3. Time is dilated in the presence of gravitational fields. Consider a planet of mass M, and radius r. A source of light

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3 Time Is Dilated In The Presence Of Gravitational Fields Consider A Planet Of Mass M And Radius R A Source Of Light 1 (172.67 KiB) Viewed 32 times

3. Time is dilated in the presence of gravitational fields. Consider a planet of mass M, and radius r. A source of light located on the surface of this planet, emitting flashes of light spaced apart by At surface, is observed at a distance far (well outside the graviational influence, where the effect no longer keeps increasing the flash spacing) from the planet. The light will appear redshifted, and the time measured at this far away place between arriving flashes (At far) will increase according to the formula: -1 At far = At surface V1 – ro/r where ro = 2MG/c2 is the Schwarzschild radius. In the following questions were are often dealing with very small effects, and if you round off answers as you go, you will come to wrong conclusions. Carry enough significant figures through at all stages! The time effects are all on the tens to hundreds of nanosecond scale, so carry enough figures to have at least 3 sig fig in nanoseconds. (i) Suppose the planet is the Earth. How much slower are the Earth clocks running relative to a clock away in free space? [2] (ii) What is the Schwarzschild radius for the Earth? What has to happen to the Earth in order for it to become a black hole? [2] A crewed spacecraft decends to the surface of a planet, then returns to the orbiting main space-craft (considered to be "at infinity" away from the planet). Clocks are compared - 3 hours passed for the spacecraft that visited the surface, compared to 10 years that passed in the space ship far from the planet. The planet orbits a black hole, at the same distance as the Earth orbits the sun. [2] (iii) Give a simple explanation as to how you know that this planet is not inside the event horizon of the black hole. (Hint: no calculation needed.] (iv) Use this time dilation information to estimate the mass of the black hole. [2] The GPS satellite system allows us to locate our position on Earth to high accuracy. The system must account for special and general relativistic effects to maintain the correct clock timing relative to clocks on the ground. In question (i), you calculated how much slower clocks far from Earth run relative to those on the surface. The GPS satellites are in orbits of 20000 km altitude above the Earth's surface, which is NOT far from Earth for the purposes of the time dilation formula. (v) How much time do clocks in GPS orbits around Earth lose relative to clocks on the ground, due to special relativistic time dilation? [4] (vi) How much time do clocks in GPS orbits gain relative to clocks on the ground, due to general relativistic time dilation? [4] (vii) Calculate the total time offset error that would accumulate in the GPS clocks if they did not account for these effects. By how much would the clocks be off after 1 minute? (2)