Calculating Time Differences in GPS Signals: A Relativistic Approach

[Blue Kangaroo]

Homework Statement

This is a lengthy problem with 3 parts, so I figured I would keep them separate. I have a diagram that corresponds with the problem as well.
GPS receivers find their locations by measuring time differences between EM wave pulses emitted from satellites at known locations and known times in the sky. If the signals are emitted simultaneously from different satellite positions, then these time differences on the Earth can give accurate position information when the satellite positions are known. However, a problem arises because simultaneous pulse emissions in the satellites' reference frame are not emitted simultaneously when viewed from the ground based GPS receiver. Consider the geometry for the satellites’ locations in the figure below with L=L’=12,000Km (γ=1 at this low satellite speed) and h=20,000 Km. The satellites are all moving at 4Km/s as shown. Suppose that at time t =t’=0 when satellite B is directly overhead of location D on the ground (x=x’=0), EM pulses are emitted simultaneously from satellite A and C in the satellites’ frame and their emission time and positions (in their reference frame) are sent to the GPS receiver on the ground.

1. First, calculate the time difference Δt between emission times from satellite A and satellite C as viewed from the GPS on the ground.

Homework Equations

Δt=γΔt0
x=γ(x'=vt')
t=γ(t'+vx'/c^2)

The Attempt at a Solution

I know that there are two frames of reference here -- one for the satellites and one for the ground. As such, the Lorentz transformation equations will be needed. I'm having trouble figuring out exactly how to use them.
Thank you very much for any help I receive!

 

[John Morrell]

It's been a while since I've done this sort of problem, so definitely check my advice against other responses, but this is how I would go about it. From the ground observers perspective, the amount of time it will take for the pulse to get there is just the distance to the satellite divided by the speed of light. To the ground observer, it doesn't matter how fast the satellites were moving when the pulse originated, because light always travels at the same speed.

But for the satellites, in their frame of reference, the ground observer is moving. They see the light traveling out away from them at the speed of light, but they also see the ground observer either moving towards the pulse of light or away from it, and so the time it will take for the pulse to arrive will be distance / (c + velocity of ground observer).

Hope that helps, you probably don't need to use lorentz transforms in this case, though as I said I may be off.