Clock In Motion
Imagine now that we are outside the ship and are back on Earth. We are equipped with a special telescope that allows us to see inside the ship as it whizzes by at a significant fraction of the speed of light. We can see the laser, the mirror, and even the laser beam itself because the occupants of the space vessel have temporarily filled it with smoke to make the viewing easy for us. (They have their pressure suits on so that they can breathe.)
What we see is depicted in Fig. 163. The laser beam still travels in straight lines, and it still travels at 3 × 10 ^{8} m/s relative to us. This is true because of Einstein’s axiom concerning the speed of light and our own hypothesis to the effect that light rays always appear to travel in straight lines as long as we are not accelerating. However, the rays have to travel farther than 3 m to get across the ship. The ship is going so fast that by the time the ray of light has reached the mirror from the laser, the ship has moved a significant distance forward. The same thing happens as the ray returns to the sensor from the mirror. As a result of this, it will seem to us, as we watch the ship from Earth, to take more than 20 ns for the laser beam to go across the ship and back.
As the ship goes by, time appears to slow down inside it, as seen from a “stationary” point of view. Inside the ship, though, the “speed of time” seems entirely normal. The faster the ship goes, the greater is this discrepancy. As the speed of the ship approaches the speed of light, the time dilation factor can become large indeed; in theory, there is no limit to how great it can become. You can visualize this by imagining Fig. 163 stretched out horizontally so that the light rays have to travel almost parallel to the direction of motion, as seen from the “stationary” reference frame.
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 1 s elapses as measured by a clock next to us as we sit in our Earthbased observatory. Let u be the speed of the ship as a fraction of the speed of light. Then
t _{ship} = (1 – u ^{2} ) ^{1/2}
The 1/2 power is an alternative and popular way to denote the square root. The time dilation factor (let’s call it k ) is the reciprocal of this:
k = 1 / [ (1 – u ^{2)1/2} ] = (1 – u ^{2)–} 1/2
Let’s see how great the time dilation factor is if the space ship is going half the speed of light. In this case, u = 0.5. If 1 s passes on Earth, then according to an Earthbound observer:
t _{ship} = (1 – 0.5 ^{2)1/2}
= (1 – 0.25) ^{1/2}
= 0.75 ^{1/2}
= 0.87 s
That is, 0.87 s will seem to pass on the ship as 1 s passes as we measure it while watching the ship from Earth. This means that the time dilation factor is 1/0.87, 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 99 percent of the speed of light. In this case, u = 0.99. If 1 s passes on Earth, then we, as Earthbound observers, will see this:
t _{ship} = (1 – 0.99 ^{2)1/2}
= (1 – 0.98) ^{1/2}
= 0.02 ^{1/2}
= 0.14 s
That is, 0.14 s will seem to pass on the ship as 1 s passes on Earth. The time dilation factor k in this case is 1/0.14, or approximately 7.1. 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 A.D. 5000. When sciencefiction writers realized this in the early 1900s just after Einstein published his work, they took advantage of it.
Practice problems of this concept can be found at: Special and General Relativity Practice Problems
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