Light Speed As Flow Of Energy Per Basic Particle

We have attempted to generalize the expression for $c$,

$c=n\sqrt { \cfrac {(32\pi ^{ 4 }-1) }{ 128\pi\theta_{\psi}ln(cosh(\theta_{\psi}))tanh(\theta_{\psi}) }  }$

$\theta_{\psi}=\cfrac{a_{\psi\,c} G}{\sqrt{2mc^2}}\sqrt [  3]{n  }$

A more useful expression for $n$ and $\theta_{\psi}$ is,

$n.a^3_{\psi\,c}=a^3_{\psi}$

by consider $n$ particles of radius $a_{\psi\,c}$ forming into one particle of radius $a_{\psi}$.

$\theta_{\psi\,c}=\cfrac{G}{\sqrt{2mc^2}}a_{\psi\,c}=0.7369$

$\theta_{\psi}=\cfrac{G}{\sqrt{2mc^2}}a_{\psi}$

$\cfrac{\theta_{\psi}}{0.7369}=\cfrac{a_{\psi}}{a_{\psi\,c}}$

$n=\left(\cfrac{\theta_{\psi}}{0.7369}\right)^3$

we will do it again with,

$c_{ adj }=c.\cfrac {2ln(cosh(3.135009)) }{ \mu _{ o }  }$

that in general,

$c_{ adj }=c.\cfrac {2ln(cosh(\theta_{\psi})) }{ \mu _{ o }  }$

$c_{ adj\,n }=n\sqrt { \cfrac { (32\pi ^{ 4 }-1).ln(cosh(\theta _{ \psi  })) }{ 32\pi .\theta _{ \psi  }tanh(\theta _{ \psi  }) }  } .\cfrac { 1 }{ \mu _{ o } }$ --- (*)

where $\mu _{ o }=4\pi \times 10^{ -7 }$.

So,

$\left( \cfrac { c_{ adj\,n } }{ n }  \right) ^{ 2 }=\cfrac { (32\pi ^{ 4 }-1).ln(cosh(\theta _{ \psi  })) }{ 32\pi .\theta _{ \psi  }tanh(\theta _{ \psi  }) } .\cfrac { 1 }{ \mu _{ o }^{ 2 } }$

A plot of ln(cosh(x))/(xtanh(x)),

and $c_{adj\,n}$ can be a constant with changing $n$, provided that,

$\theta_{\psi}$ is large and so $n$ is large.

$c$ was the rate at which energy leaves the boundary of the particle at $a_{\psi}$.  This boundary was made up of $n$ number of basic particles (each of radius $a_{\psi\,c}$), reformed into a sphere of radius $a{\psi}$

After $c$ is adjusted by expression (*), $c\rightarrow c_{adj\,n}$ we find the rate of flow of energy $E_{excess}$ out of its boundary $a_{\psi}$ increase proportionally with $n^2$, irrespective of the final boundary $a_{\psi}$ as denoted by a changing $\theta_{\psi}$.

$c_{adj\,n}=n.D$

$c_{adj\,n}$ is the flow of energy from the particle made up of $n$ basic particles and,

$D=\sqrt { \cfrac { (32\pi ^{ 4 }-1).ln(cosh(\theta _{ \psi  })) }{ 32\pi .\theta _{ \psi  }tanh(\theta _{ \psi  }) }  } .\cfrac { 1 }{ \mu _{ o } }$

is a constant.

$c_{adj\,n}$ per basic particle,

$\cfrac{c_{adj\,n}}{n}=constant$

 Irrespective of increasing $a_{\psi}$ and increasing total excess energy $E_{excess}$.  So, the total excess energy of $n$ basic particles is,

$E_{excess,\,n}=n.E_{excess,\,1}$

and the total energy flow (times per sec) out of the manifested particle boundary at $a_{\psi}$,

$c_{adj\,n}.E_{excess,\,n}=c_{adj\,n}.nE_{excess,\,1}=n^2\cfrac{c_{adj\,n}}{n}.E_{excess,\,1}$

$\because$ $E_{excess\,1}$ is a constant, $\cfrac{c_{adj\,n}}{n}$ is a constant.

$c_{adj\,n}E_{excess\,n}\propto n^2$

the total energy flow out of the manifested particle boundary at $a_{\psi}$ is proportional to its number of basic particles squared, $n^2$.

In this case, the light speed limit is interpreted as the flow of energy in a field.  Energy is imparted upon a particle until it achieve the same speed as this flow of energy.  When the particle is at less than light speed, energy flows into the particle increasing its kinetic energy.  The particle speed increases.  When its speed is at the energy flow rate, energy does not enter into the particle and its speed is constant at light speed.

No matter what the manifested particle size is (in terms of $n$, $a_{\psi}$ or $E_{excess}$), this flow rate (flow per $n$) is the same.  In other words, interactions between pairs of manifested particles of equal $n$ will produce the same light speed limit, irrespective of $n$.

But interactions between manifested particles of unequal $n$ results in different rate of transfer of energy, corresponding to particles of different charge experiencing different force in a field.  These interacting particles will still be at same constant flow rate per basic particle relative to each other when the transfer of energy between them stops.

So, an alternate view of the light speed limit is the constant flow of energy per basic particle provided the number of basic particles in the manifested particle is large.

Eventually this flow rate traces back to the flow of $\psi$ at light speed back to the time dimension at the center of each basic particle.  What is surprising is that after $n$ number of basic particles reform into a sphere of radius $a_{\psi}$ as they coalesce, the resulting energy flow rate out of the new boundary at $a_{\psi}$, is a constant per basic particle.

Light speed is mine!

Note:  Since energy flow per basic particle traces back to the flow of $\psi$ at the center of each basic particle back to the time dimension, the presentation here does not answer the question why is there a speed limit.  That was answered by entanglement as energy sharing in the time dimension, which manifests as a drag force proportional to speed squared in the space dimension.  Here, we see how manifested particles might share the same speed limit as energy flows through them at a constant flow rate per basic particle.

A manifested particle is made up of $n$ basic particles

In the starting equation equating energy emanated and excess kinetic energy, $E_{excess}$ from the post "Just When You Think c Is The Last Constant" dated 26 Jun 2016, in expression (*) repeated,

$16\pi { \cfrac { \dot { x }  }{ a_{ \psi  } }  }{ c^{ 2 }ln(cosh(\pi ))tanh(\pi) }=\cfrac{1}{2}(32\pi ^{ 4 }-1).\cfrac { 1 }{ T }$

flow rate was given by the term $\cfrac{\dot{x}}{a_{\psi}}=\cfrac{c}{a_{\psi}}$ in per sec.  In the discussion above flow rate is $c$ and $c_{adj\,n}$ in times per sec.  To fully define light speed would require the definition of a meter.  That happens when adjustments was made to $c$,

$c\rightarrow c_{adj\,n}$

by the factor

$c_{ adj,\,n=77 }=c.\cfrac {2ln(cosh(3.135009)) }{ \mu _{ o }  }$

specifically when $n=77$.

Have a nice day.