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The principle of general covariance has been explicitly described by J.-M. Souriau in his 1974 paper "Modèle de particule à spin dans le champ électromagnétique et gravitationnel"

http://www.jmsouriau.com/Publications/JMSouriau-ModPartSpin1974.pdf

It is in french, I don't know if there exists an english translation?

You can find a related paper in english, by Shlomo Sternberg, here:

Hope that helps.

Comment: The principle of general covariance gives you the so-called passive field equations, that is for example ${\rm div}(T) = 0$, that is e.g. $\nabla_\mu T^{\mu\nu} = 0$ (or the equations of geodesics, or more complex equations if you input more fields or data).

It works essentially this way. Let us say that a geometrical object is a space with a natural action of the group ${\rm Diff}(M)$, the diffeomorphisms of $M$. For example the space ${\frak M}$ of metrics with signature $(+,-,-,-)$, and ${\rm Diff}(M)$ acts by pullback, $(\varphi,g) \mapsto \varphi^*(g)$, for $\varphi \in {\rm Diff}(M)$ and $g \in {\frak M}$. Now, the principle of general covariance (its active interpretation, not with charts or frames) says that any physical object (submitted to the field $g$) belongs to the quotient ${\frak Q} = {\frak M}/{\rm Diff}(M)$. Actually it is not completely exact, we must restrict the group to ${\rm Diff}_\bullet(M)$, the group of compact supported diffeomorphisms.

Let $g \in {\frak M}$ and $\gamma = [g] \in {\frak Q}$, the "tangent space" at $\gamma$ identifies to the tangent space at $g \in {\frak M}$ (that is the space of any symmetric tensor field on $M$) modulo the tangent space to the orbit of $g$, but the tangent space to the orbit of $g$ identifies with the space of Lie derivative of $g$ by compact supported fields.

Example: Let us look for a "covector" of $\frak Q$ at the point $\gamma$, and let us assume that it is given by a smooth distribution $T^{\mu\nu}$ of contravariant symmetric tensor on $M$, according to $$\tau(\delta g) = \int_M T^{\mu\nu}\delta g_{\mu\nu}$$ This is a linear form defined on the compact supported tensor fields $\delta g$ on $M$. But to be defined on $T_\gamma \frak Q$, $\tau$ must satisfy the (Eulerian) condition $$\tau(\epsilon) = \int_M T^{\mu\nu}\epsilon_{\mu\nu} = 0 \quad \mbox{for all} \quad \epsilon = {\frak L}_\xi(g)$$ with $\xi$, any compact supported vector field. And then, you can check that this is equivalent to ${\rm div}(T) = 0$.

This may give you a taste of what contains the Souriau's paper above. So, the principle of general covariance is just a principle of invariance with respect to the action of the diffeomorphisms with compact support. It is possible to give a very precise meaning to all these heuristic considerations. It has still not been done completely.

2 Add an example from Souriau's paper; added 1 characters in body

Comment: The principle of general covariance gives you the so-called passive field equations, that is for example ${\rm div}(T) = 0$, that is $\nabla_\mu T^{\mu\nu} = 0$ (or the equations of geodesics, or more complex equations if you input more fields or data).

It works essentially this way. Let us say that a geometrical object is a space with a natural action of the group ${\rm Diff}(M)$, the diffeomorphisms of $M$. For example the space ${\frak M}$ of metrics with signature $(+,-,-,-)$, and ${\rm Diff}(M)$ acts by pullback, $(\varphi,g) \mapsto \varphi^*(g)$, for $\varphi \in {\rm Diff}(M)$ and $g \in {\frak M}$. Now, the principle of general covariance (its active interpretation, not with charts or frames) says that any physical object (submitted to the field $g$) belongs to the quotient ${\frak Q} = {\frak M}/{\rm Diff}(M)$. Actually it is not completely exact, we must restrict the group to ${\rm Diff}_\bullet(M)$, the group of compact supported diffeomorphisms.

Let $g \in {\frak M}$ and $\gamma = [g] \in {\frak Q}$, the "tangent space" at $\gamma$ identifies to the tangent space at $g \in {\frak M}$ (that is the space of any symmetric tensor field on $M$) modulo the tangent space to the orbit of $g$, but the tangent space to the orbit of $g$ identifies with the space of Lie derivative of $g$ by compact supported fields.

Example: Let us look for a "covector" of $\frak Q$ at the point $\gamma$, and let us assume that it is given by a smooth distribution $T^{\mu\nu}$ of contravariant symmetric tensor on $M$, according to $$\tau(\delta g) = \int_M T^{\mu\nu}\delta g_{\mu\nu}$$ This is a linear form defined on the compact supported tensor fields $\delta g$ on $M$. But to be defined on $T_\gamma \frak Q$, $\tau$ must satisfy the (Eulerian) condition $$\tau(\epsilon) = \int_M T^{\mu\nu}\epsilon_{\mu\nu} = 0 \quad \mbox{for all} \quad\epsilon = {\frak L}_\xi(g)$$with $\xi$, any compact supported vector field. And then, you can check that this is equivalent to ${\rm div}(T) = 0$.

This may give you a taste of what contains the Souriau's paper above. So, the principle of general covariance is just a principle of invariance with respect to the action of the diffeomorphisms with compact support. It is possible to give a very precise meaning to all these heuristic considerations. It has still not been done completely.

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The principle of general covariance has been explicitly described by J.-M. Souriau in his 1974 paper "Modèle de particule à spin dans le champ électromagnétique et gravitationnel"

http://www.jmsouriau.com/Publications/JMSouriau-ModPartSpin1974.pdf

It is in french, I don't know if there exists an english translation?

You can find a related paper in english, by Shlomo Sternberg, here: