Relativity can be developed without coordinates: Laurent 1994 (SR), Winitzski 2007 (GR).
I would normally define a vector by its transformation properties: it's something whose components change according to a Lorentz transformation when we do a boost from one frame of reference to another. But in a coordinate-free approach, we don't talk about components, and vectors are thought of as immutable. For example, Laurent describes an observer using a timelike unit vector $U$, and then for any other vector $v$, he defines $t$ and $r$ uniquely by $v=tU+r$, where $r$ is orthogonal to $U$. The $(t,r)$ pair is what we would normally think of as the coordinate representation of $v$.
In these approaches, how do you define a vector, and how do you differentiate it from things like scalars, pseudovectors, rank-2 tensors, or random objects taken from something that has the structure of a vector space but that in coordinate-dependent descriptions would clearly not transform according to the Lorentz transformation? It seems vacuous to say that a vector is something that lives in the tangent space, since what we mean by that is that it lives in a vector space isomorphic to the tangent space, and any vector space of the same dimension is isomorphic to it.
[EDIT] I'm not asking for a definition of a tangent vector. I'm asking what criterion you can use to decide whether a certain object can be described as a tangent vector. For example, how do we know in this coordinate-free context that the four-momentum can be described as a vector, but the magnetic field can't? My normal answer would have been that the magnetic field doesn't transform like a vector, it transforms like a piece of a tensor. But if we can't appeal to that definition, how do we know that the magnetic field doesn't live in the tangent vector space?
Bertel Laurent, Introduction to spacetime: a first course on relativity
Sergei Winitzki, Topics in general relativity, https://sites.google.com/site/winitzki/index/topics-in-general-relativity
Answer
Honestly, this coordinate-free GR stuff (Winitzki's pdf in particular) looks like GR as would be taught by a mathematician--very similar to do Carmo's text on Riemannian geometry. In classic (pseudo-)Riemannian geometry, vectors are defined as derivatives of affine parameterized curves, covectors as either maps on vectors to scalars or as gradients of scalar fields. Something like the Riemann tensor is defined as a map on two/three/four vectors spitting out two vectors/one vector/a scalar.
Differential geometers love defining everything as a mapping; I consider it almost a fetish, honestly. But it is handy: defining higher-ranked tensors as mappings of vectors means that the tensor inherits the transformation laws of each argument, and as such, once you establish the transformation law for a vector, higher-ranked tensors' transformation laws automatically follow.
Edit: I see the question is more how one can figure out a given physical quantity is a vector or higher-ranked tensor. I think the answer there is to look at the quantity's behavior under a change of coordinate chart.
But Muphrid, we never chose a coordinate chart in the first place; isn't that how coordinate-free GR works?
Yes, but the point of coordinate-free GR is just to delay the choice of the chart as long as possible. There is still a chart, and most results depend on there being a chart, just not on what exactly that chart is.
How does looking at a change of chart (when we never chose a chart in the first place) help us?
The transition map from one chart to another is a diffeomorphism, and so its differential can be used to push vectors forward or pull covectors back. Hence, the transformation laws that usually characterize vectors and covectors are still there. They look like this: let $p \in M$ be a point in our general relativistic manifold. Let $\phi_1: M \to \mathbb R^4$ be a chart, and let $\phi_2 : M \to \mathbb R^4$ be another chart. Then there is a transition map $f : \mathbb R^4 \to \mathbb R^4$ such that $f = \phi_2 \circ \phi_1^{-1}$ that changes between the coordinate charts.
Thus, if there is a vector $v \in T_p M$, there is a corresponding vector $v_1 = d\phi_1(v)_p \in \mathbb R^4$ that is the mapping of the original vector into the $\phi_1$ coordinate chart. We can then move $v_1 \to v_2$ by the (edit: differential of the) transition map.
But Muphrid, aren't we meant to be working with the actual vector $v$ in the tangent space of $M$ at $p$, not its expression in a chart, $d\phi_1(v)$?
You might think so, but (as was drilled to me repeatedly in a differential geometry course) we don't actually know how to do any calculus in anything other than $\mathbb R^n$. So I think there's some sleight of hand going on where "really" what we do all the time is use some chart to move into $\mathbb R^4$ and do the calculus that we need to do.
What this means is that, in my opinion, coordinate-free is a bit of a misnomer. There are still coordinate charts all over the place. We just leave them undetermined as long as possible. All the transformation laws that characterize vectors and covectors and other ranks of tensors are still there and still let you determine whether an object is one or the other, because you're always in some chart, and you can always switch between charts.
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