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Vacuum Einstein metrics with bidimensional Killing leaves. I-Local aspects
The solutions of vacuum Einstein's field equations, for the class of
Riemannian metrics admitting a non Abelian bidimensional Lie algebra of Killing
fields, are explicitly described. They are parametrized either by solutions of
a transcendental equation (the tortoise equation), or by solutions of a linear
second order differential equation in two independent variables. Metrics,
corresponding to solutions of the tortoise equation, are characterized as those
that admit a 3-dimensional Lie algebra of Killing fields with bidimensional
leaves.Comment: LateX file, 33 pages, 2 figure
Belinfante Tensors Induced by Matter-Gravity Couplings
We show that any generally covariant coupling of matter fields to gravity
gives rise to a conserved, on-shell symmetric energy-momentum tensor equivalent
to the canonical energy-momentum tensor of the flat-space theory. For matter
fields minimally coupled to gravity our algorithm gives the conventional
Belinfante tensor. We establish that different matter-gravity couplings give
metric energy-momentum tensors differing by identically conserved tensors. We
prove that the metric energy-momentum tensor obtained from an arbitrary gravity
theory is on-shell equivalent to the canonical energy-momentum tensor of the
flat-space theory.Comment: 10 pages, LaTex; misprints corrected, references added; to appear in
Physical Review
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