#### nrqed

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I am posting my question in this forum because it is about a basic conceptual aspect of LQG discussed in Rovelli's book Quantum Gravity.

He makes the following statement on page 67 (here, "e" refers to the vierbein):

I do not understand the part in boldface. First, he means that the *functional form* of [itex] e [/itex] and [itex] \tilde{e} [/itex] is the same, when he says that the two functions are equal, right? (which is different from saying [itex] e(x) = \tilde{e}(y(x))[/itex]).

If that's the case, then I don't follow the logic of the argument. First, I don't see in what way the relation with active diffeomorphisms plays a role....is he assuming that the theory is invariant under active diffeomorphisms? It seems to me that one only needs to use the freedom to make changes of coordinates to obtain the result.

A second question is:if we had a scalar function f instead of a one-form like e, then it seemes to me that we could not make the argument that we can always find a different coordinate system such that f and f' can be made equal. Am I missing something?

Thanks in advance

He makes the following statement on page 67 (here, "e" refers to the vierbein):

Now, if [itex]e[/itex] is a solution of the equations of motion, and if the equations of motion are generally covariant, then [itex] \tilde{e} [/itex] is also a solution of the equations of motion.This is because of the relation between active diffeomorphisms and changes of coordinates: we can always find two different coordinate systems on M, say x and y, such that the function [itex] e^I_{\mu} (x)[/itex] that represents [itex] e [/itex] in the coordinate system x is the same function as the function [itex]\tilde{e}^I{\mu}(y) [/itex] in the coordinate system y.Since the equations of motion are in the same in the two coordinates, the fact that this function satisfies The Einstein equations implies that [itex] e [/itex] as well as [itex] \tilde{e} [/itex] are physical solutions.

I do not understand the part in boldface. First, he means that the *functional form* of [itex] e [/itex] and [itex] \tilde{e} [/itex] is the same, when he says that the two functions are equal, right? (which is different from saying [itex] e(x) = \tilde{e}(y(x))[/itex]).

If that's the case, then I don't follow the logic of the argument. First, I don't see in what way the relation with active diffeomorphisms plays a role....is he assuming that the theory is invariant under active diffeomorphisms? It seems to me that one only needs to use the freedom to make changes of coordinates to obtain the result.

A second question is:if we had a scalar function f instead of a one-form like e, then it seemes to me that we could not make the argument that we can always find a different coordinate system such that f and f' can be made equal. Am I missing something?

Thanks in advance

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