Birkhoff's Theorem and Alternate Gravity Theories ABSTRACT The problem of whether a metric theory of gravitation violates Birkhoff's theorem is reduced to whether the field equations possess conformally flat soltions of a particular type. Several classes of theories are analysed with this method. Birkhoff's theorem plays a fundamental role in relativistic theories of gravitation. If it is satisfied thenmonopole radiation is not predicted by the theory and the theory agrees with Newtonian theory in this sense. It is a fact that most theories violate Birkhoff's theorem; a fact which does not appear to concern many authors. In my opinion the violation of the theorem should call into question the viability of the theory at the outset and it should be the responsibility of the author to discuss this matter before suggesting a new theory. Birkhoff's theorem is normally difficult to test. The usual method involve seing whether the theory in question possesses solutions for the standard metric 2 2 2 2 2 ds = A(r,t)dr + r d - D(r,t)dt (1) where the three space coordinates have some of the characteristics as in flat space. In other coordinate systems these markers do not possess such a simple physical interpretation and can lead to spurious deductions (ref). Such a calculation which questions the existence of solutions of non-linear partial differential equations and the analysis can become tedious and often intractible. The method I shall now describe has the feature whereby it is fairly simple to apply for the most complex of theories. The starting point is a conflict which occurrred in this journal some years ago concerning the gravitation theory of Yang (ref). Thompson stated thayt Yang's theory satisfied Birkhoff's theorem (ref) and this was contradicted by Ni (ref). Ni asserted that Yang's equations possesses conformally flat solutions of the form 2 2 2 2 2 ds = A(p+t)(dp + p d - dt ) (2) and hence Birkhoff's theorem was violated. I argued that (2) did not imply solutions of the form of (1) in general and proceeded to give a solution which clearly violated Birkhoff's theorem (ref). The following theorem will clear up these issues: THEOREM All conformally flat spaces of the form 2 2 2 2 2 ds = A(t)(dp + p d - dt ) 3 where A(t) is differentiable of class C may be transformed into a metric of the form 2 2 2 2 2 ds = A(r,t)dr + r d - D(r,t)dt where the dependence upon t cannot be transformed away. This theorem suggests a fairly simple method for testing gravity theories. One looks at the field equations of a conformally flat metric with a time-like conformal factor. If the equations possess a solution then Birkhoff's theorem is violated. It should be noted that if the conformal factor is space-like one cannot make a sweeping statement about Birkhoff's theorem and this was the difficulty with Ni's analysis. I shall now mention several theories to which the theorem above may be applied. First, General Relativity in vacuum does not possess conformally flat solutions and the theorem does not apply; other methods have been used to look at this question (ref RP PRD15). Consider a Lagrangian field theory with a Lagrange denstiy L constructed from the metric tensor and its first and second derivatives. We may write, in complete generality, the Lagrangian in the form L = L(R, R ,R ). If we consider a conformally flat space then by definition we may replace the Lagrangian by an equivalent L*(R, R ).