Well known result -- gravitational time dilation has to do with the
gravitational potential, not the strength of the field.
Simple gedanken: Drop a rock through a slender shaft into a spherical
hollow cut out of the center of a spherical planet. The rock has more
kinetic energy when it gets to the center of the planet.
Turn the rock (along with its kinetic energy) into photons, and beam
them back up the shaft. At the top of the shaft, catch the beam and
turn it back into a rock.
The rock must have the same mass at the end as it had to start with (or
something's very wrong), which is smaller than the mass it had at the
bottom of the shaft (due its additional kinetic energy which shows up as
a mass excess). This can only be true if the beam of light was *redder*
at the top of the shaft than the bottom. So, there must have been a
gravitational red-shift as the light climbed the shaft.
So, the /frequency/ of the light at the top of the shaft must be *lower*
than the frequency at the bottom of the shaft.
But the *total number of wave crests* in the beam of light can't
change. (You can count them, using appropriate equipment; in that sense
they behave like marbles.) A certain number of wave crests in the beam
entered the shaft at the bottom; the same number of wave crests must
have come out the top.
So, if the /frequency/ measured by an observer at the top of the shaft
is /lower/ than the frequency measured at the bottom of the shaft, the
wave crests must have taken more time to exit the top of the shaft than
they took to enter the bottom of the shaft, and so, /time must be
passing faster for the observer at the top of the shaft.
/
On 12/07/2016 12:53 AM, H LV wrote:
According to the shell theorem the gravitational force on a test mass
inside a hollow sphere is every where zero. This paper argues that
this situation is not equivalent from the standpoint of General
Relativity to the situation where gravity falls to zero far outside
the sphere. They conclude that General Relativity predicts that a
clock located inside a hollow sphere should run slower than a clock
located outside the hollow sphere. (By contrast most people are
familiar with the fact that General relativity predicts a clock should
run faster as the force of gravity approaches zero far from a
gravitational body) This could provide a laboratory test of Newtonian
gravity which predicts that both clocks should run at the same rate.
https://arxiv.org/pdf/1203.4428.pdf
Harry
