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This is a continuation of my previous thread, but I figure it's cleaner to start afresh with this topic.

I'm trying to understand why GR has only two polarizations. I've only seen treatments of this in linearized terms, so I'll start there. The reasoning is often as follows (e.g. in Carroll's notes):

You can impose harmonic gauge conditions

[tex]

\square x^\mu = 0 \Leftrightarrow \partial_\nu \bar{h}^{\mu\nu} = 0

[/tex]

which costs you four degrees of freedom. However, since this does not completely specify your coordinates, you can still perform transformations of the form

[tex]

x^\mu \to x^\mu + \delta^\mu : \square \delta^\mu = 0

[/tex]

which tells you you lose an additional four degrees of freedom.

I find this highly suspect. The harmonic coordinate conditions do not uniquely determine your coordinates, so you've not yet specified anything, only constrained the set of coordinate systems you want to use. Then the second equation actually specifies a unique set of coordinates within this restricted set. It seems to me that only after doingbothhave you actually used up your four degrees of freedom.

What am I misunderstanding here?

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# Degrees of Freedom in Linearized Gravity (and beyond)

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