- #1
Anamitra
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We consider the geodesic equation:
(1):
[tex]
\frac{ d^2 x^\alpha}{ d \tau^2} =
- {{\Gamma}^{\alpha}}_{\beta \gamma} \frac{d x^{\beta}}{d{\tau}} \frac{d x^{\gamma}}{d {\tau}}
[/tex]
For radial motion in Schwarzschild geometry
(2):
[tex]\frac{d^2 r}{d \tau^2} =
- {M / {r^2}} { (1 - {{2M} {/} {r}}) }
{( \frac{dt}{d \tau} )}^2
+ {M} {/}{r^2}
{( 1- {2M}{/}{r} )}^{-1}
{( \frac{dr}{d \tau} )}^2[/tex]
Again for radial motion we have
(3):
[tex]{d}\tau^{2} =
{(}{1}{-}{2M}{/}{r}{)}{dt}^{2}
{-}{{(}{1}{-}{{2M}{/}{r}}{)}}^{-1}{dr}^{2}[/tex]
Dividing both sides of equation (3) by [tex]{{d}{\tau}}^{2}[/tex] we have,
[tex]{1}{=}{(}{1}{-}{{2M}{/}{r}}{)}{{(}{\frac{dt}{{d}{\tau}}}{)}^{2}{-}{{(}{1}{-}{{2M}{/}{r}}{)}}^{-1}{(}{\frac{dr}{{d}{\tau}}{)}}^{2}}[/tex] ------------------------------ (4)
Using relation (4) in equation (2)
[tex]{\frac{{d}^{2}{r}}{{d}{\tau}^{2}}}{=}{-}{\frac {M}{{r}^{2}}}[/tex]
The inverse square law is valid accurately if proper time is used.Here 'r' represents coordinate distance along the radius
(1):
[tex]
\frac{ d^2 x^\alpha}{ d \tau^2} =
- {{\Gamma}^{\alpha}}_{\beta \gamma} \frac{d x^{\beta}}{d{\tau}} \frac{d x^{\gamma}}{d {\tau}}
[/tex]
For radial motion in Schwarzschild geometry
(2):
[tex]\frac{d^2 r}{d \tau^2} =
- {M / {r^2}} { (1 - {{2M} {/} {r}}) }
{( \frac{dt}{d \tau} )}^2
+ {M} {/}{r^2}
{( 1- {2M}{/}{r} )}^{-1}
{( \frac{dr}{d \tau} )}^2[/tex]
Again for radial motion we have
(3):
[tex]{d}\tau^{2} =
{(}{1}{-}{2M}{/}{r}{)}{dt}^{2}
{-}{{(}{1}{-}{{2M}{/}{r}}{)}}^{-1}{dr}^{2}[/tex]
Dividing both sides of equation (3) by [tex]{{d}{\tau}}^{2}[/tex] we have,
[tex]{1}{=}{(}{1}{-}{{2M}{/}{r}}{)}{{(}{\frac{dt}{{d}{\tau}}}{)}^{2}{-}{{(}{1}{-}{{2M}{/}{r}}{)}}^{-1}{(}{\frac{dr}{{d}{\tau}}{)}}^{2}}[/tex] ------------------------------ (4)
Using relation (4) in equation (2)
[tex]{\frac{{d}^{2}{r}}{{d}{\tau}^{2}}}{=}{-}{\frac {M}{{r}^{2}}}[/tex]
The inverse square law is valid accurately if proper time is used.Here 'r' represents coordinate distance along the radius
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