We study the field profile of a scalar field \phi that couples to a matter fluid ( dubbed a chameleon field ) in the relativistic gravitational background of a spherically symmetric spacetime .
Employing a linear expansion in terms of the gravitational potential \Phi _ { c } at the surface of a compact object with a constant density , we derive the thin-shell field profile both inside and outside the object , as well as the resulting effective coupling with matter , analytically .
We also carry out numerical simulations for the class of inverse power-law potentials V ( \phi ) = M ^ { 4 + n } \phi ^ { - n } by employing the information provided by our analytical solutions to set the boundary conditions around the centre of the object and show that thin-shell solutions in fact exist if the gravitational potential \Phi _ { c } is smaller than 0.3 , which marginally covers the case of neutron stars .
Thus the chameleon mechanism is present in the relativistic gravitational backgrounds , capable of reducing the effective coupling .
Since thin-shell solutions are sensitive to the choice of boundary conditions , our analytic field profile is very helpful to provide appropriate boundary conditions for \Phi _ { c } \lesssim O ( 0.1 ) .