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ChrisVer

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Starting from a locked thread I tried to work the gravity of a body of mass ##M## on another body starting from infinity to some distance ##d## from the gravitating body.

We have from the SR 2nd Newton law that:

[itex]\gamma^3 a = \frac{GM}{r^2}[/itex]

writting [itex]a= \frac{dv}{dt}= v \frac{dv}{dr} = \frac{1}{2} \frac{dv^2}{dr}[/itex]

Naming [itex]v^2/c^2= x[/itex] the above relation becomes:

[itex] \frac{dx}{(1-x)^{3/2}}= \frac{2GM}{c^2 r^2}dr[/itex]

Integrating:

[itex] \int_{0}^{x_d}\frac{dx}{(1-x)^{3/2}}= \int_{\infty}^{d} \frac{2GM}{c^2 r^2}dr[/itex]

[itex]\frac{2}{\sqrt{1-x_d}}-2= - \frac{2GM}{c^2 d}[/itex]

[itex]x_d=1- \Big( 1- \frac{GM}{c^2 d}\Big)^{-2}[/itex]

And so:

[itex]v(d)= c \bigg[ 1- \Big( 1- \frac{GM}{c^2 d}\Big)^{-2} \bigg]^{1/2} [/itex]

I tried plotting this solution as [itex]v(d)[/itex], the good part is that [itex]v<c[/itex] for all distances however I don't understand why for [itex]d=GM/c^2[/itex] I'm obtaining an infinity (and worse- in the imaginary regime)? By the way, that's the Schwarzschild radius...

Even worse, if I set [itex] \frac{GM}{c^2}=1[/itex] the plot of [itex] \beta = v/c = \bigg[ 1- \Big( 1- \frac{1}{d}\Big)^{-2} \bigg]^{1/2} [/itex] is given in my attachment...and doesn't seem to have a real solution away from 1?

We have from the SR 2nd Newton law that:

[itex]\gamma^3 a = \frac{GM}{r^2}[/itex]

writting [itex]a= \frac{dv}{dt}= v \frac{dv}{dr} = \frac{1}{2} \frac{dv^2}{dr}[/itex]

Naming [itex]v^2/c^2= x[/itex] the above relation becomes:

[itex] \frac{dx}{(1-x)^{3/2}}= \frac{2GM}{c^2 r^2}dr[/itex]

Integrating:

[itex] \int_{0}^{x_d}\frac{dx}{(1-x)^{3/2}}= \int_{\infty}^{d} \frac{2GM}{c^2 r^2}dr[/itex]

[itex]\frac{2}{\sqrt{1-x_d}}-2= - \frac{2GM}{c^2 d}[/itex]

[itex]x_d=1- \Big( 1- \frac{GM}{c^2 d}\Big)^{-2}[/itex]

And so:

[itex]v(d)= c \bigg[ 1- \Big( 1- \frac{GM}{c^2 d}\Big)^{-2} \bigg]^{1/2} [/itex]

I tried plotting this solution as [itex]v(d)[/itex], the good part is that [itex]v<c[/itex] for all distances however I don't understand why for [itex]d=GM/c^2[/itex] I'm obtaining an infinity (and worse- in the imaginary regime)? By the way, that's the Schwarzschild radius...

Even worse, if I set [itex] \frac{GM}{c^2}=1[/itex] the plot of [itex] \beta = v/c = \bigg[ 1- \Big( 1- \frac{1}{d}\Big)^{-2} \bigg]^{1/2} [/itex] is given in my attachment...and doesn't seem to have a real solution away from 1?

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