Figure 1-1 comes from the book: Spacetime Physics: Introduction to Special Relativity (1st ed.). San Francisco: Freeman. by Edwin F. Taylor and John Archibald Wheeler, (1966). ISBN 071670336X.
To explain the concept worldline we use the book GRAVITATION by MTW at the pages 21 and page 315.
The equation that describes the situation is the same (tau = sqrt(T^2 - R^2) but the pictures are different. It should be mentioned that the equation is confusing. IMO this should be (tau = sqrt(T^2 - (R/c)^2). However is c=1 then ....
Figure 5A shows the picture of page 21. Figure 5B shows the picture of page 315. For more detail See below.
To explain the behaviour of a moving we use
Reflection 1 - Lorentz Contraction
Reflection 1 starts with the following text:
In this experiment the train travels in the x direction with a speed v. In the train are two "mirrors". One at the bottom of the train and one at the ceiling. The two mirrors are used as a clock. The light signal is supposed to go in the y direction.
B1------B-------B2 /|\ / | \ t1/ | \ tau / | \ D L D / |t0 \ v--> / t2 | \ / -.A-.-.-.-E-.-.-.-C.-. R Figure 3
The time for a light signal to travel (v=0) from E to B and back is 2*t0
In this case t0 = BE / c = L /c
The time for a light signal to travel from A to B and back to C is 2 * t1
The time for a "train" to travel with speed v, from A to E to C is 2 * t2
If both arrive at the same time than: t1 = t2 = t
AB = D = c*t1 = c*t and AE = EC = v*t2 = v*t
Using some arithmatic we get: AB^2= AE^2 + BE^2 = c^2*t^2 = v^2*t^2+L^2
c^2*t^2 - v^2*t^2 = L^2 : t^2 = L^2/(c^2-v^2) : t^2 = t0^2*c^2/(c^2-v^2)
Now we get t^2 = t0^2/(c^2-v^2)/c^2 or t1 = t0/ SQR(1-v^2/c^2)). The factor SQR(1-v^2/c^2)) with v > 0 is smaller than 1.
You can also rewrite the last two lines and then you get:
c^2*t^2 - v^2*t^2 = L^2 or c^2*t^2 - R^2 = c^2*t0^2
Z | t+x .Z |\ | \ | \ | \ | \ t |-----.B | /. | / . | / . | / . |/ . t-x .P . | . | . A----------> x Figure 5A (p21)
Figure 5A page 21 shows two worldlines:
P is an earlier event that creates a lightray which coincides with B.
The angle PBZ is 90 degrees.
It is important to define the units of the z axis.
The time (duration) t is 2x/c in seconds or years.
^ t | T B . |\ | . | \ | . | \ | . | \ 0.5T|----.P | /. | .. | / . | . . | / . | . . |/ . . . 0 A--------->Z 0.5R Figure 5B (p315)
Figure 5B page 315 demonstrates a non straight worldline. In this case a particle moves in spacetime with uniform velocity v from A to P and back to B.
The angle APB is larger than 90 degrees because v
In this figure the worldline of a photon is not indicated. When you do that you can see that when the particle is a clock that the moving clock ticks slower than a clock at rest.
Both concepts 'Real Observer' versus 'Virtual Observer' are important concepts to study the evolution of the Universe. The concept 'Virtual Observer' is important because it allows you to study the Laws of Nature i.e. the trajectories of the stars and planets not using the speed of light. Classical Mechanics (Newton's Law) does the same. The major problem with Classical Mechanics is that forces do not act instantaneous but propagate with a constant speed.
What makes this approach IMO so powerfull is because no moving clocks, no real observers and no speed of light issues (including the bending of light) are involved. The attention is towards the physical issues directly related to how objects behave i.e. specific the forces and retarded issues.
IMO it is the best strategy to use the fixed background as your reference frame or coordinate system.
A different strategy is what SR does. Each object becomes its own inertial reference frame and the object is considered at rest in that frame. The problem is where do you draw the other objects in the reference frame of object n. The problem is horizontal lines in frame n define simultaneous events in frame n. Simultaneous events are the positions of the other objects at the same time in frame n. However these same events are not simultaneous in any other frame, which makes evertything complicated.
| 4 | 0 C 0 | . . | a b |. .| 0 a 0 . . a b |. .| 0a b0 | . . | a b | 3 | 0 b 0 | . . | a b |. .| 0 B 0 . . a b |. . 0 a 0 | . . | a b | 2 0a b0 | . . a b |. 0| b 0 . a b |. 0.|A 0 | . . a b | 1 a| 0 | . . | b |.0a .|b0 . a b |o b|0 | a b | ----X----------- L1 L2 Figure 6A
| 4 | 0 C 0 | B | 2 | B| 0 2 x . . 2 | |B 02 0 | | B 2 | 3 | x 0 | | 2 B | |20 B 0 . . | 2 0 x | 2 | | 2 0 1 0 | A | 1 | A|1 0 . 1 | 01|A 0 | 1 | | 1 | x | | | 0 | 0 . . |x |0 | | ----X----------- L1 L2 Figure 6B
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