Frame of Reference: How much physics sits in it?

Question

A naive question about nuances on meaning of the concept frame of reference and how much "intrinsically physical" information it carries about considered physical system.
To my knowledge (compare with this )naively a frame of reference is a priori a mathematical - so unphysical - procedure (because it explicitly depends on making choices) based on specifing a frame, so a picking a set of basis vectors in order to fix a welldefined coordinate system describing my given physical system or a part of it we would like to model (eg if we a dealing with "whole" system modeled inside laboratory frame and pick a relative frame of reference wrt a picked moving partial system where this object is inert).
Eg, think of priciple of gauge theory where gauge trafos swap between mathematically a priori different but physically identical configurations; keyword: invariance of laws of physics under certain trafos.

For instance a toy example, assume we pick laboratory frame (with basis $e_1=(1,0,0), e_2, e_3$ analogously in 3d where the observer stands in (0,0,0) and observes a car passing parallel to $e_1$ through point $(1,1,0)$ with constant velocity $\vec{v}=v \cdot e_1$.
Then the car's frame of reference (ie where it stays stationary at $(0,0,0)^C$ with resp car's frame of reference) is given by Galilei trafo as $e^C_1(t):=(v \cdot t,0,0), e^C_2=e_2, e^C_3= e_3$
(Note, $e_i=e_i(t)$ can clearly be time dependent)

So my understanding of a frame of reference is that as it technically corresponds to just picking a basis spanning the number of degrees of freedom of the considered physical system we are going to describe: Once we have picked frame of reference, we can model the system with respect associated coordinate system.
So at all it seems to me that picking such a frame of reference inducing coordinate system modeling my physical system is a pure mathematical construction, isn't it?

On the other hand it seems (see this question & discussion that it may be possible that a physical system may have a so called "privileged frame of reference", which seems to be an "intrinsically physical" feature of the considered system. The example there was based on an attempt to take a putative discrete spacetime & which principles would going consequently to be violated.

Therefore I have two questions:

(1) A naive one: As I elaborated above it seems so far to me that the "choice" of a frame of reference is an "unphysical" procedure as as one picks a mathematical object (a basis) to model a physical system. How can a frame of reference be "(physically) privileged"? So is essentially a frame of reference a purely mathematical construction or can it cary intrinsically physical information of the system? For instance, I could rechoose a new basis by eg rotating or shifting to old one without "changing the physics".
The issue in linked tread strongly suggests that my understand behind "physical nature" of frame of reference is wrong; namely that is not a pure mathematical construction& basing on unphysical choices.
Could somebody clarify that part how much physics really sits in concept of frame of reference?

(2): Having the problem discussed in linked question in mind: Why discrete structures (as eg there the putative discrete spacetime) tend to have a "privileged frame of reference", ie that it has a "distinguished" frame of reference reflecting intrinsically physics of the system. Is there a striking reason for this?

I noticed similar question but which not adresses exactly my concern.

Answer 1

I think you need to distinguish between the terrain and the map, to use a metaphor from philosophy, as the term 'reference frame' is often used interchangeably to refer to both, which can lead to confusion.

I suggest using the term 'coordinate system' to refer to the choice of coordinates that can be made when modelling some physical arrangement. That is essentially a mathematical choice- you can pick any coordinate system you like, as long as it has the necessary number of dimensions for the system being modelled. It is entirely 'unphysical'- as you put it- in that sense.

The physics is what it is regardless of what coordinate system you pick to model it. Where the confusion arises is that the mathematics takes a particular- and usually simpler- form when you pick what might be considered 'natural' coordinate system for the problem at hand. For example, when solving homework problems about balls rolling down ramps it makes sense to use a conventional coordinate system in which the Z axis is normal to the Earth's surface and the X and Y axes are locally parallel to it, and the origin is stationary with respect to the Earth. You don't have to do that- you can pick a coordinate system in which the origin is somewhere on the Moon, and the other axes are all tilted somehow- but that will make your job much much harder.

The coordinate systems which most closely align with the underlying physical effects are what we usually mean by frames of reference. For example, in SR when we talk about a frame of reference we mean a coordinate system which is stationary relative to something, such as a rocket or a platform or a train. We might talk about the frames of reference of an observer on a train and an observer on a platform. We could still use entirely different coordinate systems to model them, but the resulting mathematical complications would make the job much harder and mask what was actually going on.

So, loosely, you can think of coordinate systems as being the freely chosen mathematical means of referring to points in space and time, and frames of reference to mean the underlying reality that the coordinate systems are trying to map. That distinction is not one that is generally recognised, but it might help clarify the confusion you alluded to in your question.

Answer 2

Let me start with an analogy to a similar and hopefully easier to understand situation. Consider the problem of choosing $x, y, z$ axes to set up a coordinate grid in space. We are free to choose any coordinate grid we want, so there are an infinite class of possible coordinate grids related by translations and rotations that can all be used to describe the same physics. The invariance of the laws of physics under shifting between these different possible coordinate grids places interesting constraints on the form of the laws of physics. Having said that, on the surface of the Earth, there is a natural way to orient one of our coordinate axes; with the vertical direction, because that is a "preferred" direction that acts differently than the other two (we fall in the vertical but not in horizontal directions). There is a physical effect that breaks the symmetry between different possible coordinate grids, and that physical effect picks out a preferred direction.

If you understood that paragraph, then translating the concept to frames of reference should be fairly easy. There are an infinite number of possible coordinate systems on spacetime that we can choose. Physics should not depend on our choose of coordinate system. However, there can be physical objects (like, say, the Earth), that mean that in practice one frame of reference is more natural to use than others.

Your second question on discrete structures is a bit different. Again it's probably useful to think about the rotation example first. Let's just think of a 2D space, a Euclidean plane. If space is a continuum, then you can choose your axes to point in any direction and the set of allowed vectors will look the same in each one. However, if space is truly discrete, say it is made of points that form a square lattice, then coordinate systems aligned with the lattice directions will be different than coordinate systems not aligned with the lattice space. (For example, in a coordinate system aligned with the lattice vectors, you can reach any point in the space by taking integer multiples of the smallest displacement in the two coordinate directions; that property won't hold in general if you rotate your coordinate system by an arbitrary amount).

There is a similar story when talking about Lorentz transformations. If spacetime is discrete, then different frames will perceive the lattice structure differently. For example, say space is covered by a cubic lattice of points. Then the spatial volume of each cube will be a certain constant amount. However under a boost in some direction, because of length contraction, the spatial volume of the boxes of lattice points can become arbitrarily small. Reversing that, you could in principle determine your absolute speed by measuring properties of the lattice in your reference frame.