Theory of Relativity Help

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?

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).

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.

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.

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).

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

Observational Confirmation

When Einstein developed his general theory of relativity, some of the paradoxes inherent in special relativity were resolved. (These paradoxes are avoided here because discussing them would only confuse you.) In particular, light rays from distant stars were observed as they passed close to the Sun to see whether or not the Sun’s gravitational field, which is quite strong near its surface, would bend the light rays. This bending would be observed as a change in the apparent position of a distant star in the sky as the Sun passes close to it.

The problem with this type of observation was, as you might guess, the fact that the Sun is far brighter than any other star in the sky, and the Sun’s glare normally washes out the faint illumination from distant stars. However, during a total solar eclipse, the Sun’s disk is occulted by the Moon. The angular diameter of the Moon in the sky is almost exactly the same as that of the Sun, so light from distant stars passing close to the Sun can be seen by Earthbound observers during a total eclipse. When this experiment was carried out, the apparent position of a distant star was offset by the presence of the Sun, and this effect took place to the same extent as Einstein’s general relativity formulas said it should.

More recently, the light from a certain quasar has been observed as it passes close to a suspected black hole. On its way to us, the light from the quasar follows multiple curved paths around the dark, massive object. This produces several images of the quasar, arranged in the form of a cross with the dark object at the center.

The curvature of space in the presence of a strong gravitational field has been likened to a funnel shape (Fig. 16-11), except that the surface of the funnel is three-dimensional rather than two-dimensional. The shortest distance in three-dimensional space between any two points near the gravitational source is always a curve with respect to four-dimensional space. This is impossible for most (if not all) people to envision directly without “cheating” by taking away one dimension. But the mathematics is straightforward enough, and observations have shown that it correctly explains the phenomenon.

Figure 16-11. Spatial curvature in the vicinity of an object that produces an intense gravitational field.

Why Does This Matter?

We have conducted “mind experiments” in this chapter, many of which require us to suspend reality. In real life, scenarios such as these would kill anyone attempting to make the observations. So why is relativity theory important? If space is bent and time is slowed by incredibly powerful gravitational fields, so what?

The theory of general relativity plays an important role in astronomers’ quests to unravel the mysteries of the structure and evolution of the Universe. On a cosmic scale, gravitation acquires a different aspect than on a local scale. A small black hole, such as that surrounding a collapsed star, is dense and produces gravitation strong enough to destroy any material thing crossing the event horizon. However, if a black hole contains enough mass, the density within the event horizon is not so large. Black holes with quadrillions of solar masses can exist, at least in theory, without life-threatening forces at any point near their event horizons. If such a black hole is ever found, and if we develop space ships capable of intergalactic flight, we will be able to cross its event horizon unscathed and enter another Universe. We can be sure someone will try it if they think they can do it, even if they are never able to communicate back to us what they find.

According to some theorists, we need not travel far to find the ultimate black hole. It has been suggested that our entire Universe is such an object and that we are inside it.

Practice problems of this concept can be found at: Special and General Relativity Practice Problems