Relativity Lite

Gravity Lite | 53 We see then that the inertia that resists the pull of sinews or the pull of a rope or the tug of gravity on a planet is closely related to angular momentum. As the skater’s arms get closer to her body, conservation of angular momentum speeds her up. As Mercury, in its ellipti- cal orbit, gets closer to the Sun, it speeds up, and thus more effectively resists the inward pull of the stronger gravity in that region. We see that what keeps Mercury from crashing into the Sun at its closest approach is its angular momentum. That angular momentum is the left-hand “barrier” of the dotted red curve of the potential energy diagram in figure 7. This angular momentum barrier depends on distance as 1/R 2 so that it will not have much effect at large distances, where the 1/R potential energy associated with the gravitational force dominates, but it will become larger than the gravitational potential energy at small distances. To see this, divide 1 by 10 on a calculator to get 0.1. Squaring this gives 0.01, a smaller value. If you divide 1 by the larger value 100, you get a smaller result, 0.01, and squaring this gives 0.0001, a much less influential amount. What we have done in creating figure 7 is to fold into Mercury’s radial motion its motion around the Sun, manifesting the left-hand wall of the potential energy diagram that con- strains its motion toward the Sun, keeping it from getting too close. A “marble” rolling in this potential energy diagram will therefore oscillate back and forth between the inner and outer regions, modeling the radial motion of Mercury in its orbit. The solid blue curve in figure 7 is the general relativistic result, which has a new term that depends in the same way on Mercury’s angular momentum as the Newtonian term. (This is one of those times when inserting lowest-order relativistic effects into a Newtonian framework pays off.) But it has a negative sign and depends on distance as 1/R 3 so, at very small distances, it will dominate. To see this, divide 1 by 0.1 on a calculator to get 10. Squaring this gives 100, the Newto- nian term. Cubing your division of 1 by 0.1 gives you 1,000 as the much larger Einsteinian term. If you divide 1 by the smaller value 0.01, you get a bigger result, 100, and squaring this gives 10,000. Cubing your division of 1 by 0.01 gives you 1,000,000 as the much larger Einsteinian term. One may continue on in this fashion. The Einsteinian term will dominate when R gets small enough and, since it is negative in sign, bend the angular momentum barrier over into negative energies and incidentally moving the barrier inward. Even for modest distance, one sees that the inward motion of a marble rolling in the Ein- steinian potential energy curve is still constrained on the left, but not so strongly. The marble (Mercury) can come closer to the Sun than in the Newtonian case. It will spend more time in a region of stronger gravitational effects and will consequently have its orbit perturbed. The effect is not to make the orbit smaller or larger but to make it impossible for the orbit to close on itself. Mercury comes out of the point of closest approach shifted a bit to the side due to its penetration deeper into the Sun’s warping of spacetime. This precession of Mercury’s perihe- lion, which Einstein calculated in his 1915 paper to be 43 arc-seconds/century, matched the formerly unaccounted for 45 ± 5 arc-seconds/century of his day (and the more accurate 1947 value of 43.11 ± 0.45 arc-seconds/century * ), which was strong evidence in favor of his theory. * G. M. Clemence, Rev. Mod. Phys. 19 , 361 (1947).