What is the rate of time dilation in relation to velocity?

[Jeebus]

I am having some major issues grasping the concept of Einstein's proposed time dilation in respect to special relativity, specifically relative motion. I realize my questions are most likely due to a lack of my own understanding , and not a flaw in Einstein’s logic.

In using Einstein’s twins model, he says if one twin stayed on Earth, and the other was to travel at near the speed of light away from the Earth and then return, he (the traveling twin) would find his brother had aged much more rapidly than himself. The concept being that when traveling at high velocities your personal time slows.

So here is my question : Why is it that the twin on the rocket ship is the one moving? Isn’t the Earth moving away from the rocket ship just as fast as the rocket ship is moving away from the Earth? This would suggest to me an absolute motion principal, and not relative motion at all.

Lets break this down into a simpler hypothetical model. Let's say the two twins are both in rockets. The two rockets (and their contents), for the purpose of this model, are the only two objects in the universe. If one rocket starts it’s thrusters and zooms away at nearly the speed of light, it would have time dilation, and the other would not. If the moving twin cuts it’s engines, it would continue to drift away at a constant speed. So now what’s the difference between the two rockets and the two twins? We could assume either one to be motionless or traveling at great velocity according to relative motion, we just have to pick one.

Is there something I’m missing?

Problem 2

Using the same before mentioned principals, imagine a rocket traveling extremely close to the speed of light (lets say 95%). So an observer on Earth watches this rocket travel, and records the distance. According to the Earth observer, the rocket traveled .95 light years over the course of a year. If time is slower on the rocket than on Earth, due to the great velocity of the rocket, it has been less than a year to travel the distance of .95 light years. Let's say that time runs at a ratio of 3:4, on the rocket compared to the Earth. According to the rocket pilot the .95 light year trip only took .75 years. That means that he/she traveled at a rate faster than the speed of light.

So the question here is : what exactly is the rate of time dilation in relation to velocity?

Now to a degree this still irrelevant. Let's say the rocket is traveling at 99.9% the speed of light. It would only take a fraction of a percent to tip the scales over the speed of light in the perspective of the pilot.

---------------------------------------------
Time Dilation
T = time interval on clock in relative motion
t0 = time interval on clock at rest
v = speed of relative motion
c = speed of light

T = t0 / sqrroot(1-v^2/c^2)

Another useful formula is the one for the Lorentz Contraction (space along the direction traveled become shorter the closer you get to light):

l0 = length measured when object is at rest
L = length measured while in relative motion
v = speed of relative motion
c = speed of light

L = l0 * sqrroot(1-v^2/c^2)


T = 1 / sqrroot(1-0.95^2/1)
T = 1 / sqrroot(1-0.90)
T = 1 / sqrroot(.1)
T = 1 / .32

The astronaut covered the distances of .95 lightyears (as measured by the person on earth) in .31 years (as measured by himself...3 years as measured on earth).

Now, according to the astronaut, the lengh of his trip was:

L = l0 * sqrroot(1-v^2/c^2)
L = .95 * sqrroot(1-0.95^2/1)
L = .95 * sqrroot(.1)
L = .95 * .32
L = .3

So, according to the astronaut, he traveled .3 light years in in .31 years, or about 95% of the speed of light. According to the people on earth, he traveled .95 light years in 1 year, or about 95% of the speed of light.

Is that right?

 

[chroot]

Originally posted by Jeebus
The concept being that when traveling at high velocities your personal time slows... If time is slower on the rocket than on Earth, due to the great velocity of the rocket

This is your fundamental misconception. High velocities do not make time slow down. High relative velocities make time pass more slowly as measured by a stationary observer.

The "twin paradox" you reference is thoroughly explained both on this site (do a search for "twin paradox") and on other sites all over the web. Here's a good one:

http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html

- Warren