mass at fixed (you hope) radius? Schwarzschild radius is [math]m=GM/c^2[/math] where 'm' has units of length and 'M' units of mass. 'G' is the grav. constant. We are, in this circumstance, realizing larger and larger [math]M=\gamma M_o[/math]. We ask how much relativistic mass is necessary to reach BH intensity at our chosen radius and original construction mass. The factor [math]c^2/G[/math] is about [math]10^{27}[/math], so I can write: [math]\gamma M_0 = 10^{27} R[/math]. If our radius is a kilometer and the rim mass is a million kilograms, then solve: [math]\gamma \times 10^6 = 10^{27} \times 10^3 [/math], or [math]\gamma = 10^{24}[/math], rather high. We don't have to go here to see strongly relativistic effects, like a [math]\gamma=10[/math].

Elas, be careful in characterizing my photon study. It is electrodynamics and satisfies Maxwell's equations assuming that the vacuum manifests a superconducting-like response to changing A-fields. Or to put things more accurately, I entered assuming the existence of a wave packet with Gaussian falloff, and showed that the math says: [math]j_i=(-\lambda^2+\rho/U)A_i[/math]. I don't yet understand your ideas of process but it seems you are characterizing particle sizes by energy, and this is fairly comfortable as a DeBroglie wavelength, no?

I come up with an integrable expression if I integrate, from some far point, in to r=2m, the differential for proper distance: [math] dr_p=\sqrt{-g_{11}}dr[/math]. I'm not telling my answer until someone else sticks their neck out to say if this is correct thinking.

Beautiful paper, Elas. You dare to say Nature is unified. I have not yet touched on hadrons and <m=3> interpretations of loops, but I have found expression of electrons, photons, and gravitation from a common interpretation of the vacuum "availability", as I like to call it. Do you produce transverse localization of photons, as I have?

Severian, say more please. I am not well educated here yet. Elas, I worked as a student on an experiment team with the Brookhaven accelerator beam at 3 GEV. We smashed protons into protons (I think in carbon nuclei). I'll have to look up the details, but to look at the lifetime of one of the resultants, it would have been expected to decay in a very short distance even moving near c. However, it flew many meters and produced an event in our wall of neutron counters. Time dilation is given by: [math] d\tau = \sqrt{1-\beta^2}dt [/math] where t measures lab time and [math]\tau[/math] the particle frame, "along for the ride". [math]\beta[/math] is v/c. . . . . . . . .time passes . . . I started reading the lengthy wiki entry on 'renormalization'. My, my, it all hits the fan here. There is the Lagrangian I have been wondering about, and we are talking basic electron nature and virtual fields. These are things on which Singularities-R-Us has offered more than a little perspective. Speaking as Principal Instigator, I see the virtual field (as does H.Puthoff) as "loosely" an electron-positron plasma. Electrons may be seen as similar to plasma instabilities, only they are stable. This in the context of uncertainty, which points to their nature as a self-consistent process where virtual e-p pairs can be seen as tending in response to a point-like field to manifest skewing of their populations. A divergence of polarization is to be seen as charge density. The field I present in my electron study is only the first and handiest mathematical structure. In fact I have over the last few days, realized a different version characterized by a Gaussian falloff of nearfield. I chose to reject this when I worked 3-4 years ago because the [math]e^{-r}[/math] looks "more pointy". However, the squared exponent falls off faster, with a broader center. It works equally well in terms of cancelling the embarrassing orders of infinity. My colleague working on non-local QFT said he needed an inflection, so I found one. Quantum mechanics gives me a virtual field of "electron-positron pairs 'popping in and out of existence'". Though this phrase drives me up the wall, I can work with it since it manifests on the scale of Planck length and time. Think about dipole pairs in a very brief existence, given a radial E-field and central B-field. What will tend to happen to those randomly offered in either radial or transverse orientation? How will they tend to move? Given this sort of dynamic disposition we can see the monopole as nothing but an illusion of dipoles.

Profitable in that sense yes. You caught me in a cynical mood. Farsight is into beans, ask him. Then again, just what sort of magic beans?

Intriguing you damn will betcha; profitable, HORSEFEATHERS.

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.

Nice, Elas. I have to think about what is implied here. Time dilation is simply so. What is the relation of the muon to the field?

Two atoms are walking down the street and one falls down and loses an electron. It gets back up and the other one asks, "Are you OK?" "Yes, I'm positive."

Elas,, I want to read your treatment of the anomalous magnetic moment, and am downloading it.(the difference of the gyromagnetic moment from exactly two.) This is nice and is what is always touted as the greatest success of QED. Dirac called this "essentially coincidence". In 1977 he said, "People have done an enormous amount of work with the quantum electrodynamics, as it is called. They have noticed that, although attempts to solve the wave equation always lead to infinities; those infinities can be managed in a certain way. In particular, it was shown by Lamb that the infinities could be removed by a process of renormalization. This means that you assume that your parameters e and m occurring in the original equations are not the same as the physically observed quantities. In general the idea of renormaliization is quite physically sensible, but the way it is applied here is not sensible, because the factor connecting the original parameters with the new ones is inifinitely great. It is then not a mathematically sensible process at all. But still, people have worked with it, in particular Lamb. The surprising thing is that with the infinities discarded by these aritificiall renormalization rules, you get results in agreement with observation... to a very high degree of accuracy. Most physicists are very satisfied with that result. They say that all a physicist needs is to have some theory giving results in agreement with observation. I say this is not all a physicist needs. A physicist needs that his equations should be mathematically sound, that in working with these he should not neglect quantities unless they are small. Well, here again I find myself in disagreement with the great body of theoretical physicists. They are complacent about the difficulties of the QED, and I feel that kind of complacency is similar to that which people at one time had with the original Klein-Gordon equation. It is a complacency which blocks further progress. Any substantial further progress, I feel, must come from some drastic changes in the basic equations. Just where they should be I do not know but I feel this change will be rather similar to the changes the Heisenberg introduced in 1925 [non-commuting algebra]. It is a change which people will probably come to eventually only by an indirect route. The only feature of the new theory which one can be sure of is that it must be based on sound and beautiful mathmatics." Also, folks, read earlier in the paper: "One can go to 4x4 matrices [from the Pauli 2x2 matrices] and then one can easily get an expression for the square root of the sum of four squares. This led me to a new equation involving these 4x4 matrices... One can modify this equation to bring in the electromagnetic field in the same way that Schroedinger brought it to the de Broglie equation. The result is an equation for the electron moving in the EM field, in agreement with the basic requirements of relativity and quantum mechanics. It was found that this equation gave the particle a spin of half a quantum. And also gave its magnetic moment. It gave just the properties that one needed for an electron. That was really an unexpected bonus for me, completely unexpected." So, folks, you will find me in my dunce corner, meditating with dead guys. What is going round and round, Elas?

I'm so sorry I wasted my time. The physics which demands to be seen is just relativity: an energy density which is convinced to circulate is indistinguishable from mass. The relativistic mass of the DOOMSDAY WHEEL increases arbitrarily, depending upon your energy input budget. (I keep searching for my thread on the Thermonuclear Box of Manure.) Too bad if you don't know what I mean. Too much has been erased.