Theory of Relativity Help (page 2)

By — McGraw-Hill Professional
Updated on Sep 18, 2011

Spatial Curvature

Imagine that you are in a space ship traveling through deep space. The ship’s rockets are fired, and the vessel accelerates at an extreme rate. Suppose that the laser apparatus described earlier in this chapter is in the ship, but instead of a mirror on the wall opposite the laser, there is a screen. Before the acceleration begins, you align the laser so that it shines at the center of the screen (Fig. 16-6). What will happen when the rockets are fired and the ship accelerates?

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Relativity Spatial Curvature

Figure 16-6. As seen from within a nonaccelerating space ship, a laser beam travels in a straight line across the vessel.

In a real-life scenario, the spot from the laser will not move on the screen enough for you to notice. This is so because any reasonable (that is, non-life-threatening) rate of acceleration will not cause sufficient force to influence the path of the beam. However, let’s suspend our disbelief and imagine that we can accelerate the vessel at any rate, no matter how great, without being squashed against the ship’s rear wall. If we accelerate fast enough, the ship pulls away from the laser beam as the beam travels across the ship. We, looking at the situation from inside the ship, see the light beam follow a curved path (Fig. 16-7). A stationary observer on the outside sees the light beam follow a straight path, but the vessel pulls out ahead of the beam (Fig. 16-8).

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Relativity Spatial Curvature

Figure 16-7. As seen from within a space ship accelerating at an extreme rate, a laser beam travels in a curved path across the vessel.

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Relativity Spatial Curvature

Figure 16-8. When viewed from a “stationary” reference frame outside the ship, the accelerating vessel pulls away from the straight-line path of the laser beam, so the beam strikes the screen off-center.

Regardless of the reference frame, the ray of light always follows the shortest possible path between the laser and the screen. When viewed from any nonaccelerating reference frame, light rays appear straight. However, when observed from accelerating reference frames, light rays can appear curved. The shortest distance between the two points at opposite ends of the laser beam in Fig. 16-7 is, in fact, curved. The apparently straight path is in reality longer than the curved one, as seen from inside the accelerating vessel! It is this phenomenon that has led some people to say that “space is curved” in a powerful acceleration field. According to the principle of equivalence, powerful gravitation causes the same sort of spatial curvature as acceleration.

For spatial curvature to be as noticeable as it appears in Figs. 16-7 and 16-8, the vessel must accelerate at an extremely large pace. The standard unit of acceleration is the meter per second per second, or meter per second squared (m/s 2 ). Astronauts and aerospace engineers also express acceleration in units called gravities (symbolized g ), where one gravity (1 g ) is the acceleration that produces the same force as the gravitational field of Earth at the surface, approximately 9.8 m/s 2 . (Don’t confuse the abbreviation for gravity or gravities with the abbreviation for grams. Pay attention to the context if you see a unit symbolized g .) Figures 16-7 and 16-8 show the situation for an acceleration of many thousands of gravities. If you weigh 150 pounds on Earth, you would weigh many tons in a ship accelerating at a rate, or in a gravitational field of such intensity, so as to cause that much spatial curvature. In real life, no one could survive such force. No living human being will ever directly witness the sort of light-beam curvature shown in these illustrations.

Is all this a mere academic exercise? Are there actually gravitational fields powerful enough to bend light rays significantly? Yes. They exist near the event horizons of black holes.

Time Dilation Caused By Acceleration Or Gravitation

The spatial curvature caused by intense acceleration or gravitation produces an effective slowing down of time. Remember the fundamental axiom of special relativity: The speed of light is constant no matter what the point of view. The laser beam traveling across the space ship, as shown in many of the illustrations in this chapter, always moves at the same speed. This is one thing about which all observers, in all reference frames, must agree.

The path of the light ray, as it travels from the laser to the screen, is longer in the situation shown by Fig. 16-7 than in the situation shown by Fig. 16-6. This is so in part because the ray takes a diagonal path rather than traveling straight across. In addition, however, the path is curved. This increases the time interval even more. From the vantage point of a passenger in the space ship, the curved path shown in Fig. 16-7 represents the shortest possible path the light ray can take across the vessel between the point at which it leaves the laser and the point at which it strikes the screen. The laser device itself can be turned slightly, pointing a little bit toward the front of the ship; this will cause the beam to arrive at the center of the screen (Fig. 16-9) instead of off-center. However, the path of the beam is still curved and is still longer than its path when the ship is not accelerating (see Fig. 16-6). The laser represents the most accurate possible timepiece, because it is based on the speed of light, which is an absolute constant. Thus time dilation is produced by acceleration not only as seen by observers looking at the ship from the outside but also for passengers within the vessel itself. In this respect, acceleration and gravitation are more powerful “time dilators” than relative motion.

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Relativity Time Dilation Caused By Acceleration Or Gravitation

Figure 16-9. Even if the laser is turned so the light ray hits the center of the screen, the path of the ray is curved when the ship accelerates at a high rate.

Suspending our disbelief again, and assuming that we could experience such intense acceleration force (or gravitation) without being physically crushed, we will actually perceive time as slowing down inside the vessel under conditions such as those that produce spatial curvature, as shown in Fig. 16-7 or Fig. 16-9. Clocks will run more slowly even from reference frames inside the ship. In addition, everything inside the ship will appear warped out of shape.

If the acceleration or gravitation becomes far more powerful still (Fig. 16-10), the spatial curvature and the time dilation will be considerable. You will look across the ship at your fellow travelers and see grotesquely elongated or foreshortened faces (depending on which way you are oriented inside the vessel). Your voices will deepen. It will be like a science-fiction movie. You and all the other passengers in the ship will know that something extraordinary is happening. This same effect will be observed by people foolish enough to jump into a black hole (yet again ignoring the fact that they would be stretched and crushed at the same time by the force).

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Relativity Observational Confirmation

Figure 16-10. If the acceleration is great enough, the spatial curvature becomes extreme.

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