Relativity Lite

Gravity Lite | 61 dilation felt by two observers. Clearly an astronaut halfway down figure 10 (let us call him astronaut 1 ) will see a wristwatch worn by an astronaut near the bottom (let us call her astronaut 2 ) moving at a slower rate than his own. The slower frequency will cause light coming from the whites of her eyes to look reddish to him. We say that the light is gravi- tationally red-shifted . On the other hand, she will see the watch worn by astronaut 1, who is further away from the star, running faster than her own. Light from the whites of this astronaut’s eyes will look bluish to her. It has been gravitationally blue-shifted . This means that if you wear a wristwatch and an ankle watch, the latter will literally run more slowly than the former. Since your ankle is only 1 m closer to the center of the Earth (6,378,000 m away) than is your wrist, and the mass of the Earth is relatively small, the difference is only 1 part in 5,000,000,000,000,000. * That is, you will age faster by one microsecond for every story (3 m) higher you live above the surface of the Earth. The closer astronaut 2 gets to the black hole, the slower her watch will appear to run to astronaut 1. In fact, it would take an infinite amount of 1’s time for a second to pass on 2’s watch as she just approaches r s . Also, the closer she gets, the redder the light from the whites of her eyes will appear, passing in turn through red, infrared, microwave, and radio wavelengths until the wavelength would be so long as to be undetectable. Thus, we find that light cannot get out of the region inside r s , and that is why we call an object with such a sin- gularity in free space a black hole . Likewise, as astronaut 2 gets closer to the Schwarzschild radius, she sees the clocks on the spaceship far from the black hole carrying astronaut 1 spin faster. In fact, while 2 is having lunch, she would see that billions of years pass in the outside universe. The whites of astronaut 1’s eyes will become bluish, then give off ultra- violet radiation, then X-rays, and then gamma rays. Fortunately, her cone of observation simultaneously closes up so that the amount of this harsh radiation reaching astronaut 2 is diminished enough for her to survive it. The biggest problem for astronaut 2 to worry about after lunch is that she will crash into, and become part of, the singularity at r = 0 , the place where all the star’s mass is concen- trated. Einstein’s classical theory of gravity says that as she approached the singularity, it would have such a strong tidal effect that our poor astronaut would be stretched out to an infinite extent. However, quantum mechanics tells us that spacetime loses its smoothness as one experiences distances as small relative to the size of a proton as the proton is to us. Moving through distances this near the origin requires quantum jumps from one piece of a sponge-like space to another, as in figure 11c, a state of affairs not contained in Einstein’s classical theory of gravity. * We can use the second equation twice to find the algebraic expression that represents this concept. If we say that observer 1 is at distance a r 1 from the star and observer 2 is at distance r 2 from the star. Then the coordinate times are related by t GM r c GM r c t 1 1 2 2 2 2 1 2 1 2 = − − / / . Because the Earth’s mass is small, this is approximately equal to t GM r c r r r t t 1 2 2 1 2 1 2 16 2 1 2 1 2 18 10 # ª ¬ « º ¼ » u ª¬ º¼ ( ) . . The red-shifting of light can be detected in light emitted from the surface of our Sun and seen on the Earth. If we had sensitive enough instruments to measure it, light coming from the surface of the Moon would be blue-shifted when seen on the Earth’s surface.