Formula For Time Dilation:

There exists a mathematical association among the speed of the space ship in the prior "mind experiment" and the level to which time is dilated. Suppose t_ship be the number of seconds which emerge to elapse on the moving ship as accurately 1 s elapses as measured by the clock next to us as we sit in our Earth-based observatory. Assume 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 (let say it k) is the reciprocal of this; which is,

k = 1/[(1 - u²)^1/2]

   = (1 - u²)^-1/2

In such formulas, the number 1 symbolizes a mathematically precise value and is fine to any number of the significant digits.

Let us see how great the time dilation factor is when the space ship is traveling at 1.50 x 10⁸ m/s. In this situation, u = 0.500. When 1.00 s passes on Earth, then according to an earthbound spectator,

t_ship = (1.00 - 0.500²)^1/2

       = (1.00 - 0.250)^1/2

       = 0.750^1/2

       = 0.866 s

That is, 0.866 s will appear to pass on the ship as 1.00 s passes as we measure it as watching the ship from Earth. It means that the time dilation factor is 1.00/0.866, or around 1.15. Obviously, on the ship, time will appear to "flow" usually.

Just for enjoyment, let us see what ensues when the ship is going 2.97 x 10⁸ m/s. In this situation, s = 0.990. When 1.00 s passes on Earth, then we, as earthbound spectators, will see this:

t_ship = (1.00 - 0.990²)^1/2

       = (1.00 - 0.98)^1/2

       = 0.0200^1/2

       = 0.141 s

That is, 0.141 s will appear to pass on the ship as 1.00 s passes on our planet Earth. The time dilation feature k in this situation is 1.00/0.141, or around 7.09. The time "flows" more than seven times more gradually on a ship moving at 99 % of the speed of light than it "flows" on Earth-from the reference frame of somebody on Earth.

Since you can visualize, this has implications for time travel. According to the special hypothesis of relativity, when you could get into a space ship and travel fast sufficient and far adequate, you could propel yourself into the future. You may travel to a far-away star, return to Earth in what appeared to you to be only a few months, and locate yourself in the year 5000 A.D. Whenever science fiction writers understood this in the early 1900s just after Einstein published his work, and they had a bonanza with it.

PROBLEM:

Determine the necessary speed to generate a time dilation factor of k = 2.00?

SOLUTION:

By using the formula for time dilation, and assume u be the unknown. Then u can be found out, step by step, in this manner:

k = (1 - u²)^-1/2

2.00 = (1 - u²)^-1/2

0.500 = (1 - u²)^1/2

0.250 = 1 - u²

-0.750 = -u²

u² = 0.750

u = (0.750)^1/2

   = 0.866

That is, the speed is 86.6% of the speed of light, or 2.60 x 10⁸ m/s.