This question strikes at the heart of a gross misinterpretation of Special Relativity. Everyone who uses the accepted interpretation of Special Relativity will answer this can not be done. So therefore almost all physicists that answer this question will say there is no such thing as absolute motion.
The incorrect interpretation of Special Relativity actually comes from Einstein. He states early on in his famous paper, ON THE ELECTRODYNAMICS OF MOVING BODIES, that motion through space is relative just between to comparable moving frames of reference. Originally the Lorentz Transformations which comprise the theory of Special Relativity where derived using a model where there existed an absolute stationary frame of reference. The absolute frame of reference was considered to be the aether.
In his paper he denies the aether by deliberately inserting this statement.
The introduction of a “luminiferous ether” will prove to be superfluous inasmuch as the view here to be developed will not require an “absolutely stationary space” provided with special properties, nor assign a velocity-vector to a point of the empty space in which electromagnetic processes take place.
In my book “Master of Reality” I throughly investigate why Einstein very deliberately added this statement and denies the existence of the aether. As it turns out Einstein released his paper just a few short months before the results of the famous Michelson Morley Experiment were announced. The aether wind was not detected so we got the infamous announcement of the null result. It has been proven that Einstein was already aware that the Michelson Morley experiment was going to fail to find evidence of the aether wind.The null result shocked both Einstein and Lorentz as they both believed in the aether. The bottom line here is that Einstein knew that the detection of the aether would not be found by this experiment. So to save his theory from being ridiculed he disassociated the theory of Special Relativity from the aether. He made the theory all about motion relative to other moving frames of reference and not about motion relative to a stationary aether background or space.
Everyone bought into Einstein’s explanation because the M&M experiment result appeared to prove that the aether did not exist. In reality even though the measurements were correct the results and implications of Special Relativity were completely misinterpreted.
The question now arises why was the aether wind not detected? The answer is actually quite simple. The measurement of light beams at right angles was used to determine motion relative to the aether. What everyone did not realize in that time is that photons (wave-like particles of light) acquire their motion from being autonomously or self-caused mode of operation. This means that the motion of particles of light are completely independent from the source of their emission and their motion is independent from all other external frames of reference (motion). This means we cannot use the motion of particles of light to determine our motion relative to the aether. No matter how the earth moves through space we will always measure the speed of light to be the same.
Photons and in fact all particles acquire their motion as a result of the internal structure and their motion comes about as being caused by an internal particle pressure wave. Light particles are not like bullets moving through the air. A bullet gets its motion from an external force. A bullet moves because of rapidly expanding gases in the cartridge that they are fired from. That means the bullets motion is not independent from its source or other moving frames of reference.
So why am I talking so much about the M&M Experiment. The M&M experiment lead directly to a misinterpretation of our physical reality. This badly designed experiment convinced Einstein to change his mechanical model from an aether based description of reality to a description that is not aether based and when the interpretation is examined closely it does not make sense.
In the question above the I am trying to point out that something is amiss in the theory of Special Relativity. If the end result of two moving frames of reference is that one clock moves slower than the other clock that means the faster moving frame of reference will have a clock that runs slower. The utter nonsense that when observer A looks at Observer B’s clock it looks like it is running slower and when observer B looks at A’s clock it looks like it is running slower is completely untrue.
If Observer A is moving at a great rate of speed relative to the aether and observer B is moving much slower rate relative to the aether the clocks rate will be different. The rate of motion relative to the aether is what causes the clock, not time to run faster or slower. Special Relativity is really all about motion relative to the aether.
Unfortunately modern day physicists have failed to make this connection to reality.
The only way that the two observers will have the same clock rate is if both observes are moving at the same speed relative to the aether. This means that they will not be moving at all relative to each other.
So now the question arises is their a way to detect absolute motion relative to the stationary aether. Yes, of course. All we have to do is decelerate our motion relative to the aether. How will we know if we are slowing our motion down relative to the aether? Simple, the clock rate of motion will speed up as we slow down our motion relative to the aether. The clock rate who’s motion is the fastest is the frame of reference that is stationary relative to the aether.