Spacetime and Geometry · Problem 6

Gravitation — Sean Carroll

Physics — General Relativity · weak-field-metric-earth

Problem

A good approximation to the metric outside the surface of the Earth is provided by ds2=−(1+2Φ)dt2+(1−2Φ)dr2+r2(dθ2+sin⁡2θ dϕ2),(3.218)ds^2 = -(1+2\Phi)dt^2 + (1-2\Phi)dr^2 + r^2(d\theta^2 + \sin^2\theta\, d\phi^2), \tag{3.218} where Φ=−GMr(3.219)\Phi = -\frac{GM}{r} \tag{3.219} may be thought of as the familiar Newtonian gravitational potential. Here GG is Newton's constant and MM is the mass of the earth. For this problem Φ\Phi may be assumed to be small. (a) Imagine a clock on the surface of the Earth at distance R1R_1 from the Earth's center, and another clock on a tall building at distance R2R_2 from the Earth's center. Calculate the time elapsed on each clock as a function of the coordinate time tt. Which clock moves faster? (b) Solve for a geodesic corresponding to a circular orbit around the equator of the Earth (θ=π/2\theta = \pi/2). What is dϕ/dtd\phi/dt? (c) How much proper time elapses while a satellite at radius R1R_1 (skimming along the surface of the earth, neglecting air resistance) completes one orbit? You can work to first order in Φ\Phi if you like. Plug in the actual numbers for the radius of the Earth and so on (don't forget to restore the speed of light) to get an answer in seconds. How does this number compare to the proper time elapsed on the clock stationary on the surface?

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