Conservation of acceleration

In the previous post on the conservation of force and acceleration I looked at the idea that there may be an acceleration that links energy, mass and momentum. In this post I am going to have a go at putting in some numbers to the equations and see how things come out.

The equations of interest from the previous post are

$Ec = \hbar g $   ...(1)

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

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

F - force

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

m - mass

$ g = c \omega $     ...(4)

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

$ g \hbar = p c^2 $    ...(5)

p - momentum.

$I =\frac {\hbar c} {g} $   ...(6)

To open we are going to use the standard equation for calculating the minimum energy required to create a positron/electron pair

$ \hbar  \omega = 2 m_e c^2 $    ...(7)

$m_e$ - mass of the electron
the factor of two is because we are creating a positron/electron pair. Replace $\omega$ using (4) to give

$ \frac {\hbar} {c} g = 2 m_e c^2 $    ...(8)

Using standard values
$m_e - 9.109382   10^{-31}$ kg
$c - 299792458$ m/s
$\hbar  - 1.054571  10^{-34}$  J.s
giving

$g = 4.654845  10^{29} ms^{-2} $     ...(9)

Putting this value into equations (2) , (5) and (6) gives

$F = 0.848055 N$

$p = 5.461848  10^{-22}$ kg m/s

$I = 6.7919  10^{-56} kg   m^2 $

also, the photon creation/destruction time and distance is given by

$ \delta t = 1 / \omega = 6.44  10^{-22} s $
$ s = 9.653976   10^{-14} m$

So let's take a look, the only new value is the $g$, the rest can all be calculated from existing theories. $g$ is massive. Consider a neutron star with its massive gravity, this may typically have a value of $10^{12} ms^{-2}$ but this is tiny compared to the value quoted in (9). The question though is does it actually mean anything, also, can we find out anything new from this value?

The value in (9) is the minimum acceleration required to create mass.

The next section relates to something I first discussed back when I was discussing physics units and dimensions. In that post I pointed out that Energy density and pressure have the same units.

In Planck units they are identically the same

$ \rho_p^E = \frac {E_p} {l_p^3} = \frac {c^7} {\hbar G^2} = \frac {g_p^2} {G} $ ...(10)

$\rho_p^E$ - Planck energy density
$E_p$ - Planck Energy
$l_p$ - Planck length
$g_p$ - Planck acceleration
G - gravitational constant
$\hbar$ - reduced Planck constant
c - speed of light in a vacuum

$ p_p = \frac {F_p} {l_p^2} = \frac {c^7} {\hbar G^2} = \frac {g_p^2} {G} $ ...(11)

$p_p$ - Planck Pressure
$F_p$ - Planck force

$ \rho_p^E = p_p$    ... (12)

further Planck Intensity or irradiance is given by

$ I_p = \rho_p^E c = p_p c = \frac {g_p^2 c} {G} $ ....(13)

the general form of irradiance is given by

$ p = \frac {I} {c}$ ...(14)

can we then say

$ \rho^E = \frac {g^2} {G} $ ...(15)

$ p = \frac {g^2} {G} $ ...(16)

if this is valid then putting in the value of $g$ into (15) and (16) gives

$ p = 3.24664   10^{69} = F / A $  ... (17)

A - area.

Using the value of F from earlier gives an area of

$ A = 2.612   10^{-70}$ , giving a length of $l = 1.61619  .10^{-35}  = l_p$ the Planck length.

Does this imply that the general form of (11) is

$ p = \frac {F} {l_p^2} =  \frac {g^2} {G} $ ...(18)

using the definition for $l_p$

$ l_p^2 = \frac {\hbar G } {c^3} $    ...(19)

putting this into (18) and rearranging results in (2) from earlier. Applying the same result from (17) to (15) gives

$\rho^E = 3.24664   10^{69}  = E / V $   ... (20)

E - energy - $1.63742  10^{-13}$ J, this is the energy required to create e/p pair.
V - volume - $5.0434295   10^{-83}    m^3$

If we take

$ V = l_p^2 d $   ...(21)

where d is a length, then

$d = 1.930699   10^{-13}  = 2 s $

where s is the creation/destruction distance calculated earlier. The wavelength of the photon that creates the e/p pair is given by

$ \lambda = c / \nu = 1.213156   10^{-12}   = 2 \pi d $

A further result from a previous post was this

$ g l = c^2 $     ... (22)

using the value of d calculated here for $l$ and using $g$ from above the value of c is found to be

$ c = 299784999 m s^{-1} $ which is 0.999975 of c.

So, what have we learned from this post. While the physical meaning of $g$ is still not proven the equations involving this variable are found to be true, at least for the example given here. The value of $g$ is found to be incredibly high and would be so even for relatively low energy photons.

Equations for energy density and pressure have been derived (guessed!) and values calculated for a photon converting to e/p pair. The energy density and pressure have been found to be very large and have a dependency on the gravitational constant. This I find quite remarkable, a microscopic result linked to the gravitational constant.

If we assume that the electron is the smallest particle mass that can be created, the the values for acceleration, energy density and so on can be considered the smallest values that allow the creation of matter  (e/p pair) to take place.

There is of course one small problem here, momentum. The amount of energy discussed here is just enough to create the e/p pair, but these would have no momentum. If they did it would require more energy because

$ E = 2\sqrt {(pc)^2 + (m_ec^2)^2}  $

So what happened to the momentum, after all it was calculated to be

$p = 5.461848  10^{-22}$ kg m/s

Traditional thinking is that this is imparted into the local mass, a local nucleus for example. This is also the reason why a photon cannot spontaneously convert itself into an e/p pair. It needs mass. Does this imply that mass is required to accelerate/decelerate energy, yes, I think it does. It is this that will be covered in a later post (I will add a hyperlink when I publish it).