The Physical Meaning of gravitational time dilation

[yuiop]

The quotes below (in blue) are from another thread https://www.physicsforums.com/showthread.php?t=364893&page=5" , but since they are a slight digression from the topic of the original thread, I have decided to continue the conversation in this new thread.

Another example. I will call this the gravitational twin's paradox. The twins are initially together at the top of some great tower on a massive body. One twin descends to the bottom of the tower and remains there for some time. Each twin has a clock that emits signals at one second intervals. The twin at the top of the tower sees the clock at the bottom of the tower emit signals at a slower rate than his own clock or in other words he sees the signals from the lower clcok as red shifted. I have seen it argued on this forum that the this is an optical illusion brought about by the "stretching" of the light wavelength of the signal from the lower twin with the implication that the clock of the lower twin is not "really" running slower that the clock of the twin at the top of the tower. However, I guarantee that if the twin at the top of the tower were to descend down to the twin at bottom of the tower at the same rate as the first twin descended, that the twin that had been at the bottom of the tower the longest will have aged the least in a real physical sense...

I am happy to see that someone else has a problem with this common explanation of wavelength stretching due to transit up the gradient.
It is not just an idea presented in this forum but I have found it in accepted explanations of, for instance ; the red shift of light coming up the gravity well to reach earth.
Where it is proposed to be an in transit effect , happening after emission and no mention of the shift due to dilation of the frequency of the emitting electron due to position in the field. {which you seem to be talking about here]

Yes, that it what I am talking about.

I have brought this up in several threads but the response has been they were just two different descriptions of the same phenomena. It seems to me that if the gravitational effect due to locale is valid , and it seems to be verified to a great degree, the a shift due to transit is not merely superfluous but simply invalid. That if it also occurred then there should be an additive quantitative red shift at the receiver beyond what is calculated and explained by the expected shift at emission. Does this make any sense ? SO far nobody has seemed to know what I was talking about.

You might be surprised to know I do know what you are talking about and your reasoning is sound with the given information, but there is actually more to the situation than first meets the eye. Essentially, both phenomena are happening, rather than an either/or situation. The wavelength is increasing in transit AND the frequency is reduced due to physical time dilation of the emitter.

First I will describe a simple physical experiment that could be carried out in a classroom to demonstrate some basic physical principles that are required for the explanation and then I will describe what is "really" happening to a photon as it rises out of a gravitational well and what local observers in the gravitational well actually measure.

The classroom experiment.

Equipment: A flexible track, two steel balls, a tape measure, two clocks and a video camera mounted above the ramp (or a still camera that can be triggered by the steel ball).

Set up: One end of the track is mounted higher than the other end, to provide a ramp of variable curvature for the steel ball to roll down. The tape measure is laid flat on the table the ramp sits on, so that from above the tape measure runs alongside the ramp.

Method: The video camera is switched on and a steel ball is released from the top of the ramp. One second later a second ball is released from the top of the ramp. The video frame as the second ball is released will show the "wavelength" of the two balls, which is the distance the the first ball travels in the one second interval before the second ball is released as measured on the tape measure. The video frame taken as the first ball reaches the end of the track at time T1 will contain the final "wavelength distance" of the two balls. Finally the time T2 that the second ball arrives at the end of the track is recorded. If the track is well supported so that it does not flex and alter the paths as the balls roll down it, then the time (T2-T1) should always be one second, no matter how the track is set up and no matter how the "wavelength" changes during the journey. If the track is set up so that the steepness of the track increases towards the end the wavelength should increase and by some simple analysis it should be be simple to determine that the increase in wavelength is exactly proportional to the ratio of initial average velocity of the two balls to the final average velocity of the two balls.

Conclusions:

1) It is impossible to change the frequency in transit, if two successive balls or (or wave peaks) follow the same path.

2) Any change in wavelength in transit, is always accompanied by a change in velocity of the wave.

(This might make a interesting science fair project for someone).

Photon climbing out of a gravitational well.

Consider two shells in a gravitational well, such that the gravitational time dilation factor (gamma) at shell A low down in the well, is twice the gamma factor at shell B higher up. We know from the Schwarzschild metric that the coordinate speed of light is gamma² and from the experiment described above we know that the physical frequency does not change during transit. Light pulses emitted at 1 second intervals from shell A arrive at shell B at 1 second intervals and the coordinate speed of the light pulses is 4 times greater at B than at A. Therefore the wavelength at shell B is 4 times as long as when it was emitted. However, the rulers of an observer located at shell B are twice as long as the rulers of an observer located at shell A, so B only measures the wavelength to be 2 times as long as the wavelength measured by A lower down. The clocks of observer B are running twice as fast as A's clocks so B measures the interval between pulses to be twice as long as that measured by A and therefore B measures the the frequency of the light pulses to be half that measured by A. So if A measures a frequency of 1 and a wavelength of 1 then B measures a frequency of ½ and a wavelength of 2, so both measure the the local speed of light to be 1 using the equation frequency*wavelength = speed of the wave. In order for both observers to measure an equal value for the local speed of light, it is a requirement that clocks lower down in a gravitational well are physically running slower than clocks higher up and not “just appearing to run slower”.

 

[yuiop]

Thinking about it as little more, the experiment would be better set up if instead of using a tape measure on the table, marks are placed on the track at regular intervals of perhaps one centimeter. A further refinement would be to have a lauch device that launches the balls at a set velocity and a flat sections at the start of the track and at the end of the track, so that both balls have the same velocity when the "wavelength" is being measured.

 

[Jonathan Scott]

I haven't been following the other thread, but yes, provided that the layout is static, the frequency of a regularly repeating signal as seen by a fixed observer does not change in transit, as can be demonstrated in many ways. This applies regardless of the form of the signal and the method of transmission. Gravitational "red shift" is not something which happens to a signal, but is rather an effect of different observers having different clock rates at different potentials.

 

[Austin0]

The classroom experiment.

Equipment: A flexible track, two steel balls, a tape measure, two clocks and a video camera mounted above the ramp (or a still camera that can be triggered by the steel ball).

Set up: One end of the track is mounted higher than the other end, to provide a ramp of variable curvature for the steel ball to roll down. The tape measure is laid flat on the table the ramp sits on, so that from above the tape measure runs alongside the ramp.

Method: The video camera is switched on and a steel ball is released from the top of the ramp. One second later a second ball is released from the top of the ramp. The video frame as the second ball is released will show the "wavelength" of the two balls, which is the distance the the first ball travels in the one second interval before the second ball is released as measured on the tape measure. The video frame taken as the first ball reaches the end of the track at time T1 will contain the final "wavelength distance" of the two balls. Finally the time T2 that the second ball arrives at the end of the track is recorded. If the track is well supported so that it does not flex and alter the paths as the balls roll down it, then the time (T2-T1) should always be one second, no matter how the track is set up and no matter how the "wavelength" changes during the journey. If the track is set up so that the steepness of the track increases towards the end the wavelength should increase and by some simple analysis it should be be simple to determine that the increase in wavelength is exactly proportional to the ratio of initial average velocity of the two balls to the final average velocity of the two balls.

Conclusions:

1) It is impossible to change the frequency in transit, if two successive balls or (or wave peaks) follow the same path.

2) Any change in wavelength in transit, is always accompanied by a change in velocity of the wave.

(This might make a interesting science fair project for someone).

Photon climbing out of a gravitational well.

Consider two shells in a gravitational well, such that the gravitational time dilation factor (gamma) at shell A low down in the well, is twice the gamma factor at shell B higher up. We know from the Schwarzschild metric that the coordinate speed of light is gamma² and from the experiment described above we know that the physical frequency does not change during transit. Light pulses emitted at 1 second intervals from shell A arrive at shell B at 1 second intervals and the coordinate speed of the light pulses is 4 times greater at B than at A. Therefore the wavelength at shell B is 4 times as long as when it was emitted. However, the rulers of an observer located at shell B are twice as long as the rulers of an observer located at shell A, so B only measures the wavelength to be 2 times as long as the wavelength measured by A lower down. The clocks of observer B are running twice as fast as A's clocks so B measures the interval between pulses to be twice as long as that measured by A and therefore B measures the the frequency of the light pulses to be half that measured by A. So if A measures a frequency of 1 and a wavelength of 1 then B measures a frequency of ½ and a wavelength of 2, so both measure the the local speed of light to be 1 using the equation frequency*wavelength = speed of the wave. In order for both observers to measure an equal value for the local speed of light, it is a requirement that clocks lower down in a gravitational well are physically running slower than clocks higher up and not “just appearing to run slower”.

Your experiment was both informative and clear. A little thought makes it apparent that as long as the path length and conditions are the same so must be the elapsed time.
I do have a couple of related questions. WHen you use the term coordinate speed when you refer to the variance at different R's and relative gammas , does this mean there is some coordinate system of clocks and rulers that can possibly measure this change in local speed or do you simply mean that it must be calculated through factoring in the relative gamma effects and cannot be directly measured under any conditions.
Also , though I have encountered the concept of relative speed at varying Schwarzschild radii I am unsure whether this effect is limited to the radial axis or if it is locally isotropic. AS I understand it the length contraction is limited to the axis so logically I would assume that light would be too or there would seem to be a problem with light clocks changing rate if you turned them sideways. But I always like to be sure.
SO thanks for your helpful input.

 

[yuiop]

Your experiment was both informative and clear. A little thought makes it apparent that as long as the path length and conditions are the same so must be the elapsed time.
I do have a couple of related questions. WHen you use the term coordinate speed when you refer to the variance at different R's and relative gammas , does this mean there is some coordinate system of clocks and rulers that can possibly measure this change in local speed or do you simply mean that it must be calculated through factoring in the relative gamma effects and cannot be directly measured under any conditions.

Imagine you had 3 concentric rings A, B and C around a massive body with circumferances such that the calculated distance from A to B is equal to the distance from B to C with A being the smallest ring and C being the largest ring. If mirrors are placed on A and C so that they are vertically aligned then if a signal is sent from B to C and back to B it would take less time than a signal going from B to A and back to B. In other words the difference in the speed of light above B and below B could be detected by a single clock or interferometer at the mid point B. The smaller the vertical distance the less this effect can be noticed and for a very small local region the speed of light is aproximately equal and isotropic.

Although I am using distances as calculated from r = circumferance/2/pi a similar anisotrophy of the two way light speed would be noticed (to a lesser extent) using physical rulers aligned vertically.

Another way that the varying vertical speed of light could be detected is by sychronising clocks vertically, so that signals sent at one second intervals from a clock low down arrive at one second intervals as measured by a clcok higher up. When synchronised like this, observers lower down would notice that the speed of light is slower locally and that the frequency of all natural processes is also slower lower down.

Also , though I have encountered the concept of relative speed at varying Schwarzschild radii I am unsure whether this effect is limited to the radial axis or if it is locally isotropic. AS I understand it the length contraction is limited to the axis so logically I would assume that light would be too or there would seem to be a problem with light clocks changing rate if you turned them sideways. But I always like to be sure.
SO thanks for your helpful input.

Yes, the gravitational length contraction is limited to the vertical axis, but time dilation affects both the vertical and horizontal axis as you would expect from a single clock. The end result is that the vertical coordinate speed of light is slower by gamma^2 while the horizontal coordinate speed of light is slower by just gamma. Again, the local speed of light is measured to be c in all directions and appears to be isotropic, but over larger distances the anistropy would be noticeable.

 

[Al68]

In order for both observers to measure an equal value for the local speed of light, it is a requirement that clocks lower down in a gravitational well are physically running slower than clocks higher up and not “just appearing to run slower”.

This must be correct, since we would obtain this same result analyzing both clocks in an inertial (freefalling) reference frame. In an inertial frame, using only velocity based time dilation, the bottom clock would have a higher gamma than the top clock between any two given events.

As has been pointed out before, the only physical difference between gravitational time dilation and velocity based time dilation is semantical, we call it one or the other based on which reference frame we choose.

 

[Austin0]

=Al68;2521105]This must be correct, since we would obtain this same result analyzing both clocks in an inertial (freefalling) reference frame. In an inertial frame, using only velocity based time dilation, the bottom clock would have a higher gamma than the top clock between any two given events.

It appears you must be talking about an inertial frame falling in a gravitational field.
Where the bottom clock would have a higher gamma due to a lower field potential. Correct?? In an inertial frame outside of a G field this would not be the case, right?

As has been pointed out before, the only physical difference between gravitational time dilation and velocity based time dilation is semantical, we call it one or the other based on which reference frame we choose

It would seem that if you are here, talking about general inertial frames [in flat spacetime] and velocity based time dilation vs g based ,the difference is not merely semantic.
G based dilation does not have the reciprocity that inertial dilation assumes.
Clock A is definitely dilated relative to Clock B at a higher G potential but not the other way around.
Were you actually referring to accelerated systems?

 

[Al68]

It appears you must be talking about an inertial frame falling in a gravitational field.
Where the bottom clock would have a higher gamma due to a lower field potential. Correct?? In an inertial frame outside of a G field this would not be the case, right?

It would seem that if you are here, talking about general inertial frames [in flat spacetime] and velocity based time dilation vs g based ,the difference is not merely semantic.
G based dilation does not have the reciprocity that inertial dilation assumes.
Clock A is definitely dilated relative to Clock B at a higher G potential but not the other way around.
Were you actually referring to accelerated systems?

Gravitational time dilation in an accelerated frame is the same exact velocity based time dilation that would be predicted by analyzing the same two clocks from an inertial (freefalling) reference frame. This is exactly how gravitational time dilation was initially predicted by Einstein, and the basis for the derivation of gravitational time dilation equations.

From an inertial reference frame, the clocks are getting closer and closer together, the bottom clock is moving faster than the top clock. So from an inertial frame, both clocks run slow, but the bottom clock of the rocket is running slower than the top clock due to velocity based time dilation.