Travel Times & Distances: Twin Paradox & Alpha Centauri

[DAH]

Hi all
After reading this thread it got me thinking about the twin paradox, so I looked through some old notes from a module I studied on this subject. According to one of my textbooks, if the traveling twin travels to Alpha Centauri (which is 4.2 light years from Earth) at a speed of 0.9 c, then the Earthbound twin would record a time of 4.7 years to complete the journey. However, for the traveling twin in his reference frame that time would be reduced to 2 years due to time dilation.

The textbook also mentions the relativity of simultaneity, but further reading it also states that in this case the traveling twin who is essentially traveling to another star system, experiences length contraction, where the space in front of the ship contracts and therefore the distance to Alpha Centauri decreases for the traveling twin. For the Earthbound twin however, he doesn't experience length contraction therefore he sees the ship travel the full 4.2 light years.
Is this correct or is my textbook wrong?

 

[Ibix]

Is this correct or is my textbook wrong?

I don't see a problem in what you've written (although I didn't check the numbers).

 

[DAH]

Thanks for starting a new thread.
So basically I just wanted to know if the traveling twin would reach Alpha Centauri younger than the Earthbound twin due to time dilation and length contraction, without accounting for any accelerations? Or do both clocks run slowly due to the speed being relative.

 

[Nugatory]

Thanks for starting a new thread.
So basically I just wanted to know if the traveling twin would reach Alpha Centauri younger than the Earthbound twin due to time dilation and length contraction, without accounting for any accelerations? Or do both clocks run slowly due to the speed being relative.

Neither. You are not considering the relativity of simultaneity, which makes the question about which is younger when they’re separated meaningless.

The twin paradox uses a round trip to get both twins back at the same place at the same time so that there are no simultaneity problems. The acceleration is something of a red herring - it’s just that we can’t have the twins separate and rejoin without accelerating at least of one of them.

 

[Ibix]

Thanks for starting a new thread.
So basically I just wanted to know if the traveling twin would reach Alpha Centauri younger than the Earthbound twin due to time dilation and length contraction, without accounting for any accelerations? Or do both clocks run slowly due to the speed being relative.

If the traveller returns to Earth, they will be younger than the stay-at-home. If they do not return then there is no assumption-free answer. But if Earth and Alpha Centauri are mutually at rest (or close enough, as in reality) both could choose to use Einstein clock synchronisation, and then they would agree that the traveller is younger at all times (the traveller will not necessarily agree).

As Nugatory says, acceleration isn't really relevant here. It's just that you can't come back without accelerating in special relativity. The simplest explanation is geometrical: the twins follow different paths through spacetime. The "length" of a path is the interval, which will be discussed in your textbook and turns out to be the elapsed time for an observer following that path. Two paths between a pair of events need not have the same length. But if the paths don't meet up again then you can't ask which one is longer without some arbitrary decision about what constitutes "the end of the path".

 

[DrStupid]

The textbook also mentions the relativity of simultaneity, but further reading it also states that in this case the traveling twin who is essentially traveling to another star system, experiences length contraction, where the space in front of the ship contracts and therefore the distance to Alpha Centauri decreases for the traveling twin. For the Earthbound twin however, he doesn't experience length contraction therefore he sees the ship travel the full 4.2 light years.
Is this correct or is my textbook wrong?

It is correct for the distance to the destination. How else could Alpha Centauri reach the traveller with 0.9 c in just 2 years? However, the travellier itself is length contracted for his earthbound twin.

 

[Mister T]

However, for the traveling twin in his reference frame that time would be reduced to 2 years due to time dilation.

No, it would be 2 years due to length contraction.

 

[Halc]

Relative to the traveling twin, the distance between the stars is contracted to about 1.8 light years, and Alpha Centauri comes to him at 0.9c. His ship is not contracted in his own frame since he's stationary by definition in that frame.

If the textbook says that anybody 'experiences length contraction', it is wrong. It isn't something that can be experienced. I can be length contracted relative to a frame where I'm moving fast, but I'm never length contracted relative to myself, so I always measure my invariant proper length if I were to measure myself or my ship.

 

[DAH]

I often see examples (usually on You Tube) of time dilation symmetries where two observers with a relative speed of say 0.8 c will see each others clocks running slowly. One example is the spaceship (observer A) flying past Earth (observer B) with constant relative speed. According to SR, and since the speed is relative, then we can also say that observer A sees Earth fly past him/her at 0.8 c so both observers see the others clock running slowly. My question is, will length contraction make a difference to the observed time dilation's between both observers? because if I understand it correctly, observer A will see the space and the Earth in front of him contracted, whereas the Earthbound observer B will only see the spaceship contracted.

I suppose its similar to the way we observe muons traveling through the atmosphere. We on Earth see the muons travel through say 5000 m of atmosphere, but in the muon's frame of reference the atmosphere will only be around 800 m. Will the muon also see our clocks on Earth running slowly?

 

[Ibix]

Will the muon also see our clocks on Earth running slowly?

To be clear, what you see of a moving object is dominated by the Doppler effect, and direct observation of a moving clock will show it to be ticking fast as it approaches you and slow as it goes away. Once you correct for the changing light speed delay, though, you will calculate that a clock moving in any direction ticks slowly. (A lot of sources don't make yhis distinction, and that often causes confusion.)

With that clarification, yes, the muon will also calculate that Earthly clocks tick slowly.

My question is, will length contraction make a difference to the observed time dilation's between both observers?

It isn't really length contraction that's generally important, although you are correct in your analysis of the muons. To make sense of relativity, you need to be aware of the relativity of simultaneity, which is that observers in relative motion generally won't agree on the simultaneity and/or order of spacelike separated events. It's not well covered in a lot of popular sources, but it's really important.

 

[Mister T]

I suppose its similar to the way we observe muons traveling through the atmosphere. We on Earth see the muons travel through say 5000 m of atmosphere, but in the muon's frame of reference the atmosphere will only be around 800 m. Will the muon also see our clocks on Earth running slowly?

Note that the Earth-based observer needs two clocks. One at the top of the mountain and then another 5000 m below. Those two clocks are synchronized in the rest frame of the Earth-based observer. Those clocks are used to conclude that the muon's clock is running slow. But what does the muon observe? The muon would observe that not only are the two clocks running slow, they are not properly synchronized. The muon will attribute that lack of synchronization to the reason why the Earth-based observer concludes that the muon's clocks are running slow, when the muon observes that the it's the Earth-based clocks that are running slow.

This explains the symmetry of time dilation. That is, it explains how it's possible for each observer to conclude that the other's clocks are running slow.