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Earth-moon Lagrangian points as a test bed for general relativity and effective field theories of gravity
In the space surrounding two bodies that orbit about their mutual mass
center there are five points where a third body will remain in equilibrium
under the gravitational attraction of the other two bodies. These points
are called Lagrangian points in honor of Joseph Lagrange, who discovered
them in 1772 while studying the restricted problem formed by the
Sun-Jupiter system. We first evaluate the location of all Lagrangian
points in the Earth-Moon system within the framework of general
relativity. For the points L4 and L5, the corrections of coordinates are
of order a few millimeters and describe a tiny departure from the
equilateral triangle. After that we will analyze the position of
Lagrangian points in a scheme involving quantum corrections to Einstein
gravity, rather than to Newtonian gravity. Within this framework, for the
Lagrangian points of stable equilibrium, we find quantum corrections of
order 2 mm, whereas for Lagrangian points of unstable equilibrium we find
quantum corrections below a millimeter. In the latter case, for the point
L1, general relativity corrects Newtonian theory by 7.61 m, comparable, as
an order of magnitude, with the lunar geodesic precession of about 3 m per
orbit. The latter is a cumulative effect accurately measured at the
centimeter level through the lunar laser ranging positioning technique.
Thus, it is possible to study a new laser ranging test of general
relativity to measure the 7.61 m correction to the L1 Lagrangian point, an
observable never used before in the Earth-Moon system. This will then be
the basis to consider an experiment to study deviations of effective field
theories of gravity from general relativity in the Earth-Moon system.