Twin Paradox from Moving Reference Frame- Return Journey

In summary, Alice is observing a small ball of mass m in relativistic motion bouncing elastically back and forth between two parallel walls separated by a distance L with speed u. She measures the time, ##t_A##, between each "tick" of her clock to be ##t_A = 2L/u##. Bob, who is in an IRF moving with speed v in the same direction as the initial motion of the ball, measures the spatial separation between events 1 and 2 to be ##\Delta x_{12} = \gamma (\Delta x_A -v \Delta t_A)## and the temporal separation to be ##\Delta t_{12} = \gamma (\Delta t_A -\frac{v
  • #1
Bob Marsh
2
0

Homework Statement


a) Alice is observing a small ball of mass m in relativistic motion
bouncing elastically back and forth between two parallel walls separated by a distance L
with speed u. After each collision it reverses
direction, thereby creating a clock. What does Alice observe as the
time,##t_A##, between each “tick” of her clock?
(b) Add three events to the space-time diagram: the first when the
ball leaves one side of the box, the second when the ball (first) ar-
rives at the opposite side, and the third when the ball (first) returns
to its initial position. Label the events 1,2 and 3. Connect the events by the world-line
of the ball.
Bob is in an IRF which is moving with speed v in the same direction as the initial motion of the ball. Add this IRF to your space-time diagram. Assume that v<u.
(c) What is the spatial separation, that Bob measures between events 1 and 2?
(d) What is the temporal separation, that Bob measures between events 1 and 2?
(e) What is the spatial separation that Bob measures between events 2 and 3?
(f) What is the temporal separation that Bob measures between events 2 and 3?

Homework Equations


Lorentz transform

The Attempt at a Solution


a) This is easily seen to be ##t_A =2L/u##.
b) This is a triangle connecting events 1,2 and 3.
c) Using the Lorentz transform Bob would measure: ##\Delta x_{12} = \gamma (\Delta x_A -v \Delta t_A)##. Alice measures the spatial separation to be ##L## and the temporal separation to be ##L/u##. This seems correct because if I set ##v=u##, so that I'm in the rest of the ball on its out-going journey, then the spatial separation is ##0##, which makes sense.
d) Using a Lorentz transform: ##\Delta t_{12} = \gamma (\Delta t_A -\frac{v}{c^2} \Delta x_A) = \gamma (\frac{L}{u} - \frac{v}{c^2} L)##. Again checking with the ball's rest frame, I get regular time dilation which makes sense.
e) For parts e) and d) I am not sure how to account for the fact the ball's velocity is now ##-u##. How do I use this information to find the relevant intervals?
 
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  • #2
velocity reverses but so does the direction of the differential positions so you have a global sign change in the displacements but the magnitudes transform as before.
 
  • #3
So how would I implement this as equations? Something like the ball travels from ##(L,L/u)## in Alice's coordinates to ##(0,2L/u)## and then transform these points to Bob's reference frame and use the velocity of the ball as seen by Bob? Or is there something else I'm missing?
 

What is the Twin Paradox from Moving Reference Frame- Return Journey?

The Twin Paradox from Moving Reference Frame- Return Journey is a thought experiment in special relativity that explores the concept of time dilation, where the time measured by one observer can differ from the time measured by another observer due to their relative motion. It involves two identical twins, one who stays on Earth and one who travels through space at high speeds and then returns to Earth.

What is the main question that the Twin Paradox from Moving Reference Frame- Return Journey explores?

The main question of the Twin Paradox is: how can one twin age significantly less than the other, even though they are both in the same reference frame and traveling at the same speed?

What are the assumptions made in the Twin Paradox from Moving Reference Frame- Return Journey?

There are several assumptions made in this thought experiment, including the assumption that the twins are in a constant velocity and do not experience any acceleration, and that they are in the same inertial reference frame. It also assumes that the twins have accurate clocks and can measure time precisely.

What are the implications of the Twin Paradox from Moving Reference Frame- Return Journey on our understanding of time and space?

The Twin Paradox highlights the concept of time dilation, which shows that time is not absolute and can be affected by an observer's relative motion. It also challenges our understanding of space and time as separate entities, as they are intertwined in the theory of relativity.

What are some real-life examples that demonstrate the principles of the Twin Paradox from Moving Reference Frame- Return Journey?

Real-life examples of the Twin Paradox include the time dilation experienced by astronauts in space, as well as the difference in time measured by atomic clocks on Earth's surface and those on GPS satellites, due to their different velocities. Additionally, muons (subatomic particles) created in the Earth's upper atmosphere have a longer lifespan than those created in particle accelerators on Earth, again due to time dilation caused by their different velocities.

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