Special Relativity Homework

[BOAS]

Hello,

I am doing my special relativity homework and since the ideas are quite counter-intuitive it's hard to know if I'm doing the right things. So i'd like to post some work here and see if it is correct. Also, I am stuck on the final part of the question.

1. Homework Statement
A space traveler takes off from Earth and moves at speed 0.990c toward the star Vega, which is 26 ly (light-years) distant. How much time will have elapsed by Earth clocks

(a) When the traveler reaches Vega?
(b) When the Earth observers receive word from her that she has arrived?
(c) How much older will the Earth observers calculate the traveler to be when she reaches Vega that she was when she started the trip?

Homework Equations

The Attempt at a Solution

a) The proper time interval, i.e the time measured by the space traveler is found using the standard method.

$\Delta t_{0} = \frac{d}{s} = \frac{26}{0.990} = 26.26$ years.

$\gamma = \frac{1}{(1- \frac{v^{2}}{c^{2}})^{0.5}}$

Earth time, $\Delta t = \gamma \Delta t_{0} = \frac{\Delta t_{0}}{(1- \frac{v^{2}}{c^{2}})^{0.5}} = 186.17$years

I think this is correct, but that's a whole lot of time dilation...

b) 186.17 years go by, according to the Earth clock until the traveler reaches vega. It will then take a further 26 years for her signal to reach earth. So the Earth clocks will measure 206.17 years.

c) I am finding this part very confusing... Do Earth observes consider the traveler to have aged 186.17 years?

 

[BruceW]

I am not certain that you have done part a) correctly. It says the star Vega is 26 light-years distant from earth. So whose frame of reference are they implicitly talking about here? (I'll admit the problem statement is a little bit vague, so guessing this is a bit tricky).

 

[Chestermiller]

According to wiki, the distance from Earth to vega is 26 ly, as reckoned from Earth's frame of reference. So the answer to (a) is 26.2 yr, and the answer to question (b) is 52.2 yr. if the people on Earth are familiar with relativity, then they can calculate how much she had aged by the time she reached vega.

Chet

 

[BOAS]

I am not certain that you have done part a) correctly. It says the star Vega is 26 light-years distant from earth. So whose frame of reference are they implicitly talking about here? (I'll admit the problem statement is a little bit vague, so guessing this is a bit tricky).

I don't think it matters as long as I'm consistent and state my assumption explicitly.

According to wiki, the distance from Earth to vega is 26 ly, as reckoned from Earth's frame of reference. So the answer to (a) is 26.2 yr, and the answer to question (b) is 52.2 yr. if the people on Earth are familiar with relativity, then they can calculate how much she had aged by the time she reached vega.

Chet

I don't understand how this is possible.

From the Astronaut's frame of reference, she traveled to Vega in 26.2 years. She then sends a signal back to Earth which takes 26 years to arrive, so the whole thing takes 52.2 years from her perspective.

But surely this signal can't arrive at Earth before they observe her reaching the place the signal was sent from?

If you were watching from earth, you'd see her approaching Vega for 186.17 years, and then she would send a signal that would take 26 years to reach you.

Why is this wrong?

 

[PeroK]

I don't think it matters as long as I'm consistent and state my assumption explicitly.
I don't understand how this is possible.

From the Astronaut's frame of reference, she traveled to Vega in 26.2 years. She then sends a signal back to Earth which takes 26 years to arrive, so the whole thing takes 52.2 years from her perspective.

But surely this signal can't arrive at Earth before they observe her reaching the place the signal was sent from?

If you were watching from earth, you'd see her approaching Vega for 186.17 years, and then she would send a signal that would take 26 years to reach you.

Why is this wrong?

You're over-thinking part (a). From the Earth's perspective, this is a simple time = distance/speed equation. There is no time dilation or length contraction involved. If something is traveling at near the speed of light, then it will travel x light years in approximately x years (from our frame of reference on Earth).

Things are different, of course, from its frame of reference.

 

[BOAS]

You're over-thinking part (a). From the Earth's perspective, this is a simple time = distance/speed equation. There is no time dilation or length contraction involved. If something is traveling at near the speed of light, then it will travel x light years in approximately x years (from our frame of reference on Earth).

Things are different, of course, from its frame of reference.

Ahh, I think I was confusing myself about where the clocks are. The observers on Earth are at rest with respect to their clock, and the space traveler looks at the Earth clock and thinks it's 'ticking' slowly.

Is that the right way round to solve this question?

 

[PeroK]

Ahh, I think I was confusing myself about where the clocks are. The observers on Earth are at rest with respect to their clock, and the space traveler looks at the Earth clock and thinks it's 'ticking' slowly.

Is that the right way round to solve this question?

Yes. There's no relativity involved in part (a)! You're not asked what the traveller observers.

 

[BOAS]

Yes. There's no relativity involved in part (a)! You're not asked what the traveller observers.

Thanks for the help.

I'm still in a bit of a muddle regarding part C.

The astronaut, if she has a clock on board, would observe the journey to take 26.26 years, right? But if she were watching Earth's clock, she would see $\gamma \Delta t_{0}$ time go by. So is this how much she has aged with respect to the earth?

 

[PeroK]

Thanks for the help.

I'm still in a bit of a muddle regarding part C.

The astronaut, if she has a clock on board, would observe the journey to take 26.26 years, right? But if she were watching Earth's clock, she would see $\gamma \Delta t_{0}$ time go by. So is this how much she has aged with respect to the earth?

No. The Earth observers and the astronaut will measure different journey times. She measures time by her own clock!

 

[Orodruin]

The astronaut, if she has a clock on board, would observe the journey to take 26.26 years, right?

Not right. The journey took this time in the Eart/Vega rest frame. The rocket clock will be time dilated.

 

[BOAS]

No. The Earth observers and the astronaut will measure different journey times. She measures time by her own clock!

I am very confused...

The Earth observers are at rest with respect to their clock, and the Astronaut is at rest with respect to her clock. Surely this means that they both measure the same time interval on their respective clocks.

But if you were looking at the Astronaut's clock from earth, it would appear time dilated, and look as if it took her $\gamma \Delta t_{0}$ years to reach Vega.

EDIT _ I think I have it, will post in a few minutes

 

[Orodruin]

Surely this means that they both measure the same time interval on their respective clocks.

No, because they are not in rest relative to each other. Note that when we speak about time dilation, we are speaking about the time including corrections for the travel time of a signal from one spatial point to another. The times in the two inertial frames simply are not the same.

 

[PeroK]

I am very confused...

The Earth observers are at rest with respect to their clock, and the Astronaut is at rest with respect to her clock. Surely this means that they both measure the same time interval on their respective clocks.

But if you were looking at the Astronaut's clock from earth, it would appear time dilated, and look as if it took her $\gamma \Delta t_{0}$ years to reach Vega.

Okay. Confusion is part of learning SR!

We have two observers here, moving with respect to each other. The time interval between two events will be different for the two observers. So, Earth observers will measure time t between the astronaut leaving Earth and arriving at Vega. The astronaut will measure a different time (t').

This is where you need SR, to calculate t'.

 

[BOAS]

Okay. Confusion is part of learning SR!

We have two observers here, moving with respect to each other. The time interval between two events will be different for the two observers. So, Earth observers will measure time t between the astronaut leaving Earth and arriving at Vega. The astronaut will measure a different time (t').

This is where you need SR, to calculate t'.

Right.

The Earth sees the events occur at different places, so an Earth observer measures $\Delta t$, the dilated time = 26.26 years.

The astronaut sees the event occur at the same place (just outside the space ship), so she actually measures the proper time interval, $\Delta t_{0} = \Delta t \sqrt{1 - \frac{v^{2}}{c^{2}}} = 3.7 years$!

So according to the earth, she has aged 26.26 years when she reaches vega, but according to her, she has only aged 3.7 years.

 

[PeroK]

Right.

The Earth sees the events occur at different places, so an Earth observer measures $\Delta t$, the dilated time = 26.26 years.

The astronaut sees the event occur at the same place (just outside the space ship), so she actually measures the proper time interval, $\Delta t_{0} = \Delta t \sqrt{1 - \frac{v^{2}}{c^{2}}} = 3.7 years$!

So according to the earth, she has aged 26.26 years when she reaches vega, but according to her, she has only aged 3.7 years.

I didn't check your arithmetic, but it looks about right.

But, surely there is a physical reality here: either she has aged 3.7 years or she has aged 26.26 years. Which is it? If she was a child of 5 when she left, would she still be a child of almost 9 when she arrived? Or, an adult of 27?

 

[BOAS]

I didn't check your arithmetic, but it looks about right.

But, surely there is a physical reality here: either she has aged 3.7 years or she has aged 26.26 years. Which is it? If she was a child of 5 when she left, would she still be a child of almost 9 when she arrived? Or, an adult of 27?

That depends on who you ask, there is no privileged frame of reference to make an absolute measurement :)

Thanks for the help, I am a bit less confused now.

 

[PeroK]

That depends on who you ask, there is no privileged frame of reference to make an absolute measurement :)

Thanks for the help, I am a bit less confused now.

No. That's not correct. On a longer journey, if she aged according to Earth time, she would be long dead by the time she arrived, but not if she aged according to the ship's clock. She can't be both. Either she's alive and well and able to land on a planet and walk around, or she's long dead.

You do need to think about this.

 

[BOAS]

No. That's not correct. On a longer journey, if she aged according to Earth time, she would be long dead by the time she arrived, but not if she aged according to the ship's clock. She can't be both. Either she's alive and well and able to land on a planet and walk around, or she's long dead.

You do need to think about this.

hmm, I think the important time interval is the one measured by the traveler. She really does age at the slower rate, so she'd outlive everyone at mission control.

 

[Orodruin]

Correct. Just as muons created in the atmosphere survive longer than their mean life at rest.