FORMULA FOR TIME DILATION

There exists a mathematical relationship between the speed of the space ship in the foregoing “mind experiment” and the extent to which time is dilated. Let t_ship be the number of seconds that appear to elapse on the moving ship as precisely 1 s elapses as measured by a clock next to us as we sit in our Earth-based observatory. Let u be the speed of the ship as a fraction of the speed of light. Then
t_ship = (1 - u²)^1/2
The time dilation factor (call it k) is the reciprocal of this; that is,
k = 1/[(1 -u²)^1/2]
= (1 - u²)^-1/2
In these formulas, the number 1 represents a mathematically exact value and is good to any number of significant digits.
Let’s see how great the time dilation factor is if the space ship is traveling at 1.50 × 10⁸ m/s. In this case, u = 0.500. If 1.00 s passes on Earth, then according to an earthbound observer,
t_ship = (1.00 × 0.5002)1/2
= (1.00 × 0.250)1/2
= 0.7501/2
= 0.866 s
That is, 0.866 s will seem to pass on the ship as 1.00 s passes as we measure it while watching the ship from Earth. This means that the time dilation factor is 1.00/0.866, or approximately 1.15. Of course, on the ship, time will seem to “flow” normally.
Just for fun, let’s see what happens if the ship is going 2.97 × 10⁸ m/s. In this case, s = 0.990. If 1.00 s passes on Earth, then we, as earthbound observers, will see this:
t_ship = (1.00 × 0.9902)1/2
= (1.00 × 0.98)1/2
= 0.02001/2
= 0.141 s
That is, 0.141 s will seem to pass on the ship as 1.00 s passes on Earth. The time dilation factor k in this case is 1.00/0.141, or approximately 7.09. Time “flows” more than seven times more slowly on a ship moving at 99 percent of the speed of light than it “flows” on Earth—from the reference frame of someone on Earth.

As you can imagine, this has implications for time travel. According to the special theory of relativity, if you could get into a space ship and travel fast enough and far enough, you could propel yourself into the future. You might travel to a distant star, return to Earth in what seemed to you to be only a few months, and find yourself in the year 5000 A.D. When sciencefiction writers realized this in the early 1900s just after Einstein published his work, they had a bonanza with it.