The speed of light is the low-dimensional limit
The singularity in a black hole is the higher-dimensional limit
With r_S\space =\space \frac{2\space *\space l_P^2}{\lambda}, the Schwarzschild radius is directly related to the Compton wavelength
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Dimensional Physics
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Theory unifying general relativity with quantum field theories
Christian Kosmak, Germany Würzburg 2023 Version 4.1 – 05.30.2023
Binding energy as intersection of spacetime density.
Formulary for Dimensional Physics
- Formulas with derivation are listed, which are used again and again in the work.
- The collection is not complete and is intended for „quick reference“.
- The formula is usually rearranged in the sense that it is not so easy to calculate with it, but the meaning of the formula becomes clearer.
The formula for the Compton wavelength is according to the textbook (all unreduced).
\blacksquare\space\space \lambda_C\space =\space \cfrac{h}{m_C\space *\space c}
This formula is very good for calculating, but it does not explain what the Compton wavelength is actually supposed to be. If you convert it, you get a very simple invariant quantity.
Often used transformation:
h\space =\space m_P\space *\space c^2\space *\space t_P with m_P as the Planck mass, c as the speed of light and t_P as Planck time
c\space =\space \cfrac{l_P}{l_T} with l_P as Planck length
It follows:
\blacksquare\space\space\lambda_C\space =\space \cfrac{m_P\space *\space c^2\space *\space t_P}{m_C\space *\space c}\space \iff\space \cfrac{m_P\space *\space c\space *\space t_P}{m_C}\space \iff\space \cfrac{m_P\space *\space l_P\space *\space t_P}{m_C\space *\space t_P}\space \iff\space\cfrac{m_P\space *\space l_P}{m_C}\space \iff\space \lambda_C\space *\space m_C\space = l_P\space *\space m_P
As a result, one recognises:
- The Compton wavelength is, via the mass ratio, an adjusted Planck length.
- The product of mass and Compton wavelength is constant.
The formula for the fine structure constant exists in many different forms. Some of them are listed here, which are needed again and again.
The best-known representation:
\blacksquare\space\space \cfrac{e^2}{4\space *\space \pi\space *\space \epsilon_0\space *\space \hbar\space *\space c }\space \iff\space \cfrac{e^2}{2\space *\space \epsilon_0\space *\space h\space *\space c }
About a short form of k_C\space =\space \cfrac{1}{4\space *\space \pi\space *\space \epsilon_0}\space with k_C as Coulomb constant
\blacksquare\space\space \cfrac{k_C\space *\space e^2}{\hbar\space *\space c}
Via c^2\space =\space \cfrac{1}{\sqrt{\epsilon_0\space *\space \mu_0}} \space the magnetic field constant is used instead of the electric field constant
\blacksquare\space\space \cfrac{\mu_0\space *\space c\space *\space e^2}{4\space *\space \pi\space *\space \hbar}
Via Z_{w0}\space =\space \sqrt{\cfrac{\mu_0}{\epsilon_0}}\space =\space \mu_0\space *\space c as the characteristic impedance of the vacuum
\blacksquare\space\space \cfrac{e^2\space *\space Z_{w0}}{4\space *\space \pi\space *\space \hbar}
Via R_k\space =\space \cfrac{h}{e^2}\space with R_k as Von Klitzing constant
\blacksquare\space\space \cfrac{\mu_0\space *\space c}{2\space *\space R_k}
About the resolution of h with an exchange of \epsilon_0 to \mu_0 the following version comes out. This is very important for the DP.
\blacksquare\space\space \cfrac{e^2}{2\space *\space l_P}\space *\space \cfrac{\mu_0}{m_P}Part of the Planck scale is shown in two versions (not reduced):
- Textbook version (there may already be several different ones here). Here c, G and h are set
- Version for the DP via the set sizes
- m_P as Planck mass
- l_P as Planck length
- t_P as Planck time
\blacksquare\space\space l_P\space =\space \sqrt{\cfrac{h\space *\space G}{c^3}}\space \iff\space \cfrac{h}{m_P\space *\space c}
l_P\space is set in the DP
\blacksquare\space\space t_P\space =\space \sqrt{\cfrac{h\space *\space G}{c^5}}\space \iff\space \cfrac{l_P}{c}
t_P\space is set in the DP
\blacksquare\space\space m_P\space =\space \sqrt{\cfrac{h\space *\space c}{G}}\space \iff\space \cfrac{h}{l_P\space *\space c}
m_P\space is set in the DP
c\space =\space \cfrac{l_P}{t_P} is retained as an abbreviation in the formulas
E_P\space =\space m_P\space * c^2
\blacksquare\space\space h\space =\space \cfrac{l_P^2\space *\space c^3}{G}\space \iff\space m_p\space *\space c^2\space *\space t_P\space \iff\space m_p\space *\space c\space *\space l_P
\blacksquare\space\space G\space =\space \cfrac{l_P^2\space *\space c^3}{h}\space \iff\space \cfrac{l_P\space *\space c^2}{m_P}
\blacksquare\space\space G\space =\space \cfrac{l_P}{E_P}\space * c^4 This representation is important for the DP. One can recognise the dimensional limits of spacetime from the two terms.
\blacksquare\space\space T_P\space =\space \sqrt{\cfrac{h\space *\space c^5}{G\space *\space k_B}}\space \iff\space \cfrac{m_P\space *\space c^2}{k_B}\space mit k_B as Boltzmann constant
\blacksquare\space\space q_P\space =\space \sqrt{\cfrac{h\space *\space c}{k_C}}\space \iff\space \sqrt{4\space *\space \pi\space *\space \epsilon_0\space *\space \hbar\space *\space c} mit k_C\space =\space \cfrac{1}{4\space *\space \pi\space *\space \epsilon_0}