9. Both gravity and relative speed affect the passage of time. Where gravity is stronger (nearer...

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9. Both gravity and relative speed affect the passage of time. Where gravity is stronger (nearer earth's surface as compared to up higher) clocks will tick slower. According to general relativity (GR), the frequency of a clock (or any phenomenon) near earth's surface as a function of height H is given by w = wo(1+ gH/c) where wo is measured at height H=0. On the other hand, special relativity (SR) suggests that the moving clock will run slower with an expression given by w = wo1 - v/c. For GR and SR, show a first order expression for the change in frequency of a clock due to H and v. Do this using the lowest order term in a Taylor series expansion for SR. Using the results from 9, find an expression for the total difference due to both effects relate a time interval dt on a fixed clock to the interval dt' that would be measured with both SR and GR taken into account, and then show that you can arrive at the Euler-Lagrange equations by maximizing the time measured on the moving clock over a trajectory. Show that this is the equivalent of minimizing the Lagrangian as done in classical mechanics. 9. Both gravity and relative speed affect the passage of time. Where gravity is stronger (nearer earth's surface as compared to up higher) clocks will tick slower. According to general relativity (GR), the frequency of a clock (or any phenomenon) near earth's surface as a function of height H is given by w = wo(1+ gH/c) where wo is measured at height H=0. On the other hand, special relativity (SR) suggests that the moving clock will run slower with an expression given by w = wo1 - v/c. For GR and SR, show a first order expression for the change in frequency of a clock due to H and v. Do this using the lowest order term in a Taylor series expansion for SR. Using the results from 9, find an expression for the total difference due to both effects relate a time interval dt on a fixed clock to the interval dt' that would be measured with both SR and GR taken into account, and then show that you can arrive at the Euler-Lagrange equations by maximizing the time measured on the moving clock over a trajectory. Show that this is the equivalent of minimizing the Lagrangian as done in classical mechanics.