Mathematical Formulas for Presentations

Helper formulas extracted from visual index for presentation use

$$d{s}^2=c^{2}d{\tau }^2=(1-\frac{2GM}{r{c}^2})c^{2}d{t}^2-{(1-\frac{2GM}{r{c}^2})}^{-1}d{r}^2-r^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2)$$$$\frac{d{s}^2}{c^2}=d{\tau }^2=(1-\frac{2GM}{r{c}^2})d{t}^2-{(1-\frac{2GM}{r{c}^2})}^{-1}\frac{d{r}^2}{c^2}-\frac{r^{2}d{\theta }^2}{c^2}$$$$\frac{d{s}^2}{c^2}=d{\tau }^2=(1-\frac{2GM}{r{c}^2})d{t}^2$$$$d\tau =\sqrt{1-\frac{2GM}{r{c}^2}}dt$$$$d{\tau }_a=\sqrt{1-\frac{2GM}{r_a{c}^2}}dt$$$$d{\tau }_b=\sqrt{1-\frac{2GM}{r_b{c}^2}}dt$$$$\frac{d{\tau }_b}{d{\tau }_a}=\frac{\sqrt{1-\frac{2GM}{r_b{c}^2}}}{\sqrt{1-\frac{2GM}{r_a{c}^2}}}=\frac{f_b}{f_a}$$$$0=(1-\frac{2GM}{r{c}^2})d{t}^2-{(1-\frac{2GM}{r{c}^2})}^{-1}\frac{d{r}^2}{c^2}-\frac{r^{2}d{\theta }^2}{c^2}$$$$d{t}^2=\frac{{(1-\frac{2GM}{r{c}^2})}^{-1}d{r}^2+r^{2}d{\theta }^2}{(1-\frac{2GM}{r{c}^2})c^2}$$$$dt=\frac{\sqrt{{\left({\sqrt{1-\frac{2GM}{r{c}^2}}}^{-1}dr\right)}^2+{\left(rd\theta \right)}^2}}{\sqrt{1-\frac{2GM}{r{c}^2}}c}=\frac{d{x}^{\prime }}{c^{\prime }}$$$$dt=\frac{\sqrt{{\left({(1-\frac{GM}{r{c}^2})}^{-1}dr\right)}^2+{\left(rd\theta \right)}^2}}{(1-\frac{GM}{r{c}^2})c}=\frac{d{x}^{\prime }}{c^{\prime }}$$$$dt=\frac{\sqrt{{\left({(1-\frac{GM}{r{c}^2})}^{-1}dr\right)}^2+{\left(rd\theta \right)}^2+{(r\sin \theta d\varphi )}^2}}{(1-\frac{GM}{r{c}^2})c}=\frac{d{x}^{\prime }}{c^{\prime }}$$$$dt=\frac{\sqrt{{\left({\sqrt{1-\frac{2GM}{r{c}^2}}}^{-1}dr\right)}^2+{\left(rd\theta \right)}^2+{(r\sin \theta d\varphi )}^2}}{\sqrt{1-\frac{2GM}{r{c}^2}}c}=\frac{d{x}^{\prime }}{c^{\prime }}$$$$d{r}^{\prime }={\sqrt{1-\frac{2GM}{r{c}^2}}}^{-1}dr$$$$d{r}^{\prime }={(1-\frac{GM}{r{c}^2})}^{-1}dr$$$$d{R}^2+d{r}^2=d{r}^{\prime 2}$$$$dR+dr+d{r}^{\prime }$$$$c+c^{\prime }+v$$$$c^{\prime }=\sqrt{1-\frac{2GM}{r{c}^2}}c$$$$c^{\prime }=(1-\frac{GM}{r{c}^2})c$$$$f^{\prime }=\sqrt{1-\frac{2GM}{r{c}^2}}f$$$$f^{\prime }=(1-\frac{GM}{r{c}^2})f$$$$v=\sqrt{\frac{2GM}{r}}$$$$v=\sqrt{1-{(1-\frac{GM}{r{c}^2})}^2}c$$$$dr=\sqrt{1-\frac{2GM}{r{c}^2}}d{r}^{\prime }$$$$dr=(1-\frac{GM}{r{c}^2})d{r}^{\prime }$$$$dR=\sqrt{\frac{2GM}{r{c}^2}}d{r}^{\prime }$$$$dR=\sqrt{\frac{GM}{r{c}^2}(2-\frac{GM}{r{c}^2})}d{r}^{\prime }$$$$\cos {\varphi }_r=\frac{dr}{d{r}^{\prime }}=\sqrt{1-\frac{2GM}{r{c}^2}}$$$$\cos {\varphi }_r=\frac{dr}{d{r}^{\prime }}=(1-\frac{GM}{r{c}^2})$$$$\cos {\varphi }_r=\frac{dr}{d{r}^{\prime }}=\sqrt{1-\frac{2GM}{r{c}^2}}=\frac{1}{\sqrt{2}}$$$$\cos \frac{\pi }{4}=\frac{dr}{d{r}^{\prime }}=(1-\frac{GM}{r{c}^2})$$$$\cos {\varphi }_r=\frac{dr}{d{r}^{\prime }}=\sqrt{1-\frac{2GM}{r{c}^2}}=\frac{1}{\sqrt{2}}$$$$\sqrt{1-\frac{v^2}{c^2}}=\sqrt{1-\frac{2GM}{r{c}^2}}$$$$\sqrt{1-\frac{v^2}{c^2}}=(1-\frac{GM}{r{c}^2})$$$$R(r)=\int \tan \arccos \sqrt{1-\frac{2GM}{r{c}^2}}dr=2\sqrt{\frac{2GM}{c^2}}\sqrt{r-\frac{2GM}{c^2}}+C$$$$R(r)=\int \tan \arccos (1-\frac{GM}{r{c}^2})dr=\frac{2GM}{c^2}(\sqrt{\frac{2r{c}^2}{GM}-1}-\text{arctanh}\sqrt{\frac{2r{c}^2}{GM}-1})+C$$$$R^{\prime }(2\frac{2GM}{c^2})=1$$$$r_{\pi /4}=\frac{4GM}{c^2}$$$$r_{{45}^{\circ }}=\frac{4GM}{c^2}$$$$r_{{45}^{\circ }}=(2+\sqrt{2})\frac{GM}{c^2}$$$$r_{\pi /2}=\frac{2GM}{c^2}$$$$r_{{90}^{\circ }}=\frac{2GM}{c^2}$$$$r_{{90}^{\circ }}=\frac{GM}{c^2}$$$$\frac{f_{r,v}-f_{r_0,v_0}}{f_{r_0,v_0}}=\frac{f^{\prime }(1-\frac{GM}{r{c}^2})\sqrt{1-\frac{v_r^2}{c^2}}}{f^{\prime }(1-\frac{GM}{r_0{c}^2})\sqrt{1-\frac{v_0^2}{c^2}}}-1$$$$\frac{f_r-f_{r_0,v_0}}{f_{r_0,v_0}}=\frac{f^{\prime }(1-\frac{GM}{r{c}^2})\sqrt{1-\frac{v_r^2}{c^2}}}{f^{\prime }(1-\frac{GM}{r_0{c}^2})\sqrt{1-\frac{v_0^2}{c^2}}}-1$$$$\frac{f_{r,v}-f_{r_0,v_0}}{f_{r_0,v_0}}=\frac{f^{\prime }(1-\frac{GM}{r{c}^2})\sqrt{1-\frac{v_r^2}{c^2}}}{f^{\prime }(1-\frac{GM}{r_0{c}^2})\sqrt{1-\frac{v_0^2}{c^2}}}-1$$$${\overline{\lambda }}_e=\frac{\hslash }{m_{e}c}=\frac{{\hslash }_0}{m_e}$$$$\int \frac{G{m}^{\prime }m}{r_{m^{\prime }}}=\frac{GMm}{R}I=\frac{G{M}^{{}^{″}}m}{R}$$$${\int }_{S^3}\frac{G{m}^{\prime }m}{r}=\frac{GMm}{R}I=\frac{G{M}^{{}^{″}}m}{R}$$$$S{V}_{S^3}=2{\pi }^2{R}^3$$$$M=\mathrm{\Sigma }{m}^{\prime }$$$${\int }_0^{\pi }\frac{2{\sin }^2\theta }{\pi \theta }d\theta \approx 0.775929$$$${\int }_0^{2\pi }\frac{2{\sin }^2\theta }{\pi \theta }d\theta \approx 0.99133$$