variation of appropriate degrees of freedom of metric

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. Assuming that constant signature underlies the
Bianchi identity, requiring that the energy-momentum tensor be
conserved preserves the Bianchi identity in the final result and maybe
saves us from having to vary with respect to the appropriate degrees
of freedom of the metric. I would like to know if the above thoughts
are correct.

Particularly, if I try to explore what happens if the matter term in
the
action is ignored and vary with the full (inappropriate) degrees of
freedom of
the metric, I obtain solutions that violate the Bianchi identity.
Imagine a
boundary condition on G_mu_nu such that it is non-zero in some regions
of a
spacelike hypersurface then the equation resulting from this
variation: G_mu_nu=0 outside the boundary would mean the G is not
conserved. This violation seems to me more a result of varying with
respect to inappropriate degrees of freedom of metric than of ignoring
the matter term in the action.
Is this correct?

Thank you for any guidance on this.
.

Relevant Pages

  • Re: variation of appropriate degrees of freedom of metric
    ... Hilbert action with respect to the full degrees of freedom of the ... i.e. the metric variation is not restricted to preserving ... Bianchi identity, requiring that the energy-momentum tensor be ... I obtain solutions that violate the Bianchi identity. ...
    (sci.physics.relativity)
  • Re: Meet Melissa Arrington
    ... >> Henry is a fool to whom I pay no attention. ... Or there are varying degrees of freedom, ...
    (rec.bicycles.racing)
  • Re: Heinlein travelogue
    ... That would be terribly hypocritical if what we called "freedom" was, ... It does not include the ability to violate the rights of others. ...
    (rec.arts.sf.written)
  • Re: rainbew torch? was/Re: roadrunner very sick
    ... individual freewill. ... it who violate others' rights, their freedom and free will. ... flunkies) it seems that speaking out about the violation of First Amendment rights is equivalent to a dread disease. ...
    (alt.gathering.rainbow)
  • Holly war, energy crisis, mid-east crisis, followed by humanity crisis?
    ... Archival and selling of message-boards violate intimacy. ... But what happens when all books are stored, ... violated, by Google, the all so Romanian land of ignoring ... freedom and the wester spirit. ...
    (alt.politics.bush)