### Where in the World?

Here’s a conundrum: You buy a watch in Anchorage, Alaska (61°N). It keeps excellent time. Then you move to Singapore, on the Equator. Does the watch go fast or slow? For the answer to this puzzle, read on.

The Global Positioning System

In the Irish Times column this week ( TM009 ), we look at the way the Global Positioning System works. GPS is a remarkable synthesis of old and new, combining high-tech engineering and sophisticated relativistic physics with the geometry of circles and spheres developed in ancient Greece.

Relativistic Corrections

Both special relativity (SR) and general relativity (GR) have consequences for the rate at which time passes. And both have practical consequences for GPS. Indeed, the system would be useless for navigation without relativistic adjustments. First, we note that special relativity implies a dilation of time for a moving system. Since the satellites are moving relative to Earth, the satellite clocks appear to go slowly, according to

$\displaystyle d t^2_{SAT} = \left( 1-\frac{v^2}{c^2} \right) d t_{EARTH}^2$

where ${v}$ is the speed of the satellite and ${c}$ is the speed of light.  Thus, the time change due to special relativistic effects is

$\displaystyle \Delta_{\rm SR} = - \left( \frac{v^2}{c^2} \right) d t_{EARTH}^2 \,.$

This correction amounts to about   $-7{\mu s}$/day.

According to general relativity, time is slowed down by a gravitational field. The clock-tick at distance ${r}$ from the Earth’s centre is

$\displaystyle d t^2 = \left( 1-\frac{2GM}{c^2r} \right) d t_\infty^2$

where ${d t_\infty}$ is the clock-tick remote from Earth, M is the Earth’s mass and G is the universal gravitational constant. This leads to a time distortion

$\displaystyle \Delta_{\rm GR} = \frac{2GM}{c^2} \left( \frac{1}{a} -\frac{1}{r} \right) d t^2$

where ${r}$ is the distance of the satellite from the Earth’s centre and ${a}$ is the Earth radius. This is opposite in sign to the effect due to special relativity, and has a value of about 45${\mu s}$/day. Thus, the nett correction due to relativity is  38${\mu s}$/day.

Now for the answer to the conundrum: you buy a watch in Anchorage and move to Singapore. Does the watch go fast or slow?

Since you spin faster in Singapore, time should slow down. But, as the Earth is flattened, you are farther from the centre, so time should speed up!

Remarkably, on the Earth’s surface, these two effects, due respectively to special and to general relativity, cancel exactly. So, don’t worry: you can use the watch anywhere in the world. For a technical note showing this, click here: ( PDF ).