© 2024. All rights reserved.
Site created with
Hakyll.
Modified theme
lanyon-hakyll
based on
Lanyon.
© 2024. All rights reserved.
Site created with
Hakyll.
Modified theme
lanyon-hakyll
based on
Lanyon.
I am, as far as I can tell, always at some particular place, and always in some particular state of motion. As such, I have a (local) reference frame, specified by my location and momentum. I might be able to move to a different place, or change my momentum, but my reference frame – in this sense — is a consequence of my physical state.
There is another sense in which my reference frame is not a consequence of my physical state. In particular, when solving a physics problem, I can choose a reference frame. Similarly, when describing an event, I can choose a reference frame from which to describe it. I don’t need to be in that place or state of motion to describe things as if I were. In this sense, reference frames are a conceptual tool — something we use to see the world rather than a description of our place in the world.
We might say then that the concept of a reference frame is both physical (determined by one’s current state) and semantic (a tool for describing and sharing information). In this sense, the notion of a reference frame might be an extension or precisification of the notion of a perspective. When I say “my perspective”, I might simply mean where I’m sitting and where my head is turned — as in, e.g., “from my perspective, the Statue of Liberty is not visible.” But I can also use “my perspective” to mean the way that I conceive things. For example, I might say “my perspective on US politics is extremely grim”, which isn’t meant to tell you anything much about my current physical state.
The notion of a frame of reference has played an important role in physics. However, the very power and richness of the notion makes it subject to abuse. For example, it is common for people to impose a sort of criterion of reality on objects that they must be “independent of reference frame”. But what in the world is that even supposed to mean?
In the tradition of 20th century analytic philosophy, we can try to improve conceptual clarity by means of explication. To explicate a concept means something like providing a good mathematical (structural) representative of that concept. For example, we have an intuitive notion of infinity, and an explication of that concept is provided by the mathematical notion of injective and surjective functions: a set \(S\) is infinite just in case there if a function \(f:S\to S\) that is injective but not surjective.
So what then are some mathematical things that play the role of the intuitive concept of a reference frame? There are several inter-related candidates, and only some of these candidates generalize to the contemporary setting (where we can’t always depend on having global coordinate charts).
In Galilean relativity and special relativity, we can represent a reference frames as an (equivalence classes of) pairs consisting of a position (element of the manifold) and velocity (element of the tangent space at that point). In this case, \((p,\vec{v})\) uniquely determines a timelike line (geodesic) in the relevant spacetime. And each single timelike geodesic uniquely determines an entire family of parallel geodesics. In the case of Galilean spacetime, the spatial hypersurfaces are already determined, and do not depend on the frame of reference. But in Minkowski spacetime, there is a unique spatial hypersurface orthogonal to \(\vec{v}\).
In Galilean and special relativity, an instantaneous state \((p,\vec{v}\in T_p)\) also picks out a “preferred” coordinate system. The mathematics here is straightforward. What is not straightforward is what “preferred” is supposed to mean.
So long as our spacetime theory uses a manifold \(M\) and a notion of permitted trajectories of material objects, then we can say that a reference frame is a family of such lines that fill space.
In GTR, a pair \((p,\vec{v}\in T_p)\) could also be said to be a reference frame. But now the notion is being stretched, because …
There’s another mathematical idea that is (indirectly) related to reference-frame dependence: basis-dependent representations of tensorial objects. For example, lqet \(v\) be a vector in an \(n\)-dimensional vector space \(V\) over \(\mathbb{R}\). Relative to a basis, \(v\) is represented as an \(n\)-tuple of real numbers. This well known fact generalizes to all tensors over \(v\), and it seems to have led to quite a bit of confusion about how differential geometry functions. In particular, it is not uncommon to hear talk about “non-tensorial quantities”, as if there were some kind of space of quantities, some of which are tensorial and some of which are not. But in most such cases, the so called non-tensorial quantities are simply basis-dependent representations of tensors.
Basis-dependent representations of tensors are closely related to coordinate-dependent representations of tensor fields. In particular, let \(p\in M\), and let \((U,\varphi )\) be a coordinate chart such that \(p\in U\). This chart then defines a basis on \(T_p\) relative to which tensors over \(T_p\) have a numerical representation. However, this numerical representation doesn’t have any immediate ontological or epistemological significance — at least not until one says something more.