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Jun14-10, 05:06 PM
P: 3,967
Quote Quote by kev View Post
P.S. There still seems to be a problem with (13) derived from (9) but I do not have time to check that at the moment.
I have had another look at your revised document and checked all the calculations from (9) onwards and they all seem to be correct. Your final result can be simplified to:

[tex]\frac{rd\phi}{ds} = c \sqrt{\frac{GM}{rc^2 - 3GM}[/tex]

which gives the correct result that the velocity with respect to proper time (ds) of a particle orbiting at r=3GM/c^2 is infinite.

If you convert the above equation to local velocity as measured by a stationary observer at r by using [tex]ds = dt'\sqrt{1-(rd\phi)^2/(cdt')^2}[/tex] you get:

[tex]\frac{rd\phi}{dt'} = c \sqrt{\frac{GM}{rc^2 - 2GM}[/tex]

which gives the result that the local velocity of a particle orbiting at r=3GM/c^2 is c.

If you convert the equation to coordinate velocity using [tex]dt = dt'\sqrt{1-2M/(rc^2)}[/itex] you get:

[tex]\frac{rd\phi}{dt} = c \sqrt{\frac{GM}{r}[/tex]

which is the same as the Newtonian result (if you use units of c=1).

All the above are well known solutions, so it seems all is in order with your revised document (from (9) onwards anyway - I have not checked the preceding calculations).