Time dilation and length contraction question

[Glenn G]

Hi All,
Bob and Alice are both 21. I'm imagining a scenario where Alice starts on a journey at 0.8c towards a distant planet.

Bob (stays still) says this planet is 16 light years away and that it takes Alice 16/0.8 = 20 years to get there (so Bob is 41 years old when Alice gets there). So Bob says journey time t = 20 years

Alice, traveling at 0.8c. So according to her time, t0 = t x √(1-0.8^2) = 12 years
(So Bob's registered time is dilated compared to hers).

Now (I think this is right!), as Alice is traveling at 0.8c clearly can't travel 16 light years in 12 years (from her clock) so that she thinks she has only traveled L0 = L x √(1-0.8^2 = 9.6 light years, so she says "of course it only took me 12 years because I only had to travel 9.6 light years."

I drew this out on paper below and it feels right to me ..(please assist if I'm already confused)

This would be confirmed (would it not) if Alice fired a laser pulse at time zero that she records takes 9.6 years to traverse the 9.6 light years (from her perspective) but Bob who also sees the laser pulse records that it takes 16 years to traverse the 16 light years distance?

All OK with my thinking?

Kind regards,
Glenn.

 

[Ibix]

Your maths is fine, but you seem to be missing a few concepts.

How do Bob and Alice confirm that they are both 21? And if Alice sends a laser pulse to Bob, how does she determine when it arrived?

 

[Glenn G]

Hi Ibix I think I meant assume at the left of the picture assume they are both 21? The idea was that Alice sent a light pulse to the planet assume that somehow?? there was a clock of hers on that planet running at the same rate as her clock when going at 0.8c and let's say when this clock on the planet receives the pulse it records 9.6 light years ... then compared to a clock that Bob has put on the planet that runs the same as his own clock that stops when the beam from Alice arrives at the planet. This clock would record 16light years.
Is this correct?

 

[Nugatory]

The idea was that Alice sent a light pulse to the planet assume that somehow?? there was a clock of hers on that planet running at the same rate as her clock when going at 0.8c and let's say when this clock on the planet receives the pulse it records 9.6 light years ... then compared to a clock that Bob has put on the planet that runs the same as his own clock that stops when the beam from Alice arrives at the planet. This clock would record 16light years.
Is this correct?

The problem here is that there is no such thing as "a clock of hers on that planet running at the same rate as her clock". If it's on the planet it's moving relative to her, so it's not running at the same rate as her clock.

It is possible to work out when the pulse arrived using just Alice's clock, the one in her spaceship, and it's well worth the effort of figuring out how yourself. Two hints:
1) assume a mirror on the planet, so any signal Alice sends ahead will be reflected back to her.
2) think about the situation as if Alice is at rest and the planet is approaching her.

 

[Glenn G]

Thanks Nugatory,
I'd assume that the beam, traveling at c even measured by Alice would take 9.6 years to reach the planet's mirror (due to length contraction she would measure the distance to the planet as 9.6 light years)...
During this time she has been moving towards the planet at 0.8c so in 9.6 light years at 0.8c means she has moved 7.68 light years so is only 9.6-7.68 = 1.92 light years away from the planet when the light pulse bounces of the planet's mirror.
As far as she is concerned for this return event it is as though she is stationary and the planet is moving towards her at 0.8c with light that bounced off it at c so then return trip is 1.92 light years.

So on ALice's clock 9.6 years there and 1.92s years back = round trip of 11.52 years.

Bob would record 16 + 16 = 32 years round trip.

??Any truth in this at all??

 

[DrGreg]

During this time she has been moving towards the planet at 0.8c so in 9.6 light years at 0.8c means she has moved 7.68 light years

You haven't used Nugatory's hint number 2. According to Alice her own speed is zero and she has moved 0 light years...

 

[Glenn G]

Hi DrG, I did for the way back but not the way there, so that would mean she sees light shooting off at c and the planet moving towards her at 0.8c ... I'll need some paper for that, it will clearly mean it would then reach the planet even quicker now to me that feels like it's broken some sort of symmetry because if she is moving forward at 0.8 and she emits a photon at c towards a stationary planet I can't see that's the same thing as her thinking she is still and the planet moves towards her at 0.8 c. In the first case her motion has no impact on the receding light beam moving towards the planet but clearly a planet moving towards an approaching light pulse is very different. I don't see why this less intuitive (to me) interpretation is the right one?

 

[Jeronimus]

Imagine the scenario of yours, just slightly different.

Bob and Alice are astronauts floating in empty space, moving relative to each other at a speed of 0.8c. There is a planet far away, which Bob measures to be 16 lightyears away and which is at rest relative to him. Alice measures this planet to be 9.6 lightyears away.

Just when they pass by each other, they both happen to be 21 years of age. Also, coincidentally, their clocks/stopwatches happen to be in sync.
Neither Bob nor Alice can tell who is the moving one. There is no experiment they could perform which would allow them to tell who is the moving one and who is at rest. The only thing they can confirm is that they are moving relative to each other.You already calculated properly, that Alice's stopwatch clock count would display 12 years when passing by the planet, as measured by both Alice and Bob.

Also, Bob's clock count would be at 16 years when Alice passes the planet, measured by Bob.Note that the scenario above, is completely symmetric. Alice measures Bob to be moving away of her at 0.8c and Bob measures Alice to be moving away of him at 0.8c.
They both measure each others' clocks to be ticking slower.

So here are some questions to answer:

a) When Alice arrives at the planet and sees that her clock displays 12 years - How many years would Alice calculate/measure for Bob's clock to display, who according to Alice is at 9.6 lightyears away of her (assuming she hasn't de-accelerated yet)?

b) When Bob registers Alice to have arrived at the planet, his clock displays 16 lightyears, while Alice's clock only 12 lightyears. Let's call Bob with a clock count of 16 lightyears, an instance of Bob on his worldline.
Is the instance of Bob with a 16 lightyears clock count, the same instance of Bob you get when answering question a?edit: Bonus question :D

c) There is also a clock on the planet Alice arrives at, which happens to be in sync with Bob's clock and will remain in sync indefinitely as Bob and the planet are at rest, relative to each other.

Alice which arrived at the planet, de-accelerates near instantaneously to become at rest relative to Bob and the planet.

c1) what will be the clock count of the clock on the planet before and after the acceleration, measured by Alice?

c2) what will be the clock count on Bob's clock, measured by Alice, before and after the acceleration?

 

[Nugatory]

if she is moving forward at 0.8 and she emits a photon at c towards a stationary planet I can't see that's the same thing as her thinking she is still and the planet moves towards her at 0.8 c. In the first case her motion has no impact on the receding light beam moving towards the planet but clearly a planet moving towards an approaching light pulse is very different.. I don't see why this less intuitive (to me) interpretation is the right one?

It is less intuitive to you because your intuition here is wrong... and I suggested it because I suspected from the way you worded your first two posts that your intuition was still misleading you.

In particular, the confirmation you came up with in the first post: "This would be confirmed (would it not) if Alice fired a laser pulse at time zero that she records takes 9.6 years to traverse the 9.6 light years (from her perspective)" is intuitive but misleading. The speed of light is $c$ in all frames, so if Alice emits a flash of light and then waits 9.6 years, the flash of light is going to be 9.6 light-years in front of her. If the planet and Alice were 9.6 light-years apart when the flash was emitted, and the distance between Alice and the planet is getting smaller as they approach one another, the flash of light has to reach the planet in some time less than 9.6 years.

And here is the easy way to work it out, using just Alice's clock. Alice emits the flash of light at time zero. The reflection comes back to her at time $T$. Because we're using the frame in which Alice is at rest, we know that the light hit the mirror at time $T/2$ - the distances traveled by the outgoing and the return pulse are the same in this frame, $T/2$ light-years out and $T/2$ light-years back. The fact that the mirror/planet was moving is irrelevant; all that matters is where the mirror was at the moment that the light reflected from it. Note also that we don't need Bob, or Bob's numbers, or Bob's observations... Alice is perfectly capable of setting her clock to zero at the moment that the planet is 9.6 light years away from her without any help from Bob.

For this particular example (planet/mirror is 9.6 light years away when the flash is emitted, relative velocity of planet/mirror and Alice is .8c), it turns out that $T$ is 32/3 years, $T/2$ is 16/3 years, the flash of light reaches the mirror when it is about 5.3 light-years from Alice and 5.3 years have passed for her.

(This might be a good time to mention the relativity of simultaneity. In Bob's frame, the events "light hits mirror" and "Bob's clock reads ten years" happen at the same time - that's what it means to say that the light hits the mirror after ten of Bob's years. In Alice's frame the "light hits mirror" event happens at the same time as the event "Alice's clock reads 5.3 years". However, it does not follow that the events "Bob's clock reads ten years" and "Alice's clock reads 5.3 years" are simultaneous in either frame - they are not.)

 

[Glenn G]

Thanks Jeronimus and Nugatory, I still need to work on this ( I'm not surprised my intuition was wrong ).