Subject: | Coordinates, reference frames, and general relativity Date: | 09 May 2011 Message-ID: | iq9d5s$9dl$1@speranza.aioe.org

This post flows from years of discussion on the claim by one of Talk Origin's
resident creationists that a fixed earth is permitted within relativity theory.

There's been some confusion here about coordinate systems, reference frames, general relativity, and geocentrism. Some of the issues are a bit subtle, so I thought I might try to clarify a little.

*Coordinate systems*: coordinates are human-made labels of points
in space or spacetime. There are a few technical restrictions, but
in general, coordinates are almost completely arbitrary. Points don't
come wearing little name tags; we can call them pretty much what we
want.

Obviously, Nature doesn't care about the choices we make for naming
points. So no real physical process can depend on the choice of
coordinates. The way we *describe* a process may depend on what
coordinates we use for the description, but the actual process cannot.
This is not just true in general relativity; it holds for any sensible
physical theory. (It's possible, for instance, to rewrite ordinary
Newtonian gravity in a way that makes no reference to coordinates;
Cartan did this in 1923.)

Can we make a choice in such a way that the coordinates labeling the Earth's position don't change in time? Certainly. We can also make a choice in which the coordinates labeling the Earth and the coordinates labeling the Moon dance a foxtrot. This means nothing about the actual motion; it's just a statement about our ability to choose creative names for points.

(While coordinates don't affect physics, there is a sense in which physics can affect coordinates. Real physical processes can be easier to describe in some coordinates than others. If you want to actually calculate the motion of the planets, you'd be foolish to use anything other than a barycentric coordinate system. But even this is really a statement about us – our ability to do the math – and not about Nature.)

*Reference frames*: a reference frame is not just another name for a
coordinate system. A reference frame is a collection of imaginary
observers, spread throughout space and moving along predetermined,
nonintersecting trajectories, each carrying a standard clock. Talking
about a physical process in a particular reference frame means
describing what such a collection of observers would see. This may
be a somewhat anthropomorphic formulation – another definition
refers to a system of ideal "rods and clocks" – but the point is that
a reference frame labels what a real, physical observer could actually
observe.

Every reference frame determines a coordinate system. We can simply label points by the observers at those points. The converse is not true, though: not every coordinate system determines a reference frame.

For instance, we can choose coordinates such that the coordinate values of points on the surface of the Earth are not changing in time. (The shorthand is that this is a coordinate system in which the Earth is "not rotating," but keep in mind that this is a statement about the coordinates, not the Earth.). In such a coordinate system, however, distant objects will have rapidly changing coordinates ("rotating around the Earth"). You don't have to go very far – just as far as Neptune – to get to a place where the "coordinate speeds" are faster than light. Since no physical observer can move faster than light, such a coordinate system does not determine a reference frame.

In short, coordinates are imaginary; reference frames must be at least potentially real.

*Causality*: it can sometimes be hard to disentangle real physical
effects from effects of the choice of coordinate. In the early days of
research on gravitational waves, for instance, there were debates
about whether the waves were real or just "coordinate artifacts."

There are some cases, though, in which the distinction is clear. There's a popular saying (well, popular in certain narrow circles) that physics propagates at the speed of light, but coordinates can propagate at the speed of thought.

For coordinates in which the Earth is not rotating, for instance, it is certainly not true that Neptune is physically moving faster than light. We can imagine coordinates moving faster than light, but if we do, can be sure that the coordinate description will give us artificial results that do not reflect the real physics.

For an Earth-centered coordinate system, in which the Earth is rotating but not moving around the Sun, the situation is slightly – but only slightly – trickier. We observe aberration of starlight, a regular change in the direction the light from stars reaches us. (Simple analogy: if you walk in the rain, the direction the drops hit you depends on which way you are walking.) In a heliocentric coordinate system, this variation comes from the orbital motion of the Earth. In an Earth-centered coordinate system, on the other hand, the change must instead come from the motion of the stars. (Simple analogy: you could get the same effect of slanting rain if you were standing still and the clouds were moving.)

If light traveled at an infinite speed, this would be no problem. But in fact, light travels at a finite speed. In an Earth-centered coordinate system, the aberration of light coming from a star 100 light years away would have to reflect the motion of the star 100 years ago; the aberration of light coming from a star 1000 light years away must reflect the motion of the star 1000 years ago. While one can choose such a description, it does not reflect the real causality: there is no physical mechanism by which the motion of the Earth today can affect the motions of stars in the past. This is especially true because the Earth's orbit varies; the "cause" of a change in motion now cannot have the "effect" of changing stars' motions hundreds or thousands of years ago.

There is an even more dramatic instance of this issue of causality. The Universe is filled with cosmic microwave background radiation (CMBR), basically the afterglow from the period just after the Big Bang when the Universe was very hot and dense. When we observe the CMBR, we see an annually varying Doppler effect that precisely matches the Earth's orbital speed and direction. But this radiation has been traveling freely in space for some 14 billion years. If one tries to physically explain this Doppler shift in an Earth-centered coordinate system, one must claim that the 14-billion-year-old plasma in the very early Universe, ten billion years before the Earth even existed, somehow exactly anticipated the Earth's orbit, with all its local variation. Call this what you like, it is not physics.

[By the way: we really know that the CMBR pervades the Universe, and doesn't just surround the Earth. We can observe its effect in distant galaxies – it can produce observable low-energy transitions among molecular energy levels – and we can observe the effects of distant galaxies on the CMBR – they can cause small but measurable shifts in its spectrum. Note also that this argument does not rely on any particular cosmological model: it's enough to know that the CMBR is reaching us from the very distant Universe, and we can tell that by the fact that it affects and is affected by distant galaxies.]

*Relative and absolute motion*: There is an old argument over whether
it is sensible to talk about absolute motion at all. The discussion
is commonly expressed in terms of Mach's principle, which says in
some form or another that local properties of matter, such as inertia
and rotation, are determined by distant matter, and are fixed only
relative to a particular distribution of matter in the Universe.

The short answer is simply that we don't know. In particular, the question of whether general relativity implies/is consistent with Mach's principle is not settled. The longer answer is that we don't even know how to formulate the question properly. A 1997 paper by Bondi and Samuel, for instance, listed ten formulations of Mach's principle, some of which gave contradictory predictions for certain experiments.

Note, though, that whether or not some version of Mach's principle
is correct, it makes no sense to claim that *all* relative descriptions
are physically meaningful. One still needs consistency with the
requirement of causality that I described above. In particular:

– If absolute motion does not exist, then any correct description must be relative. But this does not mean any relative description must be correct. A relative description must still not allow events in the present to affect the past.

– If there is some absolute description of motion, it, too, must obey the condition of causality. Again, such a description must not allow events in the present to affect the past.

In either case, a heliocentric description is very clearly ruled out.

Steve Carlip

addendum

Alextangent asked this supplementary question related to frame-dragging:

> Now, can I ask one further favour; *rotation*. The argument still rages elsewhere.

Dr. Carlip replied in message iqh4jp$bjs$1@speranza.aioe.org:

There are at least two distinct issues: can we detect "absolute" rotation with local measurements, and can we detect rotation if we are allowed to look at the distant Universe?

For the first question, my answer is, quite firmly, "I don't know." Local experiments such as the Foucault pendulum or Gravity Probe B, see effects that are most easily attributed to rotation. But it's an open question whether general relativistic effects of distant rotating matter could reproduce the same results.

There are hints in both directions. For instance, in the solution of the Einstein field equations for an isolated spherical mass in an otherwise empty universe, we can unambiguously tell whether the mass is rotating. This can't be an effect of other matter, because in this solution there is no other matter. On the other hand, if we take this same isolated spherical mass – say, nonrotating – and put it inside a massive rotating shell, in an otherwise empty universe, we will see effects such as a Coriolis force that are "induced" from the rotating shell and that mimic rotation of the mass.

At the very least, the answer depends on the setting (in particular, on boundary conditions). It remains possible that general relativity is "Machian" – unable to locally identify absolute rotation – if the Universe is spatially closed. As I said in my previous post, it's not entirely clear even how to pose the question.

For the second question – can we detect rotation if we are allowed to look at the distant Universe? – the answer is certainly "yes," as long as we are allowed to assume causality. In particular, changes in local rotation have identifiable local causes, and these can be attributed to the distant Universe only if we allow these local causes to act backwards in time (or if you reject causality altogether).

Steve Carlip

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