- #1

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## Main Question or Discussion Point

Hello all,

I've been puzzled by this problem for some time now and was wondering if anyone here could help me out. Textbooks on GR (specifically when going into gravitational waves) tend not to elucidate this. It's often taken for granted that through the gauge diffeomorphism invariance (or general covariance), you can choose a 'gauge' by choosing coordinates on your manifold and that this choice deprives your metric of four degrees of freedom.

I don't quite understand this. A metric is just a set of ten functions

[tex] g^{\mu\nu} : M \to \mathbb{R} [/tex]

where [itex]M[/itex] is your manifold. Choosing coordinates amounts to choosing four more functions

[tex] x^{\mu} : M \to \mathbb{R} [/tex]

so that the [itex]x^{\mu}[/itex] label the points on your manifold. Having labelled those points, we can work with our coordinates instead of with the abstract manifold:

[tex] g^{\mu\nu} : \{x\} \to \mathbb{R} [/tex]

How does doing this, i.e. just labelling your points, constrain the components of your metric?

I've been puzzled by this problem for some time now and was wondering if anyone here could help me out. Textbooks on GR (specifically when going into gravitational waves) tend not to elucidate this. It's often taken for granted that through the gauge diffeomorphism invariance (or general covariance), you can choose a 'gauge' by choosing coordinates on your manifold and that this choice deprives your metric of four degrees of freedom.

I don't quite understand this. A metric is just a set of ten functions

[tex] g^{\mu\nu} : M \to \mathbb{R} [/tex]

where [itex]M[/itex] is your manifold. Choosing coordinates amounts to choosing four more functions

[tex] x^{\mu} : M \to \mathbb{R} [/tex]

so that the [itex]x^{\mu}[/itex] label the points on your manifold. Having labelled those points, we can work with our coordinates instead of with the abstract manifold:

[tex] g^{\mu\nu} : \{x\} \to \mathbb{R} [/tex]

How does doing this, i.e. just labelling your points, constrain the components of your metric?