Petri Lievonen

Work in progress

Dynamic examples

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Zero-energy principle (Arthur Haas, Dennis Sciama, Pascual Jordan, Richard Feynman, Tuomo Suntola)

$$m{c}_0^2-\frac{GMm}{R}=0\text{with}{c}_0(t)=\frac{dR}{dt}\text{and}\hslash ={\overline{h}}_0{c}_0=m_P{l}_P{c}_0$$

$$m{c}_0^2-\frac{GMm}{R}=0$$$$c_{0}m{c}_r=\frac{GMm}{R}-\frac{G{M}_{r}m}{r}=\frac{GMm}{R}(1-\frac{M_r/M}{r/R})=\frac{GMm}{R}(1-\frac{G{M}_r}{r{c}^2})=\frac{GMm}{R_r}=cm(c\cos \theta )$$$$c^2=c_r^2+v_{esc}^2={\left(c(1-\frac{GM}{r{c}^2})\right)}^2+v_{esc}^2$$$$F_r=\frac{G{M}_{r}m}{{(r/\cos \theta )}^2}\cos \theta =\frac{G{M}_{r}m}{r^2}{(\cos \theta )}^3$$$$f\propto m_v{c}_r=mc\sqrt{1-\frac{v^2}{c_r^2}}(1-\frac{GM}{r{c}^2})$$$${\lambda }_m=\frac{h}{m_{v}c}=\frac{h_0}{m_v}$$$${\lambda }_p=\frac{h}{p}=\frac{h_0}{m\gamma \beta }$$

$$\tanh w=\beta =\frac{v}{c}=\sin \varphi$$$$\cosh w=\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}=\sec \varphi$$$$E=|c{p}_4|=c\sqrt{p^2+(m{c}_r{)}^2}$$$$\frac{1}{2}(\frac{m\sqrt{1-\frac{v^2}{c^2}}}{1-\frac{v}{c}}-\frac{m\sqrt{1-\frac{v^2}{c^2}}}{1+\frac{v}{c}})=\frac{m}{\sqrt{1-\frac{v^2}{c^2}}}\frac{v}{c}=m\gamma \beta$$$$\frac{1}{2}(\frac{mc\sqrt{1-\frac{v^2}{c^2}}}{1-\frac{v}{c}}-\frac{mc\sqrt{1-\frac{v^2}{c^2}}}{1+\frac{v}{c}})=\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}=m\gamma \beta c$$$$\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}(\frac{GMm}{R^2}-\frac{m{v}^2}{R})=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}(\frac{m{c}^2}{R}-\frac{m{v}^2}{R})=\sqrt{1-\frac{v^2}{c^2}}\frac{m{c}^2}{R}=\sqrt{1-\frac{v^2}{c^2}}\frac{GMm}{R^2}$$

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