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Mar2-12, 12:02 PM
P: 3,967
See post #1 of this old thread

(Note that the second equation given for the acceleration measured by a local observer is the same as the one given by Peter, just expressed in a different format.)

Just to give you an idea where the equation comes from, start with the equation given by mathpages for the local acceleration in terms of the proper time and coordinate distance:

[tex] \frac{d^2 r}{d\tau^2} = \frac{GM}{r^2} [/tex]

and given that from the Schwarzschild metric the relation between proper time and coordinate time is:

[tex] \frac {d\tau}{dt} = \sqrt{1-2GM/rc^2} [/tex]


[tex] \frac{d^2 r}{d\tau^2}\frac{d\tau^2}{dt^2} = \frac{GM}{r^2}\frac{d\tau^2}{dt^2} [/tex]

[tex] \frac{d^2 r}{dt^2} = \frac{GM(1-2GM/rc^2)}{r^2}[/tex]

which is the acceleration in purely coordinate terms (as given by Bill_K).

Similarly if you convert coordinate distance (dr) to proper distance (dr') you obtain the proper acceleration as measured locally as:

[tex] \frac{d^2 r'}{dt^2} = \frac{GM}{r^2 \sqrt{1-2GM/rc^2}}[/tex]

(as given by PeterDonnis).