In physics, symmetry breaking has an important significance because the mechanism reveals the hidden secrets of the nature. For example, it gives mass to the leptons by Higgs mechanism in the Standard Model.
So, what is the symmetry breaking in general relativity?
Because of the tradition from Newtonian mechanics that a projectile motion of a macroscopic object can be mathematically described by the motion of the center of mass of the object, spherical object has been favored typically as an ideal sample mechanical system to start with.
The problem with this idealized spherical example was that it hid the important dipole gravity effect in the linearized weak field limit of general relativity.
This can be seen from observing the shift of the center of mass in the two different cases of a rotating hemisphere and a rotating sphere.
The relativistic mass increase effect shifts the center of mass of a rotating hemisphere but not that of a rotating sphere.
Here, we see that the symmetry is broken in relativity in the rotating hemispherical system and that it divulges what has been concealed. It means that a rotating hemisphere has to be dealt with in a totally differently manner in comparison to a rotating spherical system.
The whole basis of the theory of Dipole Gravity is in this symmetry breaking feature of its rotating systems.
In effect, the presence of the shift of the relativistic center of mass is more of a natural and general phenomenon than that of a perfectly carved unnatural spherical system that totally conceals the effect of the shift of the relativistic center of mass.
The fundamental building block of general relativity for a starting model mechanical system is a symmetry broken object like a cone or a hemisphere not a perfect sphere. A rotating sphere can certainly be created by two opposing rotating hemispheres but not the other way around.