Re: variation of appropriate degrees of freedom of metric

On Feb 15, 7:11 am, Tom Roberts <tjroberts...@xxxxxxxxxxxxx> wrote:
babalu...@xxxxxxxxx wrote:
In Carroll's text Einestein's equations are obtained from varying the
Hilbert action with respect to the full degrees of freedom of the
metric, i.e. the metric variation is not restricted to preserving
constant signature.

Maintaining constant signature is absolutely essential. But note the
variational technique is a CONTINUOUS technique, and both the manifold
and the metric are continuous as well. For any given metric, there is a
neighborhood in which the signature remains unchanged, and the variation
must occur within that neighborhood -- this is implicit in using such
variational techniques. IOW: the technique is valid only for SMALL
variations, and the definition of "small" must include not changing the
metric signature.

So is it correct to say that one cannot continuously change the
signature of a metric?

Is a signature change a discontinuous process like a parity inversion?

[...]
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