Time Dilation Help (page 3)
The relative location of an observer in space affects the relative readings of clocks located at different points. Similarly, relative motion in space affects the apparent rate at which time “flows.” Isaac Newton hypothesized that time flows in an absolute way and that it constitutes a fundamental constant in the universe. Einstein showed that this is not the case; it is the speed of light, not time, that is constant. In order to understand why relativistic time dilation occurs based on Einstein’s hypothesis, let’s conduct a “mind experiment.”
A Laser Clock
Suppose that we have a space ship equipped with a laser/sensor on one wall and a mirror on the opposite wall (Fig. 20-2). Imagine that the laser/sensor and the mirror are positioned so that the light ray from the laser must travel perpendicular to the axis of the ship, perpendicular to its walls, and (once we get it moving) perpendicular to its direction of motion. The laser and mirror are adjusted so that they are separated by 3.00 m. Because the speed of light in air is approximately 3.00 × 10 8 m/s, it takes 1.00 × 10 −8 s, or 10.0 nanoseconds (10.0 ns), for the light ray to get across the ship from the laser to the mirror and another 10.0 ns for the ray to return to the sensor. The ray therefore requires 20.0 ns to make one round trip from the laser/sensor to the mirror and back again.
Fig. 20-2 . A space ship equipped with a laser clock. This is what an observer in the ship always sees.
Our laser emits pulses of extremely brief duration, far shorter than the time required for the beam to get across the ship. We might even suppose that the beam emits just a few photons in each burst! We measure the time increment using an extremely sophisticated oscilloscope so that we can observe the pulses going out and coming back and measure the time lag between them. This is a special clock; its timekeeping ability is based on the speed of light, which Einstein proposed is constant no matter from what point of view it is observed. There is no better way to keep time.
Suppose that we start up the ship’s engines and get moving. We accelerate with the eventual goal of reaching nearly the speed of light. Suppose that we manage to accelerate to a sizable fraction of the speed of light, and then we shut off the engines so that we are coasting through space. You ask, “Relative to what are we moving?” This, as we shall see, is an important question! For now, suppose that we measure speed with respect to the Earth.
We measure the time it takes for the laser to go across the ship and back again. We are riding along with the laser, the mirror, and all the luxuries of a small spacecraft. We find that the time lag is still exactly the same as it was when the ship was not moving relative to Earth; the oscilloscope still shows a delay of 20.0 ns. This follows directly from Einstein’s axiom. The speed of light has not changed because it cannot. The distance between the laser and the mirror has not changed either. Therefore, the round trip takes the same length of time as it did before we got the ship moving.
If we accelerate so that the ship is going 60 percent, then 70 percent, and ultimately 99 percent of the speed of light, the time lag will always be 20.0 ns as measured from a reference frame , or point of view, inside the ship.
At this point, let’s add another axiom to Einstein’s: In free space, light beams always follow the shortest possible distance between two points. Normally, this is a straight line. You ask, “How can the shortest path between two points in space be anything other than a straight line?” This is another good question. We’ll deal with it later in this chapter. For now, note that light beams appear to follow straight lines through free space as long as the observer is not accelerating relative to the light source.
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 pressure suits on so that they can breathe.)
What we see is depicted in Fig. 20-3. The laser beam still travels in straight lines, and it still travels at 3.00 × 10 8 m/s relative to us. This is true because of Einstein’s axiom concerning the speed of light and the fact 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 3.00 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 20.0 ns for the laser beam to go across the ship and back.
Fig. 20-3 . 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.
As the ship goes by, time appears to slow down inside it, as seen from a “stationary” point of view. Inside the ship, however, time moves at normal speed. 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. 20-3 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.
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 − u 2 ) 1/2
The time dilation factor (call it k ) is the reciprocal of this; that is,
k = 1/[(1 − u 2 ) 1/2 ]
= (1 − u 2 ) −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 × 10 8 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 − 0.500 2 ) 1/2
= (1.00 − 0.250) 1/2
= 0.750 1/2
= 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 × 10 8 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 − 0.990 2 ) 1/2
= (1.00 − 0.98) 1/2
= 0.0200 1/2
= 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 . When science-fiction writers realized this in the early 1900s just after Einstein published his work, they had a bonanza with it.
Time Dilation Practice Problems
Why don’t we notice relativistic time dilation on short trips by car, train, or airplane? When we land, the clocks are still synchronized (except for time-zone differences in some cases). Why isn’t the traveling clock behind the earthbased ones?
Theoretically, they are. However, the difference is too small to be noticed. The time dilation factor is exceedingly small unless a vessel travels at a significant fraction of the speed of light. Its effect at normal speeds can’t be measured unless atomic clocks, accurate to a minuscule fraction of 1 s, are used to measure the time in both reference frames.
What speed is necessary to produce a time dilation factor of k = 2.00?
Use the formula for time dilation, and let u be the unknown. Then u can be found, step by step, this way:
k = (1 − u 2 ) −1/2
2.00 = (1 − u 2 ) −1/2
0.500 = (1 − u 2 ) 1/2
0.250 = 1 − u 2
−0.750 = − u 2
u 2 = 0.750
u = (0.750) 1/2 = 0.866
That is, the speed is 86.6 percent of the speed of light, or 2.60 × 10 8 m/s.
Practice problems of these concepts can be found at: Relativity Theory Practice Test
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