Diffeomorphism Invariance in Einstein's Gravitation Theory Explained

[eVulcanon]

Why is the Hilbert Action and the matter actionof the Einstein's Gravitation theory diffeomorphism invariant, as Wald said in his textbook General Relativity on Page 456 and Sean Carroll said in his Spacetime and Geometry on Page 435.

In other words，why do we have to set
$$\delta S_{M}$$ to be zero?

Thx!

 

[eVulcanon]

Nobody knows the reason？

 

[samalkhaiat]

Because it is a scalar.

 

[eVulcanon]

Because it is a scalar.

Is any scalar diffeomorphism invariant?

 

[xepma]

Diffeomorphism invariance (or better: covariance) is the statement that fundamental laws do not depend on the coordinate system chosen. This is, in some sense, quite natural to assume.

By this assumption it also very reasonable to use a mathematical language in which such a co-/invariance is manifest. This is why General Relativity is completely based upon differential geometry. Physical quantities are described by tensors, which are, in some sense by definition, diffeormophism invariant.

A scalar is just a special version of a tensor (the simplest one). Since we assume that our physical laws are diffeormorphism invariant, and all these laws are in one way or the other connected to the action, the action itself better be invariant as well.