Answer 1:
Gravitational time dilation and standard
relativistic time dilation are actually measurably
different. Nonetheless, it is possible to
determine the velocity required to create a
special relativistic time dilation equivalent to
the time dilation due to a gravitational
potential. This played a critical role in one of
the early precision tests of General Relativity
performed by Pound and Rebka. The time
dilation due to motion at a velocity v is
given by
tSlow = Sqrt( 1 
v^{2}/c^{2}
)tFast
while the time dilation due to
being a distance r from an object of mass
Mis given by
tSlow = Sqrt( 1 
2GM/rc^{2} ) tFast In their
experiment, Pound and Rebka balanced the
gravitational blue shift of a photon falling down
an elevator against the doppler redshift of a
target moving away from the photon by adjusting
the velocity of the target. By finding the
velocity at which no shift in frequency
occurred,they were able to measure the
gravitational blue shift. The
gravitationally blue shifted frequency (after
falling down an elevator of height h) is given
by
fGravBlue = f0 Sqrt( 1 
2GM/(r+h)c^{2} ) / Sqrt( 1 
2GM/rc^{2} ) The doppler
redshift due to moving at a velocity v is given
by
fDopplerRed = f0 Sqrt( 1  v/c ) /
Sqrt( 1 + v/c ) Thus, Pound and Rebka
needed to find the velocity v such
that
1 = ( Sqrt( 1 
2GM/(r+h)c^{2} ) Sqrt( 1  v/c ) ) / (
Sqrt( 1  2GM/rc^{2} )Sqrt(1 + v/c )
) They found that they needed v =
gh/c, which confirmed the prediction of
General Relativity.
