Conservation of Force (Acceleration)

Had a bit of a surprise earlier.  It dawned on me today that the entire universe may actually be governed by one major principle. That of force, well actually acceleration, so as usual I do the Google thing and low and behold there is nothing new under the sun.

Hermann Ludwig Ferdinand von Helmholtz: On The Conservation Of Force, 1863!

'According to Helmholtz, the primary curator of this principle, the "law of conservation of force", as he called, had been enunciated prior to him by Isaac Newton, Daniel Bernoulli, Benjamin Thomson, and Humphry Davy. Likewise, in the 1670s the theory of vis viva or “living force” of German mathematician Gottfried Leibniz was prominant. In 1837, German pharmacist Karl Mohr gave one of the earliest statements of the conservation of force...'

The law of conservation of force, according to Helmholtz, states:

“The quantity of force which can be brought into action in the whole of nature is unchangeable, and can neither be increased nor diminished.”

If anyone has read my previous posts on the nature of photons and the moment of inertia of a photon you would have seen that I have been doing a little playing around with the Planck Units. Out of this came the equation

$ E c = \hbar g $     ...(1)

E - energy
c - speed of light
$\hbar$ - Reduced Planck constant
g - acceleration

From this equation I suggested that maybe energy has an associated, intrinsic, acceleration given by equation (1). I went further to suggest that  photons are created in a finite time given by

$ \delta t = 1 / \omega $    ...(2)

$\omega$ - frequency of the photon

and that the creation process actually takes place over a distance of

$  s = \lambda / 4 \pi $    ...(3)

$\lambda$ - wavelength of the photon created.

There are a number of issues with the idea, but lets say for now that it may have some merit. Continuing on the same idea it is possible to derive the following from Planck Units

$ F_p = \frac {\hbar} {c^3} g_p^2 $    ...(4)

where
$g_p$ - Planck acceleration, defined below
$F_p$ - Planck Force, defined below

$ F_p = \frac {c^4} {G}   ,   g_p = \sqrt \frac {c^7} {G\hbar} $   ...(5)

In a post I will publish later it is possible to show that there are real world equivalents to many of the equations you can derive using the Planck units. This is were we get a little dodgy because if we apply this idea then from (4) we have

$ F = \frac {\hbar} {c^3} g^2 $    ...(6)

Where $g$ is the acceleration described in (1). In fact, using (1) and the standard Planck relationship

$ E = \hbar \omega$   ... (7)

it is possible to derive

$ E = F \frac {\lambda} {2\pi} $    ...(8)

The maths is not that difficult, but if you are struggling I am happy to post it in detail. Further if (6) is correct then using Newton's version

$F = m a$    ...(9)

with a = $g$ and

$m = \frac \hbar {c^3} g$     ...(10)

using this value of m in

$ E = mc^2 $    ...(11)

just gives us

$ E = \frac \hbar {c^3} g c^2 = \frac {\hbar} {c} g$     ...(12)

which just boils down to (1). From the previous post another equation was derived

$g = c    \omega $    ...(13)

$\omega$ - $2 \pi \nu$
$\nu$ - frequency

It is this last equation, along with (1) and (10) that this post is really about. Is it possible that

equation (1) really does represent a relationship between energy and acceleration?
equation (10) indicates that mass also as a direct relationship with the very same acceleration?
equation (13) a photon can also exist that is directly dependent on acceleration?

If there is any truth to any of this then is the implication that energy and mass are related not only by (11), but also (1), (10) and (13), namely, acceleration. Is acceleration really that significant?

Further, are the conservation of energy and the conservation of momentum laws actually a single law - the conservation of acceleration?

Hold on one minute. Conservation of momentum, we haven't even mentioned that yet, how did that sneak in? Take (12) and apply it to momentum

$ E = p c = \frac {\hbar} {c} g $   ... (14)

p - is momentum of a photon, so

$ g \hbar = p c^2 $     ...(15)

again a direct relationship between acceleration and momentum. Subbing this back into (6) gives

$ F = p \frac {g} {c} $    ... (16)

but using (13) this just gives

$ F = p   \omega $    ...(17)

using (2)

$ F \delta t = p $    ...(18)

which is the more traditional representation of momentum and force.

The final part of this post on acceleration concerns electric charge, from Planck units

$ V_p = m_p c^2 / q_p  = \frac {\hbar g_p} {q_p c} $     ...(19)

$q_p$ - Planck charge
$V_p$ - Planck voltage
$m_p$ - Planck mass

and the equivalent

$ V = m_e c^2 / e  $     ...(20)

$m_e$ - mass of the electron
c- speed of light
e - charge on the electron

This is just the unit conversion of electron mass into electron volts. Using (10) and rearranging slightly this gives

$ e V = \frac {\hbar} {c} g  $    ...(21)

This time implying a relationship between charge and acceleration. Equation (21) is a bit of an after thought and needs more investigation.

IMPORTANT NOTE. There is nothing here to prove that the two laws are actually one, and the assumption that energy may under go acceleration is unproven and without corroborating evidence as they say in the movies. This is a house of cards built on the assumption used to give (6). There are a number of instances where this is assumption is valid, (19) and (20) are certainly true and is one example.

I can't help thinking that there is something not quite right in all of this. After all it seems to imply that energy, momentum, photons, mass and charge are actually related to some intrinsic acceleration. Further that this acceleration is conserved and seems to show that there is no need for two laws, namely the conservation of energy and conservation of momentum.

Though Helmholtz was keen on the idea so it does have some historical merit.

I'm going to try and apply this idea to the creation of an electron / positron pair from a photon and see what numbers I get, will publish this in a future post.

Amendment: Got to pondering this one and there is more to this than meets the eye. will report soon on this one.