# Introductory Relativity, Time Dilation

## Homework Statement

Heide boards a spaceship and travels away from Earth at a constant velocity of 0.45c toward Betelgeuse (a red giant star in the constellation Orion). One year later on Earth clocks, Heide’s twin, Hans, boards a second spaceship and follows her at a constant velocity of 0.95c in the same direction.
a) When Hans catches up to Heide, what will be the difference in their ages?
b) Which twin will be older?

Hint given: Work primarily in the Earth's frame, and compute the a) the headstart, in Earth's frame Heidi has before Hans leaves, b) the time in Earth's frame it takes for Hans to catch Heidi. Then apply time dilation carefully or Lorentz transformations to compute how much time passes in each of their frames, keeping in mind that Hans is in the Earth's rest frame for one year.

## Homework Equations

Δt=γΔt0
γ= 1 / √(1-v2/c2)

## The Attempt at a Solution

For part a) Using the hint I got that Heide's headstart is .45 cy (light years). After the one year period, Hans is moving at .5c relative to Heide, in Earth's frame. Moving at this speed, again in Earth's frame, it would take him 0.9 years to catch up to Heide. Now I calculated gamma for Hans to be 1.1547, and I am just not sure what to do next. I think that Heide will measure the proper time because now she is in Earth's rest frame as Hans moves towards her. So is this simple enough that I can just plug numbers into the first equation?

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not surewhat to do next. I think that Heide will measure the proper time because now she is in Earth's rest frame as Hans moves towards her.
As I read the problem, Heide is still moving at .45c in the Earth's rest frame when Hans passes her.

As for what to do next... You might try drawing a space-time diagram showing the three events involved, and the trajectory followed by each twin. Figure how long in the earth frame each twin spent on each section of their respective paths through space-time to the meeting point, then use time dilation to figure out how much each twin ages on the same section.

Our prof has not told us how to solve any problems using a space-time diagram, so I need a different way to go about this. And yes I see you are right about Heide moving. Does this mean neither of them will measure the proper time then?

SammyS
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## Homework Statement

Heide boards a spaceship and travels away from Earth at a constant velocity of 0.45c toward Betelgeuse (a red giant star in the constellation Orion). One year later on Earth clocks, Heide’s twin, Hans, boards a second spaceship and follows her at a constant velocity of 0.95c in the same direction.
a) When Hans catches up to Heide, what will be the difference in their ages?
b) Which twin will be older?

Hint given: Work primarily in the Earth's frame, and compute the a) the head start, in Earth's frame Heidi has before Hans leaves, b) the time in Earth's frame it takes for Hans to catch Heidi. Then apply time dilation carefully or Lorentz transformations to compute how much time passes in each of their frames, keeping in mind that Hans is in the Earth's rest frame for one year.

## Homework Equations

Δt=γΔt0
γ= 1 / √(1-v2/c2)

## The Attempt at a Solution

For part a) Using the hint I got that Heide's head start is .45 cy (light years). After the one year period, Hans is moving at .5c relative to Heide, in Earth's frame. Moving at this speed, again in Earth's frame, it would take him 0.9 years to catch up to Heide. Now I calculated gamma for Hans to be 1.1547, and I am just not sure what to do next. I think that Heide will measure the proper time because now she is in Earth's rest frame as Hans moves towards her. So is this simple enough that I can just plug numbers into the first equation?
Heidi is not in Earth's rest frame.

What is gamma for Heidi ?

How did you calculate gamma (ɣ) for Hans? That ɣ is for v = 0.5 c .

The given Hint states:
Work primarily in the Earth's frame, ... Then apply time dilation carefully or Lorentz transformations to compute how much time passes in each of their frames, ...​

What is Hans' ɣ as computed in Earth's frame?

What is Heidi's ɣ as computed in Earth's frame?

Ok, I see that I was wrong about Heide's being in Earth's rest frame. In Earth's frame Heide and Hans are both still moving relative to Earth. The gamma for Hans and Heide in Earth's frame, respectively are 3.203 and 1.120. I am still not sure where to go from here because I don't know who measures the proper time, if anyone.

Edit: I think the proper time is in earth's frame since it is spatially coincident (stays in the same place the whole time) so .9 years should be the proper time. I think i should just solve for delta t for each of Heide and Hans... is this correct?

Last edited:
SammyS
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Ok, I see that I was wrong about Heide's being in Earth's rest frame. In Earth's frame Heide and Hans are both still moving relative to Earth. The gamma for Hans and Heide in Earth's frame, respectively are 3.203 and 1.120. I am still not sure where to go from here because I don't know who measures the proper time, if anyone.

Edit: I think the proper time is in earth's frame since it is spatially coincident (stays in the same place the whole time) so .9 years should be the proper time. I think i should just solve for delta t for each of Heide and Hans... is this correct?
0.9 years is the proper time for what? In other words: 0.9 years is a time interval in the Earth's frame. What two events are separated by 0.9 tears in Earth's reference frame? You need to be very specific and work this stuff out very carefully.

Every time someone goes into their spaceship and, for the purposes of your problem, instantly accelerates to a velocity you should make a new event. Break this problem up into a bunch of events/sub-events. Perhaps something like this..
1) Heidi goes for a year of earth time. How much of Heidi's time is this?
2) Hans takes off and his frame is no longer earth frame.
3) If Hans has a buddy on earth, Franz, he will age different from both Hans and Heidi. For Franz, how long does it take for Hans to meet Heidi? What does Franz think of the new ages? Events 3 and 1 can be compared once the times are sorted out.

0.9 years is the proper time for what? In other words: 0.9 years is a time interval in the Earth's frame. What two events are separated by 0.9 tears in Earth's reference frame? You need to be very specific and work this stuff out very carefully.
The two events separated by 0.9 years are 1) Hans leaves Earth and starts traveling towards Heidi, who is at .45 cy, and 2) Hans catches up with Heidi.

Every time someone goes into their spaceship and, for the purposes of your problem, instantly accelerates to a velocity you should make a new event. Break this problem up into a bunch of events/sub-events. Perhaps something like this..
1) Heidi goes for a year of earth time. How much of Heidi's time is this?
2) Hans takes off and his frame is no longer earth frame.
3) If Hans has a buddy on earth, Franz, he will age different from both Hans and Heidi. For Franz, how long does it take for Hans to meet Heidi? What does Franz think of the new ages? Events 3 and 1 can be compared once the times are sorted out.
1) She is moving for exactly one year so I should multiply her Lorentz factor by 1 year, so I get that one year of Earth time is 1.12 years for Heidi? Or is it opposite, Should I divide 1 by 1.12? Reason I ask this is because I get confused with the statement "moving clocks tick slower"...
3) Franz is in Earth's rest frame, so he should age 0.9 years, which is the time it takes Hans to reach Heidi.

1) Well, when you're done with this problem maybe you can just relate in your mind that moving clocks "age" slower (that's how I think about it at least). So someone in a super fast spaceship will have much less time (aged much less) wrt to earth. So, yeah, divide by 1.12. At this point Hans is ____ and Heidi is ____.

3) Okay, I'm sleepy so I'll assume your .9 years is right (sounds reasonable). .9 years corresponds to ___ years in Hans frame and ___ years in Heidi frame.

Total Hans is ____ and Heidi is ____

You're close!

So just double checking, diving by 1.12 (gamma) means that I'm actually solving for the proper time, since delta t / gamma = proper time.

So in the year that Heidi traveled before Hans left, she aged .89 years while Hans aged 1 (he was at rest in Earth's frame). Now since it takes .9 years in Earth's frame for Hans to reach Heidi, Hans will have aged 0.28 years on his voyage (Lorentz for Hans is 3.203, and I used .9 as the time interval), while Heidi ages another 0.8 years as Hans approaches her. In total Hans aged 1.28 years (this includes the year he waited) and Heidi has aged 1.69 years. So their difference in age will be 0.41 years and Heidi is the older twin. How does this sound?

SammyS
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