We study spherically symmetric solutions to the Einstein field equations
under the assumption that the space-time may possess an arbitrary number of
spatial dimensions. The general solution of Synge is extended to describe
systems of any dimension. Arbitrary dimension analogues of known four
dimensional solutions are also presented, derived using the above scheme.
Finally, we discuss the requirements for the existence of Birkhoff's theorems
in space-times of arbitrary dimension with or without matter fields present.
Cases are discussed where the assumptions of the theorem are considerably
weakened yet the theorem still holds. We also discuss where the weakening of
certain conditions may cause the theorem to fail.Comment: 14 pages with one fugure. Uses AMS fonts and Prog. Theor. Phys. style
files. Added section on neutron star and anisotropic fluid star as well as
Comments on Buchdahl's theorem and more analysis regarding the Birkhoff's
theorem. Accepted for publication in Prog. Theor. Phy