Reissner-Nordstrom-? metric

[Norman Albers]

Consider the Reissner-Nordstrom GR metric solution for a charge mass-point. This is a free-space solution assuming those two characteristics are "vanishingly small" as to the region of divergence. We see the expected term in m/r; also we see the electric term in G/r^2, which we may interpret as the gravitation effect of the homogeneous (r^-2) electric field energy density. Thus people say it is not possible to understand mass! I assume a near-field inhomogeneity which is not bound to this relationship, and I am working to show an inhomogeneous solution where such a field produces the 1/r term.

[Norman Albers]

I mean to say here that homogeneous field terms do not supply enough OOMPH to constitute mass in the far gravitational sense. I suspect I might solve things differently assuming the point of DIVERGENCE is not so. This, simply , is where we swept the problem under the rug. Did you happen to notice the little low-lying cloud of dust when you walked into the room? The devil is in the divergence.

[Norman Albers]

I have achieved the solution of this metric assuming the addition of the inhomogenous near-field I have discussed, (r^-2 + r^-1)e^-r. Just as this approach relieves the infinities at the origin of the electric field, so is the case with the metric. The "mass" term of 2m/r is produced equally well by my inhomogeneous representation.

[Norman Albers]

I have achieved a few points of clarity in this solution. The association of the inverse-square (in radius) term in the metric as electromagnetic energy density of the field is bogus. This has confused many discussions, and one should not so easily relate different quantities. When you posit all the divergence of an electron to the center point, you get a theory showing that. Whether or not you do that, or allow it to be spread out a bit as I have, you get whatever variation of the near field, blending into the far field, reflected as a variation on the 1/r^2 term previously taken to solve the metric. If you assumed a point, the field energy you calculate as a function of radius becomes infinte near zero, clearly, because the model is not adequate. It is stupid to associate the integral of the field energy of an r^-2 field with the source, because you can only integrate the energy outside of your radius. The integral blows up at the origin and is not defined. THEREFORE, it is not reasonable to say the original solution shows mass to be something separate from the energy in the field. The metric solutions offer possible physical responses of both first and second order, though the second-order term vanishes for my assumptions, near the origin. I started to freak out when I thought the sign of my near-field term in 1/r might cancel the -2m/r, but happily this in not the case; it reinforces and is of the same magnitude, assuming roughly classical radius where the farfield yields.

[Norman Albers]

Perhaps it is useful to say that gravitational and electromagnetic disturbances are the first and second multipole orders of a common neutral field.

[Norman Albers]

Both gravitation and charged particles involve increases in vacuum polarizability. Electric fields involve divergence of the polarization state of this field, and this includes, or may be a thinning of radially-oriented dipoles which oppose the sense of the net field.

[Farsight]

I'm listening, Albers.

[Norman Albers]

Consider the electric field in this model: [math]E=-r^{-2} + (r^{-2}+r^{-1)}e^{-r}[/math]. The first term may be thought of as a divergenceless polarization. In the far field this is all there is, effectively. In the near field it is balanced by the opposite, diverging polarization. In the limit at the origin they cancel in orders [math]<r^{-2}, r^{-1}>[/math] so there is a finite E-field, namely -1/2. Remember that a polarization reflects a net field of oriented dipoles, and in both the positive and negative parts identified separately here there must be strong [math]r^{-2}[/math] behavior at the origin. This can be associated with an increase of the polarizability of the vacuum to a value of 3 as per dielectric theory. The relatively smaller value of the inhomogeneous part can be linked to favored annihilation of that population orientation. If you think about the fate of a randomly oriented dipole introduced here, there will be some rotation regardless, and I have not done specific analysis here, but the radial population will be biased as the oppositely-oriented dipoles are repelled and speeded to annihilation by virtue of being drawn closer together. At this point one needs a statistics of the vacuum.

[Farsight]

Sometimes I wonder if that polarizability of the vacuum is what space is, Albers. The vacuum would truly be nothing without it.

[Norman Albers]

There must be some way out of here, said the joker to the thief... Good one. Farsight. As far as I know, cosmologies have no outside It does seem that no one gets out alive. . . . . time passes...... multiple cosmologies have separate spaces, I guess.

[Norman Albers]

One should distinguish discussions of small Reissner-Nordstrom systems, such as my electron theory, from listings to be seen on Google for R-N black holes. This is an important characteristic of the metric; for the mass of an electron, the Schwarzschild radius is something like twenty magnitudes smaller than the Planck length. Thus I call them degenerate horizons.

[WaTTo]

Hi, I'm a brazilian graduating student and my current 'research' is related to this topic.. So I think we got commom business =).

 

I recently studied (sort of 'deeply') Klein's original paper (1926) and I'm trying to find Reissner-Nordstrom metric arising from such formalism by my own (not original , but still a challenge for my begginer profile).

 

I'm going to keep reading this topic and maybe posting some doubts (not stupid doubts, I promisse).

 

Nice work Norman =).

 

(contact: domingos_salazar@hotmail.com - please, i NEED to talk to other physicists.. ordinary chat and 'existencial' stuff +_+.. email me =D )

[Norman Albers]

WaTTo, nice to hear from you. This stimulates me to finish doing the magnetic part. I am wondering about the simple original choice of the metric form. It should become clear if this is not consistent. FURTHER OPEN QUESTION: I am informed that deep inelastic scattering gets us to the realm of effectively [math]10^{-21}[/math] meters (correspondende with H.Puthoff). From this perspective the electron is still "pointlike", and in fact we experience an effective fine structure constant maybe 5% or so higher. How does this match with considerations of electromagnetic field energy densities? Integrated from infinity to the classical radius, we already have mass-energy in the range of a half-MEV. The integral is simply equal to 1/r, evaluated wherever you think the physics holds. This relationship must yield at smaller magnitudes, where we are deep into inhomogenous physics.

[Norman Albers]

Five frustrating days and many pieces of paper with matrices later, this is starting to make sense. I don't have examples of magnetic terms in spherical form being treated in the Minkowski tensor and it is a disaster if you don't follow the correct formalisms. I think it is coming out nicely like my spread-out electric field did: there is another near-field term in the current equation, and a metric coefficient multiplies the current term on the RHS. More when I convince myself this is correctly done. Is anyone out there experienced in contravariant representations?

[Norman Albers]

FINALLY I SEE A PATH, and have the current eq. I knew had to come out. I am entering with a contravariant 4-vector potential, and the Minkowski field tensor is mixed. I'm not making it at the moment in the LaTex, so I'll say that if we define that tensor in terms of covariant, not simple derivatives, we come out with the extra terms such as needed to describe curl in spheric coordinates. Now I may proceed to work out the altered metric equations with the magnetic terms.

[Norman Albers]

I am working from the wikipedia page "Maxwell's equations in curved spacetime" and would appreciate someone explaining the partial differential operator with a superscript, seen in the definition of the contravariant Minkowski field tensor, [math]F^{ab}[/math]. Good, [math] is back.

[Norman Albers]

Watto, what regimes are you studying? I have worked only in the small in the electron context. Are you doing massive bodies and quantum mechanics? Can you tell me about the regimes available and useful that are being investigated?

[Norman Albers]

When I do electrodynamic accounting I consider the energy densities of both source and field terms. In a closed system they will be equal, but not distributed equally, in their totals. It turns out it is not necessary to deal with this upon entering the Einstein field equations as I have with my electron nearfield, and this is an important fundamental statement. I am dealing with the alleviation of "point sources"; in the inhomogeneous E&M representations I use I may alternately consider densities as the square of electric or magnetic fields, or else as source terms like [math]\rho U[/math]. However, the gravitational field equations are written as functions of the "stress-energy tensor", fundamentally as the fields rather than sources. This is important in the general understanding of what we are structuring into which theories, as well as telling me that I don't need to process a charge/potential term through the R-N metric. Thus the observations offered by my electric field substitution cancelling the gravitationally repulsive nature of the metric field may stand.

[Norman Albers]

I have finally found how to represent the magnetic vector potential in spheric coordinates. The Minkowski tensor cannot be simply constructed in terms of [math] A_{r,\theta,\phi}[/math]. It is an expression of the curl operation which is defined only on covariant vectors, and the arbitrary A-vector does not qualify. One must construct a multiple of the usual A-vector with the local metric scale, which in spheric coordinates is a 3x3 matrix whose diagonal is [math]<1,r,rsin\theta>[/math].

[Farsight]

I have finally found how to represent the magnetic vector potential in spheric coordinates. The Minkowski tensor cannot be simply constructed in terms of [math] A_{r,\theta,\phi}[/math]. It is an expression of the curl operation which is defined only on covariant vectors, and the arbitrary A-vector does not qualify. One must construct a multiple of the usual A-vector with the local metric scale, which in spheric coordinates is a 3x3 matrix whose diagonal is [math]<1,r,rsin\theta>[/math].

 

I've written CHARGE EXPLAINED, Albers. I explain electric fields, magnetic fields, magnetic materials, superconductors, why the electron is stable, and why charge is not fundamental. But I'm not posting it on a public forum.

[Norman Albers]

Farsight, I'll contact you about reading your presentation. I have gone through the numbers of the Reissner-Nordstrom fields, over in the QM section, and it seems to me that the gravitational energy of all measured "particles", even in the GEV range, is insignificant compared to electric energies and thus they must be explained as electromagnetic resonances. I see charge as a nearfield of divergence which is always quantized outside of nuclear states.

[Norman Albers]

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.

[Norman Albers]

This time Nature is being more demanding and testing me. The diagonal terms in the stress-energy tensor contain [math]B_\theta^2, B_r^2[/math] which have [math]\theta[/math] dependencies, even in the far-field sum. Thus I must come up with a more involved metric tensor in the near field and the original Reissner-Nordtsrom form is not sufficient here. Furthermore there are off-diagonal terms which must be matched by further Christoffel symbols in the construction of the Einstein tensor on the LHS. You'll find me in my dunce corner. . . . . .I am wondering if I can approach with the Kerr-Newman metric. There we have an angular momentum term, and I certainly can identify that in my field. In fact this could be most interesting against the observation that the Minkowski tensor terms reflect only fields, whereas the mechanical stress-energy tensor part accounts for usually masses moving, with terms like [math]\sigma v_a v_b[/math]. Electromagnetically I can make a statement in sources, as opposed to fields, that momentum density is: [math]\rho A_\phi[/math] and that angular momentum is the appropriate projection of this. Is anyone out there hot on the Kerr-Newman metrics?

[Norman Albers]

I thought at first I am a victim of my own cartoon when I cranked the tensor machinery to get a trace of the contracted stress-energy tensor [math]{T^a}_b[/math]: it is perfectly nothing. After getting reassurance from my book (Adler, Bazin, Schiffer) that it is indeed zero, I am emboldened to offer for comments my rendering of [math]T_{ab}[/math] for the fields in my electron model, [math]E_r [/math] and [math]A_\phi[/math]: [math]T_{ab}=\left( \begin{matrix} T_{00} & 0 & 0 & -e^{-\lambda} rsin\theta E_rB_\theta \\0 & T_{11} & -rB_rB_\theta & 0\\ 0 & -rB_rB_{\theta }&T_{22} & 0\\ -e^{-\lambda}rsin\theta E_rB_{\theta} &0&0&T_{33} \end{matrix}\right) [/math], where the diagonals are: [math]\left(\begin{matrix} 2T_{00}= e^{-\lambda}E^2 + e^{\nu - \lambda}B_\theta^2 + e^{\nu}B_r^2 \\2T_{11}=-e^{-\nu}E^2 - e^{\lambda}B_r^2 + B_{\theta}^2 \\2T_{22}= e^{-(\nu+\lambda)}r^2E^2 - e^{-\lambda}(rB_{\theta})^2 + (rB_r)^2\\2T_{33} = e^{-(\nu+\lambda)}(rsin\theta)^2E^2+e^{-\lambda}(rB_{\theta})^2 + (rsin\theta B_r )^2 \end{matrix}\right) [/math] . All this assumes a diagonal form of the metric tensor as in the Schwarzschild solution, and I do not expect this will be adequate. It is useful, though, to ascertain the form of expressions to this point.

[Norman Albers]

I am considering the Kerr metric as the exterior solution to my electron problem, but with the limit taken for small mass but significant geometric angular momentum. When we allow both 'mass' and 'angular momentum' to be small, we get the expansion for the Lense-Thirring form. In particle scales, however, Schwarzschild radius is very small compared to geometric angular momentum (E-57 to E-13), so we can choose to ignore the terms involving 'm' and obtain a form describing metric anisotropies governed by rotation, and its expression in the vacuum. The regime is described as a~r>>2m.