Kepler, Newton and the fine structure constant

One of the things I love about physics is the little surprises it shows up. I did a post where I had "discovered" an interesting result that appears to have something to do with Kepler's 3rd law. This post follows on from one that extends a result about Newton's law of gravity.

In this case I consider the gravitational force when a positron/electron pair are created with a balancing Coulomb force. This may seem a little odd, since both force are attractive, but go with me on this one.

For Newton

$F_{Newton} = $ $ \frac {G m_e^2} {l_p^2}$   ...(1)

$F_{Newton}$ - Resulting force - $0.212013  N$
G- Gravitaional constant - $6.67384     10^{-11}  m^3   kg^{-1}   s{-2}$
$m_e$ - mass of the electron - $9.109382    10^{-31}  kg $, $2m_e$ - e/p pair
$l_p $ - Planck length - $1.616199    10^{-35}  m$
m - a value that was not stated


For Coulomb

$F_{Coulomb} =$ $ \frac{e^2} {4 \pi \epsilon_0 r^2}$     ...(2)

e - charge on the electron - $1.602176565    10^{-19}  C$
$\frac {1} {4 \pi \epsilon_0}$ - Coulombs constant - $8.98755178    10^{9}   kg  m^3   s^{-2}    c^{-2}$
r - separation of the electrons

What we are going to consider is the point where the two forces are actually equal in order to determine the value of r, the separation of the electrons.

$F_{Newton} = F_{Coulomb}$   ...(3)

so

$\frac { G m_e^2} {l_p^2} = \frac{e^2} {4 \pi \epsilon_0 r^2}$     ...(4)

rearrange to give r, so we have

$ r^2 = $ $ \frac {e^2 l_p^2} {4 \pi \epsilon_0 G m_e^2} $     ...(5)

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Aside: it is possible to replace the $4 \pi \epsilon_0 G$ with the Planck equation

$4 \pi \epsilon_0 G = $ $\frac {q_p^2} {m_p^2}$     ...(A1)

resulting in

$ r^2 = $ $ \frac {e^2 l_p^2 m_p^2} {q_p^2 m_e^2} $     ...(A2)

or

$ r = $ $ \frac {e l_p m_p} {q_p m_e} $     ...(A3)

which gives

$ \frac {r m_e} {e} =  \frac {l_p m_p} {q_p} $     ...(A4)
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putting in the numbers gives a value of 

$ r = 3.29874  10^{-14} m$

Now in a post on photons I derived a relationship (equation 27 in the post)

$ s = \lambda / 4 \pi$   ... (6)

In a later post I calculated a value for this for a photon with enough energy to create an electron/positron pair, it was

$s = 9.653976    10^{-14}   m  $    ...(7)

dividing r by this gives

$r / s = 0.341698$

divide by 4 and square gives and 

$(\frac {r} {4s})^2$  $ = 0.0072973$   ...(8)

the reciprocal of this is 137.035854, which is pretty close to the inverse fine structure constant. This is just

$(\frac {r \pi} {\lambda})^2$  $ = 0.0072973$   ...(9)

Also, if we take the classic electron radius

$r_e = 2.81794    10^{-15}$

divide this by r and square we get

$(\frac {r_e} {r})^2 $ $= 0.007300$    ...(10)

Let's continue this flight of fancy by saying that (8) and (10) are close enough to say they are equal, we ill revisit this later, so we end up with

$(\frac {r} {4s})^2 = (\frac {r_e} {r})^2$    ...(11)

rearranging this gives

$r^2 = r_e 4s$    ...(12)

which is also

$r^2 = r_e \lambda / \pi$    ...(12a)

if we then use the value for s given by 6 and

$r_e = 2 \pi \lambda \alpha$    ...(13)

we can show

$r^2 = 2 \alpha \lambda^2$    ...(14)

or

$r^2 =$  $ \frac {r_e^2} {2 \pi^2 \alpha}$   ... (15)

OK, so what does all this mean? if anything? We have defined a new value r that lies somewhere between the radius of an electron and the wavelength of a photon capable of creating an electron/positron pair.

Further to this let's take a look at the first part the gravitational force. This is the force that would occur between two electrons separated by a Planck length. Is that even valid to consider gravitation at the Planck length. Let us try something else, if we replace the $l_p$ with its definition

$l_p = $ $ \sqrt \frac {\hbar G} {c^3}$     ...(16)

then (1) becomes

$F_{Newton} = $ $ \frac {G m_e^2 c^3} {\hbar G}$

cancelling the G gives

$F_{Newton} = m_e a = $ $ \frac {m_e^2 c^3} {\hbar }$  ...(17)

a - acceleration, where

$ a = $  $\frac {m_e c^3} {\hbar}$    ...(18)

which does NOT involve the gravitational constant.  Does this help, I'm not sure.

If I am honest what appeals to me here is the fact that a new distance has been defined in equations (14) and (15) that lies in between the size of a photon (with enough energy to create an electron/positron pair) and the classic electron radius. It was derived by considering some balance point between the Coulomb force and, what appears to be, a gravitational force.

One problem though is if it is an electron/positron pair then both the Gravitational force and the Coulomb force are attractive, so how can there be a balance point? Even if we consider a pair of electrons in the vacuum of space, does this "balance point" actually tell us anything? Do we actually have pairs of electrons in a common orbit around each other at the distance calculated here? I find that hard to believe. Though I would think that it would be relatively easy to confirm by experiment.

One problem with orbiting electrons though is that Coulombs law is for static point like charges that are stationary relative to each other. If they are in orbit around each other are they stationary relative to each other?

If these electron pairs did exist then what would be their nature? In superconductivity there are "bound" Cooper pairs, would these electrons behave in a similar manner? The value of r is less than the wavelength of the individual electrons, so would they have to be bound in a "low orbit". In superconductivity, the coherence length, the distance between the electrons in a Cooper pair, is far far larger, typically 3 - 100 nm.

According to quantum mechanics electrons in a smaller orbit than the wavelength of each electron would have a total angular momentum of zero. This would the indicate that they may behave similarly to the ground state electron in a hydrogen atom and we could then take a look at the Schrödinger equation for that scenario. Will try this in a later post.

I'm going to finish this post here, though I can't help thinking I am going to revisit the idea in a later post once I have a better understanding of what has gone on.

I wrote this while listening to this.