Coordinate systems in GR

No, a coordinate system is not fundamentally a team of observers. However, there is a very powerful and intuitive analogy between them that is often used for physical interpretation.

Let's break this down.

1. What a Coordinate System Actually Is (Mathematically)

In general relativity, a relativistic spacetime is a 4-dimensional smooth manifold. A coordinate system (or chart) is simply a way of labeling each event in some region of this manifold with a set of four numbers $(t, x, y, z)$ . There is no inherent physical meaning to these numbers.

They are just labels, like street addresses.
They can be highly non-linear, singular, or behave in counter-intuitive ways (e.g., Schwarzschild coordinates at the horizon).
The physics (encoded in the metric tensor $g_{μ ν}$ ) tells you how to calculate proper times, proper distances, and causal relationships from these labels.

2. The "Team of Observers" Analogy

The analogy arises when you consider a coordinate system that is adapted to a foliation of timelike worldlines. Here's how it works:

Choose one coordinate, say $x^{0}$ , to represent a time label.
The worldline defined by $(x^{1}, x^{2}, x^{3}) = constant$ can then be interpreted as the path of a hypothetical observer.

In this picture:

The set of all such worldlines (for all constant values of $x^{1}, x^{2}, x^{3}$ ) forms a "congruence" or a "team" of observers. Related: latticework.
The coordinate time $x^{0}$ is the time measured by some chosen clock (often not the proper time of any of the observers).
The spatial coordinates $(x^{1}, x^{2}, x^{3})$ are like "comoving" labels for each observer in the team.

Common Examples:

Schwarzschild Coordinates: The team is observers who are "stationary" with respect to the black hole, firing their rockets endlessly to hover at fixed $(r, θ, ϕ)$ . Their proper time is not the Schwarzschild time $t$ .
Cosmological (FLRW) Coordinates: The team is the "comoving observers" — galaxies following the Hubble flow, moving with the average expansion of the universe. Here, the cosmic time $t$ is their proper time.
Rindler coordinates: The team is a set of observers in flat spacetime undergoing constant proper acceleration, each maintaining a constant "distance" from a Rindler horizon.

3. Key Differences and Why the Analogy Breaks Down

Not All Coordinate Systems Admit This Interpretation: You can define perfectly valid coordinates (e.g., null coordinates like Eddington-Finkelstein, or this worked example) where one coordinate is light-like, not time-like, or spatial-like or both depending on the point.
The "Observers" May Be Unphysical: The worldlines defined by holding spatial coordinates constant might require infinite acceleration or be otherwise non-physical. (e.g., inside the event horizon in Schwarzschild coordinates, the "stationary observer" worldlines become space-like—they'd have to move faster than light!).
Simultaneity is Coordinate-Dependent: In the analogy, "constant coordinate time" defines simultaneity for the team. But GR has no universal simultaneity. A different coordinate system (used by a different "team") will slice spacetime into different "simultaneous" hypersurfaces. Neither is more "real"; simultaneity is a gauge choice.
The Core Difference: An observer is a physical entity with a worldline and a clock (measuring proper time $τ$ ). A coordinate system is a mathematical map of spacetime. An observer can use many different coordinate systems to describe their surroundings.

4. The Bridge: vielbeins or tetrads

The true link between coordinates and observers is the concept of a tetrad (or vierbein). A tetrad is a set of four orthonormal vectors $(e_{0}, e_{1}, e_{2}, e_{3})$ at each point along an observer's worldline.

$e_{0}$ is the observer's 4-velocity (points along their time direction).
$(e_{1}, e_{2}, e_{3})$ define their local spatial axes (their "laboratory").
This tetrad is the observer's local reference frame. Any coordinate system can be used to calculate the components of this tetrad, but the tetrad itself is a coordinate-independent representation of what the observer actually measures (proper time, local distances, local speeds).

Conclusion: While it is extremely useful to think of certain coordinate systems as corresponding to a particular team of observers for building intuition (e.g., "what would a hovering observer near a black hole see?"), it is a pedagogical and interpretive crutch, not a fundamental definition. The foundation of GR treats coordinates as arbitrary labels, and the real physics is in the invariant quantities measured by actual observers using their local tetrads.