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 ).