The Principle of General Covariance

[Metric]

Partly to get my mind off some recent absurd arguments generated by race-baiting, I present here an introduction to the principle of "general covariance" -- one of the most fundamental and commonly misunderstood principles of nature.

"General covariance" is a pretty controversial topic in physics, and differing views of covariance have for example been responsible for the rift between superstring theory and loop quantum gravity. Below, I'll give a bit of an introduction to what covariance is about.

One meaning of covariance has to do with Lorentz transformations. A theory is said to be covariant if the equations and predictions transform in a well-defined way under Lorentz transformations (moving to a new frame of reference in relative motion with the first one). Electromagnetism, the standard model of particle physics, etc. is all "covariant" in this sense.

But there is a deeper principle of "general covariance" associated with general relativity. It is this property which made the theory so difficult to understand -- both Einstein and the great mathematician Hilbert struggled independently for years to find the equations of general relativity, and although Hilbert had far greater mathematical ability, Einstein had a unique ability to see and understand the fundamental principles (though he himself had abandoned general covariance for some time in the face of predictions which he misinterpreted).

The principle is this: A generally covariant theory makes no distinction between "dependent" and "independent" observables. To see this, let's consider an example. Traditional electromagnetism is covariant in the special relativistic sense -- the theory knows how to handle Lorentz transformations. But it is not generally covariant. Why? The theory is designed to tell us, for example, the value of the electric field "E" at each spacetime point "x". To test the theory, I need a device that measures "E", but I also need a device to tell me where I am -- what point "x" we are talking about. Without both of these measurements, the theory cannot make predictions. Now, the equations of the theory determine "E" -- but they do not determine "x"! The spacetime points are assumed to live in the background, and are not effected by the dynamics of the theory. They simply exist, and can be measured in order to make predictions of electromagnetism possible. The measurable quantity "E" depends on "x", but the measurable quantity "x" does not depend on "E"!

General relativity, which posesses the property of general covariance, is quite different. Predictions of the theory are not the values of various fields at various spacetime points "x". If you try to make the theory work like this, predictions become non-unique (this is what initially confused Einstein and baffled Hilbert). Covariance is essential. In GR, to make predictions, we can only compare dynamical variables to other dynamical variables. Coordinates have no meaning -- they are used only as a mathematical book-keeping device. To make a prediction, we have to "solve away" dependence on coordinates.

For example, let's say that there are n dynamical variables F_1 ... F_n in the theory, and four space-time dimensions (x,t). The equations tell us how to solve for F_1(x,t)...F_n(x,t) -- but as stated before, predictions of the theory cannot refer to the coordinates, which are arbitrary and not observable. Instead, we have to pick out four dynamical variables to serve as a "material reference frame" and solve away (x,t) in the following way:

F_1(x,t)...F_4(x,t) --> x(F_1...F_4), t(F_1...F_4)

Then, we can express the remaining n-4 variables in terms of the first four as follows:

F_m(F_1...F_4) = F_m(x(F_1...F_4),t(F_1...F_4)) (here m = 5...N)

THESE are functions which can be compared directly to experiment. As Carlo Rovelli has written, "The world is made up of fields. Physically, these do not live on spacetime. They live, so to say, on one another. No more fields on spacetime, just fields on fields."

It is this principle which makes quantum gravity hard -- the idea that predictions should be "background free" in some sense. As I mentioned before, loop quantum gravity theorists take a "hard line" approach to this principle, making it covariant from the very start -- while superstring theorists tend to do perturbation theory about a particular background (though they can still go from one background to another).

My personal feeling is that this is probably one of THE principles of nature. The idea of physically measurable fixed background structure which cannot be effected seems very unnatural to me.

[Borodog]

Very nice.

[Metric]

I suppose I will give this post a bump in the hopes of generating a question or comment before it is inevitably buried under the "meaning of life" stuff.

[bunny]

Well my comment is I enjoyed reading it. Unfortunately, I dont feel qualified to even ask an intelligent question. Perhaps one will come in time...

[cambraceres]

Is GR the only theoretical structure that does not need this "x" coordinate to make a "real" prediction?

What I mean is, in the first description of covariance, the one active in special relativity, there needs to be a spacetime measurement to have a valid prediction. In the second this is not the case, but is there any other artificial structure in science that follows this second line?

Sorry for the noobish question, but I'm noobish ya know?

Much Love

Cam

[Metric]

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Is GR the only theoretical structure that does not need this "x" coordinate to make a "real" prediction?

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One can construct other models or new theories that have this property, but GR is of course the one that people believe is a realistic theory. I should also point out that other field theories like electromagnetism etc. become generally covariant when combined with GR. So strictly speaking general covariance is not just a property of the gravitational field -- it is a property of all of physics the moment that gravity is included.

[ QUOTE ]
What I mean is, in the first description of covariance, the one active in special relativity, there needs to be a spacetime measurement to have a valid prediction. In the second this is not the case, but is there any other artificial structure in science that follows this second line?

[/ QUOTE ]
One can actually formulate ordinary mechanical theories etc. in a covariant way, but the essential point is that they don't really need to be formulated in this way for us to use them effectively. Their dynamics takes a particular form which effectively says "you can pretend that there is a background structure without any loss of generality." But with GR, we can't do this anymore -- it is a theory in which we are forced to come to terms with covariance in order for the theory to make any sense at all.

But to answer your question a little more directly, no, I can't think of another well established physical field theory that explicitly requires general covariance quite apart from being connected to GR (though of course many "cutting edge" theories want to include it due to the lesson that GR has taught us).

[cambraceres]

But would you consider one of these cutting edge theories to be viable?

Cam

[evil twin]

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Very nice.

[/ QUOTE ]

[Metric]

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But would you consider one of these cutting edge theories to be viable?

[/ QUOTE ]

Sure... Superstring theory and loop quantum gravity are two examples, though as I mentioned before they have a subtly different take on covariance. LQG is TOTALLY background free -- states are made up of abstract "spin networks" that sort of build up spacetime from nothingness (the downside is that it is extremely difficult to make predictions starting out this way). Superstring theory lives with a background spacetime, but the theory doesn't exactly care which background is used (within reason) -- it retains the freedom to go back and forth between different ones (this approach is easier to live with from the point of view of making predictions, but at the cost of living with an arbitrary background).

But anyway, these are both certainly viable and they are both desperately trying to make testable predictions, though without much success.

[Skidoo]

Bump to the present.

Well written post. It chimes in with my own general observation of how many absolute frameworks have fallen in recent years mostly due to unexamined assumptions about their empirical nature being discovered to be incorrect. Even space itself isn't a sure thing to hang your hat on anymore.