When Einstein’s gravitational waves meet newton’s tides

The Moon and the Sun pull on the oceans and on the solid Earth, and both deform in response. These movements can be recorded with high-precision instruments and calculated very accurately with Newtonian gravity. However, a fuller description of gravity is provided by Einstein’s general theory of relativity, according to which mass and energy alter spacetime, and changes in spacetime can travel as gravitational waves. The new study brings these two perspectives together in one mathematical model. It extends the classical model of lunar tides by an additional term that is based on the cross polarization of a gravitational wave, which is one of the characteristic ways in which gravitational waves can deform spacetime.

“We are not claiming to have discovered a new gravitational effect. What we show is how one could look for it in a targeted way,” says Muhittin Eser, a mathematician in the Department of Mathematics and Computer Science at Freie Universität Berlin and the author of the study. “The Earth–Moon system is particularly interesting for this because we can already describe and measure its tidal effects very precisely.”

One difficulty is distinguishing possible additional signals from known uncertainties. Deviations from the predicted tidal signals can come from the choice of reference frame, from uncertainties in computing the positions of the Earth and Moon, or from systematic measurement errors. The study therefore shows which influences would have to be controlled before a remaining signal could be examined as a possible additional physical effect.

A Basis for Future Precision Measurements

The study offers a quantitative basis for future searches for very small deviations from the classical description. Such searches could draw on high-precision measurements of gravity and tides, on geodetic observations, and on lunar laser ranging. In lunar laser ranging, laser pulses are sent to reflectors on the Moon to measure the Earth–Moon distance very precisely.

“Isaac Newton explains the tides extremely well. Albert Einstein shows us that gravity is also dynamic and that changes in spacetime can propagate as waves,” says Eser. “Our work brings both perspectives together and asks what traces of this might be found in high-precision measurements.”

This could open a new way of testing general relativity. So far, gravitational waves have mostly been studied through large cosmic events such as mergers of black holes or neutron stars. The new method instead looks for possible tiny signatures in our immediate astronomical neighborhood.