Sorry if I make your headache worse, but... Special relativity actually implies that the mass of an object is not an absolute quantity. That is the mass of an object depends on its reference frame and the reference frame of the observer who measures its mass. The mass of the object in a reference frame in which that object is at rest is called its proper mass, or more commonly (and surprisingly enough) its rest mass. In a reference frame in which that object is moving its mass will be measured to have a larger value than the rest mass. This is a consequence of the fact that, at least in ordinary circumstances, there doesn't seem to be any such thing as negative energy. So the answer to your question, based on relativity theory is that both the object and its environment are moving. If you are moving with respect to the environment in the same direction as the object and at the same speed, you will measure the rest mass of the object. If you are in any other reference frame you will measure a higher mass.
AAAAAIIIIGH!!!