A reference frame is aptly described by Einstein as a lattice of rigid rods with clocks attached such that all rods are identical and all clocks synchronised. It embodies the basic concept of relativity in the Galilean sense – to measure something is to compare it with something else of the same type.
Once the need for a reference frame is settled, the question arises – to what can we attach this imaginary construct? Must always a reference frame be attached to a real entity?
To answer this question, we step back and ponder on the relationship of the other physical quantities with reality. In every analysis of a system, what humans do is that we create a mental model of the system based on our observations, i.e. sensory perceptions.
For each set of observations, we bind them together in a single ‘mental’ entity in our mental model. In this manner, we establish a fine isomorphism between physical and mental imagery. Then we talk about energy, momentum and their associated interactions for the mental particles. It is important to note that these quantities are linked to physical reality only through the ‘mental’ particles, they have no image in the correspondence mirror. We notice here that, except for certain postulates, every entity here has a direct or indirect link to the real objects, with indirect links being through the mental objects (solely). Moreover, the postulates in question must be a part of a falsifiable theory.
One can now conclude that if reference frames are a part of physics, they must be linked, directly or indirectly, to a real entity. This conclusion seems consistent with instances from special relativity wherein if we move into frames which are moving at with respect to another frame or object, we encounter absurd or non-real results. This can be explained by thinking of a frame as being attached to a particle which is itself bound by the constraints of special relativity.
On the other hand, we feel no problem in taking a reference at a speed v to analyse a system of particles moving at where . This suggests that we are attaching the reference frame to an ideal virtual particle that doesn’t interact with the other particles (analogous to virtual displacements in D’ Alembert’s Principle of Virtual Work) but is constrained by the laws of the spacetime.
At this point, we realise that our virtual particle has no direct or indirect link to real entities contradicting our hypothesis. The final nail in the coffin is driven when we turn to quantum mechanics. What do we mean here by “attaching a reference frame to an object”? Till now, we had been assuming the origin of the coordinate frame to be situated at a particular point on the object, at all times. How do we bring a position of a quantum particle into the picture here? Clearly, this route seems to be a dead end.
Since we have eliminated matter as the basis of reference frames, we must turn now to the only other entity left, i.e. space itself. The frame then relies upon the attributes of space (or spacetime) in order to be defined. One would then say that it is the hyperbolic spacetime of special relativity that constrains reference frames. But such a definition, relying upon the spacetime, has not yet been found by this author.
Should we build a postulate about reference frames and leave ‘reference frame’ itself undefined? But then there would be no way to falsify that postulate, for we rely upon reference frames in order to measure and observe reality. This circularity leaves us nowhere. We are forced then, to come back to having a reference frame defined on the basis of spacetime. Such a calculus of reference frames based on spacetime may solve our question.