We show succinctly that all metric theories with second order field equations
obey Birkhoff's theorem: their spherically symmetric solutions are static.Comment: Submitted to CQ

Charmousis C

Deser S

J Franklin

S Deser

Zegers R

English

arXiv

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Birkhoff for Lovelock redux

We show succinctly that all metric theories with second order field equations obey Birkhoff's theorem: their spherically symmetric solutions are static

Deser, S.

Franklin, J.

Caltech Authors - Main

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BRX TH-565
Birkhoff for Lovelock Redux
S. Deser1 and J. Franklin2
1Department of Physics, Brandeis University
Waltham, MA 02454, USA
deser@brandeis.edu
2 MIT Kavli Institute for Astrophysics and Space Research
Cambridge, MA 02139, USA
jfrankli@mit.edu
Abstract
We show succinctly that all metric theories with second order field equations obey Birkhoff’s
theorem: their spherically symmetric solutions are static.
The special, “Lovelock”, metric theories with second derivative order field equations (relevant
1
details, and the original references, may be found in [1], whose notation we use.) obey Birkhoff’s
theorem: their spherically symmetric solutions are static [2, 3]. In this note, we provide an alterna-
tive derivation of the theorem, using the approach of [4], where historical references-and justification
of our shortcuts-are given. We also show that the apparent counterexamples to Birkhoff in tuned
combinations of Lovelock actions found in [2, 3] are merely deSitter vacua, such as can always be
introduced by tuning any action polynomial in curvatures.
We will focus on the single relevant, (0r), “radial energy flux” field equation, using the metric
ansatz
ds2 = −ab2dt2 + a−1dr2 + r2dnΩ+ 2fbdrdt (1)
where (a, b, f) depend on (r, t) and dnΩ is the unit n-sphere interval. As we shall see, the seemingly
superfluous f ∼ g0r component’s variation provides precisely the required field equation. The
Lovelock actions of order k in D = n+ 2 dimensions are defined as
Ik =
∫
dDx Lk ≡
∫
dDx(−g)−1/2ǫµ1...ǫν1...Rµ1µ2.. . . . Rνk−1νk.. g.. . . . g.. (2)
There are k curvatures, whose 4k indices contract with the 2D ǫ-indices; the indicated metrics
soak up the rest. Lk vanishes for D < 2k and Ik is a topological invariant at D = 2k. The
2
Palatini, δR ∼ DDδg, and Bianchi, Rµ[ναβ;λ] ≡ 0, identities guarantee that only the variations of
the explicit metrics (but not those of the R’s) survive, leading to field equations Gµν = 0 algebraic
of order k in the curvatures but without explicit derivatives. The leading, k = 0, 1, 2 actions are the
cosmological, GR and Gauss–Bonnet terms. The form of every Ik[a, b], upon inserting the above
metric, first at f = 0, is [1]
Ik[a, b] ∼
∫
drb[rn+1ψk]′, r2ψ = (1− a) (3)
where primes denote radial- (and dots will mean time-) derivatives. Note first that, even though we
have not assumed time independence of (a, b), only spatial derivatives (that there is only one prime,
rather than two, is peculiar to Schwarzschild gauge.) survive [5]. Further, (3) will be unaffected by
subsequent inclusion of f , since the latter will only be used as a Lagrange multiplier to deliver the
(0r) equation, then set to zero, to recover the 2-function (f = 0) metric. We learn from varying (3)
with respect to a that b = b(t), which means it can be removed entirely by using the residual time-
gauge choice b(t)dt = dt˜. [Thence our unusual, but legitimate, parametrization of ds2.] Varying
with respect to b fixes a(r, t), e.g. a = 1 − 2m(t)/rn−1 in GR; the notation emphasizes that m’s
3
time-dependence is as yet undetermined. For that, we need the final, f -field equation,
δIk[a, f ]
δf
∣∣∣∣
f=0
≡ Fk−1(a)a˙ = 0 (4)
where we can use the now known (a, b) to remove b altogether and as explained, f has also been
set to 0 after variation. We have-legitimately-anticipated the form of this equation, simply because
it must have exactly one time derivative and only depends on a(r, t). So Birkhoff is proved, unless
Fk−1(a) = 0, which is certainly impossible for any one k or we would have lost an entire field
equation! We will show below that this is also the case for all linear combinations,
∑
k ck Ik. So
the only work required is to justify that (4) follows from (2) at b = 1, f = 0. In varying the
action to get its (0r) part, we need not keep the explicit (−g)−1/2, since δln(−g) = gµνδgµν . The
(D − 2k) gµν factors of (2) all vary identically, so we find
δIk
δg0r
≡ G0r ∼ ǫ0...ǫr...g . . . g R . . . R
∣∣∣∣
f=0
= 0. (5)
Because g0i = 0, the remaining gαβ must all be spatial (since only one “0” index is available); then
all curvatures but one are purely spatial, the remaining one being ∼ R0ijk.
For orientation, let us briefly consider (5) in Riemann normal coordinates: the Rijkℓ are
proportional to a and its radial derivative, while R0ijk ∼ (g˙ij,k− g˙ik,j) has a single ∂/∂t. Hence, we
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are morally assured that the “guess” (4) is correct, and indeed that Fk−1(a) is just the (k − 1)th
power of F1 = (1 − a), the Einstein value. The Appendix confirms this: all spatial curvature
components depend on (1− a) and (1− a)′.
Finally, we turn to generic Lovelock combinations whose actions are of the form [1]
I =
N∑
k=0
ckIk =
N∑
k=0
ck
∫
drb(rn+1ψk)′, r2ψ = (1− a) (6)
plus terms linear in f . Clearly, b = b(t) again and gets absorbed by a change of t while ψ obeys
the algebraic equation
∑
ckψ
k = m(t)r−(n+1). Finally, as predicted, the “Birkhoff equation”, is
(
N∑
k=1
dkψ
k−1
)
ψ˙ = 0; (7)
the dk differ from the original constants ck by constant numerical factors. As we noted, a cosmo-
logical term (k = 0) would not contribute explicitly. So ψ is time-constant unless the prefactor in
(7) vanishes. This vanishing was interpreted in [2, 3], in a different gauge, to imply that there are
Birkhoff-violating solutions. We now see instead that the theorem always holds, and that these
exceptional geometries are nothing but (anti) deSitter vacua that generalize the original flat one
by fine-tuning the ck: From (7), the zeros of the polynomial are its (necessarily time-constant)
roots, ψi = Ri, namely the constant curvature metrics ai = 1 + Rir
2. These deSitter spaces not
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only obey Birkhoff, but the effective cosmological constants can be written, upon inserting the ai
into the a-field equation, in terms of the ck coefficients, including (if any) the original c0 = Λ.
Their significance as additional vacua is understood as follows: Suitable choices of the constants
in any I =
∑
γkR
k, Lovelock or not, can be fine-tuned to allow constant curvature spaces for any
desired Λ. These vacua actually agree with the seemingly time-dependent spaces of [2, 3] upon
transforming the latter to Schwarzschild gauge, namely choosing the coefficient of dnΩ to be r
2.
Finally, this also shows that we have not lost a field equation here either-it has just “solved itself”
to yield deSitter!
Appendix
In this Appendix, we first provide the form of the Riemann tensor for our time-dependent metric
ansatz; it depends on five functions, reducing to [1] for a˙ = b˙ = 0:
Rµναβ = 4Aδ
0
[µδ
r
ν]δ
0
[αδ
r
β] + 4BZiδ
0
[µδ
θi
ν]δ
0
[αδ
θi
β] + 4CZiδ
r
[µδ
θi
ν]δ
r
[αδ
θi
β] + 2r
4ψZiZjδ
θi
[µδ
θj
ν] δ
θi
[αδ
θj
β]
+ 4DZi
(
δ0[µδ
θi
ν]δ
r
[αδ
θi
β] + δ
r
[µδ
θi
ν]δ
0
[αδ
θi
β]
)
, (8)
6
with
A =
b
2
[
3a′b′ + a′′b+ 2ab′′
]
+
1
2a2
[
− a˙b˙
b
+
aa¨− 2(a˙)2
a
]
(9)
B =
1
2
(rab)
[
a′b+ 2ab′
]
(10)
C = −a
′r
2a
, D = − a˙r
2a
, ψ =
1− a
r2
(11)
and Zi is the i
th angular component of the metric.
We are interested in relating G0r(k) and G0r(k−1) to exhibit their “polynomial in ψ” nature,
outlining the steps here. We find that effectively, the Riemann tensor appears in the field equation
in the combination
R˜αβγρ = 2ψgαβgγρ + 4Dr
−2gγρδ
r
αδ
0
β . (12)
The first term above is G0r(k−1) while the second is (in odd dimensions) k−1G0r(k) when replacing
the kth Riemann tensor in the field equation (5). After some algebra, this leads to
G0r(k) = 2ψG0r(k − 1) + k−1G0r(k)→ G0r(k) = 2kψ
k − 1G
0r(k − 1), (13)
and so
G0r(k) = (const.)× ψk−1G0r(1); (14)
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the final term here is Einstein, and contains a factor of D and a product of all the angular depen-
dence but no ψ.
Our other task here is, for concreteness, to show explicitly in D=5 how the seeming obstruc-
tions to Birkhoff are really deSitter vacua, as shown generically in text.
For the metric ansatz:
ds2 = −ab2dt2 + a−1dr2 + 2bfdtdr + r2(dψ2 + sin2 ψ(dθ2 + sin2 θdφ2)) (15)
we form the cosmological, Einstein, and Gauss-Bonnett actions
I =
∫
d5x
√−g
(
Λ+R+ α(R2 − 4RµνRµν +RµναβRµναβ)
)
, (16)
keeping only terms linear in f and its derivatives. The field equations at f = 0 read:
δI
δa
∣∣∣∣
f=0
= 3(4α(1 − a) + r2)b′ = 0 (17)
δI
δb
∣∣∣∣
f=0
= Λr3 + 6r(1− a)− 3r2a′ − 12αa′(1− a) (18)
G0r ≡ δI
δf
∣∣∣∣
f=0
= −3(r2 + 4α(1 − a))a˙a−1. (19)
Now consider the obstruction, 4α(1 − a) + r2 = 0, to Birkhoff. This is just a = 1 + r24α , vacuum
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deSitter. The constant 4α is determined by inserting into the b equation:
δI
δb
∣∣∣∣
f=0,a=1+r2/(4α)
=
r3(2αΛ− 3)
2α
= 0. (20)
The Lovelock coefficients determine the deSitter cosmological constant according to the tuning
α = 3Λ/2; in the absence of Λ, there are no roots, except flat space.
We thank B. Tekin for a discussion. This work was supported by NSF grant PHY-04-01667.
References
[1] S. Deser and A.V. Ryzhov; gr-qc/0505039.
[2] C. Chermousis and J.F. Dufaux, Class. Quant. Grav. 19, 4671 (2002); hep-th/0202107.
[3] R. Zegers; gr-qc/0505016.
[4] S. Deser and J. Franklin, Am. J. Phys. 73, 261 (2005); gr-qc/0408067.
[5] S. Deser and B. Tekin, Class. Quant. Grav. 20. 4877 (2003); gr-qc/0306114.
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Birkhoff for Lovelock Redux

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