Chapter 1 discusses two frames a laboratory frame and a rocket frame. The rocket frame has a speed v towards the right. My understanding of the lattice is that when at the start of the experiment both lattices are 'the same', that each time when two grid points pair (considering movement in the x direction) all the other grid points also pair. When that is the case the clock readings don't have to be the same. However when the clocks readings in the two frames are 10 and 8 then the clock readings of all the other clock pairs are also 10 and 8.
The problem with two or more reference frames is that you only can compare physical processes within one reference frame. One reference frame means generally speaking one clock. When you consider a second clock you can define a second reference frame and declare the clock at rest in that frame. But that does not make much sense. The best way is to describe and compare its behaviour (both) within the same frame, only than it makes sense to claim that one clock runs slower or faster than the other.
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The Figure at the left demonstrates clock synchronisation in the laboratory frame or a frame at rest.
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*z2
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| *y2
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*z1 \
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| *y1
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* | O--|---*---|---*---
| x1 x2
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* -
Figure 3
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Figure 4 at the left demonstrates clock synchronisation in the rocket frame or a moving frame. This Figure only applies for the x direction..
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r1-->
A-->
x x
x Earth x Sun
x x
<--B
r2-->
Figure 5
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Figure 5 shows two objects: The earth and the Sun. The Earth turns clock wise.
At the surface of the (Equator) there are two observers: A and B. There are at least three inertial frames (three lattices with rods and clocks which require synchronization in 3D):
The question is will both rockets reach the sun simultaneous. Figure 1 shows the initial state at the start of the experiment.
This experiment raises certain problems. Gravity is not considered. |
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It is important that this clock works with two parallel mirrors perpendicular in the direction of movement. In the example at page 22 the mirrors are parallel in the direction of movement.
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In my humble opinion this equation is more practical than the lorentz transformation because it is purely based on observations.
For further reading select this link: Reflection 3 - Are all moving rods at rest?
Mentioned above: the total length of the light path in Figure 6 for each observer is exactly the same. What that means is that in principle both clocks should show the same time. Because they don't in reality, moving clocks should not be used to measure speeds. Or there time indications should be corrected.
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The figure on the left represents the x,y plane of the experiment with two parallel mirrors.
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Figure 8 on the left represents the x,y,t spacetime frame of the experiment with two parallel mirrors.
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The figure on the left shows clock synchronisation in the x and y direction. (ground plane) The movement is towards the right in the x direction. Figure 9 shows two mirrors.
In the rocket experiment the clock signal moves in the ground frame from point A towards point R (is reflected) and returns towards the observer at point B. In the clock synchronisation experiment this signal is not reflected but continues and reaches clock y1 at point R'' in the ground plane. in the space time diagram this the point B'. The line A, R', B', x' shows the world line of the origin. The line A', X shows the world line of the second mirror. The line A, D', E' X shows world line of the synchronisation signal in the x direction. This signal reaches the origin at point X' |
The problem with the book "Spacetime Physics" is that the behaviour of clocks is not directly (separately) discussed in the first chapter.
When you want to understand the behaviour of clocks you should study them in one coordinate system. That means you should compare the behaviour of two clocks within the same frame. You should make this frame as large as possible and by means of experiments establish which clock runs the fastest.
When you have two clocks (in relatif movement) it is important to consider that the speed of a specific light ray is the same for both clocks (locally). The same for a light ray going in the opposite direction (locally). That does not mean that the speed of a light ray (travelling from the Sun towards the earth) is everywhere the same.
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Figure 6 (above) and Figure 10 shows the results of this experiment.
Clock A has counted 8 ticks and clock B 6 ticks. That means the average count rate is 4 versus 3 per year. A year being defined as a complete revolution of the Earth around the Sun. That means the moving clock ticks slower. The idea behind figure 10 is to do the same experiment again but with a slight modification: Each time when a clock tick it will also send a signal towards the other clock. The idea is to emphasize the differences between the two clock. It is important to remember that both clocks are the same and each uses two parallel mirrors. The direction of the mirrors are perpendicular to the direction of movement
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But now comes the most important exercise of this experiment.
The only conclusion is also that performing an experiment like in figure 6 and figure 10 and if the results are such that at the end of the experiment clock B shows less counts than clock A that clock A has the lowest speed.
You can also do the same experiment when the mirrors are parallel in the direction of movement. Clock A will still count 8 counts, but the count of clock B will be different (when they meet)
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