1 Time-rate change in relatively moving frames

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 frame 1 too, although the 2 'k clocks' are identical, which means that material clock shows slower time-rate when moving while keeping the same mechanical structure.

If we really want to find what object causes time to slow, we would say the culprit is our standpoint. The path of the photon is straight upward when we see it in frame 2. The path of the same photon is slanted when we see it in frame 1. So, it is our standpoint that makes the path to appear slanted and longer, which makes it to contain more ticks. In consequence, the intrinsic mechanical structure of clocks and time itself do not change, only their appearance changes depending on our standpoint.

2 Time-rate change in relatively moving frames

This is incorrect. Using standard English words as they apply to physics, the "time-rate" is the same in every locally inertial frame -- it never "changes".

[It is best to avoid such wishy-washy phrases as "time rate". Talk instead about definite, unambiguous, and directly measurable quantities such as clock tick rates.]

Einstein's first postulate, solidly confirmed experimentally, implies that clocks always tick at their usual (standard) rate, regardless of where they are located or how they might be moving (because the laws of physics that govern their ticking are the same). Since "Time is what clocks measure [Einstein and others]", this also applies to "time rate".

The rest of your article is useless because it fails to recognize this very basic and fundamental aspect of relativity.

Tom Roberts

3 Time-rate change in relatively moving frames

Regarding above. There exists no usual standard (clock) rate. The rate of the clock being defined based on the number of clock counts divided by elapsed time. Elsapsed time being defined, as measured by an optical atomic clock.
Each clock ticks as designed. Two identical clock tick at the same rate keeping both their same position. Two identical clocks, using lightsignals, moving from A to B following a different path (in length) will not.

• The operation of a nuclear optical clock based on the energy of Thorium-229 is discussed in Nature Vol 573 of 12 September 2019.
  1. At page 202 in News & Views: "One tick closer to a nuclear clock" by Jason T Burke.
  2. At page 238 in the article: "X-ray pumping of the Th-229 nuclear clock isomer" by Takahiko Masuda ea.
  3. At page 243 in the article: "Energy of the TH-229 nuclear clock transition" by Benedict Seiferle ea.
• A different type of optical clock is discussed in article "Operation of an optical atomic clock with a Brillouin laser subsystem" by William Loh e.a. in Nature Vol 588, page 244, 10 December 2020.

A more important comment is that the internal functioning of an the atomic clock is not governed by the laws of physics. The same is true for any process including a an atomic reactor or the LHC. It is the purpose of the LHC to unravel the details of the chemical reactions between elementary particles, when they collide . Often these details can be described using mathematics, but the mathematics pure, are no explanation why reactions are the way they are.

The functioning of an atomic clock is an internal coordinated result of an accurate design of many intricate moveble components, backed up by years of physical research and experiments. The results of this research is a mechanical and physical evolution in clock design at the same time followed by improvents in our basic understanding of the physics on which these designs are based, including a certain amount of mathematics. The two articles in Nature mentioned are a reflection of this understanding. It should again be mentioned that the physical operation is not governed by the mathematics involved. If an optical clock is governed by anything than it is a photon generator which requires energy, otherwise the clock would stop.

What we humans call time (in hours, minutes and second) is that what a clock indicates. Time is not something physical. A clock is a physical object. It is a physical process.