A method is described for constructing, from any source‐free solution of Einstein's equations which possesses a Killing vector, a one‐parameter family of new solutions. The group properties of this transformation are discussed. A new formalism is given for treating space‐times having a Killing vector.
REFERENCES
1.
2.
J. Ehlers, in Les théories relativistes de la gravitation (CNRS, Paris, 1959).
3.
4.
E. T.
Newman
, L.
Tamburino
, and T.
Unti
, J. Math. Phys.
4
, 915
(1963
).5.
Since we are permitted by (3) to add a gradient to and these fields can always be chosen to satisfy (4). Note, however, that even (3) and (4) together do not determine the fields uniquely: There remains the freedom to add the gradient of any scalar field which is constant along the trajectories of While the addition of such a gradient does formally change the resulting metric (5), the change is of a trivial sort: It can be effected by a diffeomorphism on M.
6.
That is a consequence of (3), (4), and the following identity. If is any vector field and any skew tensor field, then
7.
Our conventions for the Riemann and Ricci tensors are
8.
We are ignoring a trivial transformation:
9.
Since K is compact, the curls do not, of course, contribute to the integrals. The second integral (19) is well known. The third appears to be new.
10.
11.
Note, however, that the transformation as originally given by (5) does not require any assumptions whatever on
12.
Note that we use Latin indices for both tensors on M and tensors on S. The reason for this will emerge shortly.
13.
See, for example, S. Kobayashi and K. Nomizu, Foundations of Differential Geometry (Interscience, New York, 1963).
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© 1971 The American Institute of Physics.
1971
The American Institute of Physics
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