Relativity Lite

52 | Relativity Lite Figure 7. A diagram of the potential energy curve for the radius of Mercury’s orbit. The dotted curve is the Newtonian model, with the constraining peak near the origin arising from the square of Mercury’s angular momentum, ℓ . It also depends on the inverse square of the planet’s orbital radius, R , so the term gets bigger the smaller R becomes. To see this, divide 1 by 0.1 on a calculator to get 10. Squaring this gives 100. If you divide 1 by the even smaller value 0.01, you get an even bigger result, 100, and squaring this gives 10,000. One may continue on in this fashion. The solid blue curve is the general relativistic result, which has a new and negative term that likewise depends on the square of Mercury’s angular momentum. But this term depends on the inverse cube of the planet’s orbital radius, R . If you cube your division of 1 by 0.1 you get 1,000, and if you cube your division of 1 by the smaller value 0.01, you get 1,000,000. Since these terms have a larger size and are negative, they will at some small radius exactly cancel the Newtonian angular momentum term and for still smaller radii take this effect negative.This termdominates at small distances, bending the angularmomentumbarrier over intonegative energies and thereby moving the barrier inward. Thus, under general relativity, Mercury spends more time at smaller radii, where it moves faster and therefore slides out of purely elliptical motion. Consider the ice skater twirling with her arms outstretched. She will feel an outward force on her arms that her sinews must counteract. What happens when she pulls her arms in toward her body? Indeed, her spin will increase. But why is that? Just as with linear momentum, m v , which we explored in chapter 2, there is a quantity called angular momen- tum that is a conserved quantity. It is conventionally labeled with a script ℓ and defined as, ℓ = mvR , where R is the size of the object’s orbit. As the ice skater pulls her arms in toward her body, R becomes smaller, and in order to conserve angular momentum, v has to become larger to compensate (since m does not change). Thus, her spin will increase.