How a reference frame relates to observers and charts?

Question

Recently I've been watching the General Relativity lectures from the "International Winter School on Gravity and Light" by Frederic Schuller. In those lectures he made the following two definitions:

1. A coordinate system on spacetime $M$ is a chart $(U,\phi)$ with $U\subset M$ and $\phi : U\to \mathbb{R}^4$ a homeomorphism.

2. One observer is a timelike, future pointing worldline $\gamma : I\subset \mathbb{R}\to M$ on spacetime, together with four vector fields $e_\mu : I\subset \mathbb{R}\to TM$ along $\gamma$, that is, $e_\mu(\lambda)\in T_{\gamma(\lambda)}M$ such that $e_0 = \gamma'$ and such that $g_{\gamma(\lambda)}(e_\mu(\lambda),e_\nu(\lambda))=\eta_{\mu\nu}$, in other words they are orthonormal.

My question is what reference frames becomes in this setting. Actually in the lectures reference frame was never defined. I just know by intuition that a "reference frame" represents a point of view, and is what we use to assign components to tensors, that is, we can think of them as sets of axes. As far as I know a reference frame is a section of the frame bundle, but I fail to see how this relates to observers and charts.

However we can see that in some sense the observers carries with them a reference frame. But this is extremely local: it is defined just on points of his worldline and this is what makes me confused.

In SR, observers, coordinate systems and reference frames are all identified by the fact that spacetime is flat. One just considers cartesian coordinates, which are the same as sets of axes, and those are global to the whole manifold. In that case it is common to talk about the whole dynamics of a particle for example in "the reference frame of an observer", meaning just "to use the set of axes of a coordinate system where the observer's evolution is $\tau \mapsto(\tau, x_0,y_0,z_0)$".

Now in General Relativity what we really mean by reference frames and how they relate to these ideas of "observers" and "charts" as I presented? Are they just those "basis carried by observers"? And if so how they are actually used if they are defined just on the worldline of the observer?

I'll give one example. Suppose we have a particle of mass $m$ and we want to discuss its dynamics. We certainly need a reference frame if we are to write down for example its four momentum and equations of motion. Indeed if $\gamma$ is its worldline, the four momentum is $p = m\gamma'$ but we want to resolve in components.

What would we do? Pick one observer $\alpha,e_\mu$? But in that case we could only expand $p = p^\mu e_\mu$ on the coincident events where both the observer and the particle are there, that is, $\alpha(\tau_1)=\gamma(\tau_2)$. This is just one spacetime point. It is certainly not like that that we should work.

Changes of frames also gets confused in this setting, since we could only perform a change on the event where two observers are together. This is also strange, comparing to SR.

So what is actually a reference frame in GR? How does it relate to observers and charts? And how they get used in practice?

Answer 1

[If] one observer is a timelike, future pointing worldline ...

There are indeed different interpretations of the notion "observer" in use; as concisely contrasted by this Wikipedia page as

(a) "observer referring to an (inertial) reference frame", or

(b) "observer referring to an individual gathering observations".

Arguably, the latter has a close correspondence to Einstein's notion of "material point"; e.g. as relating this to that.

[then] what do reference frames become in this setting ?

Therefore the question is also which definition or intuition on the notion of "reference frame" to consider other than (a).

As far as flat spacetimes are concerned, the applicable notion appears spelt out by Rindler:
"An inertial frame is simply an infinite set of point particles sitting still in space relative to each other."
.

More generally it might be required of a reference frame $\mathcal F$ in a spacetime (set of events) $\mathcal S$ that

• each member of $\mathcal F$ is (strictly, or at least piecewise) timelike,

• the members of $\mathcal F$ are disjoint, and

• the union of all members of $\mathcal F$ equals the entire spacetime $\mathcal S$ under consideration.

This describes frame $\mathcal F$ as a partition of $\mathcal S$, and as a timelike congruence (at least in some sense or generalization).

To each individual member (constituent, participant, point $P$) of frame $\mathcal F$ there is usually (cmp. Einstein 1905) associated a notion of "good time-coordinate $t_P$":

$$ t_P : (P \times P) \times (P \times P) \rightarrow \mathbb R, \qquad t_P[ \, P_A, P_B, P_J, P_K \, ] \mapsto \frac{\tau P[ \, \_A, \_B \, ]}{\tau P[ \, \_J, \_K \, ]}, $$

where $\tau P[ \, \_A, \_B \, ]$ is $P$'s duration between $P$'s indication $P_A$ (of having participated in event $\varepsilon_{P A} \in \mathcal S$) and $P$'s indication $P_B$ (of having participated in event $\varepsilon_{P B} \in \mathcal S$); and $\varepsilon_{P J} \not\equiv \varepsilon_{P K}$, thus $\tau P[ \, \_J, \_K \, ] \ne 0$.

Arising from further requirements associated with the notion of "reference frame", additional geometric relations between members of frame $\mathcal F$ might be defined and determined due to

• identifiable material points (participants, timelike worldlines) who are not members of frame $\mathcal F$ but who met selected members of $\mathcal F$ (usually "in passing"; necessarily "only one at a time"), with the associated coincidence determinations, described as coincidence events in spacetime $\mathcal S$; or

• lightlike paths between such coincidence events (with possible coincidence determinations).

The members of frame $\mathcal F$ may accordingly evaluate "boomerang" experiments, or (even primarily) "ping" experiments in order to characterize their geometric relations between each other; possibly determining (ordered neighborhood relations by) their "radar distance" between each other; possibly even determining simultaneity relations between their individual indications.

As far as a thus obtained system of ("conical", "causal diamond") neighborhoods comprises a topological space it may then be determined which assignments of (subsets of) of $\mathbb R^n$ as coordinate tupels to (subsets of) spacetime $\mathcal S$ are homeomorphisms, and correspondingly are providing a coordinate chart on each neighborhood; and which are not.