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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?

Related:

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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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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$\let\lam=\lambda \let\th=\theta \def\ns#1#2{#1_{\rm#2}} \def\le{\ns\lam e} \def\lr{\ns\lam r} \def\te{\ns te} \def\tr{\ns tr}$ I found Ben Crowell's self-answer very clear and most useful to many people having confused ideas on the matter. However there are some points I don't fully agree on. I'll limit the present answer to just one point, the most relevant IMO.

@BenCrowell writes:

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.

I think this isn't exact. Let's briefly review the matter. The accepted structure of spacetime at cosmological scale is described by a Robertson-Walker geometry. I leave aside the other two founders, Friedmann and Lemaitre, as I'm only concerned with the geometrical structure, with no consideration of what causes it: no hypotheses on matter quality and distribution, no Einstein equations. I only need the cosmological principle, i.e. the assumption that space (not spacetime!) is homogeneous and isotropic. This implies - as I already said - that spacetime has a R-W geometry.

There are several coordinate systems in use for R-W geometry. I'll choose the following: $$ds^2 = dt^2 - a(t)^2 (dr^2 + \cdots) \tag1$$ (the dots stay for the angular part of the metric I'll not use).

Looking at metric (1) one point shows itself about the choice of coordinates. It's perfectly true that in GR coordinates are entirely arbitrary and are not bound to have a physical interpretation. This notwithstanding, in several cases a judicious choice of coordinates can reaveal important features of a geometry. In our case we can clearly see that there is a choice of the time coordinate (called cosmic time) such that equal-time sections (what we can summarily call space) are endowed with a rich symmetry - the mathematical counterpart of the homogeneity and isotropy requisites.

Homogeneity reflects itself in space having constant curvature (not to be confused with spacetime curvature). As to this point R-W geometry can be of three kinds:

  • positive curvature, space being a hypersphere
  • null curvature - a flat, euclidean space
  • negative curvature, i.e. hyperbolic geometry

In eq. (1) the three kinds are not apparent, as their difference is in the angular part of metric. It's well known that at present the accepted model is a flat space, but the model choice is not relevant for my argument.

The coefficient $a(t)$ whose square multiplies the space part of metric in eq. (1) is called the scale factor. Its value is a measure of how space curvature depends on time and the exact expression of function $a(t)$ can only derived by cosmological dynamics, i.e. Einstein equations together with hypotheses on the kind of matter present in the universe. I'll not deal with that. I'll assume $a(t)$ is known and am going to state some consequences of its not being a constant. The scale factor is usually assumed to equal 1 at present time.

It can be shown that lines with $r$, $\th$, $\phi$ constant are geodesics of spacetime. Then a free object (a galaxy) can stay put in those (comoving) coordinates. If $r$-origin is in our position then the distance of that object from us is $a(t)\,r$. A variation of $a$ with time entails a proportional variation of that galaxy's distance.

If light coming from a faraway object reaches us at present time, we observe a cosmological redshift: received wavelength $\lr$ is greater than emitted wavelength $\le$ and a simple law holds: $${\lr \over \le} = {a(\tr) \over a(\te)}$$ where $\te$ is the time when light was emitted, $\tr$ is present (received) time.

It seems clear IMO that in this setting the most natural interpretation is that space is expanding (scale factor $a(t)$ is increasing) whereas galaxies occupy fixed position in comoving coordinates. In order to embrace the alternative view - space not expanding, galaxies moving in it - it would be necessary to exhibit a suitable coordinate system where metric is static (metric coefficients independent of time) and coordinates of all galaxies depend on time. I don't know of such system and suspect it doesn't exist. I'm waiting to be proved wrong.

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