Relativity Lite

68 | Relativity Lite the proportion of helium to hydrogen that is observed in the universe. They also estimated that this cooling would have continued for the subsequent 15–20 billion years until today, when one should see the ghost of this explosion as microwave radiation with a temperature of about 5 degrees above absolute zero, or 5 Kelvins (K; room temperature is 300 K). * In 1965, Arno Penzias and Robert Wilson observed this background microwave radiation at a temperature of 3 K, † so the Big Bang appears to be confirmed. You actually know enough now to see how a Big Bang would work. Throughout this book, we have been using a pattern in which we relate proper time to coordinate time and coordinate distance. In chapter 1’s discussion of special relativity, we saw that the Pythago- rean theorem applied to the light path in a ship moving at speed v relative to the Earth gave the full spacetime relation for proper time, ‡ τ 2 2 2 2 1 = − t c r (1) For an accelerating rocket ship, or the equivalent observer in a gravitational field, the equivalent relation is τ 2 2 2 2 2 2 1 2 1 1 2 = −    − −    + GM rc t c GM rc r angular terms (2) where a full accounting would include angular spatial terms § as well as the radial one we concentrated on in chapter 4, but those play no part in the discussion that follows. Comparing this with time dilation in special relativity, you may see a symbolic pattern emerging in the shape of the equations, even had you not gone through the details of the idea in pictures. For the Robertson-Walker relation that describes the expansion of the uni- verse, we might expect to see some quantity that changes with coordinate space and time multiplying the t 2 term minus another quantity multiplying the r 2 term. It turns out that the correct relation is τ 2 2 2 2 2 2 1 1 = − − + t R kr c r ( ) angular terms (3) R is a quantity that changes with time (often expressed by the notation R(t) ) and so shows us how the intergalactic distances involving r and angles change with time. This cos- mic scale factor is most easily thought of as the radius of a four-dimensional cosmic balloon, a hypersphere , that has as its outer surface our three-dimensional space. Now we do not have the ability to draw a four-dimensional balloon, but we can get an idea of how this would * R. A. Alpher and R. C. Herman, Nature 162 , 774 (1948); Phys. Rev. 75 , 1089 (1949). † A. A. Penzias and R. W. Wilson, Astrophys. J. 142 , 419 (1965). ‡ We can set v t = r in τ 2 2 2 2 2 2 2 2 1 1 = − = − t c v t t c r . § See, for instance, Ronald Adler, Maurice Bazin, and Menachem Schriffer, Introduction to General Relativity (McGraw- Hill, New York, 1975), p. 225.