Quick scenario

[havok20x]

Let's say that Bert and Steve are racing for 100 light years. Bert is travelling 99% the speed of light and Steve is travelling at half that speed. From Bert's frame of reference, when will each arrive at the finish line? What about from Steve's? What about from a stationary Observer?

 

I am having difficulty understanding the time dilation aspect of relativity. From Bert's perspective, won't Steve arrive at the finish line first, since time appears to speed up in the space outside of his ship?

[swansont]

Let's say that Bert and Steve are racing for 100 light years. Bert is travelling 99% the speed of light and Steve is travelling at half that speed. From Bert's frame of reference, when will each arrive at the finish line? What about from Steve's? What about from a stationary Observer?

 

I am having difficulty understanding the time dilation aspect of relativity. From Bert's perspective, won't Steve arrive at the finish line first, since time appears to speed up in the space outside of his ship?

 

While relativity has implications for simultaneity, it can't change events that happen, so if you consider that the winner breaks a tape at the finish line, everyone has to agree who breaks it. The stationary observer sees Bert break the tape after ~101 years, and Steve crosses after 200. Bert breaks the tape in all frames of reference.

 

As long as there is no acceleration involved, all observers see each others' clocks as running slow. Nobody sees time speed up outside of their frame.

 

From Bert's frame, since gamma is ~7, he sees the trip as being ~14.3 LY and take just under 14.5 years.

Steve has a gamma of ~1.15, so his trip is 86.6 LY, and it takes just over 173 years.

 

Speeds do not add linearly; to find relative speeds you use s = (u + v)/(1 + uv/c^2) From that you can do an accounting of other events

[Didymus]

I think the most interesting aspect, though, is how each of them see each other traveling.

 

Say Bert has a navigator named Velanthria that is more interested in the race than the finish line and the whole time is watching their opponent. Traveling aimlessly through space, she has a speed relative to Steve of less than .5C... meaning, while she's comparing her speed to Steve, space is contracted by a factor of less than 1.15 for the duration of their race.

 

... Now, Steve thinks Velanthria's cute and is paying attention to her this whole race. Although they're moving through space in the same direction, his speed relative to Velanthria's ship is exactly equal to her speed relative to him (in opposite directions because one is pulling ahead at the same speed as the other is falling behind)

 

... Care to take a swing at that mess?

Some theories look beautiful, until you see all the variables that have been swept under the rug.

I'm sure you'll disagree... but, please be specific as to which part. You do agree that the ships have a speed relative to each other, correct? And that this frame of reference is equally valid as a comparison to any other point of reference (such as the finish line)

[elfmotat]

I think the most interesting aspect, though, is how each of them see each other traveling.

 

Say Bert has a navigator named Velanthria that is more interested in the race than the finish line and the whole time is watching their opponent. Traveling aimlessly through space, she has a speed relative to Steve of less than .5C... meaning, while she's comparing her speed to Steve, space is contracted by a factor of less than 1.15 for the duration of their race.

 

... Now, Steve thinks Velanthria's cute and is paying attention to her this whole race. Although they're moving through space in the same direction, his speed relative to Velanthria's ship is exactly equal to her speed relative to him (in opposite directions because one is pulling ahead at the same speed as the other is falling behind)

 

... Care to take a swing at that mess?

 

Some theories look beautiful, until you see all the variables that have been swept under the rug.

 

I'm sure you'll disagree... but, please be specific as to which part. You do agree that the ships have a speed relative to each other, correct? And that this frame of reference is equally valid as a comparison to any other point of reference (such as the finish line)

 

As swansont mentioned at the end of his post, relativistic velocity addition. If I observe you moving at speed v relative to me, and you throw a baseball at speed u relative to you, then I observe the baseball moving at (u+v)/(1+uv/c²) relative to me. For u & v much less than c, this is approximately equal to u+v.

[Didymus]

true. by the U+V formula, their relative speeds would be .49c. That formula compresses the number the closer either number gets to C, it drops to a relative speed of .327...c

 

(i retract my previous statement about something interesting happening. Dyslexia attack.)

Edited March 16, 2013 by Didymus

[swansont]

Oddly enough, this particular formula is interesting using c as a unit of measurement instead of first converting to a smaller unit like miles per second or kilometers per second.

 

It's generally easier to use fractions of c than e.g. km/s

[md65536]

I think the most interesting aspect, though, is how each of them see each other traveling. [...]

Care to take a swing at that mess?

 

Some theories look beautiful, until you see all the variables that have been swept under the rug.

A relativistic Doppler analysis will tell you what anyone sees.

Sometimes that makes it a lot easier to understand and cleaner, sometimes it's harder and more of a mess (see my recent posts for examples!).

But nothing's swept under the rug. The relativistic Doppler factor can be derived from the Lorentz factor just by including the travel time of light. It should not be surprising that everything is consistent, whatever variables you want to use.