spacetime - Meaning of general covariance

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?

Answer

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.