From: titang78@gmail.com   
      
   a)	Material clock   
      
   What is time? This question is tricky because in relativity time-rate   
   changes when frame of reference changes. Time-rate changing is puzzling   
   because it is in conflict with our intuition that time is the flow of   
   ticks of clocks of which the mechanical structure does not change. Then   
   how to clearly explain the contradiction between the constant flow of   
   ticks delivered by clocks and the relativistic time dilation?   
      
   In order to grasp the essence of time-rate, we have to understand the   
   fundamental property of time. In still image, there is no time. When we   
   see scenes of cinema time emerges. Time emerges in moving scenes because   
   objects in the scenes change. So, the fundamental property of time is   
   the ability of objects to change state in moving image as well as in   
   reality.   
      
   For recording the rate of change of objects, human has invented clock,   
   the work of which is the change of state of the clock itself. For   
   example, in Figure the hands of the clock change position, the pendulum   
   changes position. Clocks record time by counting the number of times   
   that one part passes a specific state, for example, the big hand at the   
   number 12 on the dial. If the big hand has passed n times this state, we   
   say that the recorded time is n hours.   
      
   From the principle of work of clock, we extract the fundamental function   
   of clocks: counting the number of times an oscillating object passes by   
   a fixed point in space. This is true for archaic sundial as well as for   
   modern quartz clock which makes a quartz tuning fork to vibrate around   
   its neutral position.   
      
   So, all material chocks can be represented by the abstract clock in   
   Figure, which is formed by a material point k oscillating between the   
   ends of the short rod a and b. The motion of k is characterized by the   
   length of the rod. Let us refer to this abstract clock as “k clock”. The   
   time recorded by the k clock is the number of times that k strikes the   
   point a. We define one tick of time delivered by this clock to be one   
   strike.   
      
   Below, we will show how the time-rate of the k clock changes while ticking at   
   the same rate. For doing so, we pair it with a light clock.   
      
   b)	Paired with a light clock   
      
   In relativity light is the reference to all motion, so we make the light   
   clock in Figure which is formed by a photon bouncing back and forth   
   between the two mirrors Ma and Mb at the end of the long rod. In order   
   to calibrate the rate of the k clock, we synchronize it with the light   
   clock by matching the length of the short rod with that of the long rod   
   such that if k starts from the point a simultaneously with the photon   
   from Ma, k gets back to “a” simultaneously with the photon back to   
   Ma. So, the k clock is synchronous with the light clock and they make   
   one pair of “k clock - light clock”.   
      
   The time recorded by the light clock is the number of strikes the photon   
   makes on the mirror Ma. As k clock is synchronous with the light clock,   
   the number of strikes k makes on the point a always equals the number of   
   the photon’s strikes. The lengths of the rods and the identically   
   repetitive motion of k stay the same for whatever motion they are   
   in. This way, when the pair of “k clock - light clock” of Figure is   
   brought into motion, they are always synchronous.   
      
   The flow of ticks is the intrinsic tick-rate of a clock. Because the   
   length of the short rod and the motion of k do not change, the intrinsic   
   tick-rate of a material clock does not change either. But the time-rate   
   they show can change due to motion, which we will see below.   
      
   c)	Time-rate change   
      
   Let us take 2 frames of reference frame 1 and frame 2, frame 2 moves at   
   constant speed in frame 1. In order to show the relativistic change of   
   time-rate of frame 2, we will put one pair of “k clock - light clock” in   
   frame 1 and an identical one in frame 2, see Figure. If we stand in   
   frame 1 and look the pair of this frame, then we stand in frame 2 and   
   look the pair of this frame, we will not detect any difference, which   
   shows that material clock does not change when jumping frame.   
      
   Then, why is the time-rate of frame 2 different from that of of frame 1?   
   Let us see Figure in which a pair “k clock - light clock” moves with   
   frame 2 in frame 1. In frame 2 the photon goes straight upward. But due   
   to the motion of the light clock, the path of the same photon is slanted   
   in frame 1. Let us denote the length of the path (back and forth) in   
   frame 1 with L1 and that in frame 2 with L2. Because the path in frame 1   
   is slanted, L1 is longer than L2.   
      
   One strike of the photon indicates that it has done the distance L2 once   
   in frame 2. Meanwhile, the same photon has done the distance L1 in frame   
   1, see Figure. Suppose that we have counted n2 strikes, then the photon   
   has done n2 times the distance L1 in frame 1, which makes the length of   
   its total path to equal S1=n2L1, see equation.   
      
   For counting the time passed in frame 1 during the n2 strikes, we count   
   the ticks given by the identical pair “k clock - light clock” in frame   
   1, see Figure. Within the same frame, light travels simultaneously the   
   same distance in all direction. Then, during the n2 strikes the photon   
   of frame 1 will also do the distance S1. Because the length of the long   
   rod is also L2 in frame 1, this photon will strike n1=S1/L2 times and S1   
   also equals n1L2, see equation. Then, we find in equation that n1 = n2   
   L1/L2. As L1>L2, the number n1 is bigger than n2.   
      
   Notice that n1 and n2 concern only the length of the photon’s paths, not   
   time. For knowing the time-rate in frame 1 and 2, we define the quantity   
   of time passed as the number of ticks delivered by light clocks which   
   equals the number of strikes by their respective photons. As n2 ticks is   
   delivered by the one of frame 2, the quantity of time passed in frame 2   
   equals n2 ticks. Simultaneously, the photon of the light clock of frame   
   1 has struck n1 times, so the quantity of time passed in frame 1 equals   
   n1 ticks, see equation and.   
      
   So, when the light clock of frame 2 delivers n2 ticks, simultaneously   
   the light clock of frame 1 delivers n1 ticks. If 2 clocks deliver   
   different number of ticks simultaneously, we say that the one that   
   delivers fewer ticks is slower. Using this image, we say that time is   
   slower in frame 2 than in frame 1 because n2 is smaller than n1. But   
   “time slowing” is only an image to describe this phenomenon, it is not   
   an appropriate term and it confuses people for understanding relativity.   
      
   Notice this difference: the n2 ticks are delivered by the light clock of   
   frame 2 but we count them in frame 1, the n1 ticks are delivered by the   
   light clock of frame 1 and also counted in frame 1.   
      
   d)	Moving material clock   
      
   What about the moving k clocks? As it is synchronized with the paired   
   light clock, the number of ticks it delivers equals that of the paired   
   light clock and the k clock of frame 2 delivers fewer ticks than that of   
      
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