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The LIGO interferometers are 2 dimensional, they have $4\rm\ km$ legs that are perpendicular to each other. Could a 3rd $4\rm\ km$ leg either with a straight up tower or deep into the earth, provide directional information about the source when gravitational waves occur?

I know a $4\rm\ km$ tower or hole isn't really feasible and maintaining the same level of stability of the mirrors would be difficult. But perhaps the proposed satellite version in space could use a 3 dimensional approach.

Or am I misunderstanding something about gravitational waves to think that a 3rd dimension would give us any extra information?

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Multiple LIGO detectors are used to determine direction and position by means of triangulation. This technique uses the fact that the waves travel at the speed of light and will reach different detectors at different times depending on their source direction. Although the differences in arrival time may be just a few milliseconds, this is sufficient to identify the direction of the origin of the wave with considerable precision.

Only in the case of GW170814 were three detectors operating at the time of the event, therefore, the direction is precisely defined. The detection by all three instruments led to a very accurate estimate of the position of the source, with a 90% credible region of just 60 degrees, a factor 20 more accurate than before.

The proposed LISA mission would also use this technique. It does not require more than 3 "arms" because there is to be assumed no Earthly noise, and it would still be able to use triangulation.

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