I am reading A Brief History of Time by Stephen Hawking, and in it he mentions that without compensating for relativity, GPS devices would be out by miles. Why is this? (I am not sure which relativity he means as I am several chapters ahead now and the question just came to me.)
General Relativity – Why GPS Depends on Relativity: Understanding the Science Behind
general-relativitygpsspecial-relativity
Related Solutions
I am a beginner in the topic of special relativity so I apologize for any lack of understanding on the subject.
I can't improve on Samuel's comment except to expand a little bit, in light of your comment above, on what special relativity describes. It is a more sophisticated method (and accurate description) of physical phenomenona in 4 D spacetime. But, although for example, this site is packed with apparent paradox after apparent paradox based on this new, better explanation, such as the one you brought up about the ordering of events, they all turn out to have explanations that do not contravene any existing physical laws that we previously accepted, other than the mixing of space and time that SR allows for.
SR did permit the discovery and explanation of lots of new ideas and observation, but they fit in with what we previously established, again once we accept the SR postulates.
In the diagram the interval AB is 'time-like'; i.e., there is a frame of reference in which events A and B occur at the same location in space, separated only by occurring at different times. If A precedes B in that frame, then A precedes B in all frames. It is hypothetically possible for matter (or information) to travel from A to B, so there can be a causal relationship (with A the cause and B the effect).
The interval AC in the diagram is 'space-like'; i.e., there is a frame of reference in which events A and C occur simultaneously, separated only in space. There are also frames in which A precedes C (as shown) and frames in which C precedes A. If it were possible for a cause-and-effect relationship to exist between events A and C, then paradoxes of causality would result. For example, if A was the cause, and C the effect, then there would be frames of reference in which the effect preceded the cause. Although this in itself won't give rise to a paradox, one can show that faster than light signals can be sent back into one's own past. A causal paradox can then be constructed by sending the signal if and only if no signal was received previously.
Image Source and Extract: Causality and SR Wikipedia
Best Answer
Error margin for position predicted by GPS is $15\text{m}$. So GPS system must keep time with accuracy of at least $15\text{m}/c$ which is roughly $50\text{ns}$.
So $50\text{ns}$ error in timekeeping corresponds to $15\text{m}$ error in distance prediction.
Hence, for $38\text{μs}$ error in timekeeping corresponds to $11\text{km}$ error in distance prediction.
If we do not apply corrections using GR to GPS then $38\text{μs}$ error in timekeeping is introduced per day.
You can check it yourself by using following formulas
$T_1 = \frac{T_0}{\sqrt{1-\frac{v^2}{c^2}}}$ ...clock runs relatively slower if it is moving at high velocity.
$T_2 = \frac{T_0}{\sqrt{1-\frac{2GM}{c^2 R}}}$ ...clock runs relatively faster because of weak gravity.
$T_1$ = 7 microseconds/day
$T_2$ = 45 microseconds/day
$T_2 - T_1$ = 38 microseconds/day
use values given in this very good article.
And for equations refer to HyperPhysics.
So Stephen Hawking is right! :-)