Seiryuu

2008-Oct-12, 01:04 AM

A small math question. Can anyone tell me if I'm allowed to do the following:

Starting from a formula from Wikipedia

http://upload.wikimedia.org/math/a/8/2/a82acb0200bb91d7dabec509a3c5f0d4.png

where

t is time as calibrated with a clock distant from and at inertial rest with respect to the Earth,

r is a radial coordinate (which is effectively the distance from the Earth's center),

θ is the latitudinal coordinate, being the angular separation from the north pole in radians.

phi is a longitudinal coordinate, analogous to the latitude on the Earth's surface but independent of the Earth's rotation. This is also given in radians.

m is the geometrized mass of a central massive object, being m = MG/c2,

M is the mass of the object,

G is the gravitational constant.

When standing on the north pole, we can assume dr = dtheta = dphi = 0 (meaning that we are neither moving up or down or along the surface of the Earth)

Giving the following formula

http://upload.wikimedia.org/math/2/1/7/217d81c5cd35bcf723bd60b769c21f4a.png

So we have

dτ = dt(1 - 2GM/rcē)^1/2

dτ/dt = (1 - 2GM/rcē)1/2

I use the variable Td for the left part. So

Td = dτ/dt

And we get

Tdē = 1 - 2GM/rcē

2GM/rcē = 1 - Tdē

G = (1 - Tdē)rcē / 2M

Inserting it in F = GMm/rē we get

F = (1-Tdē)rcēMm/2rēM

F = (1-Tdē)cēm/2r

Which leads to

F = mcē(1-Tdē)/2r

Is this allowed or am I making some stupid errors here?

Starting from a formula from Wikipedia

http://upload.wikimedia.org/math/a/8/2/a82acb0200bb91d7dabec509a3c5f0d4.png

where

t is time as calibrated with a clock distant from and at inertial rest with respect to the Earth,

r is a radial coordinate (which is effectively the distance from the Earth's center),

θ is the latitudinal coordinate, being the angular separation from the north pole in radians.

phi is a longitudinal coordinate, analogous to the latitude on the Earth's surface but independent of the Earth's rotation. This is also given in radians.

m is the geometrized mass of a central massive object, being m = MG/c2,

M is the mass of the object,

G is the gravitational constant.

When standing on the north pole, we can assume dr = dtheta = dphi = 0 (meaning that we are neither moving up or down or along the surface of the Earth)

Giving the following formula

http://upload.wikimedia.org/math/2/1/7/217d81c5cd35bcf723bd60b769c21f4a.png

So we have

dτ = dt(1 - 2GM/rcē)^1/2

dτ/dt = (1 - 2GM/rcē)1/2

I use the variable Td for the left part. So

Td = dτ/dt

And we get

Tdē = 1 - 2GM/rcē

2GM/rcē = 1 - Tdē

G = (1 - Tdē)rcē / 2M

Inserting it in F = GMm/rē we get

F = (1-Tdē)rcēMm/2rēM

F = (1-Tdē)cēm/2r

Which leads to

F = mcē(1-Tdē)/2r

Is this allowed or am I making some stupid errors here?