General covariance

(Not to be confuse with my reflection general covariance and contravariance)

From a purely mathematical perspective, the idea of invariance under coordinate transformations (diffeomorphisms) seems almost self-evident. However, the historical and conceptual development in physics, especially within general relativity, made "general covariance" a crucial and sometimes contentious principle. Here's a deeper look at why it became a named concept:

Newtonian Physics and Special Relativity:
- Classical Newtonian physics often relied on absolute space and time, implicitly favoring certain coordinate systems.
- Even special relativity, while a significant step towards relativity, still often used inertial frames, which are a specific class of coordinate systems.
- Therefore, the idea that any coordinate system should be equally valid was a radical departure.
The Struggle with General Relativity:
- Einstein's journey to formulating the Einstein field equations was long and arduous. He initially struggled with the concept of general covariance, even temporarily abandoning it.
- He initially thought that general covariance would lead to too many solutions, making the theory physically meaningless.
- It took considerable effort and the insights of mathematicians like Marcel Grossmann to realize the importance and necessity of general covariance for a consistent theory of gravity.
The Hole Argument:
- Einstein's "hole argument" was a famous thought experiment that challenged the physical significance of generally covariant theories. It seemed to suggest that general covariance could lead to indeterminism.
- The resolution of the hole argument clarified the interpretation of general covariance and the role of spacetime points as events rather than fixed locations (would parenthood exist if there were no fathers and children?).
Emphasis on the Manifold vs. Coordinates:
- Physicists, especially in the early 20th century, often worked more with coordinate-based calculations than with the abstract concept of a manifold.
- General covariance shifted the focus from specific coordinate representations to the intrinsic properties of spacetime, which are independent of coordinates. This conceptual shift was significant.
Distinguishing True Physics from Coordinate Artifacts:
- General covariance serves as a powerful tool for distinguishing genuine physical effects from coordinate artifacts. If a phenomenon disappears under a coordinate transformation, it's not a real physical effect.
- This is a very powerful check to make sure the physics is correct.

In essence, while the mathematical idea of diffeomorphism invariance might seem obvious, its application and interpretation in physics, particularly in the context of gravity, required a significant conceptual leap and careful analysis. The term "general covariance" became a shorthand for this crucial principle and its implications.

My approach

Definition

Theory: It is a way to select some fields. Given a mapping $f : M \to R$ from the set $F = Maps (M, R)$ , we aim to select a subset $S \subseteq F$ by means of any criteria. This are the fields of our theory.

Example. Suppose a coordinate system $x : M \to R$ . For $f \in F$ , $f$ is selected if $f_{1} := f \circ x^{- 1} : R \to R$ satisfies the condition:

\frac{d f_{1}}{d x} = f_{1}

With this setup, this theory passive general covariant, since given a new coordinate $y : M \to R$ related to $x$ by a smooth invertible function $x = ϕ (y)$ , in the new coordinate $y$ , the field $f$ is expressed as $f_{2} = f \circ y^{- 1}$ . The relationship between $f_{1}$ and $f_{2}$ is:

f_{2} (y) = f_{1} (ϕ (y)) .

Now, the original selection condition in the $x$ -coordinate is $\frac{d f_{1}}{d x} = f_{1}$ . We need to express this condition in the $y$ -coordinate. Using the chain rule:

\frac{d f_{2}}{d y} = \frac{d f_{1}}{d x} \cdot \frac{d ϕ}{d y} = f_{1} (ϕ (y)) \cdot ϕ^{'} (y) = f_{2} (y) \cdot ϕ^{'} (y) .

Thus, in the $y$ -coordinate, the condition becomes:

\frac{d f_{2}}{d y} = f_{2} (y) \cdot ϕ^{'} (y) .

The theory is not passive generally covariant.
If we focus on coordinates $y$ satisfying $x = ϕ (y) = y + k$ , with $k \in R$ , then the criteria still works, so we can say that this theory is passive translation covariant.