Global Positioning System / Clocks in Space

[Markus Kahn]

Homework Statement

The Global Positioning System (GPS) consists of at least 24 satellites that are orbiting the Earth at
a distance of $h = 26'600$ km from its center with a velocity of $v\approx 3.9$ km/s. All the satellites carry atomic clocks which are synchronized such that they all show the same (GPS) time. At certain time intervals, the satellites simultaneously emit a signal which carries their orbital data and the time $t_e$ when the signal was emitted.

Due to relativistic effects, the satellite clocks will run with a different speed than clocks on Earth. Assuming $v^2,\phi << c^2$ we have $g_{00}=1+\frac{2 \phi}{c^{2}}, g_{i j}=-\delta_{i j},$ up to $\mathcal{O}(\frac{1}{c^{3}})$ in the Newtonian limit. Thinking as an observer far away from Earth (neglecting Earth's motion), compute the relation between infinitesimal elements $d\tau_E$ of Earth coordinate time and $d\tau_S$ of satellite coordinate time. By expansion up to $\mathcal{O}(\frac{1}{c^{3}})$, find by how much a satellite clock runs faster/slower than a clock on Earth? What is the absolute error after one day? Compare SR and GR effects.

Relevant Equations

Nothing given..

I'm a bit lost at how to exactly start this exercise... As far as I understand we need to first determine $d\tau_E$ and $d\tau_S$.

First question: Since we can neglect the Earth's movement, can I also neglect the movement of the satellite with respect to the far away observer? If so, I don't really get this exercise, since then the signal will not be delayed, just redshifted by the gravitational field.

If I can't neglect the movement of the satellite I still think the following should hold:
$$d\tau_E = \sqrt{g_{00}(\vec{r}_E)}dt_E\quad \text{and}\quad d\tau_S = \sqrt{g_{00}(\vec{r}_S)}dt_S,$$
which means we express the proper time in terms of the times in the rest frame of Earth and satellite. The issue is, that I don't really know where to take it from here.. Is this even the right idea, or do I need to start somewhere completely different?

 

[PeroK]

First question: Since we can neglect the Earth's movement, can I also neglect the movement of the satellite with respect to the far away observer? If so, I don't really get this exercise, since then the signal will not be delayed, just redshifted by the gravitational field.

I take that to mean that you can ignore the fact that the whole system is orbitting the Sun and that the Earth is spinning. You certainly can't ignore that the satellite is in orbit and not stationary wrt the Earth.

I'm a bit lost at how to exactly start this exercise... As far as I understand we need to first determine $d\tau_E$ and $d\tau_S$.

It depends on what you've been studying. You could compute the proper time in Schwarzschild coordinates of

a) something at rest on the surface of the Earth (assuming non-rotating).

b.1) something in the given circular orbit.

or

b.2) You could take gravitational time dilation and velocity-based time-dilation as complementary and simply calculate the two separately for the satellite.

You might expect, up to the accuracy you've been asked for, that these two approaches would yield the same answer.

 

[Markus Kahn]

Thank you for the answer and the hints. Up until now we have only been doing special relativity, slowly starting to move towards general relativity by introducing acceleration in inertial frames (e.g. the exercise you helped me solve last weekend about orbits of particles with constant acceleration and an external observer). So I don't really know what "Schwarzschild coordinates" are (I just checked Wikipedia, and I really don't..). If I don't read your suggestions wrong this only leaves b.2) as an option, namely calculating the velocity-based time dilation.

For the gravitational time dilation I have (for a clock at rest)
$$d\tau =\frac{ds}{c}= \sqrt{g_{\mu\nu}(x)dx^\mu dx^\nu} = \sqrt{g_{00}(x)}dt \approx \sqrt{1+\frac{2\phi(x)}{c^2}}dt.$$
This would have been my answer for the proper time of the earth, since it is at rest with respect to the mentioned observer in the exercise. But I'm a bit confused on what exactly this now has to do with the velocity of the satellite, since as far as I understand the same expression would describe the proper time of the satellite as well..

 

[PeroK]

Thank you for the answer and the hints. Up until now we have only been doing special relativity, slowly starting to move towards general relativity by introducing acceleration in inertial frames (e.g. the exercise you helped me solve last weekend about orbits of particles with constant acceleration and an external observer). So I don't really know what "Schwarzschild coordinates" are (I just checked Wikipedia, and I really don't..). If I don't read your suggestions wrong this only leaves b.2) as an option, namely calculating the velocity-based time dilation.

For the gravitational time dilation I have (for a clock at rest)
$$d\tau =\frac{ds}{c}= \sqrt{g_{\mu\nu}(x)dx^\mu dx^\nu} = \sqrt{g_{00}(x)}dt \approx \sqrt{1+\frac{2\phi(x)}{c^2}}dt.$$
This would have been my answer for the proper time of the earth, since it is at rest with respect to the mentioned observer in the exercise. But I'm a bit confused on what exactly this now has to do with the velocity of the satellite, since as far as I understand the same expression would describe the proper time of the satellite as well..

The satellite additionally has uniform circular motion.

I assume you are expected to calculate the gravitational and velocity-based dilation separately.

 

[TSny]

For the gravitational time dilation I have (for a clock at rest)
$$d\tau =\frac{ds}{c}= \sqrt{g_{\mu\nu}(x)dx^\mu dx^\nu}$$

For the satellite, pick a point on the orbit where the satellite is moving in the x-direction direction such that during the coordinate time interval $dt$, you have $dy = 0$ and $dz = 0$. Write out $\sqrt{g_{\mu\nu}(x)dx^\mu dx^\nu}$ in terms of $dt$ and $dx$.