General Question

davidgro's avatar

At what speed would you experience a year when traveling a light-year?

Asked by davidgro (609points) August 5th, 2010

I know that due to time dilation, when traveling very close to the speed of light, you would experience less than a year of time to travel one light-year of distance, (You’d feel no time at all if you could travel At the speed of light)

Of course when traveling well under the speed of light you would feel it taking more than a year to get that far.

So, somewhere close, but not too close, to the speed of light there must be a speed where you would experience one year to travel a light-year (or a second to go a light-second, etc.)
Of course from the perspective of the people you left at home you would be taking more than a year, since your speed is under that of light.
I don’t know what math is involved, so I can’t figure it out myself, but I’m curious just what that speed is (say as a fraction of C)

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15 Answers

Kraigmo's avatar

mattbrowne, wake up, there’s a question here we need you to answer

wgallios's avatar

I believe you would experience for 1 year traveling at as close to light speed as you could, 100 years would pass on earth.

But as you stated you are correct, most people on earth would never see you again. So if you got into space shuttle today, and traveled around the earth at light speed you would come back to roughly the year 3010.

Unfortunately I cant find any reference to site this, although I have heard it before from theoretical physicist’s. But obviously it’s all based upon the theory of relativity.

davidgro's avatar

@wgallios I think you are really close to understanding my question. Yes, there is a speed close to light-speed where you would feel 1 year for every 100 light-years you travel. (Remember, a light-year is a distance, it’s how far light goes in a year)
To the people back on earth it would take a little more than 100 years for you to make the trip, because you’re still going a tiny bit slower than light.

If you go even faster, (closer to the speed of light) then there is a speed where you would feel 1 year for every 1,000 light-years you go. faster still and it could be 1-to-10,000, etc.

But if you go slower (but still fast), you might feel it take a year to go only 10 light-years,
If you go slower still, 1 year for only 2 light-years.
Call this speed A.
If you go half the speed of light, you would feel it taking almost 2 years to go one light-year, while to the earthlings you are taking exactly two years.
Call this speed B.

Somewhere between A and B is the speed I am asking about, where traveling 1 light-year would take you (on the ship) one year, and those on earth see you taking a bit more than a year. I want to know how fast this is exactly (with the equations).

I doubt I’m the first person to think of this, so I wouldn’t be surprised if this speed already has a name. But I don’t know how to find it. (Seriously, try to google it ;-) )

Zyx's avatar

This is what you were looking for, thank me very much.

davidgro's avatar

@Zyx, I do thank you, but not very much, because you didn’t find the answer to the question. However, I do think that I have found the answer, by playing with the equation there: T=1/√(1-v^2/c^2) (simplified a little) (v being the speed as fraction of c, which is the speed of light, the result T is the time dilation factor)

I figured it might work to find a speed that equals it’s time dilation factor (the reciprocal of the time dilation factor really, so both are between 0 and 1)

So I tried to find a value that it converges to when you feed it the output as an input (I removed the “1/” from the beginning so this would work) and found that for each value I give it, it bounces between that and another value. Taking the average got me a value who’s partner was much closer to it, and so I found it quickly heading toward .707something times lightspeed as the right speed. This looked familiar to me.

The reciprocal of the square root of 2 (also known as half the square root of 2) is about 0.7071067811865475244… and plugging in that as the speed (to the unmodified equation) results in the square root of 2 (about 1.4142135623730950488…) as the time dilation, which matches what the travel time would be to the earth viewpoint.

So I think the answer is that if you travel at
The (reciprocal of the square root of two) times the speed of light =
(1/(2^(½)))c =
about 0.7071067811865475244c =
about 211,985.28 kilometers per second =
about 131,721.546 miles per second =
about 763,147,008 kilometers per hour =
about 474,197,566 miles per hour
then you will experience as much travel time as the light-distance measured before you start the trip.

I’d love to have someone confirm if I’m right, and let me know if this speed has a more searchable name.

mattbrowne's avatar

Check out @davidgro‘s answer. It really depends on whether it 99% or 99.99% or 99.9999999999999999999% speed of light. 100% is impossible because your mass would be infinite. You can’t travel at the speed of light for one year.

davidgro's avatar

@mattbrowne, But can you tell me if I am I right about 70.71% of light speed being the ‘break-even’ speed from the point of view of the travelers? (as in, at that speed they would feel and age about 4.2 years on a journey to reach Proxima Centauri which is about 4.2 light years away as measured from earth. The earthlings would measure (but not see exactly) the ship taking about 5.9 years)

That is after all my original question ;-)

Note: I am assuming something like a “Park Shift” – instant speed changes (I just read the first two “Ender’s Game” books) so that there is no acceleration or deceleration to complicate things.

Regarding infinite mass traveling at c, As I understand it, from the point of view of the traveler, it’s the rest of the universe that would have infinite mass/energy (including the cosmic microwave background radiation and starlight. Hmm. An engineering problem…) However the trip would be over instantly as the destination Lorentz-contracts to being infinitely close. (and thin) (I suppose if you aimed for empty space then the trip would take NaN time as you multiply 0 by infinity. I have no idea what that means to the traveler.)

mattbrowne's avatar

@davidgro – I’m not sure what you mean by break even?

davidgro's avatar

@mattbrowne I genuinely don’t know how to explain it any clearer than I did in the question’s description or in my first comment to the question.
For that matter the question’s title seems like it should convey the idea: what is wrong with “At what speed would you experience a year when traveling a light-year?”?

The question seems perfectly reasonable to me, but am I really writing gibberish due to completely misunderstanding relativity?
(Certainly possible, but I thought my grasp of it is decent for a layperson… This is especially disconcerting because the answer I think I’ve found appears to fit perfectly and makes sense to me, so I think the question is valid)

mattbrowne's avatar

@davidgro – Experience in relation to what? Maybe it could help to talk about clocks and calendars on Earth and on that spaceship. So your idea is that the astronaut starts his stopwatch and rangefinder. Both are set to zero before the thought experiment begins. He also got a speedometer. You want the stopwatch to show 1 year has passed and the rangefinder that the ship has travel 1 light year. Then check the display of the speedometer?

Or is it about how the people on Earth left behind? Their stopwatch and their speedometer observing the ship with a telescope?

davidgro's avatar

Oh, yeah I guess I didn’t make the frames of reference explicit. I mean that before the trip the astronauts measure the distance from Earth to the destination and come up with X light-years. I’m ignoring minor things like Earth’s orbit or the sun’s relative motion to the destination
Then the astronauts start traveling at about .7071c in Earth’s frame.
(Of course to the people on the ship they are not moving and the rest of the universe is. Related Question: would they measure (after accounting for light propagation delays) the movement of say the Sun and the destination as each being ~.7071c, but distances Lorentz-contracted by ~1.4142 so the distance and time comes out even in their frame? Or would they see the speed as different?)

Is that the speed (in Earth’s frame at least) where the atomic-clock controlled calendar on the ship would mark that X years have passed when they reach the destination?
(I think most stopwatches won’t run for more than a day without looping)

In Earth’s frame, I would expect the trip to take about 1.4142*X years
(With Earth receiving the “We made it” message 2.4142*X years after the ship left, because it was sent at a point X light-years away. Unless the ship has an ansible ;-) )

davidgro's avatar

@mattbrowne (sorry, I had forgotten to @ you.)

Cirbryn's avatar

@davidgro wrote: “I know that due to time dilation, when traveling very close to the speed of light, you would experience less than a year of time to travel one light-year of distance, (You’d feel no time at all if you could travel At the speed of light)”

I think this is where you go off the rails. I may be “traveling” using conventional language, but from my point of view I’m always standing still and everything else is moving around me. So for me to travel a light year in (what to me is) a year, I’d have to go the speed of light.

davidgro's avatar

I don’t know what you mean. Sure “I’m always standing still and everything else is moving around me.” – that’s what traveling is at any speed (especially when you think in terms of relativity) so the only way reasonable way to talk about it is already ‘conventional language’. What else would you call it?

As far as the second point, What my question is about is when you travel a distance that you measured before you started traveling, using time measured As you travel. (Not the time “back home”) Of course using the clocks at home you can only go a light year in a year by going the speed of light, but that’s trivial, and isn’t a useful answer to the people on the ship who have clocks (and bodies and brains) running at a different rate due to time dilation.

Of course one can also measure both distance while traveling and the time while traveling, but then the star system that was say 4.2 light years away before you left is now only 2.97 from Lorentz contraction, so it would still come out to a speed of .7071c for a 4.2 year (ship time) trip.
(I’m pretty sure this is right)

Cirbryn's avatar

Well my point was that if you’re traveling in your ship, time dilation due to speed won’t affect you because (from your point of view) you’re always standing still. You’re right about the spatial contraction though. You may also be subject to time dilation due to acceleration once you reach your destination and attempt to land.

So ignoring acceleration effects and applying the Lorentz equation you linked, a speed of .7071 c produces a contraction of .7071. So in a year (your time) you’d travel what you thought was .7071 light years, but it would turn out to be an entire light year due to contraction.

Very cool. I hadn’t realized that before.

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