How does time dilation affect travel time at different speeds?

[John Fluharty]

Forgive me if this is the wrong place to ask this, but I am trying to do a mental exercise to understand time dilation relative to an observer's frame of reference. In my though experiment I am on a space station and someone is coming to meet me here. GST or Galactic Standard Time is always relative to this particular space station. My friend begins from a point 10 light hours away from me and travels at some fraction of C. I used the formula from the wiki page simplified by expressing V in terms of C: T' = T/SQRT(1-V**2). Below I have a chart calculating T' given V at various fractions of C to travel over this fixed distance to me.

T V T'
50.0 0.20 51.03
40.0 0.25 41.31
33.3 0.30 34.94
28.6 0.35 30.50
25.0 0.40 27.28
22.2 0.45 24.88
20.0 0.50 23.09
18.2 0.55 21.77
16.7 0.60 20.83
15.4 0.65 20.24
14.3 0.70 20.00
13.3 0.75 20.16
12.5 0.80 20.83
11.8 0.85 22.33
11.1 0.90 25.49
10.5 0.95 33.71

So if I did this correctly, the quickest trip in terms of GST from a distance of 10 light hours is at 0.7C. The fact that it took my friend 14.3 hours his time is irrelevant to me. He can reach me in 20 hours, my time, if he travels at 0.7C. This might be miles off base. I am not a physics person. Did I miss something here?

 

[phinds]

He can reach me in 20 hours, my time, if he travels at 0.7C.

If you travel 10 light hours at 70% the speed of light, how can this take exactly 20 hours? That's not physics, it's arithmetic.

 

[russ_watters]

If you travel 10 light hours at 70% the speed of light, how can this take exactly 20 hours?

By taking the time in one frame and the distance in the other. Just because you and your friend are 10 light hours apart before you start your trip, that doesn't mean the trip covers 10 light hours!

[Er...still an issue though.]
By my math, the 10 light hour trip takes 5 hours, not 20. Dividing by 2 (1.4²), not multiplying.

 

[John Fluharty]

Wait, how can a trip which takes light 10 hours take the ship 5 hours? Seems like my friend is traveling at twice the speed of light!

 

[russ_watters]

Wait, how can a trip which takes light 10 hours take the ship 5 hours? Seems like my friend is traveling at twice the speed of light!

By having the trip time dilated and length contracted. That's the trick of interstellar travel in a human lifespan: you can travel anywhere you want in the galaxy in your lifetime as long as you don't mind everyone you know being dead when you return.

 

[PeroK]

Wait, how can a trip which takes light 10 hours take the ship 5 hours? Seems like my friend is traveling at twice the speed of light!

You've got to learn at least something about Special Relativity before you start doing calculations! Who says it takes light 10 hours? Someone on the space station. But, your friend would measure it differently. Time to reach for an SR textbook!

 

[John Fluharty]

As I said in my post, the only frame of reference I care about is that of the space station. Is the formula I used not relevant in that context? Let me ask another way to make sure I am asking my question clearly. Suppose I have 16 drone ships leaving from the same point and moving toward me at various fractions of the speed of light. Which of these drones will I observe to arrive first? The faster they go the more time dilates, right?

 

[PeroK]

As I said in my post, the only frame of reference I care about is that of the space station. Is the formula I used not relevant in that context? Let me ask another way to make sure I am asking my question clearly. Suppose I have 16 drone ships leaving from the same point and moving toward me at various fractions of the speed of light. Which of these drones will I observe to arrive first? The faster they go the more time dilates, right?

The formula for a journey is:

Time = distance/speed

There is no time dilation in a single frame of reference.

 

[ZapperZ]

As I said in my post, the only frame of reference I care about is that of the space station. Is the formula I used not relevant in that context? Let me ask another way to make sure I am asking my question clearly. Suppose I have 16 drone ships leaving from the same point and moving toward me at various fractions of the speed of light. Which of these drones will I observe to arrive first? The faster they go the more time dilates, right?

This is very puzzling. What does time dilation have anything to do with which one gets to you first? The one that is traveling FASTER will get to you first! You measure time in YOUR reference frame. You don't measure your time in someone else's reference frame (do you look at the clock and wonders what time it is on Alpha Centauri?).

Zz.

 

[John Fluharty]

So I will observe the drone moving at 0.2C to arrive in 50 hours but the ship will observe less time passing! I think I get it!

 

[Mister T]

T' = T/SQRT(1-V**2). Below I have a chart calculating T' given V at various fractions of C to travel over this fixed distance to me.

For $v<1$ you will always have $T'$ greater than $T$. In your spreadsheet that is not always the case, so evidently there's something wrong with your calculations.

 

[John Fluharty]

Yep. I was standing on my head, as Russ pointed out.

 

[russ_watters]

You don't measure your time in someone else's reference frame (do you look at the clock and wonders what time it is on Alpha Centauri?).

If by Alpha Centuari you mean San Francisco, I do it all the time. If you want to set up a meeting with someone in a different reference frame it is common courtesy to calculate the time shift for them when suggesting the meeting time. But when you send the meeting invite, Outlook does it for you.

So yeah, I would suggest that if you are traveling to visit someone on Alpha Centuari, you should calculate and tell them what time their clock will read when you get there, not what your clock will read when you get there.

 

[John Fluharty]

I was lacking a basic understanding of reference frames, but you all have helped. Thanks a bunch!

 

[sydneybself]

I'm not quite sure what you're trying to get at, but let me have a shot at it.
To begin with, calculations are simpler if you use either a speed of 60% of the speed of light or 80% of the speed of light because in either case you're dealing with a 3 x 4 x 5 triangle, so traveling at 80 % of the speed of light it takes your friend 12.5 hours your time to arrive. However he experienced time dilation for 60% of those 12.5 hours or 7.5 hours. In other words, although his trip took 12.5 hours on your clock it only took 7.5 hours by his clock. Does that help any?

 

[Vitro]

However he experienced time dilation for 60% of those 12.5 hours or 7.5 hours.

To be rigorous, he did not experience time dilation but length contraction such that he only had to travel a distance of 6 light-hours which at a speed of 0.8c took him 7.5 hours.

 

[Ibix]

To be rigorous, he did not experience time dilation but length contraction such that he only had to travel a distance of 6 light-hours which at a speed of 0.8c took him 7.5 hours.

To be more precise, he experienced neither length contraction nor time dilation. He felt as normal. However, the distance between his origin and destination was length contracted in his rest frame to 6 light hours, so it took 7.5 hours for his destination to reach him at 0.8c.

Compare with the description in the rest frame of the origin and destination, in which case it takes the guy 12.5 hours to cross 10 light hours at 0.8c. However his clocks tick slower due to time dilation so he only experiences 7.5 hours.

 

[John Fluharty]

Yep, it definitely helps. I just had a hard time finding explanations online that clicked for me. The twins example just confused me and high school physics class and calculus were nearly 30 years ago so I was quickly lost on the sites that jumped into any kind of math.