# The Laufenberg Rhine Bridge Construction Problem

Laufenberg Rhine Bridge

Some time ago, Germany and Switzerland decided to build a bridge over the Rhine between their cities on both sides, both named Laufenburg. According to the agreement, each country would begin construction on its side of the river and meet in the middle. The bridge was almost completed in 2003, when both countries realized that half of the bridge was 54 centimeters (21 inches) higher than the other.

The mistake is due to the way each country has defined the term “sea level”. Most countries have different methods for determining sea level, considering that it is not the same everywhere. Germany uses the North Sea to define the sea level, while Switzerland prefers the Mediterranean Sea.

There was a difference of 27 centimeters between the respective sea levels of the countries. Germany and Switzerland knew it and took it into account in their calculations. However, someone did it in such a way that the disparity was doubled, which caused a rise of one side of the bridge by 54 centimeters more than it should have been.

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To rule out such errors in the future, geodeticists want to recalculate sea level based on the Earth’s gravitational force and they want to have an accuracy of 1 cm – but that is expanding the limits of conventional measurement procedures or GPS technology via satellite.

Optical atomic clocks offer a new approach, because the clock rate is affected by gravity. This famous little effect was measured with unprecedented precision in 2010 using two optical clocks – located in the same institution. Now, up to 2000 km may lie between them; Using commercial optical fibers and advanced amplifier techniques, the frequency of one atomic clock can be transported to another where the frequencies can then be compared.

The highly sensitive method of interferometry allows long-distance transfers and comparisons to be made with an accuracy of 19 digits. Optical atomic clocks offer a perspective for realizing frequencies with such accuracy that even small frequency deviations, caused by differences in height by a few centimeters, eventually become apparent. What is behind this is Einstein’s general theory of relativity, called the gravitational redshift: If a clock is farther from Earth, time actually goes a little faster for it. For a difference in height of one meter, the speed (ie frequency) of a clock changes by 1 × 10.

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Many things have been investigated in this field: Atomic clocks, for example, have been transported halfway around the world by plane – and after that it was actually discovered that their time passed was slightly different from atomic clocks on Earth. And three years ago, Chou et al. (Science 2010) installed two optical aluminum clocks in a neighboring laboratory with a height difference of 33 cm – and they were able, in fact, to measure the effect of this small difference in height on a two-hour frequency.