An Extended Gravitoelectronic Equivalence Principle

[sb635]

This topic is both gravitational and electronic in nature. It could easily be in the relativity forums, but also has significance (maybe) in the electronic world of QM. Einstein described his gravitational EP by envisioning a "man floating in an elevator." The man is at first floating because no accelerations are "present" inside the elevator.The man has a cop's radar gun, and at first, the radar does not register any Doppler shifts off the interior walls of the elevator. They "look" stationary, and do not appear to be moving either towards or away, because the radar sees no Doppler shifts. Suddenly, the radar does register a Doppler shift, and one of the walls appears to be moving towards the radar. One side is "blue shifted" and the opposite side is "red shifted." Some relative motion has been physically detected by the radar.

The man is a good physicist, and ponders the "truth" behind the observations. He considers Einstein's quote:

"The belief in an external world, independent of the perceiving subject, is the basis of all natural science."

He realizes he is Einstein's "independent perceiving subject," and his radar observations, because they are observationally independent (isolated) from the "external world," will never be able to discern the actual "physical truth" of Einstein's "external world." It may have been that the exterior walls have rocket engines attached (that run *real* quiet <g>) and the walls actually, physically moved towards and away from the radar. Or maybe when the radar gun showed Doppler shifts, the elevator was actually sitting on the surface of a large solid spherical mass. The man understands that he and his radar gun have attractive mass, and then, it was actually he and the radar gun that physically experienced a "gravitationally-field-induced" acceleration towards and away from the interior walls of the elevator. Or even, the observed relative motion resulted from some combination of both of the two fundamental types of accelerations in the Universe: inertial and field induced.

To me, there is an extremely important fact to walk away with here: There is always some assumed physical motion in Einstein's "external world" responsible for the observed nonzero relative motion. Otherwise, what was Einstein referring to by an "external world" that is "independent of the perceiving subject"?

Now assume the floating man inside the elevator is electronically charged, either positive or negative, it does not matter. Assume the walls of the elevator are neutral, and do not electronically interact with the man and his also neutral radar gun. (But he is charged, and grasping the gun.) The radar gun sees a Doppler shift, indicating some "physical motion" has occurred, and he ponders what it was. Like before, perhaps some exterior electronically-neutral rocket engines inertially accelerated the walls. He is charged, but they are neutral, and the observed motion is not from electronics. Or perhaps, just as before, some big gigantic neutral mass is responsible. The man still has mass, so the neutral radar gun cannot tell. It could have been all gravitationally-induced motion, given the exterior "external world" is electronically neutral. What if it is not? Imagine a big "massless" but charged, exterior body. It will pull (or push, it doesn't matter) the charged man, just like gravity, and produce the observed Doppler shifts. Or maybe the observations are the result of a combination of all three: inertially-induced (rocket engine) accelerations, gravitational-field induced accelerations, or electronic-field induced accelerations. Assuming the interior body is charge, interior observations can't tell the difference between gravitational, electrical, or inertial accelerations. Or any combinations thereof. Hence electricity has been conceptually "brought iinto" Einstein's gravitational EP.

The full mathematical development that Einstein produced after envisioning the Equivalence Principle, described an essential required nature of all field theory, quantum or not: The theory must be coordinate-frame/coordinate-system independent. The type of field mathematics that satisfies this requirement, and is most general, occurs when the field is modeled by a nonEuclidean differential geometry. From this, in the gravitational world, came the wealth of macroscopic orbit theory, first Schwarzschild and then Kerr. By bringing electricity up into the Equivalence Principle, creating an extended "Gravitoelectonic Equivalence Principle," the logic demands there must exist a nonEuclidean differential geometry representation of a simple spherically symmetric electronic Coulomb field, such as that which is generated by the proton in hydrogen, with all of the general time dilation characteristics of such a nonlinear field.

A nonEuclidean description of a spherically symmetric Coulomb field can be formulated, and now the topic goes "new theory." The description is mathematical, and the forum managers can shift this thread/topic to wherever deemed appropriate. I assume I'll be able to follow the thread, if there are any replies.

[ajb]

The root of the equivalence principal, stated in this way, is that the inertial mass and gravitational mass are the same thing. But when we look at the motion of test particles in an electromagnetic field we see that the electric charge, which allows the coupling of the particles to the EM field, is different from the inertial mass. Therefore we have no direct EM equivalence principal.

[sb635]

I agree charge and mass are not the same "physical thing." Charge is charge and mass is mass. I am glad you mentioned the charged based coupling. The nonEuclidean model for the electronic field in hydrogen has a type of "coupling" in it. It is a very strange differential geometry, not in its identity, that is, the first model is pure Schwarzschild in form, but both the rest mass and charge of the electron themselves must partake in defining the curvature (the strength of the field) in the generalized spacetime mathematics. The electron's rest mass and charge themselves (actually the electron's charge to rest mass ratio) partake in defining the geodesic upon which it coasts.The electron's charge to rest mass ratio resides in the equations for the elements of an electronic Schwarzschild metric tensor. In the final representation of this Schwarzschild metric tensor, Newton's G does not appear anywhere. It is purely electronic in form. But it is for a system of two particles, (e.g., the electron and proton in hydrogen). When the orbiting body changes (has a different charge to mass ratio) like in "muonic hydrogen," the entire metric structure of the electronic Schwarzschild field shifts in accord.

[sb635]

Envision the electron in the ground state in hydrogen to be Einstein's "independent perceiving subject." Imagine a neutral atom-sized elevator somehow rigidly fixed with the center of its inside volume at a point along the electron's orbit. At first, think "classically, deterministically" here, and view the electron as coasting along its smooth Bohr orbit while inside this fixed elevator. The electron has a sub-atomic sized radar gun, and takes Doppler shift readings off the inside walls of the elevator while coasting through. Based on this data, the electron concludes its pathway through this fixed elevator was curved. The electron concludes there must be something in Einstein's physical "external" world that forced it along this curved pathway.

 

The electron somehow knows the elevator was fixed, and discounts inertial-based accelerations to have caused the apparent curved pathway. The elevator did not move. The electron knows it has mass and charge, and their amounts. Based on the observed curvature of the orbit inside the elevator, the electron pinpoints the center of this curvature at a specific point "out there" in the external world. The electron now knows where the center of its orbit is at, but does not know the physical cause of the attraction. Since the electron knows it has mass, it maybe gravitational. Maybe there is a neutral central mass at the center, which produced gravitational field-induced accelerations curving the electron along its orbit. Or maybe the induced accelerations were electronic. Maybe there was an oppositely charged central body present, with no mass. Or maybe the central body has both mass and charge, and the total accelerations came from a combined source. The electron does not know, and cannot tell from the observed Doppler shifts.

 

The electron assumes the physical source of the attraction is electronic. Using classic Coulombic theory and its known charge, mass and the observed orbital characteristics, the electron computes there must have been 1e worth of massless positive charge at the center of its orbit, identical to its charge, but opposite in sign. The electron then assumes the physical source of the attraction is gravitational. Using its mass (its charge is irrelevant, assuming the central body is neutral) and the observed orbital characteristics, it computes an amount of "effective central mass" which produces identical observed forces and accelerations inside the elevator. The computed value, in the mks system, is a whopping ~10¹² kg. The electron is surprised, and concludes it must have been subjected to an electronic force, not a gravitational force, but conceptually, it cannot tell.

 

How did the electron compute the required ~10¹² kg for the "effective central mass"? Assuming the forces are electronic, they are (using absolute value)

 

[math]{F_e} = \frac{1}{{4\pi {\varepsilon _0}}}\frac{{\left| {{e_e}{e_p}} \right|}}{{{r^2}}}[/math]

 

Set this proportional to a gravitational force (allowed by the gravitoelectronic EP) so that

 

[math]\frac{1}{{4\pi {\varepsilon _0}}}\frac{{\left| {{e_e}{e_p}} \right|}}{{{r^2}}} = G\frac{{\chi {e_p}{m_e}}}{{{r^2}}}[/math]

 

where [math]\chi [/math] is a proportionality constant producing equality of electronic and gravitational forces. This proportionality constant is a mass-to-charge ratio, and transforms the amount of central charge, computed from the electronic-only attraction, to an amount of "effective central mass" which produces the equality of forces. Modeling the forces as such is completely allowed by the extended gravitoelectronic EP. Solving for [math]\chi [/math] produces

 

[math]\chi = \frac{1}{{G4\pi {\varepsilon _0}}}\left| {\frac{{{e_e}}}{{{m_e}}}} \right|[/math]

 

For the hydrogen atom, [math]\chi {e_p}[/math] = ~10¹² kg. Note the statement must be "for the hydrogen two-body system/atom." The electron's own charge-to-mass ratio partakes in the definition of how much "effective central mass" is needed to produce gravitational-electronic force equality. The product [math]\chi {e_p}[/math] is an amount of central mass. This central rest mass can now serve as the mass-value used to parametrize a nonEuclidean Schwarzschild geometrical representation of the proton's electrostatic Coulomb field. An electronic Schwarzschild radius can be defined as

 

[math]{r_S} = \frac{{2G\chi {e_p}}}{{{c^2}}}[/math]

 

[math] = \frac{1}{{2\pi {\varepsilon _0}}}\frac{{\left| {{e_e}{e_p}} \right|}}{{{m_e}{c^2}}}[/math]

 

This Schwarzschild radius is completely electronic in form. The diagonal elements of a 4x4 electronic timelike Schwarzschild metric tensor G can be defined, all based on this electronic Schwarzschild radius for the "central body" of hydrogen. These diagonal elements are (in spherical polar coordinates)

 

[math]{{g_{rr}} = - \frac{1}{{{c^2}}}{{\left( {1 - \frac{{{r_S}}}{r}} \right)}^{ - 1}}}[/math]

 

[math]{{g_{\theta \theta }} = - \frac{{{r^2}}}{{{c^2}}}}[/math]

 

[math]{{g_{\phi \phi }} = - \frac{{{r^2}}}{{{c^2}}}{{\sin }^2}\theta }[/math]

 

[math]{{g_{tt}} = 1 - \frac{{{r_S}}}{r}}[/math]

 

The off-diagonal elements are zeros. The electronic Schwarzschild total orbital energy is

 

[math]E = \frac{{\mu {c^2}}}{2}\left[ {{{\left[ {\left( {1 - \frac{{{r_S}}}{r}} \right)\frac{{dt}}{{d\tau }}} \right]}^2} - 1} \right][/math]

 

This total orbital energy equation incorporates a nonCoulombic Schwarzschild system potential. The time dilation is now generally defined as (using matrix algebra)

 

[math]\frac{{dt}}{{d\tau }} = {\left( {\frac{{d{{\bf{x}}^T}}}{{dt}}{\bf{G}}\frac{{d{\bf{x}}}}{{dt}}} \right)^{ - 1/2}}[/math]

 

where [math]{\bf{x}} = {(r,\theta ,\phi ,t)^T}[/math] (I put coordinate time as the fourth spacetime element.)

 

When no central body is present, then [math]{{r_S}}[/math] is zero, and the electronic Schwarzschild metric tensor is Euclidean/Minkowski. Electronic special relativity, as now applied to hydrogen, is the special case. The above time dilation is then the familiar one as in special relativity, which is a function of only velocity. This more geometrically general theory predicts an added component to time dilation, given as a function of the position in the field, just as in gravitation. This is really the key, is that true, or not? I am sure I will get many "no's." <g>

 

Quantization of this orbital theory can be accomplished by relying on relativistic de Broglie "matter-wave" relationships. The result is a quantized electronic Schwarzschild total orbital energy equation. This can be inserted into Schroedinger theory, and Schwarzschild radial wave solutions produced. These then go to the probability theory, and define Schwarzschild radial pdfs. This theory predicts stronger relativistic effects than special relativity, and the means and modes should should shift even closer to the proton than in special relativity. The shell shrinks, gets less variant, and becomes more tightly bound than in special relativity.

[sb635]

Since the time of Bohr, physicists have known the speed of the electron in, for example, hydrogen's ground state is about 0.7%c, about 1% the speed of light. Only a percent dictates, perhaps, a statement of "low relativistic effects," and that is true. Nonrelativistic Bohr theory does remarkably well given its simplicity, when predicting hydrogen transition frequencies. But as well know, it fails miserably for more complex systems (atoms). The electronic [math]{r_S}[/math] for hydrogen equals ~10^-15 m, about the size of the proton. (This is probably just coincidence, but maybe not.) A Schwarzschild geometry is "parametrized" or "fixed in metric structure" given a value of the geometry's Schwarzschild radius. The ground state radius is about 10^-11 m, or about 10⁴ electronic Schwarzschild radii from the center of the field. This is also a "low relativistic effect" region for any Schwarzschild field, macro or microscopic. The electronic Schwarzschild field described here is an atomic-sized "tightly curved" spacetime, at least "tightly curved" enough for closed-loop-binding of a 1%c orbiting body.

An important product has emerged in the equations, the quantity [math]G4\pi {\varepsilon _0}[/math]. This is the square of the Planck charge-to-mass ratio, which is [math]{e_P}/{m_P} = \sqrt {G4\pi {\varepsilon _0}} [/math]. A "particle" with the Planck mass [math]{m_P}[/math] and the Planck charge [math]{e_P}[/math] has this charge-to-mass ratio. I don't think any observed, nor theoretical particle, has this charge-to-mass ratio, but two of these (identical) particles interact in a most force-symmetric way. The gravitational force a distance r apart, is

[math]{F_g} = G\frac{{m_P^2}}{{{r^2}}}[/math]

Given the Planck charge-to-mas ratio, the square of square [math]{m_P}[/math] equals

[math]m_P^2 = \frac{{e_P^2}}{{G4\pi {\varepsilon _0}}}[/math]

Substitution into the gravitational force produces

[math]{F_g} = G\frac{{m_P^2}}{{{r^2}}}[/math]

[math] = \frac{1}{{4\pi {\varepsilon _0}}}\frac{{e_P^2}}{{{r^2}}}[/math]

[math] = {F_e}[/math]

In this two-Planck-body system, the force of gravity exactly equals the force of electricity. In a general n-Planck-body system, the gravitational contributions to collapse exactly equal the positive and negative electronic attractions and repulsions (assume a mix of positive and negative charges). In this type of Planck particle system, gravity is an "equal partner" and has been "unified" with electricity in terms of field strength.

[Strange]

Since the time of Bohr, physicists have known the speed of the electron in, for example, hydrogen's ground state is about 0.7%c, about 1% the speed of light.

 

But the Bohr model is wrong.

[sb635]

 

 

But the Bohr model is wrong.

 

Yes, strictly speaking, the electron cannot be orbiting on a smooth Bohr-like orbit. But if you want to associate a "velocity" with a shell, both nonrelativistic theory and relativistic theory (even this more general theory) predicts the electron's "velocity" in ground state hydrogen is about 1% the speed of light. Schroedinger's theory is completely deterministic (no random variables) up the derivation of the wave equation solutions, like the radial wave equation solution. From a deterministic standpoint, the electron is viewed as "everywhere in configuration space at once," with the amplitudes of the electron's deterministic standing "matter-wave" over this space, defining "how much" of the electron is "here or there." Of course, this is an extremely difficult mental picture, so probability is introduced. Born interpreted the deterministic amplitudes as proportional to the probability of the electron "being" at the position (location) part of configuration space. The introduction of probability theory demands actualizations occur. After all, what's the point in deriving a pdf if then no "draws" are ever made from it. So probability actually demands the electron sometimes "takes on" a physical presence. But where was it when it was going from the physical "here" to "there"? It looks like (to me) Feynman decided it is traveling along a "virtual path integral," which if it were real, is a type of very "fractured, jittery" motion. The analog in the physical world is a white noise 3-D Weiner stochastic process.

 

Deterministic orbital theory cannot be absolutely correct for an atom. The primary obvious reason, to me, has never been mentioned in any text book I've read. Say an electron did in fact reside in a classic orbit. It would always stay in the plane of this orbit, until disturbed. That would mean hydrogen would be extremely flat, and show no 3-D structure. We know this is not true, atoms are 3-D. I personally believe the Feynman path integrals are physically true, and the electron is always "here" undergoing some type of deterministically nonlinear chaotic and fractured motion when in a shell. The deterministic orbit theory sets the deterministic characteristics of the shell, while nature "fractures" it. This "fracturing" process pushes the electron out of its current orbital plane, and causes it to stochastically wander around in the 3-D shell (but not jump shells), all the while amazingly maintaining exactly what the deterministic theory dictates for its orbital energy.

 

 

[sb635]

To expose the field contribution to the electronic Schwarzschild time time dilation, set [math]\theta = \pi /2[/math] for an equatorial orbit. Then the general matrix algebra simplifies to

 

[math]\frac{{dt}}{{d\tau }} = {\left( {1 - {{\left( {\frac{v}{c}} \right)}^2} - \frac{{{r_S}}}{r}} \right)^{ - 1/2}}[/math]

The dimensionless ratio [math] - {r_S}/r[/math] is the nonEuclidean contribution, and subtracts more from 1, increasing the time dilation beyond special relativity. A neutral clock placed in an electronic field obviously would not electronically time dilate. But what about a charged clock? The electron can be considered a type of charged clock. Does a charged clock run slower in an electronic field, even if it is not moving through this field? Then v = 0, but the nonEuclidean term is still there, producing a non-unit time dilation.

There has already been set the precedent of charge curving spacetime. I think it was back in the 60s, the charged Kerr metric was derived, although introducing charge into GR has been going on for decades. In this final statement of the union of gravity and electricity, the amount of central charge enters the metric and contributes to the Schwarzschild radius, and produces a different curvature if assumed nonzero compared to neutral. The nonEuclidean Kerr time dilation is technically different, depending on the charge or neutrality of the central body. In other words, charge itself "curves' or dilates time according to this well-accepted metric.

 

There is no logical reason why the charged Kerr metric could not be applied to the field surrounding a proton. If it is, the contribution to the total field curvature is pathetically low, and does not produce the degree of curvature to bind a 1%c orbiting body. This why the electromagnetic field equations are always "tagged along" as auxiliary. But even in these relativistic Lorentz equations, where time dilation is also represented in the mathematics, the time dilation is not as in special relativity, because the overall curvature of spacetime is not flat if any central mass or charge is present. The effects of a much more strongly curved spacetime resides in the auxiliary electromagnetic equations, not in the metric itself. In the charge Kerr metric, in the actual metric itself, electricity is as "weak" as gravity. Application of the charge Kerr metric would simply produce essentially special relativistic predictions, using the auxiliary electromagnetic equations just as they are in special relativity. In the nonEuclidean atomic mechanics presented here, all of the force of binding comes entirely from the metric structure of the Schwarzschild spacetime. No auxiliary electromagnetic equations are need.

 

There also exists an unrealistic nature to the charged Kerr metric. If the central charge is nonzero then, say, a particular geodesic pathway is produced. If the central charge is simply turned off, a completely different geodesic results. The body coasting on the geodesic could be neutral, and yet it would travel on different geodesics, which is in fact an electronic interaction that should not be there. Neutral and charge bodies should not interact in the theory. Note in the nonEuclidean theory presented here, if either the central or orbiting charge is zero, the entire electronic metric structure goes flat. There are no electronic interactions, as there should not be.