That is exactly what the Michelson-Morley experiment looked for and
did not find.
The MMX looked for a stationary aether the Earth moves through. The
MMX looked for the aether wind.
The state of the aether as determined by its connections with the
matter and the state of the aether in neighboring places is the state
of displacement of the aether.
Take a mesh bag of marbles and put it into a tank filled with a
supersolid. The mesh bag of marbles will displace the supersolid. Now
spin the mesh bag of marbles as you move the mesh bag of marbles
around in the tank. The state of the supersolid is going to be
determined by its connections with the mesh bag of marbles and the
state of the supersolid in neighboring places. It is going to be very
difficult for any apparatus on the surface of the mesh bag to detect a
change in state with respect to the state of the displaced supersolid.
Watch the following video starting at 0:45 to see a visual
representation of the state of the aether. What is referred to as a
twist in spacetime is the state of displacement of the aether. What is
referred to as frame-dragging is the state of displacement of the
aether.
http://www.youtube.com/watch?v=s9ITt44-EHE
The aether connected to and neighboring the Earth is in the same
state, or almost the same state, throughout the Earth's rotation about
its axis and orbit of the Sun.
The state of which as determined by its connections with the Earth and
the state of the aether in neighboring places is the state of
displacement of the aether.