...
GR explains the difference in observations of clock tick rates between two different frames by transforming the measurements of distance from one frame to the other. The equations include the constant speed of light as a common denominator and thus, they calculate what the difference in distance measurements would have to be between frames in order to make the speed of light come out the same, and thus to explain the observed difference in clocked time by a difference in the distance light has to travel. Yes?
...
I got no help with this explanation and so either it is perfect or no one cares if it is wrong.
I'm still testing my understanding of it though, so I will expand the explanation in my own words to see if anyone will correct or acknowledge it.
In this example, the existence of two inertial reference frames simply means that there is relative motion between the frames. We can have a single clock that exists in both frames. We can have an observer in each frame who observes that one clock. Each observer is in motion relative to each other, but in this example observer A is at rest relative to the common clock and observer B is in motion at a constant velocity away from the common clock as he observes it.
They both have wrist stopwatches and observer A's watch stays at rest relative to the common clock while observer B's watch moves with him in his frame. They pre-plan the distance that observer B is going to travel during the experiment and calculate how long he will have to travel to go that distance. The mile markers are laid out along the planned path in advance. Both observers will stop their watches when the planned time has elapsed in their local frame. For observer B, that elapsed time is expected to correspond to the planned distance and will occur at the planned mile marker.
Observer B sets out on his planned motion relative to the common clock.
When they compare the passing of time on their watch to the passing of time on the common clock, observer A will see no difference while observer B will notice that the common clock appears to be ticking slower than his wristwatch.
During his motion relative to the common clock, observer B measures the wavelength of the light coming from the common clock and assigns it a value x and he measures the wavelenght of the light coming from his watch and assigns it value y. The difference between x and y corresponds to the redshift in the light coming from the common clock vs the light coming from is local wristwatch.
Observer A who is standing next to the clock and whose wristwatch is the same distance from his eye as the common clock is from his eye, measures the wavelength of the light from both the clock and his watch and assigns a value x and y respectively. In his case x = y because there is no relative motion and thus no redshift between his watch and the common clock.
When observer B reaches the agreed planned mile marker, observer B stops his wrist stopwatch, he stops and sets it down on the pavement at the agreed mile marker. Also when the agreed planned time has elapsed according to observer A's wrist stopwatch, he stops his watch as well.
Leaving his stopwatch behind but taking his recored wavelength values back with him, observer B then goes back to the common clock and meets up with observer A. They compare their measured wavelength values of the light coming from the common clock. Observer B's difference in measured wavelength coming from the common clock is attributed to red-shifting. They evaluate the results as follows:
They conclude that time didn't slow down and that light travels at the same velocity in all frames.
Observer B observed a slower tick rate of the common clock and states that it appeared to run slower while he was moving away from it.
Observer A confirmed that the common clock did not slow down compared to his wristwatch, and so they jointly concluded that the appearance of slowing of the common clock from observer B's frame could be explained by the relative motion.
They had data to quantify that relative motion because observer B has the wavelength difference recorded during his period of relative motion, and observer A observed no wavelength difference.
They decide that the wavelength difference recorded by observer B must correspond to the distance travelled during B's relative motion, and they decide to divide the distance by the change in wavelength to quantify the relationship, giving them a meaningful equation that relates motion to distance and gives them the expected difference in wavelength that will be observed during that motion.
They write it up and explain that since the light leaving the clock face had further to travel to meet observer B's eye, then during his motion away from the clock the common clock appears to slow down. Observer A standing by the common clock sees the time correspond with his watch and knows for a fact that the common clock did not slow down, so they attribute the difference in observed tick rate of the common clock to the change in the wavelength that was recorded by observer B.
Question: Will the elapsed time on the stopwatch left behind by observer B agree with the elapsed time on observer A's stopwatch?
If not, why not?
