GEEE PEEE ESSS

Did you know that the first time GPS (Global Positioning System) was used was for military purposes? None other than 'MURICAAAA. It was based on two other systems that were developed in World War II, something about war brings about extreme innovations, albeit perhaps innovations we would rather not have. 

Situation

Ok ok, as soon as I landed on my planet I designed 2 satellites and put them to orbit such that they help me with navigating and making a map of the new planet. 

As we speak, the satellites are sending me signals about their position and time (their time silly, not mine), basically telling me when and where the signals are being sent. The satellites are in perfect circular orbit around the equator, meaning we can use a 2 dimensional system of (x,y) to describe positions. The origin is at the centre of the planet and we made sure that both clocks are synchronised at the beginning (t=0, if t = 0 is not agreed upon we get problems with SR).

I can picture you looking like this rn 

But what we're interested in is: Does relativity affect satellites that help us with navigation? What happens if we don't consider the effects of relativity? I mean the usual thought goes like, you don't have to worry about relativity in everyday life but is that true?

Principles 

• Basic vector knowledge, namely the cosine law of vectors or sometimes it's called the dot (inner) product rule. Nothing you haven't encountered at school
• Schwarzschild (had to google that) line element (or metric). This allows us to do 2-in-1. Otherwise we would have to measure Special relativity time dilation and General Relativity time "contraction" separately, but because the line element captures velocity as well as the height above the gravitational centre we get the 2-in-1
• Celestial mechanics: The velocity of a body in circular motion
• That's actually it.  

The bread and butter

As usual, the maths will be really toned down, don't come at me, these are orders from up above, I'm just following orders.

To work with GR (general relativity from now on) we must find the height above the centre of gravitation. So we do just that:

Okay time to put this thing in motion. Given only (x,y) coordinates we can find the height of the satellites, this is because we assumed a perfect circular orbit, or in other words that the radius (or the height) of the orbit is constant throughout the orbit. We found that our height is 14 973 km above surface. This is quite a bit lower than earth's GPS satellites (20 000 km) but that's because our planet is smaller.

Any work with special relativity requires knowing the velocity, and in this case you should know how to find this if you've been paying attention to physics class! 

Where G is the gravitational constant, M is the mass and r is the radius of the orbit. Inserting numbers, we get v = 3.7875 km /s

The next part is math heavy but the concepts are important regardless. We would like to find our position using the data we have already. This is accomplished by finding angles (alpha1, alpha2) that in turn help us find the one angle (Phi) at which I am standing, and because we know the radius we can easily find exactly where I stand. 

But how do we find these angles? It's beyond the scope a bit, but using cosine law of vectors we can find the pink angles but thanks to trig properties we can't immediately know if the angle is positive or negative, so to find out we use our Beta angles (easily found using basic trig from x,y coords of the spaceships), and we try all the possible combinations (4, check the light brown text and angle). The two combinations that give us the same angle also tell us that that same angle is the actual angle, using polar coordinates shift, we obtain the following coords: [3837.478 ,4237.355], neat eh? That's how GPS would work in a perfect world without goddamn relativity, but here comes the least intuitive physics concept banging on the door. 

!!! ATTENTION: I just got orders from up above that what I'm doing is not necessary and I should simply present my final answer, sorry fellas but this is gonna be cut short, we'll cut the crap and look at the effect of relativity on our measurements  !!!

So taking into account relativistic effects at t = 372.975s (planet frame) put us about 30000 meters off of where we actually were!! Can you believe it? 380 seconds is just a few minutes. These numbers sounds too high to be true, relativity moment (where you can't use intuition), imagine driving around and the GPS is showing you 30 kilometres away, imagine all the systems that are dependent on accurate position. 

Now let's look at t = 15027s, we get a deviation of 1280 KILOMETERS!!! Jesus Christ, who knew shitty relativity was so important to daily life? Well our numbers are probably exaggerated but the point of this is not accurate numbers, it's to demonstrate how general relativity overtook special relativity and affects our daily lives that strongly! GPS would be useless within some time

On a side note, why is the deviation larger as we go through time? We can think of it two ways, either mathematically or in "intuitive physics". Think of it like this, the satellites and me start at t = 0 in agreement, but as one second goes for me, 0.9999997 goes for the satellite, this is not that noticable at first but then at the seconds increase, the difference adds up and the world line diverges more and more. If that didn't make sense think of a car that is driving at 1.000001 your speed, in the first meters the difference will be unnoticeable but if after a loooooong time (years) the car will be muuuuch more ahead than you, because that small difference adds up.