- #71
PeterDonis
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WannabeNewton said:First just to respond to your previous post, I meant the full text. I vaguely recall from some other thread that you said you had the text?
No, unfortunately I don't, I only have the online lecture notes.
WannabeNewton said:That isn't what he means although his wording is terrible. What he is saying is that if M and N are diffeomorphic and any properties regarding the smooth structure of M hold true for M then they also hold true for N.
This makes sense, but then what's the difference between M and N? He puts all the additional structure (metric, fields, etc.) into other objects, not M or N.
WannabeNewton said:The concept of integral curves do not require any kind of Riemannian structure.
Agreed. I wasn't really thinking about that when I read the notes, but you're right, everything he says in this passage is valid without any metric.
WannabeNewton said:And for that final point, yes the derivation of the local conservation of energy comes as a consequence of the invariance of the matter field action under diffeomorphisms. It again uses the concept of infinitesimal diffeomorphisms and the lie derivative of the metric tensor under the associated flows generated. The same argument can be used on the Hilbert action to derive the contracted Bianchi identity independent of the field equations.
Ok, so basically, the LHS of the EFE has zero covariant divergence because of the Bianchi identities; the RHS of the EFE has zero covariant divergence because of diffeomorphism invariance; and these items serve as a sanity check on the EFE itself.