# 6

Today's post, Mach's Principle: Anti-Epiphenomenal Physics was originally published on 24 May 2008. A summary (taken from the LW wiki):

Could you tell if the whole universe were shifted an inch to the left? Could you tell if the whole universe was traveling left at ten miles per hour? Could you tell if the whole universe was accelerating left at ten miles per hour? Could you tell if the whole universe was rotating?

Discuss the post here (rather than in the comments to the original post).

This post is part of the Rerunning the Sequences series, where we'll be going through Eliezer Yudkowsky's old posts in order so that people who are interested can (re-)read and discuss them. The previous post was My Childhood Role Model, and you can use the sequence_reruns tag or rss feed to follow the rest of the series.

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The amazing thing is that this is a scientifically productive rule—finding a new representation that gets rid of epiphenomenal distinctions, often means a substantially different theory of physics with experimental consequences!

This "Anti-Epiphenomenal Physics" is well known by a less fancy name of symmetry. Looking for hidden symmetries (and applying the Noether theorem to them, whenever possible) is the basic tool theorists use all the time. If anything, they go farther than that by reconstructing broken symmetries or even designing theories with hard-to-imagine symmetries.

I don't think Eliezer is right when he says that Mach's principle (the way he interprets it) is widely accepted. It's true that the general theory of relativity is formulated so that there is no privileged coordinate frame. However, Mach's principle goes beyond this, saying that there is no privileged state of motion. On the usual interpretation of GR, this latter claim is false. Inertial motion can be distinguished from other states of motion, in a coordinate-independent way. Inertial worldlines are just the ones that follow geodesics.

Now Eliezer points out that by changing the space-time curvature, we can change inertial motion to non-inertial motion. This is true, but relativists don't usually treat curvature the way they treat coordinate frames. A coordinate frame is conventional, something that we apply to the universe for convenience. Space-time curvature, on the other hand, is out there. There is a genuine fact of the matter about the curvature of space-time. And it follows that there is genuine fact of the matter about which worldlines are inertial.

Maybe Eliezer is right that we should treat curvature as conventional, but this is not the way most relativists think of it. Also, it doesn't seem like a very compelling position. If curvature is conventional, then so is the space-time metric, which means so is geometry. This leads to a thorough-going Poincare-esque instrumentalism, which is a consistent world-view but one that I find unattractive. And knowing what Eliezer says about quantum mechanics, I suspect he would find it unattractive as well.

Putting aside the specific definition of "Mach's principle" (which seems to mean different things to different people), as well as sociological questions about what relativists usually do or don't do, the way I understand it is as follows (and I would welcome corrections from anyone in a position to offer them):

In general relativity, spacetime is a Lorentzian manifold -- that is, a manifold modeled on Minkowski space, which is just like Euclidean space except with a funny inner product such that the "norm" is constant on hyperboloids instead of spheres -- which is itself determined, according to a system of hyperbolic PDEs known as the Einstein equations, by a particular tensor field (representing "matter") on it. (It's not clear to me whether what is being "determined" here is just the Lorentzian structure, or if one somehow "solves for" the underlying topological and differentiable structures of the manifold as well; but leave this point aside for now.)

A "frame of reference" is actually the same thing as a "state of motion": they are both physicists' jargon for a chart of the manifold, i.e. a mapping that serves to identify a particular open set (or "locality") of the manifold with a corresponding open set in the model space, which in our case is Minkowski space.

Part of what it means to be a manifold of a given type (e.g. topological, differentiable, Lorentzian) is that any two charts defined on the same locality are considered equivalent, so long as the resulting mapping between the two corresponding open sets in the model space (called a "transition map") preserves the structure in question (topological, differentiable, Lorentzian, etc). In our case, then, any two charts defined on a locality are equivalent, provided that the transition map is "Lorentzian" -- which, one has to assume, must mean that the derivative of the transition map, at each point, is a Lorentz transformation (a linear operator on Minkowski space which preserves the "Minkowski norm" defined by the funny inner product mentioned earlier).

In other words, every "change of coordinates" (transition map) which is locally a Lorentz transformation is to be considered "legal". When physicists say that there is "no privileged frame of reference", they are not saying anything not already contained in the statement that spacetime is represented by a Lorentzian manifold. (Of course, it is not true that there is no privileged class of reference frames: to be a member of the privileged class, a chart must be "Lorentz-compatible" with all the other charts in the privileged class.)

Now, what about curvature? Well, note that in order to even be able to say that a Lorentzian manifold is capable of having curvature, the notion of "curvature" must be chart-independent -- otherwise, it's only the charts that have "curvature", and not the manifold itself. It turns out -- so mathematical legend has it -- that there is such a "global" notion of curvature that makes sense for Lorentzian manifolds; and in fact it is (more or less) the very thing that the Einstein equations say is determined by the "matter" fields on the manifold.

(Roughly speaking, what it means for the spacetime manifold to be "curved" is that you can end up "heading in a different direction" after a while despite being "at rest" the whole time.)

So, the point is that there is no way a "legal" change of coordinates can ever change the curvature of spacetime. This isn't a matter of the convention of physicists, except insofar as they have decided to represent spacetime as a Lorentzian manifold rather than some other type of manifold (or some other kind of mathematical structure altogether). If you want to be able to change the curvature, you have to allow non-Lorentz local coordinate changes, and thus work in some other kind of mathematical structure different from a Lorentzian manifold.

(And I don't think that's what Eliezer was suggesting.)

When I said "state of motion", I was talking about whether motion is inertial or non-inertial. This is indeed frame-independent in general relativity. There are privileged states of motion.

In Newtonian mechanics and the special theory of relativity, the way they are usually formulated, inertial motion is just motion that is non-accelerated relative to an inertial frame. So privileging inertial motion amounts to privileging a set of frames - the inertial frames. Both of these theories are ordinarily formulated so that their equations of motion only hold in inertial frames.

General relativity, in its ordinary formulation, is generally covariant. There is no set of privileged reference frames such that the equations only hold in those frames. Incidentally, this amounts to more than just the fact that spacetime is represented by a Lorentzian manifold (also, your definition of Lorentzian manifold is incorrect). Spacetime is represented by a Lorentzian manifold in special relativity as well. What makes GR generally covariant is that its kinematical and dynamical equations are tensor equations. Tensors are coordinate-independent objects, so tensor equations are true in all frames of reference (that satisfy certain continuity conditions).

So now the theory doesn't just hold in inertial frames of reference. But it turns out that there is still a privileged notion of inertial motion, one that can be expressed in a coordinate-independent manner. Basically, inertial trajectories are trajectories in freefall under gravity, and these trajectories are just the ones that satisfy the geodesic equation, which is also a tensor equation. So yeah, GR does have privileged states of motion.

It's worth noting that the existence of privileged inertial trajectories does not correspond to the existence of privileged inertial frames that extend across the entire manifold. Suppose we construct a frame that is "adapted" to a particular inertial trajectory, i.e. one in which the object following that trajectory is stationary. If space is variably curved, then relative to this frame, some other free-fall trajectory will be accelerated. So, unlike Newtonian mechanics or the special theory, we don't have frames where all inertial motion is unaccelerated.

As for what Eliezer was suggesting, perhaps I have misinterpreted him, but in his discussion of GR he brings up the principle of equivalence a lot. He says, for instance:

This meant you could never tell the difference between firing your rocket to accelerate through flat spacetime, and firing your rocket to stay in the same place in curved spacetime.

Coupled with his subsequent claim that epiphenomenal distinctions are, as a rule, illusory, he seems to be strongly suggesting that there is in fact no difference between these two states of affairs, which would imply that there is no objective fact of the matter about whether spacetime is flat or curved. This is the claim I was disputing.

When I said "state of motion", I was talking about whether motion is inertial or non-inertial.

What does this mean in terms of the mathematics? My understanding (which you seem to confirm) is that "inertial motion" refers to geodesic paths. But those are precisely the paths which (the theory says) describe all motion! In other words, there's no such thing as "non-inertial motion". (Remember that GR is a theory of gravity alone -- as far as it's concerned, the other forces of nature don't exist, so everything is always "in freefall under gravity" at all times.)

your definition of Lorentzian manifold is incorrect

Spacetime is represented by a Lorentzian manifold in special relativity as well

I thought that in special relativity, spacetime was represented by a specific Lorentzian manifold: Minkowski space itself. (Or, perhaps more precisely, an affine space over Minkowski space.) In other words, the manifold is required to be flat (have zero curvature). Whereas in general relativity, the curvature is determined by the equations of motion.

Special relativity is supposed to be what general relativity reduces to in the local limit: it's what goes on in the tangent space at a point. Right?

[Eliezer] says, for instance:

This meant you could never tell the difference between firing your rocket to accelerate through flat spacetime, and firing your rocket to stay in the same place in curved spacetime.

Coupled with his subsequent claim that epiphenomenal distinctions are, as a rule, illusory, he seems to be strongly suggesting that there is in fact no difference between these two states of affairs, which would imply that there is no objective fact of the matter about whether spacetime is flat or curved

It seems to me that you're mixing up the local and global structures of spacetime. There is no fact of the matter about whether spacetime is flat or curved locally, because many of the permissible coordinate changes turn straight lines into curves and vice-versa. However, there is a fact of the matter about the global curvature of the manifold.

Consider the twin paradox: the twin who leaves Earth has the right to say that he/she was at rest the whole time (thus traveling along a path that appeared locally "straight"), but must admit that the region of spacetime through which he/she traveled had nonzero global curvature. (Here, of course, we're assuming that the journey was caused by gravity rather than a rocket ship, in order for GR to be strictly applicable.)

My understanding (which you seem to confirm) is that "inertial motion" refers to geodesic paths. But those are precisely the paths which (the theory says) describe all motion! In other words, there's no such thing as "non-inertial motion". (Remember that GR is a theory of gravity alone -- as far as it's concerned, the other forces of nature don't exist, so everything is always "in freefall under gravity" at all times.)

Some worldlines satisfy the geodesic equation, others don't. The ones which do are geodesics. It's not true that GR cannot incorporate other forces of nature. It can, as long as these forces are amenable to a field-theoretic formulation. See here, for instance.

You're right! I'm sorry, I read your definition wrong the first time.

I thought that in special relativity, spacetime was represented by a specific Lorentzian manifold: Minkowski space itself. (Or, perhaps more precisely, an affine space over Minkowski space.) In other words, the manifold is required to be flat (have zero curvature). Whereas in general relativity, the curvature is determined by the equations of motion.

That's right. The point I was making is that representing spacetime as a Lorentzian manifold (even a Lorentzian manifold with arbitrary curvature) is insufficient (and unnecessary) to get "no privileged reference frame". What that requires is that the laws in the theory are formulated in such a way that they do not presume anything about the reference frame in which they hold. Incidentally, both special relativity and Newtonian mechanics can be re-formulated in this way, it's just that they usually are not. That a space-time theory is generally covariant does not express a constraint on its content, only on its formulation. On the other hand, whether or not the manifold is Lorentzian is a matter of content. See here.

It seems to me that you're mixing up the local and global structures of spacetime. There is no fact of the matter about whether spacetime is flat or curved locally, because many of the permissible coordinate changes turn straight lines into curves and vice-versa. However, there is a fact of the matter about the global curvature of the manifold.

I'm not sure what you mean by whether or not a spacetime is curved locally. If the Riemann curvature tensor at a point vanishes in one frame of reference, then it must vanish in every frame of reference. So one cannot go from non-zero to zero curvature tensor locally by a change of coordinates.

It is true that one can change coordinates so that the metric at a point is Minkowski. This is what people usually mean when they talk about using a locally flat coordinate frame. But the metric reducing to Minkowski at a point does not mean that the curvature at that point vanishes. Curvature has to do with the second derivatives of the metric.

Some worldlines satisfy the geodesic equation, others don't. The ones which do are geodesics. It's not true that GR cannot incorporate other forces of nature

This is a matter of terminology, but I would maintain that a physical theory is defined by its equations of motion, and that GR is defined by the Einstein equation(s); and since the other forces do not appear in the Einstein equation(s), they are not part of GR. (There is no question of being able to incorporate them; the theory either does or does not incorporate them.) If you're working with Maxwell's equations in curved spacetime, you're not working in GR; you're working in a hybrid of GR and Maxwell's theory.

Thus defined, GR itself does not (as far as I know) allow non-geodesic paths to be worldlines. (An Einstein-Maxwell hybrid theory, on the other hand, might; but I would be tempted to suspect in that case that it isn't formulated "properly".)

The point I was making is that representing spacetime as a Lorentzian manifold (even a Lorentzian manifold with arbitrary curvature) is insufficient (and unnecessary) to get "no privileged reference frame". What that requires is that the laws in the theory are formulated in such a way that they do not presume anything about the reference frame in which they hold.

But that is the same thing, only expressed in old-fashioned physicist's language instead of modern mathematician's language. If you translate "the laws in the theory are formulated in such a way that they do not presume anything about the reference frame in which they hold" into modern mathematician's language, what you get is "the equations of motion must be expressed in terms of objects which are well-defined on a manifold (of the appropriate type)".

It is true that one can change coordinates so that the metric at a point is Minkowski. This is what people usually mean when they talk about using a locally flat coordinate frame. But the metric reducing to Minkowski at a point does not mean that the curvature at that point vanishes. Curvature has to do with the second derivatives of the metric.

I could hardly have said it better myself. The ability to use a locally flat coordinate system at a point (regardless of the value of the Riemann tensor at that point) is all that

you could never tell the difference between firing your rocket to accelerate through flat spacetime, and firing your rocket to stay in the same place in curved spacetime.

means.

I could hardly have said it better myself. The ability to use a locally flat coordinate system at a point (regardless of the value of the Riemann tensor at that point) is all that

you could never tell the difference between firing your rocket to accelerate through flat spacetime, and firing your rocket to stay in the same place in curved spacetime.

means.

I don't see how it could mean that. Rockets are extended objects, as are people. While we can always find coordinates that make the metric Minkowski at a single point, it is not true that we can always find coordinates that make the metric Minkowski over a finite region, no matter how small.

While we can always find coordinates that make the metric Minkowski at a single point, it is not true that we can always find coordinates that make the metric Minkowski over a finite region, no matter how small.

Indeed; this is why the concept of an "inertial frame" does not exist in general relativity, except in the infinitesimal limit.

But as long as we're going to permit ourselves to speak about the motion of an entire rocket or person, rather than the motion of its parts (thus in effect modeling the object as a point-particle), we can equally well describe the same rocket or person as being at rest.

This leads to a thorough-going Poincare-esque instrumentalism, which is a consistent world-view but one that I find unattractive.

Why do you find it unattractive?

I don't think all instrumentally equivalent theories are epistemically equivalent. There are other considerations, like simplicity, that play a role in determining which theory we should regard as true. For instance, we could posit that the geometry of our universe is actually Euclidean, and apparent non-Euclidean measurements are accounted for by the fact that the size of our measuring sticks changes depending on where they are in space. This theory is instrumentally equivalent to the theory that our universe is non-Euclidean, but the choice between them is not merely conventional. The Euclidean account is less simple, and so inferior.