Symmetrical twin paradox without changing direction

Question

If I understood well, in the special relativity

1- A stationary observer sees other moving observer's clock works more slowly than the stationary clock.

2- Motion is relative, the moving observer thinks the stationary observer is moving.

I cannot understand how it is possible then. Because each observer claims the other observer's clock works more slowly which obviously cannot be true. It is like to have two numbers each smaller than the other, it is not possible.

Suppose observers A and B are at rest in the origin and simultaneously move in two opposite direction with the same acceleration until they reach the velocity 0.9c and then stop accelerating. Then after some time one observer in the origin send two flashes of light in opposite direction toward A and B. When they receive the light, A and B record what their clock shows and send the result to the origin.

Because the situation is symmetric their clock must show the same number, but because each of them sees the other as moving one each thinks the other clock ticks more slowly.

What is wrong in this paradox?

Answer 1

It is like to have two numbers each smaller than the other, it is not possible.

It's not like that at all. Consider the following true statement:

• B observes A's clock to run slowly while A observes B's clock to run slowly.

This is not a contradiction due to the crucial word observes. Consider an additional true statement:

• The observer at the origin observes A's clock and B's clock to run at the same rate as each other but slower than his clock while both A & B observe the origin's clock to run slower.

To see that this is not a contradiction requires clearly thinking about what it means to observe (which doesn't mean see) - how does one determine the rate of a moving clock?

Here's one way:

• Position two clocks at rest in your lab and spatially separated along the line of motion of the moving clock.
• Synchronize the two clocks
• As the moving clock passes the first clock, record both the time on the moving clock and the first clock
• As the moving clock passes the second clock, record both the time on the moving clock and the second clock

Carefully note that it requires two, synchronized clocks in the lab to observe the elapsed time on the moving clock.

According to the lab technician, the elapsed time in the lab is $\Delta t = (t_2 - t_1)$ (since the two lab clocks are synchronized) which is greater than the elapsed time of the moving clock. Thus, the lab technician can validly claim that the moving clock is running slowly compared to the lab clocks.

However, while the two lab clocks are synchronized in the lab frame, the two lab clocks are not synchronized in the moving clock's rest frame (relativity of simultaneity).

This is the key to the 'paradox'; according to the moving clock, the lab clocks are not synchronized and thus, $(t_2 - t_1)$ is not a valid elapsed time.

To better understand how this works 'both ways', carefully consider the cute spacetime diagram from this answer:

Image credit

Answer 2

Your confusion comes from the fact that you are IMPLICITLY reasoning using the concept "absolute simultaneity", that is: "if two events in different points of the space are simultaneous for one inertial observer then they are simultaneous to all inertial observers".

One of the most important features of Special Relativity is precisely that there is NO "absolute simultaneity".

Suppose observers A and B are at rest in the origin and simultaneously (for a third observer also at rest in the origin) they move in two opposite direction with the same acceleration until they reach the velocity close to c and then, simultaneously (for a third observer, at rest in the origin), they stop accelerating.

After they stopped accelerating, observer A reads his/her own clock (Event 1) and simultaneously (for A) A reads B's clock (Event 2). We might have, FOR INSTANCE:

(Event 1) A's clock: 10 hours at A position

(Event 2) B's clock: 5 hours at B position

Observer A is sure both readings were simultaneous! However, observer B DISAGREES. Observer B saw observer A accessing B's clock when B's clock was 5 hours, however, as seen by B, "observer A reading A's clock" was not simultaneous (for B) with "observer A reading B's clock". In fact, observer B sees

(Event 1) A's clock: 10 hours at A position

(Event 3) B's clock: 20 hours at B position

as simultaneous events. Of course, it is NOT what is seen by A.

Important: It is somehow similar to what happens in every day life spatial perspective. Suppose you and I are standing in a plain. We are quite away from each other. Using a rule I measure you and, for me, you are only 2 inches tall. I measure myself and find I am 70 inches (or similar) tall. You have a rule exactly like mine, and you measure me, and for you I am only 2 inches tall, and you measure yourself and find you are 70 inches (or similar) tall.

Remark: You wrote: "each observer claims the other observer's clock works more slowly which obviously cannot be true. It is like to have two numbers each smaller than the other, it is not possible". As you may realize now, it is NOT like "to have two numbers each smaller than the other", because the observer who sees " $a<b$ " is not the same observer who sees " $b<a$ ".

Second Part of the question: After some time the third observer, let us call observer C, in the origin send two flashes of light in opposite direction toward A and B. When they receive the light, A and B record what their clock shows and send the result to the origin.

According to C, the flash sent to A got to A at exactly the same time the flash sent to B got to B, but this simultaneity is seen by observer C only.

Event 4: flash 1 got to A

Event 5; flash 2 got to B

For observer C those events are simultaneous. However, A saw flash 2 getting to B AFTER Flash 1 had got to A. Observer A could measure, FOR INSTANCE:

Event 4: flash 1 got to A. A's clock shows 10 hours (and B's clocks seen from A shows 5 hours).

Event 5; flash 2 got to B. A's clock shows 20 hours (and B's clocks seen from A shows 10 hours).

Observer A sends the information "10 hours" (the hour from A's clock) back to observer C. And observer B? Observer B saw flash 1 getting to A AFTER Flash 2 had got to B. Observer B could measure, FOR INSTANCE:

Event 4: flash 1 got to A. B's clock shows 20 hours (and A's clocks seen from B shows 10 hours).

Event 5; flash 2 got to B. B's clock shows 10 hours (and A's clocks seen from B shows 5 hours).

Observer B sends the information "10 hours" (the hour from B's clock) back to observer C.

Note that A and B see Event 4 and Event 5 in oposite order in time, while observer C see them as simultaneous.