# Can anyone explain this possible paradox?

Okay, lets say 2 ships start a voyage to Alpha Centauri, about 4 light years away. Ship A travels at 0.99 x C and reaches Alpha Centauri but because of time dilation let's say 100 years pass on Earth. Ship B travels at 0.1 x C and gets to Alpha Centauri in about 40 years according to an Earth observer because the time dilation is minimal.

Could a slower ship get somewhere faster in certain frames of reference?

would an earth observer see the slower ship passing the faster ship?

An Earth observer would see the faster ship travel 4 light years in 100 years. since speed = distance by time that would equal to the ship moving at 4% the speed of light.

Thanks

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Doc Al
Mentor
Okay, lets say 2 ships start a voyage to Alpha Centauri, about 4 light years away. Ship A travels at 0.99 x C and reaches Alpha Centauri but because of time dilation let's say 100 years pass on Earth.
At that speed it would take slightly over 4 years for the trip, according to Earth observers. Not 100 years.

HallsofIvy
Homework Helper
Okay, lets say 2 ships start a voyage to Alpha Centauri, about 4 light years away. Ship A travels at 0.99 x C and reaches Alpha Centauri but because of time dilation let's say 100 years pass on Earth.
"let's say"? How did you arrive at 100 years? If we are just making up numbers why not say "one tenth second"?
If the ship is moving at just slightly below the speed of light it will cross 4 light years in just slightly more than 4 years. That's pretty simple arithmetic, isn't it? Both your "4 light year" distance and "0.99 c" speed are from the earth's frame of reference so there is NO time dilation.

Ship B travels at 0.1 x C and gets to Alpha Centauri in about 40 years according to an Earth observer because the time dilation is minimal.

Could a slower ship get somewhere faster in certain frames of reference?

would an earth observer see the slower ship passing the faster ship?

An Earth observer would see the faster ship travel 4 light years in 100 years. since speed = distance by time that would equal to the ship moving at 4% the speed of light.

Thanks

I didn't think the precise numbers are important to the idea.

At that speed it would take slightly over 4 years for the trip, according to Earth observers. Not 100 years.
Really? I thought the whole idea of relativity was that a time passes more slowly for you if your traveling closer to light speed. The person in the cockpit moving at 0.99 C would expierience 4 years and travel ~ 4 light years but on earth more time would pass. Isn't that the whole idea behind the twin paradox? that more time passes on earth because the person is "stationary" with respect to the ship?

So if you plug 0.99 into this calculator http://www.1728.org/reltivty.htm?b0=0.99 you get a time dilation factor of 7.088 which means a person on earth would experience ~ 7.088 x 4 years = ~ 28 years.

If we're going to quibble about exact numbers, make that 0.999C that becomes 22.36 x 4 years = ~ 88 years that pass according to an earth observer

Whereas if a person is traveling at 0.1 C the time dilation factor is 1.005037 (negligible) and a person in the slower ship will experience 40 years to travel 4 light years and a person on earth observing him will experience 40 x 1.005 = 40.2 years.

In the above scenario the person on Earth would see the slower ship going at 0.1 C arrive at the destination first, 40.2 years after the ship launched.

Doc Al
Mentor
Really? I thought the whole idea of relativity was that a time passes more slowly for you if your traveling closer to light speed. The person in the cockpit moving at 0.99 C would expierience 4 years and travel ~ 4 light years but on earth more time would pass. Isn't that the whole idea behind the twin paradox? that more time passes on earth because the person is "stationary" with respect to the ship?

So if you plug 0.99 into this calculator http://www.1728.org/reltivty.htm?b0=0.99 you get a time dilation factor of 7.088 which means a person on earth would experience ~ 7.088 x 4 years = ~ 28 years.
Earth observers will say the ship took about 4.04 years to travel the distance. According to the ship's clock, the trip only took 4.04/γ = 0.57 years.
Whereas if a person is traveling at 0.1 C the time dilation factor is 1.005037 (negligible) and a person in the slower ship will experience 40 years to travel 4 light years and a person on earth observing him will experience 40 x 1.005 = 40.2 years.
At a speed of 0.1c, earth observers will say the trip took 40 years. According to the ship's clock, the trip only took about 39.8 years.
In the above scenario the person on Earth would see the slower ship going at 0.1 C arrive at the destination first, 40.2 years after the ship launched.
Why in the world would you think that a person on earth would see the slower ship reach the destination first?

mfb
Mentor
According to the ship's clock, the trip only took 4.04/γ = 0.57 years.
@shatteredmerc: You can get the same result in the frame of the ship: There, the length is contracted. Therefore, the ship does not need 4 years for its own clocks, but just 0.57 years (I did not check the number).

Wow, I can see why this hard to grasp :)

Earth observers will say the ship took about 4.04 years to travel the distance. According to the ship's clock, the trip only took 4.04/γ = 0.57 years
Wouldn't this mean that the pilot in his frame of reference crossed 4 light years in 0.57 years? In essence traveled faster than light?

I am aware that during his trip when he looks out the window he will see light racing away from him at C regardless of his speed, but the fact remains he crossed those 4 light years of distance in 0.57 years......

Wouldn't this mean that the pilot in his frame of reference crossed 4 light years in 0.57 years? In essence traveled faster than light?
You are missing the fact that not only the measurement of time changes, but the measurement of distance also changes. The distance to Alpha Centauri is 4 light years as measured in the Earth rest frame, but to the pilot the distance is length contracted to 0.56427 light years. Since he measures the time of the trip to be 0.57 years, he calculates his speed to be 0.56427/0.57 = 0.99c.

Ah, I thought the ship contracts not the space around it as well. Thanks guys I understand it now :) cheers

Ah, I thought the ship contracts not the space around it as well. Thanks guys I understand it now :) cheers
The ship does contract as measured in the Earth reference frame, but in the rocket reference frame the Earth to Alpha Centauri distance contracts. In the rocket rest frame the rocket is still the same length as it was when it at rest on the Earth.

so, just how far away is alpha centauri then? you are saying that at 0.99C, we can travel 4 LY in just over .5 yr - either we have just traveled 8 x C, or alpha cent is really only a half a LY away from us... the closer you get to C, the crazier this sounds (like at 0.99999999999C, alpha centauri is only about 1 mile away from earth?).

ie, if we really can build a ship that can reach V of near C, we will not ever need FTL speeds to travel the universe. of course, the results of our travels will never be of use to the earthlings we left behind.

mfb
Mentor
Alpha Centauri is 4 ly away in our system.
Alpha Centauri is 0.5 ly away in another system.

There is no contradiction here, the distance between two objects depends on the reference frame.

if we really can build a ship that can reach V of near C, we will not ever need FTL speeds to travel the universe
This is possible. If your ship can accelerate as long and as much as you want, you can reach every point in the obserable universe (and beyond) within your lifetime. However, if you would return to earth afterwards, you would see that the sun died long ago.