Time Travel: How Far & Fast to Reach Alien Neighbors?

[Saerryan89]

I was discussing with a friend about speed of light and time traveling and I realized something bizarre. I could be wrong so please correct my mistakes.
So let's assume we want to go to our alien neighbour whose planet is 100 light years away. When we travel there with near light-speed, we should be going forward in time too. Which means if we travel, let's say, 120 years, 150 years could have been passed in the world and in our neighbours planet. This is a problem if we want to arrive there as soon as possible relative to them. I did some calculations and;

let's say we are going at;

0,6 c which means we would arrive to our destination in 166,67 years (for traveler). But on the outside it would be ~208,33 years which means we traveled ~41,66 years forward in time.

0,8 c which means we would arrive to our destination in 125 years (for traveler of course). But on the outside it would be ~208,33 years (wow same as 0,6 c) which means we traveled ~83,33 years forward in time.

0,99 c which means we would arrive to our destination in ~101,01 years(again, for traveler). But on the outside it would be ~5075,879 years! It is because we are so close to speed of light it literally fast-forwards the universe!

So by going faster we actually arrived almost 5000 years late! But if you travel at more reasonable speeds(0,8 c) we can travel 100 light years in 125 years while the universe around us age more than 200 years which is still reasonable(a lot more than 0,99 c at least)

Here are my calculations -> http://www.mediafire.com/view/1tq82330kffsby3/IMG_0471.jpg

 

[PeroK]

Your calculations are all wrong. As measured by Earth and the destination planet, the time taken to travel 100 light years is simply the distance divided by the speed of the rocket. So, at 0.99c, only 101 years have passed on Earth and at the destination.

The time that passes on board the ship will be less than this, according to the time dilation formula.

 

[HallsofIvy]

I was discussing with a friend about speed of light and time traveling and I realized something bizarre. I could be wrong so please correct my mistakes.
So let's assume we want to go to our alien neighbour whose planet is 100 light years away. When we travel there with near light-speed, we should be going forward in time too. Which means if we travel, let's say, 120 years, 150 years could have been passed in the world and in our neighbours planet. This is a problem if we want to arrive there as soon as possible relative to them. I did some calculations and;

let's say we are going at;

0,6 c which means we would arrive to our destination in 166,67 years (for traveler). But on the outside it would be ~208,33 years which means we traveled ~41,66 years forward in time.

No, if you are moving at 0,6c relative to "the outside world" the it will take you 100/.6= 166,67 years "on the outside world". To the traveler, taking time dilation into account, that would be 133.33 years. In any case I not clear on what you mean by "travel into the future". We all "travel into the future" even if we are sitting still.

0,8 c which means we would arrive to our destination in 125 years (for traveler of course).

Again, no. 100/0,8= 125 would be the time to people who are not moving relative to the two stars.

But on the outside it would be ~208,33 years (wow same as 0,6 c) which means we traveled ~83,33 years forward in time.

To the traveler, the time required would be 75 years.

0,99 c which means we would arrive to our destination in ~101,01 years(again, for traveler). But on the outside it would be ~5075,879 years! It is because we are so close to speed of light it literally fast-forwards the universe!

So by going faster we actually arrived almost 5000 years late! But if you travel at more reasonable speeds(0,8 c) we can travel 100 light years in 125 years while the universe around us age more than 200 years which is still reasonable(a lot more than 0,99 c at least)[/quote]
What? Where did you get 5000? At 0.99c, you can go 100 light years in 100/.99= 101 years. Again, that is relative to people at rest relative to the stars. To the traveler it would take 14,25 years.

Here are my calculations -> http://www.mediafire.com/view/1tq82330kffsby3/IMG_0471.jpg

 

[Ibix]

As others have noted, you are mixing up who measures what, which is why you are getting odd answers. It's important in relativity to keep track of who is measuring distances and times, because people will not in general agree.

You said the alien planet was 100ly away. Presumably that's measured by the people on the planets. If the ship travels there at 0.6c, then according to clocks on the planets it takes distance/speed=100/0.6=166.67 years to get there. Clocks on the spaceship are ticking slowly seen from this perspective, so the travel time according to people on the ship is 133.33 years.

From the perspective of the ship, however, it is stationary and the planets are moving at 0.6c. Its clocks are ticking normally from this perspective - but the distance between the planets is length contracted to 80ly. The time for the destination planet to come to the ship is distance/speed=80/0.6=133.33 years.

There is a third element to relativity, in addition to length contraction and time dilation, which is the relativity of simultaneity. I haven't mentioned it above, but you do need it to paint a complete and coherent picture of this journey.

 

[Herman Trivilino]

Let's take a look at your first calculation:

0,6 c which means we would arrive to our destination in 166,67 years (for traveler).

Not for the traveler. This calculation is done using coordinates in Earth's frame of reference. Imagine an x-axis stretching from Earth to the distant planet. It's marked off in units of light years. If x=0 at the location of Earth, then x=100 at the location of the distant planet. It would indeed take 166.67 years for the rocket to reach the distant planet, in this frame of reference. Note that we are looking at two events here. Event 1 occurs when the rocket leaves, Event 2 occurs when the rocket arrives.

You then calculated correctly that $\frac{1}{\sqrt{1-0.6^2}}=0.8$. However, you made a mistake when you divided by this number, you should instead have multiplied:$$(0.8)(166.67)=133.33.$$

The best way I know to keep track of this is to look at things from the frame of reference of the rocket. Imagine an x'-axis with the rocket at the origin. As the rocket moves it carries this x'-axis with it, so when it gets to its destination it's still at the origin! In other words, it hasn't changed position in its frame of reference. In this frame of reference Events 1 and 2 occur at the same place, so that the time elapsed between them is by definition a proper time, called $\Delta \tau$.

$$(\Delta \tau)^2 = (\Delta t)^2-(\Delta x)^2 = (166.67)^2-(100)^2 = (133.33)^2$$

In your next two calculations you made this same error, but you also made mistakes in the arithmetic you did. If you fix those, you'll get the right answers.