On Birkhoff's Theorem in Horava Gravity

We study Birkhoff's theorem, which states the absence of time-dependent, spherically symmetric vacuum solutions in four-dimensional Horava gravity, which has been proposed as a renormalizable quantum gravity without the ghost problem. We prove that the theorem is still valid for the usual type of solutions which admit the general relativity limit in the low energy (IR) region. However, for the unusual type of solutions, it can be violated in high-energy (UV) region, due to the non-linear effects. This implies that the scalar graviton can emerge as the results of non-linear UV effects but is decoupled in IR regime. An important implication of the non-linear, UV scalar graviton in Big Bang cosmology is also discussed.

Birkhoff's theorem, which states the absence of time-dependent, spherically symmetric vacuum solutions, is an important consequence of general relativity (GR) [1,2]. This implies that the uniqueness of a spherically symmetric solution as the static one, given by the Schwarzschild solution. Furthermore, it also implies the absence of gravitational radiation for pulsating or collapsing, spherically symmetric bodies, which can be stated as the absence of spin-0, or "scalar gravitons" in modern terms. The Birkhoff's theorem is also quite important for the study of cosmological as well as astrophysical problems [3].
On the other hand, it is also well known that GR would not be a UV complete theory due to lack of renormalizability. Several years ago, Hořava proposed a renormalizable, higherderivative gravity theory, without the ghost problem in the usual covariant higher-curvature gravities, by considering different scaling dimensions for space and time [4]. So, it would be an important question whether the Birkhoff's theorem can be still valid or needs to be modified for this quantum gravity model. Actually, since there have been long standing debates about the scalar graviton mode and the recovery of GR in IR, the study of Birkhoff's theorem is also directly related to the fundamental question about the consistency of Hořava gravity [5][6][7].
To this ends, we start by considering the ADM decomposition of the metric and the Hořava gravity action 1 , which is power-counting renormalizable in four dimensions [4], ) is the Cotton tensor, κ, λ, ν, µ, Λ W are coupling parameters, and ω an IR-modification parameter which breaks softly the detailed balance condition in IR [4,9,10] so that Newton's gravity or GR limit exits, without changing the improved UV behaviors. [ ǫ ikℓ is the Levi-Civita symbol, R ij and R are the three-dimensional (Euclidean) Ricci tensor and scalar, respectively ] In addition to these standard terms of the action, for completeness, we will also consider the extension terms which depend on the proper acceleration a i = ∂ i lnN [7], which have been introduced to avoid the problems raised in [5,6]. But, since the key role of the extension terms is in IR, we will consider the modification of V[g ij ] by which is the only relevant term in IR [7]. In order to study Birkhoff's theorem, let us consider the spherically symmetric, time dependent metric with the ansatz, ds 2 = −e 2α(t,r) dt 2 + e 2β(t,r) dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ).
Then, the Hamiltonian and momentum constraints are reduced to and the field equations of g ij are reduced to a single equation (see Appendix A for the details) In GR case, where λ = 1, σ = 0, and the higher-derivative terms ({· · ·} terms in (5) and (7)) are absent, it is easy to see that one can obtain the unique solutionβ = 0 from H i = 0, andα ′ = 0 fromḢ = 0, which tells the time-independence of metric (4) and so proves the Birkhoff's theorem:α ′ = 0 implies α(t, r) = a(t) + b(r) but a(t) can be removed by redefining the time coordinate t.
Generally, however, there could exist solutions which may break the theorem due to, either the IR Lorentz violation from λ = 1 or σ = 0, or the UV Lorentz violation from higherderivative terms. We accordingly classify the solutions largely by the time dependency of β, i.e.,β = 0 orβ = 0.
A. Caseβ = 0: For λ = 1, this is the only possible solution of the momentum constraint (6), as in GR case, though not the unique solution for λ = 1. However, either λ = 1 or λ = 1, one can provė α ′ = 0, i.e., admitting Birkhoff's theorem, for generic values of coupling parameters, except one particular set of parameters which relates UV and IR. To prove this, we consider the time derivatives of (5) and (7), which reduce tȯ where we have used (8) in the last step of (9).
On the other hand, in the absence of the extension, i.e., σ = 0, (9) gives the usual solutioṅ α ′ = 0, or an unusual solution for β(t, r), which makes {· · ·} = 0 in (9), even without knowingα ′ , where C is an integration constant, which corresponds to the mass for the static case. In general, the second, unusual solution (10) is not compatible with the Hamiltonian constraint (5), which reduces to However, there exists one exceptional, compatible solution when the conditions are satisfied. 2 There are two possible solutions for the second condition in (12), i.e., ω = 0 or ω = 2Λ W . The first case, ω = 0 corresponds to the solution without the IR-modification term [11]. The second case, ω = 2Λ W is the corresponding " new " solution but, at this time, with the IR modification. A curious property of these solutions is that " α(t, r) is not constrained " by the equations of motions so that α(t, r) can be an arbitrary function of space and time. It is this later property that allows the time dependence in the metric and so could violate the Birkhoff's theorem, even thoughβ = 0 from (10) and (12). However, we note that these solutions do not have the GR limit, λ → 1, µ → 0, ω → ∞, Λ W → ∞ with ' µ 2 ω, µ 2 Λ W ∼ fixed ' [14] so that Birkhoff's theorem could be violated but only in the non-GR branch. 3 Another peculiar property of the solution for ω = ω(Λ W ) is that it requires some correlations between IR and UV terms so that only their combined equations have the solution, though separate ones do not.
In order to see whether this is really a feasible solution or not, we need to see whether it is compatible with the Hamiltonian constraints and other field equations. Actually, in this case α(t, r) can be determined as from (6), and be integrated as with one undetermined, time-dependent function a(t). Then, after some computations with this general solution (14), one can find that Hamiltonian constraint (5) reduces, for the standard case of σ = 0, where we have introduced f (t, r) ≡ e −2β(t,r) for convenience. Once we get the solutions for (15), one can prove or disprove the Birkhoff's theorem by checking whether it can still satisfy the remaining field equation (7), or not. To this ends, instead of getting the "explicit" solutions for the full equations with both UV and IR parts, which is a formidable task for our case of λ = 1, we consider the limiting solutions for the UV and IR equations separately. First, for the Hamiltonian equation (15) with only the UV terms ({· · ·} terms in the right hand side), which corresponds to the limiting case of ω, Λ W → 0 of the full equation (15), one can solve the equation explicitly and finds that there are two general solutions but only one solution is compatible with the field equation (7), which is given by 4 , where ǫ ≡ sign(3λ − 1). This solution is peculiar in that there is no integration constant and this is due to a factorization of an algebraic equation, called Abel's equation, in the UV limit of ω, Λ W → 0 [16]. From our starting assumption, we findβ ∝ȧ = 0, i.e., a(t) can not be trivially a constant so that the solution manifestly violates the Birkhoff's theorem in the UV regime. 5 4 In IR region of large r, the solution approaches to f = e −2β = 1, N 2 = e 2α = 0, which is consistent with the case of λ = 1 in Sec. A. In other words, there is no discontinuity at λ = 1 in IR, as in the Vainshtein mechanism for the massless limit of massive gravity [15]. 5 For the case of λ = 1/3, where a separate analysis is needed, we find that there is no corresponding timedependent solution in UV [17]. This is the only result which is qualitatively different from the case of λ = 1/3 and it would be probably due to the additional (anisotropic) Weyl symmetry in UV [4,8].
On the other hand, from the relation (13) or (14), α(t, r) is given by Moreover, it is interesting to note that the solutions (16) and (17) exist only for the de-Sitter branch with a positive cosmological constant Λ ∝ −µ 2 > 0, like our current accelerating Universe ! [18]. Second, for the Hamiltonian equation (15) with only the IR terms (the first two terms in the right hand side), one can also get the explicit solution but find that it is not compatible with the field equation (7) either. So, even for the λ = 1 case with an IR Lorentz violation, there is no time-dependent, i.e., Birkhoff's theorem violating, solution in IR regime.
Finally, with the extension terms in (3), the Hamiltonian equation (15) has additional contributions where we have introduced U(t, r) ≡ (lnβ) ′ , J(t, r) ≡ −2β ′ so that the original PDE problem is reduced to an ODE problem with respect to r, with a fixed time t. With the extension terms, solving even the IR equation of (15), which actually shows the key role of the extension terms, is a difficult task and its general solution is not available. However, there exists a simple situation that allows an exact solution, which is now compatible with the equation (7) [we will omit the detailed derivation, which is quite cumbersome], with λ = −1, Λ W = 0, σ = −µ 2 κ 2 ω/8, a(t) = −ln −κ 4 µ 2 ω/64 and two arbitrary functions C 1 (t) and b(r). From our stating assumption ofβ = 0, C 1 (t) can not be just a constant so that this solution violates the Birkhoff's theorem manifestly even in IR. This situation is quite different from the case ofβ = 0 and the other cases ofβ = 0 with σ = 0, where the violations of Birkhoff's theorem may occur as the UV effects (or combined UV/IR effects for the case ofβ = 0). For other values of parameters, we have obtained the (time-dependent) solutions "numerically", by solving about β(t, r) andβ(t, r) using Mathematica. (Fig.1) In conclusion, we have proved that, for the standard form of Hořava gravity, Birkhoff's theorem is satisfied in IR but can be violated in UV. In relation to the gravitational radiations, this implies that the scalar gravitons can exist as the results of UV effects but are decoupled in IR regime. Here, it is important to note that we have considered the problems with the full non-linearity by obtaining the exact (time-dependent) solutions for the non-linear equations of motion. This result is consistent with the (fully non-linear) constraint analysis [24], which has been thought to be inconsistent [25,26], but in contrast to the absence of the scalar gravitons for the whole UV and IR energy ranges in the linear perturbation analyses [19][20][21][22][23]. This implies a remarkable fact that the UV emergence of time-dependent solutions, i.e., violation of Birkhoff 's theorem, or equivalently scalar gravitons, are the "non-linear" effect in UV ! Actually, if we consider small a(t), the UV time-dependent solution (16), (17) can be expanded as where λ−1 . This shows explicitly that a(t) does not appear at the leading, linear orders but emerges only at the sub-leading, i.e., non-linear orders, in consistently with the constraint analysis [24]. [Note that, at the linear order, theȧ 2 (t) factor in (20) can be removed by redefining the time as dt → dt ′ = dt/ȧ(t).] On the other hand, for the extended Hořava gravity with the term of (3), the Birkhoff's theorem is still satisfied forβ = 0, i.e.,α ′ = 0 so that there is no time-dependent solutions for the full theory with both the UV and IR terms. However, we have shown an explicit timedependent solution which violates the theorem in IR forβ = 0, λ = 1. This is consistent with the perturbative [7] as well as non-perturbative analyses [27] but this seems to be "potentially" problematic since it implies the existence of gravitational radiations even for pulsating or collapsing, spherically symmetric bodies in IR, which have not been detected yet; even more, it does not reproduce the GR or Newton's gravity limit in IR, which has been well tested [28].
On the contrary, the existence of a scalar graviton mode, which seems to be represented by an arbitrary function a(t) in the general solution (14), would have an important role in cosmology. Usually, we need (at least one) primordial scalar matter field in order to accommodate the observed (nearly scale invariant) scalar power spectrum in CMB data within the inflationary theory [29]. But, now the scalar degree of freedom which is inherent in the non-linear UV regime of Hořava gravity could have a similar role of the primordial scalar field in the early Universe ! It would be an outstanding question whether Hořava gravity can provide a consistent framework for the Big Bang cosmology without introducing the artificial primordial scalar field and inflationary scenario [23]. where The equations of motion from variation of δg ij are given by [11][12][13] E ij ≡ 2 where E (1)