Principle of general covariance follows from principle of equivalence?

I am reading the book gravitation from Weinberg. On page 93 he states the principle of covariance. I attach a screenshot below. I think Weinberg is saying (below the enumeration) that the principle of covariance implies the principle of equivalence. I understand that. But I feel like that the principle of equivalence does not imply the principle of covariance. Would you agree to that? Is the principle of covariance more general than the principle of equivalence? If yes then I don't understand why one uses the principle of covariance all the time, since the only thing that I can imagine being obviously true is the principle of equivalence.

• What a fantastic book this is! I don't quite understand your question - Weinberg is explaining how covariance follows from equivalence. Do you object to his argument? Commented Mar 30, 2023 at 7:53
• He's explaining how equivalence follows from the principle of covariance. I think it should be the other way around. But I think this is not obvious Commented Mar 30, 2023 at 16:36
• I think you need to reread - there is no mistake, Weinberg is explaining how covariance follows from equivalence. Commented Mar 30, 2023 at 17:06
• I agree that this took me a bit aback on first reading and I've had to delete and repost this comment thinking about it! The way it is worded makes it seem almost(!) as if Weinberg is saying that the principle of covariance (POC) is defined by conditions (1) and (2). But that is not what he is saying - instead POC is defined (for him) to be the property that whenever (1) and (2) hold then the prospective equation is true. So indeed, he wants to begin by supposing a) equivalence and b) conditions (1) and (2) and then(!) show that the principle of covariance must hold. I hope this makes sense. Commented Mar 30, 2023 at 17:06

I do think at first reading this can be a little confusing, so let me rephrase Weinberg's argument with the slight possibility that it will make things a little clearer.

First, let us trivially rephrase the definition of general covariance:

Definition. Suppose that we are given an equation such that

1. The equation holds in the absence of gravity, i.e. in a frame where $$g_{\alpha\beta} = \eta_{\alpha\beta}$$ and $$\Gamma^\alpha_{\beta\gamma} = 0$$.
2. The equation is generally covariant, that is, it preserves its form under a general coordinate transformation $$x\rightarrow x'$$.

Then the principle of general covariance states that such an equation holds generally.*

Note the difference between general covariance of an equation, and the principle of general covariance. It is possible that this is what may have you confused.

Weinberg states the following:

Claim. The principle of equivalence implies the principle of general covariance.

Proof (paraphrasing Weinberg). Suppose the principle of equivalence holds, as do conditions (1) and (2) above with respect to some physical equation. We have to show that this equation holds generally.

Because of the principle of equivalence, we know that there is an inertial frame in which gravitation is (locally) absent. Perform a coordinate change to this frame, which by (2) preserves the form of the prospective equation. Then by (1) the equation is true in this frame. And hence, by equivalence and (2), the equation is true in all frames.

*Hopefully you spot an enormous danger in this definition! As Weinberg states at the end of this section, in general there are many generally covariant equations which match onto a given equation in the absence of gravity - you can always add on terms involving higher derivatives of the metric $$g$$. So secretly Weinberg needs equations to only involve $$g$$ and its first derivatives (even then uniqueness is not completely obvious). Physically, this is also related to what he talks about in the penultimate paragraph of the section, about general covariance only holding locally (in most cases infinitesimally). If I remember correctly, in his book Wald includes this limitation on derivatives of $$g$$ in his definition of general covariance, which can seem rather opaque if you don't realise why.

In reality, one can add these terms and indeed they are expected to exist. They are the reason why you can't $$`$$prove' the Einstein field equations from the principle of equivalence - you have to include an extra simplicity assumption. Similarly for electromagnetic equations etc. In an effective field theory of gravity, higher order terms are included naturally. So it is ultimately up to experiments to pick which possible general covariant equation is correct. Luckily most of us don't live near black holes, so these higher order gradients are very small and we can safely ignore them. And when these gradients are important we probably need a quantised theory anyway.

• Thanks for your answer! However I still feel like that the POC is more general than the principle of equivalence (POE). The POE just says that we can find a coordinate system in which the effects of gravity are absent. The POC adds that the equation should have the same form in all coordinate systems. Commented Mar 30, 2023 at 17:27
• Only if it satisfies conditions (1) and (2)! You are right but only in the sense that POE does not mean that all physical equations (apriori at least) must satisfy conditions (1) and (2). But this does not contradict the fact that the POC does indeed follow from the POE, as Weinberg argues. The reason for introducing the POC is that in practice, conditions (1) and (2) are easy to check, making it useful for developing the subject. Commented Mar 30, 2023 at 17:38
• I have a feeling that one probably can reverse the argument (POC implies POE), perhaps with some subtleties. But I haven't really thought about it. And it's not really essential to the development of the subject. Commented Mar 30, 2023 at 17:40
• You may also like to consult the other excellent book on GR by Wald, for a complementary perspective. Commented Mar 30, 2023 at 17:40
• Thanks! I'll look more into the book that you recommended. In general I don't see why the equations should be invariant under general coordinate transformations. How does this follow from the POE? As an example I tried to derive the Maxwell equations on curved space time from the POE. I want to see that this agrees with the equations that one gets from the POC. Until now I haven't been successful ... Commented Mar 30, 2023 at 17:55