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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 16-3 The laser beam still travels in straight lines, and it still travels at 3 108 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 16-3 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
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There exists a mathematical relationship between the speed of the space ship in the foregoing mind experiment and the extent to which time is
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Position of laser/sensor Figure 16-3 This is what an external observer sees as the laser-clock-equipped
space ship whizzes by at a sizable fraction of the speed of light
Special and General Relativity
dilated Let tship 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 Earth-based observatory Let u be the speed of the ship as a fraction of the speed of light Then tship (1 u2) 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 u2) 1/2 ] (1 u2)
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 05 If 1 s passes on Earth, then according to an Earthbound observer: tship (1 (1 052) 1/2 025) 1/2
0751/2 087 s That is, 087 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 1/087, or approximately 115 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 099 If 1 s passes on Earth, then we, as Earthbound observers, will see this: tship (1 (1 002 0992) 1/2 098) 1/2
014 s That is, 014 s will seem to pass on the ship as 1 s passes on Earth The time dilation factor k in this case is 1/014, or approximately 71 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