Interesting question. Since the frequency of a photon increases as it gains energy on the way into the hollow gravitational sphere one might expect time to speed up for it. If it is allowed to pass through another hole on the other side the time rate would return to the original value once it reaches the same distance away from the sphere in that direction.
This appears to be a paradox of some type. It is common to speak of time slowing down, but a bit strange to think of it as speeding up under some conditions. Wonder where I went wrong with this arguement? Perhaps the photon could bounce around inside the hollow reflective sphere for a long time before exiting an offset hole. Since its frequency is higher while trapped inside it appears that many more cycles of oscillation would take place for this photon than for a brother photon reflecting between two mirrors outside the sphere for the same elapsed time. Would a phase detector comparing the two show anything? Dave -----Original Message----- From: H LV <[email protected]> To: vortex-l <[email protected]> Sent: Fri, Dec 9, 2016 1:42 pm Subject: Re: [Vo]:Newtonian Gravity and General Relativity inside a spherical shell. On Wed, Dec 7, 2016 at 4:04 PM, Stephen A. Lawrence <[email protected]> wrote: 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
