This may be an easy or a very difficult question, depending how you looked at it:
1. The easy way - ignore the gravitational mass of the Moon. Apply the Schwarzschild solution
[math](d\tau/dt)^2=(1-r_s/r) -(v/c)^2[/math]
Where r is the distance between the Earth and Moon centers, r_s=9mm is the Earth Schwarschild radius, v is the tangential speed of the Moon wrt the Earth
2. The very difficult way, use the "two body solution" for the EFE's. This solution has become available only recently. The same steps apply as above.
