Gravity Effects on Rod Length - Reference Sources

[Passionflower]

The energy of a test observer in a Schwarzschild solution is constant as long as it travels on a geodesic.

You can use the following equality:
\Large E=\sqrt {1-{\frac {{\it r_s}}{r}}}{\sqrt {1-{v}^{2}}} ^{-1}
So if you know the energy at one location it is very easy to calculate either the velocity or position at another location or velocity.

For instance it is easy to see that the energy of an observer free falling at escape velocity is exactly 1, just plugin v_escape in the equation above.

From the energy we can get the apogee of a test radial observer, because when the velocity is zero we get:
\Large E=\sqrt {1-{\frac {r_{{s}}}{r_{{{\it ap}}}}}}<br />
Or alternatively we can get the apogee from the velocity at a given r value:
\Large {\frac {r_{{s}}}{r_{{{\it ap}}}}}=- \left( {\frac {r_{{s}<br /> }}{r}}-{v}^{2} \right) \left( {v}^{2}-1 \right) ^{-1}<br />

 

[jk22]

I think you're missing the point.

In general relativity, the metric coefficent for g_00 can be apprxoimated as (1-2U), where U is the Newtonian potential.

It's the metric coefficeint which causes time dilation and it's not velocity dependent.

In sepcial relativy, the metric coefficeints are always unity. So the time dilation in SR isn't coming from the metric coefficients.

If 1-2U is an approximation, Do we know the exact form of g_00 ?

 

[pervect]

The exact form of g_00 is (1-2M/r) in Schwarzschild coordinates, and geometric units. Of course it's worth remembering that often other coordinates are used. For instance isotropic coordinates, which make the speed of light the same in all directions for small M/r and are used in the PPN approximations would make

g_00 = [(1-M/2r) / (1+M/2r) ]^2

This still approaches 1-2M/r if you series expand it in 1/r, however ...

(1-2M/r + 2M^2/r^2 - (3/2) M^3 / r^3 + ...)

(1-2U) is symbolically identical to the first expression in Schwarzschild coordinates, but the r in that expression stands for the Newtonian radius, not the Schwarzschild r coordinate, so the resemblance is formal and not an identity.