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Quoting from Wald's GR:

In the context of special relativity, the principle of general covariance states that the spacetime metric $\eta_{ab}$, is the only quantity pertaining to spacetime structure which can appear in any physical law.

What other quantities "pertaining to spacetime structure" are there, besides the metric? What would be an example of a (false) law or equation that would violate this principle?

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  • $\begingroup$ And what about the Levi-Civita alternate symbol $\varepsilon_{abcd}$ ? This is another geometric structure that may appear in physical equations. $\endgroup$ – Cham Dec 23 '18 at 1:25
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Wald is a first rate relativist, and as such he is phrasing the concept of general covariance in terms of purely geometrical quantities, rather than resorting to the somewhat imprecise notion of coordinate transformations. In the discussion on pg. 57, he goes on to give an example of what it means to violate the principle of general covariance.

In his example, he supposes that you have, in addition to the metric, a preferred vector field $v^a$. This vector field defines a preferred direction in spacetime, and hence encodes additional geometric structure. You could think of $v^a$ as a type of aether. Then, in this theory there would be a preferred coordinate system in which the vector $v^a$ has components $v^\mu = (1,0,...,0)$. The resulting theory would therefore not be generally covariant.

To take this example a little further, we can consider a scalar field. An action you could write is $$S = \frac12\int(\eta^{ab}\partial_a\phi\partial_b\phi - \alpha (v^a\partial_a\phi)^2) $$ and the equation of motion is then $$\eta^{ab}\partial_a\partial_b\phi - \alpha v^b\partial_b(v^a\partial_a\phi) = 0,$$ In a more transparent form, assume $\partial_b v^a = 0$, and let $v^a$ be pointing in the time direction. Then this equation becomes $$(1-\alpha)\partial_t^2\phi - \nabla^2\phi=0,$$ i.e. a wave equation with a speed $c_s^2 = (1-\alpha)^{-1}$. However, this speed could easily be greater than $1$ by letting $\alpha$ be positive, so this clearly violates special relativity. The reason this happened is because we included $v^a$, a geometrical structure beside the flat metric $\eta_{ab}$.

So in general, any fixed vector or tensor field can act as an additional geometrical structure. You can also do other things like having a fixed derivative operator $\nabla_a$, which means you would then be able to write Christoffel symbols $\Gamma^a_{bc}$ explicitly in equations.

The point of making the distinction from coordinate invariance is that it is sort of empty to say that an equation is valid in all coordinate systems. This is because if I have an equation, and then I change coordinates, I still get the same equation, but just in a different coordinate system. The way to write the equation in a coordinate invariant way is to identify all the additional geometrical structures in the theory (i.e. $v^a$ as well as $\eta_{ab}$ in our above example). If, after doing this, the only geometrical structure needed to write the equations in a generally covariant was was the flat metric $\eta_{ab}$, we say that the theory is a generally covariant special relativistic theory.

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  • $\begingroup$ Helpful example. Now, this follow-up question is probably much beyond my current level, but out of curiosity I'll ask it anyway: is there currently a proof of the claim that a theory is generally covariant if (and only if?) the only geometric entity appearing in the equations is the metric? $\endgroup$ – yjc Nov 25 '14 at 7:44
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General covariance basically means you can change your coordinate system arbitrarily and express the laws of physics in the new coordinates. Because of this freedom, the relationship between coordinate distances, angles, etc. and physical distances, angles, etc. is variable and is expressed by the metric.

So the quoted statement is basically saying that you can't take your coordinates to mean anything physically, and laws of physics shouldn't be formulated in a way that requires you to use a particular coordinate system. You always have to translate coordinates to physical values through the metric.

For example, you can't just use the Euclidean distance formula $\sqrt{x^2+y^2+z^2}$ for distances between points when calculating a force or some such, as that's only valid in flat space in Cartesian coordinates.

The other side of this, and the deeper physical meaning, is the equivalence principle, which states that acceleration is equivalent to a gravitational field. This can be stated as saying that the laws of physics can't "know" anything about spacetime beyond what's in the metric. An accelerating reference frame and a gravitational field give the same metric locally, so they're completely equivalent for all physics purposes. The metric doesn't know the difference, so the laws of physics don't know the difference.

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