Justifying the Approximations

We calculated the speed of a satellite in circular orbit and the speed of the clock on Earth's surface using Euclidean geometry and Newtonian mechanics with its "universal time." Now, the numerator in each expression for speed, namely rdty, is the same for Euclidean geometry as for Schwarzschild geometry because of the way we defined r in Schwarz-schild spacetime. However, the time dt in the denominator of the speed is not the same for Newton as for Schwarzschild. In particular, the derivation of equation [3] assumes that the speeds in that equation are to be calculated using changes in far-away time dt. Think of a spherical shell constructed at the radius of the satellite orbit and another "shell" that is the surface of Earth. Then our task boils down to estimating the difference between far-away time dt and shell time dts^eü in each case, which can be done using our equation [C] in Selected Formulas at the end of this book.

QUERY 10 General-relativistic times vs. Newton's universal time. For comparison, equate Schwarzschild far-away time with Newton's universal time and see what difference there is between this time and "shell time" either at the radius of the satellite orbit or the surface of Earth. Use equation [13] and the approximation equation [5] to set up an approximate relation between two measures of velocity in each case:

dt shell where q is a small number. Find an algebraic expression for q. Then find numerical values of q both for Earth's surface and at the orbital radius of the satellite. Use these results to estimate the difference that changed velocity values will make in the numerical result of Query 9. Is this difference significant?

Two Notes

Note 1: The approximate analysis in this project also assumed that the radius rsatellite the circular orbit of the satellite is correctly computed using Newtonian mechanics. The Schwarzschild analysis of circular orbits is carried out in Chapter 4. When you have completed that chapter, you will be able to show that this approximate analysis is sufficiently accurate for our purposes.

Note 2: Our analysis assumed the speed z>Earth °f the Earth clock to be that of the speed of the equator. One might expect that this speed-dependent correction would take on different values at different latitudes north or south of the equator, going to zero at the poles where there is no motion of the Earth clock due to rotation of Earth. In practice there is no latitude effect because Earth is not spherical; it bulges a bit at the equator due to its rotation. The smaller radius at the poles increases the M/rEarth term in equation [12] by the same amount that the velocity term decreases. The outcome is that our calculation for the equator applies to all latitudes.

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