Dale said:
The time dilation formula says that a moving clock's proper time is slower than the coordinate time of a set of synchronized clocks. So the orange clock's proper time is slow compared to the apple system of synchronized clock's coordinate time and the apple clock's proper time is slow compared to the orange system of synchronized clock's coordinate time.
Proper time is time in observer’s reference frame or rest frame, it is invariant. It is better to say that time interval measured by single moving in reference frame ##K## clock will be shorter than time interval measured by two (at least) Einstein – synchronized clocks of reference frame ##K##. Sure, in every point ##x## of reference frame ##K## moving clock will show less time than any clock of reference frame
pic upload
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Also good picture is there (chapter time dilation) http://www.pstcc.edu/departments/natural_behavioral_sciences/Web%20Physics/Chapter039.htm
because it shows two clocks in the point of departure and point of arrival of light clock. This is small but IMHO very important detail. Quite often popular books omit this detail.
Moving clock C starts at clock A and arrives to clock B. If readings of clock C and A were the same (for example 12) , readings of clock B and clock C (when the clocks are in immediate vicinity) will be different (for example C will show 3PM and B will show 6PM).
If observer C wants to measure time dilation of clock A, he must hire an assistant and add another clock C2 at another spatial position in his reference frame and synchronize that clock. He can use Einstein convention (i.e. velocity of light to that clock will be c). Then time interval measured by clock A will be shorter than time interval measured by clocks C and C2.
Animation is here:
https://en.wikipedia.org/wiki/Time_dilation#/media/File:Time_dilation02.gif
This set of clock is often named as a REST FRAME of observer. Velocity of light in the rest frame is the same in all directions.
Every observer in Special Relativity conducts measurements by introducing his own rest frame, by means of hiring assistants and filling space with Einstein - synchronized clocks.
Thus, time dilation can be measured by a TEAM of observers.
We can imagine a row of synchronized clocks of reference system K. each denoted by letter - A, B, C, D ….. Z. Then a person with single clock on his wrist (A’ for example) moves in this reference system K and compares readings of his clock with these clock A-Z successively. When he comes to the clock Z, his clock A’ shows gamma times less time, than clock Z. Thus, clock Z shows gamma times more time, than his own.
At this point Z this clock A’ immediately turns back and starts traveling in reverse direction, passing by clock Z, Y, X ….. C, B and finally arrives into point A of reference frame K. Clock A’ compares readings with clocks Z-A successively again and sees, that it dilates itself gamma times, i.e. every clock on the way shows gamma times more time. When clock A’ arrives into point A, clock A’ shows gamma times less time than clock A, and clock A shows gamma times more time than clock A’.
This discrepancy of clock readings when they meet again is often called clock paradox or Twin paradox.
Mister T said:
One observation can't cancel out another. They are separate observations made by separate observers. Each will observe the other's clock to be running slow compared to theirs. This seems like a paradox with no solution, but the solution lies in an understanding of simultaneity. If the apple is to measure the rate of the orange's clock, then the orange will need two clocks. This is because you need two events so you can measure the amount of time that elapses between them. Orange will need one clock next to the apple for the first event, and since the orange is moving it'll need another clock that will be next to the apple for the second event. Orange will have to synchronize these clocks, but the apple will not agree that he has done so correctly because events that are simultaneous for the orange are not simultaneous for the apple. The difference explains why apple observes orange's clocks to run slow when orange knows that apple's clocks are running slow.
By symmetry you can do the same with the apple and consider it in motion so that it will need two clocks.
Vice versa. If apple will do as you proposed, apple will see that orange clocks run faster, as I mentioned above. If apple wants to measure dilation of orange’s clock, apple should do that from its own rest frame, i.e. by means of putting two synchronized clocks at points of departure and arrival. It is the source of confusion very, very often. We say that object moves if he changes spatial position in our frame.
All that goes straight from the Lorentz transformations.
## T = \frac {t'_{x'}+ \frac {v'} {c^2} x'} {\sqrt {1-( \frac {v} c)^2}} ## (1)
##T## is clock readings that belongs to reference frame ##K##, taken in point ##x'## at moment of time ##t'_{x'}## of reference frame ##K'##, and ##t'_{x'}## reading of clocks that belongs to reference frame ##K'## in the point ##x'## of reference frame ##K'##
How to interpret Lorentz transform for time?
Transformation demonstrates, that time ##T## of reference frame ##K## (in which it does not depend of ##x## coordinate or any other coordinate) is universal in reference frame ##K## and each point of this frame.
Now let's fix point ##x'##, for example ##x'=0##. In this case this transformation will look like that:
## T= \frac {t'_{0'}} {\sqrt {1-( \frac v c)^2}}## (2)
##T## is clock reading of reference frame ##K## taken in point ##x'=0## (in the origin ##O'## of reference frame ##K'##), and ##t'_{o'}## is time in the reference frame ##K'##, in particular in the origin ##O'##.
We can take ## \frac {dT} {dt'}## when ##x'## is fixed and will get ##{dT}/{dt'} = \frac 1 {\sqrt {1- \frac {v^2} {c^2}}} ##
According to (2) it is not time ##t'_{o'}## which is showed by single clock in the point ##O'## runs slower, but time ##T## , which is "distributed" through all reference frame ##K## and taken in the origin ##O'## of reference frame ##K'## runs faster (relatively to time ##t'_{o'}## that is in the origin ##O'## of frame ##K'##). Time dilation comes by means of transfromation of (2) into:
##t'_{o'}=T \sqrt {1- {\frac {V^2} {c^2}}} ##
It is correct that ##T>t'## and ##t'<T##. It is also true that ##T'>t## and ## t<T'##. But that ##t<t'## and ##'t<t## from different points of view is nonsense.
Orodruin said:
No it isn't. In order to understand this (taking away the "travelling at the speed of light part" and replacing it for "moving very fast relative to") you need to understand the relativity of simultaneity. Events that are simultaneous for the apple are not (necessarily) simultaneous for the orange. Not taking relativity of simultaneity into account is the main source of many of the apparent paradoxes that you will find in relativity (which really are not paradoxes once you understand the theory).
But, you can measure ticking rate of clock which is very, very far away from you if that clock emits radiation (light in visible spectrum, for example). You have just to look at it (or direct your measuring device, or tube) when this source of radiation and you are at points of closest approach.
You have to do nothing special, just to direct you tube (or gaze) transversely to direction of motion of the source and wait. For example, you stay at point Y of y-axis and look into origin.
Soon or later source passes the origin and you see a brief flash in your tube. This flash will be of different color than proper color of the source, it will be reddish due to dilation of source’s clock, i.e. moving clock oscillates slower. This is so called transverse Doppler Effect.
Of course, due to light – time correction apparent position of the source is different from its actual position and when you see the light the source has already moved into another place.
This measurement can be (and had been) actually done and doesn’t need consideration of spatially separated and synchronized clocks.
Interesting to note that time dilation had been measured with high precision by means of Mossbauer Effect based rotor experiments (Champeney and Moon, Hay et all, Kundig, Kholmetskii and Yarman, Fridman et all). Clock hypothesis says that clock dilates purely due to relative motion and not acceleration.
In these experiments they place source of radiation in the center, and absorber on the rim of rotating disc (or vice versa) and measure transverse Doppler Effect. Results show, that if absorber rotates it measures blueshift of radiation, if absorber is in the center it measures redshift of radiation. That means, that rotating clock (absorber or source) ACTUALLY dilates.
Distance between source and absorber doesn’t change. Relative to what absorber moves and dilates?
Good to note, that if source and absorber are placed on opposite sides of the disc, they will not measure any frequency shift (any time dilation), since both dilate at the same magnitude. That was confirmed by Champeney and Moon in 1961. Though it is 100% clear. Source and absorber have equal linear velocities, and energy of photon cannot vanish to nowhere.
The same effect (absence of Transverse Doppler effect) will be in case of rectilinear motion, if source and observer think that they move at equal relative velocities. In this case source has launch a flash when the both source and observer will be at equal distances from the normal between their patches of motion. In this case source launches a photon at oblique angle to direction of its motion backward. An observer tilts detector forward at the same angle. They have to do that because of aberration of light. These angles are connected by aberration of light formula,
Moving observer will see, that source appears in the front and radiation frequency becomes higher (more blue). The faster he moves, the more oblique angle at which source appears to him will be. Thus, observable clock runs faster and faster because observer's own clock dilates. This effect is often called relativistic beaming.
