Relativity Lite

58 | Relativity Lite We can run this film forward and declare that 11.2 km/s is also the speed with which we would have to throw the pen to have it escape to infinity—its escape velocity . Now suppose all the Earth’s mass were contracted to 1/100th its size, 64,000 m. What difference would this make in these calculations? The required velocity is 10 times bigger, v GM R GM R m s m E E 64 000 2 1 100 100 100 2 100 10 11 200 , , / = = = × = 112 000 , / m s (We are allowed to multiply any expression by one without changing its nature, and in the square root, I multiplied by a convenient form, 1 = 100/100 , which helped cancel the 1/100 in the denominator by which R E was multiplied to get this compacted mass. What remains is a factor of 100 in the numerator that we then pull out of the square root.) Suppose all the Earth’s mass were contracted to 1/100th of 1/100th its original size, 640 m. What difference would this make in these calculations? v 640 m → 1,120,000 m/s Suppose all the Earth’s mass were contracted to 1/100th of that size, 6.40 m. v 6.4 m → 11,200,000 m/s Suppose all the Earth’s mass were contracted to 1/100th of that size, 64 mm. v 64 mm → 112,000,000 m/s Suppose all the Earth’s mass were contracted to 1/100th of that size, 0.64 mm. v 0.64 mm → 1,120,000,000 m/s Do you notice anything about the size of this velocity? What is the speed of light? We’ve just exceeded it! We had better back up and find the smallest possible size into which the Earth could be crammed so that its escape velocity would be precisely c . It turns out to be about halfway between the last two examples, 8.89 mm. The corresponding distance at which a photon could just escape a pinpoint containing all the Sun’s mass is the ratio of masses larger: r mm mm km S = = × = 8 89 8 89 333 333 2 96 . . , . M M S E This is called the Schwarzschild radius—r GM c s = 2 2 = 2.953 km—found by Karl Schwarzschild in 1916, * a year after Einstein published his general theory of relativity. † It * K. Schwarzschild, Sitzber Deut. Akad. Wiss. Berlin, Kl. Math. Phys. Tech. , 189–96 (1916); 424–34 (1916). † The final theory appears in his fourth paper of that year: A. Einstein, Preuss. Akad. Wiss. Berlin, Sitzber. , 844–47 (Decem- ber 2, 1915).