In both Newtonian gravity and special relativity, every frame of reference can be described by a coordinate system covering all of time and space. How does this work in general relativity? When an observer watches matter falling into a black hole, or observes a distant galaxy receding from ours, can we describe what they see in terms of a certain coordinate system that is tied to that observer?


Inertial frames of reference

The following are very similar questions but at a fancy mathematical level, and focusing on the technical details such as charts and manifolds:

How a reference frame relates to observers and charts?

What does a frame of reference mean in terms of manifolds?

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    $\begingroup$ Someone downvoted both the question and the answer within a few minutes after I posted them, but I don't see any comments about why. Could the downvoter explain? If there's something inappropriate about my question, or incorrect about my answer, I'd like to know so I can learn from my mistakes or make edits to improve what I wrote. $\endgroup$ – Ben Crowell Feb 4 at 17:31
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    $\begingroup$ I'm glad you posted this Q/A. Attempts to apply the "frames" idea in GR seems to be a common source of confusion on Physics SE, so I hope to see lots of links to this post in the future. Great idea (+1 Q, +1 A)! $\endgroup$ – Chiral Anomaly Feb 5 at 1:08

General relativity only has local frames of reference, not global ones. In general relativity, coordinate systems are entirely arbitrary, and we can't typically take a coordinate system and relate it in any meaningful way to a the frame of reference of some observer.

In Newtonian gravity, there is an implicit assumption that an observer can and does know about the current state of all matter in the universe. Without this information, there would be no way to apply Newton's laws, since gravity is a long-range force acting instantaneously at a distance, and there would also be no way to determine what was an inertial frame of reference. Traditionally, the frame of the "fixed stars" was considered to be a good enough inertial frame of reference for all practical purposes, and it was implicitly assumed that we could observe the stars instantaneously, neglecting any possible delay due to the time taken for light to propagate. Other frames of reference in uniform motion relative to this frame were also valid. Each these frames of reference was described by a certain coordinate system, and the different coordinate systems were connected by rotations and Galilean boosts.

In special relativity, it gets harder. We can't instantaneously observe all of space, but we don't need to, because in order to make predictions about our own neighborhood, we only need to know the conditions inside our own past light-cone, i.e., at events that are close enough in space and far enough back in time so that signals have had time to get from them to us. For convenience, we still usually go ahead and extend this description to include all of spacetime, which implies a sort of elaborate surveying system whose results are known to us only much later. For example, the surveyors might have to place clocks in various positions and synchronize the clocks by exchanging radio signals. Time is relative, but for an observer in a certain state of motion, we can define a notion of simultaneity. Each inertial observer corresponds to a set of Minkowski coordinates, which are the result of the surveying process.

In general relativity, basically all of this goes out the window. Coordinate systems may not be able to cover all of spacetime, for the same reason that we can't put a coordinate system on the earth's surface without having it misbehave in certain places such as the poles. Even for spacetimes, such as FLRW cosmological spacetimes, for which it is possible to have such a global coordinate system, these coordinates cannot be identified with observers or frames of reference. Frames of reference exist only locally, i.e., at scales small compared to the scale set by the curvature of spacetime. When we discuss what an observer "sees" in general relativity, we mean exactly that: the optical signals that they receive.

Example: An observer outside a black hole will never see infalling rock pass through the event horizon. This is trivially true, because the event horizon is defined as the boundary of the region that is externally unobservable. This does not mean that the observer's frame of reference corresponds to some set of coordinates, or that what the observer sees can be explained by such coordinates. What the observer sees is simply explained in terms of the trajectories of the light rays that travel from the rock to the observer's eye.

Example: General relativity doesn't tell us whether distant galaxies are "really" moving away from us, or whether they're "really" at rest while the space in between fills up. We only have a local coordinate system, not a global one that would allow us to define and measure velocity vectors for distant objects. We can define things like coordinate velocities, but they're not particularly meaningul, because coordinates are arbitrary.

Example: Given a flat spacetime with Minkowski coordinates $(t,x)$, we can define new coordinates $(t,u)$, where $u=ax+(1/4)\sin ax$ and $a$ is a constant. There is nothing wrong with these coordinates, because the transformation is one-to-one and smooth, but these coordinates clearly don't correspond to the frame of reference of some observer.

People often get confused about this kind of thing because of historical treatments of general relativity, including Einstein's own popularizations. Einstein's original inspiration for general relativity had to do with a set of concepts including Mach's principle and the notion of extending the set of allowable frames of reference to include accelerated ones. General relativity is now over a century old, and many of Einstein's original vague inspirations have not turned out to be the best way to think about these things.

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    $\begingroup$ I would put your first paragraph in 48 point bold - in flashing colours! Beginners to GR almost always fail to understand that there need not be any physical significance to a coordinate system. $\endgroup$ – John Rennie Feb 5 at 7:25

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