Monday 21 October 2013

Planck equations

In a number of recent posts I have been taking a look at Planck units and have found a number of equations that have real life equivalents. By this I mean that there are a number of equations that you can derive purely from Planck units that also have recognizable every day version, for example

$E_p = m_p c^2$    ...(1)

$E_P$ - energy
$m_p$ - Planck mass
c - speed of light

this is the well known

$E = m c^2$    ...(2)

yet (2) could have been derived from the Planck units without any knowledge of relativity and mass energy equivalence. This is not the only case, here are a few more

Planck equation General equation
$F_p = m_p g_p$ $ F = m a$
$V_p = T_p k_b / q_p$ $V = T k_b / e$
$T_p k_b = m_p c^2$  $T k_b = m c^2$
$E_p = \hbar \omega_p$ $E = \hbar \omega$
$Z_p = \hbar / q_p^2$ $R_k = \hbar / e^2$

The question must arise as to whether it is possible to derive other Planck equations that do not currently have an every day equivalent? If so, should these equations already exist but it is just a case that we have not found them yet? For example,


Planck equation General equation
$g_p l_p = c^2$ $ g \lambda = c^2$
$E_p c = \hbar g_p$ $E c = \hbar g$
$g_p l_p^2 = m_pG $ $g l_p^2 = m G$
$g_p = m_p c^3 / \hbar $  $g = m c^3 / \hbar$
$F_p = \hbar g_p^2 / c^3$ $F = \hbar g^2 / c^3$
$E_p I_p = \hbar^2$ $E I = \hbar^2$


There are a fair number of these and I have mentioned some of them in previous posts with possible interpretations of what they may be. No doubt I'll cover some of the others in future posts.

Great. Well yes, potentially it is, because we may have a bunch of answers looking for questions which will then tell us what the answer really means. A bit like "42"!

Let's take those listed here;

$ g \lambda = c^2$    ...(1)

does this imply that there is an acceleration associated with a wavelength of a photon just in the same way

$E = h \nu$    ...(2)

links energy with frequency, but if that is the case then does

$E = $ $ \frac {\hbar g} {c} $   ...(3)

imply that there is a relationship between energy and acceleration. The acceleration of what? Photons? I discuss this one in a previous post in more detail.

What about 

$g  =$  $ \frac {m G} {l_p^2}$   ...(3) 

Does this give us insight into Newton's law of gravitation? It throws up at least one surprise for me that I cover here.

Then 

$E I = \hbar^2$    ...(4)

where $I$ has the same units as moment of inertia. Is it the moment of inertia of a photon?  Again, I cover this elsewhere

I genuinely don't know if these are of value or if my interpretations have any validity, but I can't believe that they are not completely without worth. The equations above must mean something, it is really just a question of determining exactly what that is. I shall continue to post on these and give my view on what they may mean. 

Even though this approach yields some interesting results in the form of "new" equations, it does not give us any insight into why the fundamental constants are what they are. Take charge for example, we have

$ \alpha =$  $\frac {e^2} {q_p^2}$   ...(5)

$\alpha$ - fine structure constant
$e$ - charge on the electron
$q_p$ - Planck charge

$q_p = $ $\sqrt {4 \pi \epsilon_0 \hbar c}$    ...(6)

$\hbar$ - reduced Planck Constant
$ 1/ 4 \pi \epsilon_0$ - Coulomb constant

This shows us a relationship between the Planck charge and the charge on an electron, which turns out to be the fine structure constant, but it does not give us the reason why.

There are a number of equations that link mass, length and charge, so if an explanation for one of them can be found then the rest will fall into place. Though that is far easier to say than do.

The Planck units give us a number of Planck equations that may be giving us some new insights that we have not previously seen, but they do not appear to be giving out any clues as to why the electron has the charge, mass and radius it does have. Or is it just that I am missing something?

I shall continue to ponder.

I wrote this while listening to this.

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