This is NOT the fine structure constant

I came across something the other day, won't put a link to it. An alternative theory of everything type thing. You can't help stumble upon them if you are on the web looking at physics sites. Now this one had come up with an alternative idea where they did all these calculations to prove this and that.

The problem was that, although the figures appear to be right, the physics was just plain wrong. I won't give the example, but I will give you one of my own. Try this one on for size,

$ \alpha^{-1}$ $ = \frac {e^2 G^2} {m_e^2}$   ...(1)

$\alpha$  - to be calculated
$e$ - charge on the electron - $1.602176 10^{-19} C$
$G$ - Gravitational constant - $6.67384   10^{-11} m^3 kg^{-1} s^{-2}$
$m_e$ - mass of the electron - $9.109382  10^{-31} kg$

put in the numbers and what do you get?

$ \alpha^{-1} = 137.8876$    ...(2)

Hold on a minute that is real close to the fine structure constant, which is

$\alpha^{-1} = 137.035999$   ...(3)

In fact my value from (2) divided by (3) is 1.00621! I'm onto something here right. In fact if we just tweak the Gravitational constant a little, use

$G$ - Gravitational constant - $6.65573   10^{-11} m^3 kg^{-1} s^{-2}$

then it comes out exact. We all know that the gravitational constant is difficult to measure so who is to say I am not right with my value? We have all read that gravity is different if you are on the top of a mountain, or at the poles due to the shape of the earth right?

Not only that but I can go further, the real fine structure constant is just given by

$\alpha =$ $ \frac {e^2} {4 \pi \epsilon_0 \hbar c}$    ...(4)

using my value from (1) I can show

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

re-arrange this and you have

$m_e^2 = G^2 4 \pi \epsilon_0 \hbar c$   ...(6)

Oh my word, the mass of the electron is just the Gravitational constant multiplied by the other universal constants. This is a MAJOR result.

This is also completely WRONG!

Anyone checking this will notice I pulled a trick when I compared (4) to (1) and said they were equal, they are not. (1) is the inverse of (4), so (5) and (6) are mathematically wrong.

Now you may be thinking ok, but there is still something to (1) it is so close. Yes it is close, but it is not close enough. It is not good enough just to argue that the experimenters have it wrong and then dream up some daft excuse on why they have it wrong.

Acceleration due to gravity does vary across the earth, but the Gravitational constant does not. The Gravitational constant, as the name suggests, is constant!

Finally, even if they did have the value of the Gravitational constant wrong, and say they admitted it on the news tomorrow. "The actual Gravitational constant is wrong it should be ....", drum roll,

$G$ - Gravitational constant - $6.65573   10^{-11} m^3 kg^{-1} s^{-2}$

the value from (1) will equal the fine structure constant in numerical value, but it will still NOT be the fine structure constant!

Eh? how can it be numerically equal and not be the same? After all 2+2=4, 2+2 is the same as 4. Taught me that early on. So if equation (1) gives 137.035999 and the inverse of the fine structure constant is 137.035999 then they are the same. Have to be. So

$ \alpha^{-1}$ $ = \frac {e^2 G^2} {m_e^2}$   .

is true if $G$ - Gravitational constant - $6.65573   10^{-11} m^3 kg^{-1} s^{-2}$, right?

WRONG! and this is the point of this post. Although the numbers are the same, the equation does not balance when it comes to dimensions. In physics the dimensions have to balance. The fine structure constant has no units/dimensions, it is dimensionless. but the right hand side of the above has the units

$ Q^2 L^6 M^{-4} T^{-4} $  ... (7)

$Q$ - charge
$L$ - Length
$M$ - Mass
$T$ - time

which is far from dimensionless.  So even though the numbers are close numerically, they are not the same dimensionally. If you multiply a velocity by mass it is no longer a velocity, it is momentum.

So while equation (1) may look intriguing it is wrong. This result is just one of things.

Dimensions seems to be forgotten in many of the online Theories of everything.

The next time you come across an online Theory of everything check the dimensions, if they are wrong, then guess what? yep, it's wrong!

In order to make equation (1) unitless we would have to find something, lets call it K, with units of

$ Q^{-2} L^{-6} M^{4} T^{4} $  ... (8)

does such a thing exist? Actually it does

$K =$ $\frac {c Z_p} {G^3}$ $= 3.02352   10^{40}$ ...(9)

where $Z_p$ is the Planck Impedance, but this is not even close to 1 so while it would make the dimensions correct it would also change the value of $\alpha$ in (1) by a large amount!

Wrote this while listening to this.