Yes.So this notion of "at rest" is actually w.r.t. a coordinate system.
In flat space-time, there is relative velocity -- the velocity of the one body in the [momentary, inertial] rest frame of the other. If this is zero then the objects are at rest relative to each other.What about the notion of "being at rest" w.r.t. another body ?
In curved space-time, the notion of relative velocity gets slippery. Non-local comparison of velocities becomes ambiguous and one needs to "parallel transport" the velocity of the one into the local frame of the other. The result can depend on the path over which this "parallel transport" is performed.
There is a different notion which might be used. Have you Googled "born rigidity"?