Special Relativity for Beginners: Clock Comparison

[docnet]

TL;DR Summary

Seeking curt answers for special relativity for a beginner. One sentence answers would suffice.

Hi, I have no education in physics beyond the intro undergrad level. How do the clocks compare in the following scenarios?

In the beginning, two clocks are placed next to each other and synchronized. Then they are moved in opposite directions, traveling an equal distance from the beginning with equal acceleration. An observer in the center who is equidistant to the clocks would observe the same time between the clocks, right?

Scenario 1: One of the clocks are moved to the other clock which stayed in place.

Scenario 2: The two clocks are moved to the center. The two clocks started moving the same time, but one got there first.

Scenario 3 : The two clocks are moved to the center. They started moving at the same time and got there at the same time. One clock took a detour, twisting and turning and tracing out a longer path.

 

[Dale]

Summary:: Seeking curt answers for special relativity for a beginner. One sentence answers would suffice.

An observer in the center who is equidistant to the clocks would observe the same time between the clocks, right?

Right.

For all of the scenarios, what is the reference used in the description and what is the question?

 

[docnet]

what is the reference used in the description and what is the question?

I was wondering, do the clocks display the same time for the observer in the center, after the clocks have been brought back to him/her?

 

[Nugatory]

Scenario 1: One of the clocks are moved to the other clock which stayed in place.

By "the other clock" do you mean the one in the center, or the one that was accelerated in the opposite direction? And either way, when you say "Stayed in place", what frame are you using? The one in which the central clock is at rest, or some other one?

Scenario 2: The two clocks are moved to the center. The two clocks started moving the same time, but one got there first.

"At the same time" using which frame? If the two accelerated clocks turn around and move back towards the center at the same time using the frame in which the center clock is at rest, the turnarounds will not happen at the same time using any other frame. You have to specify the frame any time you use those words "at the same time" when talking about events that don't happen at the same place.

Scenario 3 : The two clocks are moved to the center. They started moving at the same time and got there at the same time. One clock took a detour, twisting and turning and tracing out a longer path.

The first use of "at the same time" has the same problem as in the previous scenario. The second one is OK because the center is a single point in space - all frames will agree that either thetwo clocks smashed into one another at that point or they didn't.

Until you answer these questions, your question is not completely specified and cannot be answered.

 

[Ibix]

Special relativity isn't enormously complicated, but there are some catches. A big one is that you must always be specific about what you are regarding as "at rest", since distance, time, and speed measurements all depend on that information.

In this case, I think we all suspect that you mean for all measurements to be made assuming the middle clock to be at rest. However, one of the simplest things we can do to help you understand relativity is to insist that you tell us so explicitly. Accidentally mixing measurements made in different frames is one of the big causes of confusion when learning relativity, so we're teaching you a useful discipline, not being awkward for the fun of it.

 

[PeroK]

Summary:: Seeking curt answers for special relativity for a beginner. One sentence answers would suffice.

Hi, I have no education in physics beyond the intro undergrad level. How do the clocks compare in the following scenarios?

In the beginning, two clocks are placed next to each other and synchronized. Then they are moved in opposite directions, traveling an equal distance from the beginning with equal acceleration. An observer in the center who is equidistant to the clocks would observe the same time between the clocks, right?

Scenario 1: One of the clocks are moved to the other clock which stayed in place.

Scenario 2: The two clocks are moved to the center. The two clocks started moving the same time, but one got there first.

Scenario 3 : The two clocks are moved to the center. They started moving at the same time and got there at the same time. One clock took a detour, twisting and turning and tracing out a longer path.

For any scenario, the time shown on the clock when it returns to the starting point (the origin of an inertial frame of reference) is a function of the clock's speed (as measured in the said reference frame) throughout its journey. If a clock moves at a constant speed it's easy to calculate. If the clock moves at a variable speed you have to integrate:
$$\tau = \int_0^t \sqrt{1 - \frac{v(t)^2}{c^2}}dt$$
Where $\tau$ is the time shown on the traveling clock, $t$ is the time on the inertial clock at the origin and $v(t)$ is the traveling clock's speed as a function of $t$.

Acceleration, in itself, has no effect other than changing the speed of the traveling clock.

 

[Dale]

I was wondering, do the clocks display the same time for the observer in the center, after the clocks have been brought back to him/her?

And what reference frame is used to describe all of the scenarios? I assume the frame is the central observer’s

 

[docnet]

Thank you to all the experts for taking the time to answer my questions.

Special relativity isn't enormously complicated, but there are some catches. A big one is that you must always be specific about what you are regarding as "at rest", since distance, time, and speed measurements all depend on that information.

I think it will start making more sense when I learn it properly from a textbook. I've been reading popular literature in physics and complicated concepts like relativity are described without the mathematical framework. It just increases confusion to the layperson without concrete instruction.

If the clock moves at a variable speed you have to integrate:
τ=∫0t1−v(t)2c2dt
Where τ is the time shown on the traveling clock, t is the time on the inertial clock at the origin and v(t) is the traveling clock's speed as a function of t.

The fact that this equation is a function of velocity as well as time is pretty elucidating. What is the name of this equation and is it a simplified version of a more complicated equation?

Using this equation and an integrating software I was able to work on answering my own questions.

Scenario 1: One of the clocks are moved to the other clock which stayed in place.

Scenario 2: The two clocks are moved to the center. The two clocks started moving the same time, but one got there first.

Scenario 3 : The two clocks are moved to the center. They started moving at the same time and got there at the same time. One clock took a detour, twisting and turning and tracing out a longer path.

Assuming the distance from the center to each clock was a billion meters and the clocks traveled with only constant velocity,

Scenario 1: the clock that moved at 1 m/s would display a time around 3.2 seconds "earlier" than the clock that stayed in place. To any observer who is stationary with respect to the final state of the system.

scenario 2: Compared to the clock that traveled at 1 m/s, a clock that traveled at 2 m/s would display a time around 2.3 seconds "earlier" to the observer at the center.

Scenario 3: The clock that took a curved path at a higher rate of speed would show an earlier time, in accordance with the equation provided by the expert.

Is this what physicists refer to as time dilation due to special relativity?

In real life, would acceleration also need to be considered for time dilation?

 

[PeroK]

The fact that this equation is a function of velocity as well as time is pretty elucidating. What is the name of this equation and is it a simplified version of a more complicated equation?

Using this equation and an integrating software I was able to work on answering my own questions.

It's just the equation for time dilation, but integrated to cover the case of changing velocity. Starting with:
$$d\tau = \sqrt{1 - \frac{v^2}{c^2}}dt$$
Which gives the tick rate ($\tau$) of a clock measured in a reference frame where it is moving at speed $v$, compared to the tick rate ($dt$) of a clock at rest in that reference frame. If $v$ is constant, then we get:
$$\Delta \tau = \sqrt{1 - \frac{v^2}{c^2}}\Delta t$$
And if we $v$ is a function of time, then we get:
$$\Delta \tau = \int_0^t \sqrt{1 - \frac{v(t)^2}{c^2}}dt$$
(Technically I should use another dummy variable in the integral, but let's keep it simple.)

 

[Nugatory]

Is this what physicists refer to as time dilation due to special relativity?

No. It's a different phenomenon with a different cause, and is a variant of the so-called Twin Paradox (this online FAQ, and about 93 kilobajillion threads here).

In real life, would acceleration also need to be considered for time dilation?

If you and I are moving relative to one another, you will consider my clock to be running slow and I will consider your clock to be running slow. That's time dilation, and it happens even with constant speeds, no acceleration (in fact, many intro textbooks use only the constant speed case in their example because the math is less demanding - you can get by with ordinary algebra instead of the integral that @PeroK uses above).