Exact Solutions and Scalar Fields in Gravity - Instituto Avanzado de ...

REFERENCES 13
be shown that the analog of the Newtonian result that straight lines parallel
to the axis of rotation have at most two points of intersection with
the bounding 2-surface (we may call this “vertical convexity” for short)
also holds for the Kramer solution. Obviously, we have to generalize
appropriately the concept of “straight line parallel to the axis”; we do
it in the obvious way, by considering geodesics which intersect the equatorial
plane orthogonally and possess a constant azimuthal coordinate.
Once this is done, vertical convexity can be proven, by looking at how
the pressure varies along such geodesics, and making use of the remarkable
fact that the pressure is harmonic (for a 2-dimensional flat-space
Laplacian) in the Kramer solution.
Acknowledgments
It is for me an honor and a pleasure to participate in the celebration
for Prof. Dehnen and Prof. Kramer. I thank Dr. T. Matos for
his invitation to do so. The work described here has been or is currently
supported by Projects PB89-0142, PB92-0183, PB95-0371, and
PB98-0772 of Dirección General de Enseñanza Superior e Investigatión
Científica, Spain. I am indebted to other members of some of those
Projects (L. Fernández-Jambrina, L.M. González-Romero, C. Hoenselaers,
F. Navarro-Lérida, and M.J. Pareja) for discussions.
References
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[14] F.J. Chinea and M.J. Pareja, Class. Quantum Grav. 16 (1999) 3823.