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Dicke [17], and certain vector-tensor theories, such as units in which c = 1 and the conventions of [25] in which Einstein-Æther theory [18, 19], are two examples where Greek letters in tensor indices stand for spacetime quan- a priori it is not obvious that all methods will yield tities. the same GW SET. In scalar-tensor theories, the GW SET was first computed by Nutku [20] using the Landau- Lifshitz method. In Einstein-Æther theory, the symmet- II. GENERAL RELATIVITY ric GW SET has not yet been calculated with any of the methods discussed above; Eling [21] derived a non- In this section, we review how to construct the GW symmetric GW SET using the Noether current method SET in GR and establish notation. We begin by describ- without applying perturbations. Foster [22] perturbed ing the perturbed action method, following mostly [9]. the field equations a la Isaacson and then used the We then describe the perturbed field equation method, Noether charge method of [23, 24] to find the rate of following mostly [25] and the perturbation treatment change of energy carried by all propagating degrees of of [7, 8, 26]. We then discuss the Landau-Lifshitz method, freedom. following mostly [10, 27] and conclude with a discussion We here use all four methods discussed above to cal- of the Noether method, following mostly [26]. culate the GW SET in GR, scalar-tensor and Einstein- In all calculations in this section, we work with the Æther theory. In the GR and scalar-tensor cases, we Einstein-Hilbert action find that all methods yield exactly the same GW SET. In the Einstein-Æther case, however, the perturbed field 1 4 SGR = d x √ g R (1) equations method, the perturbed action method and 16πG − Z the Landau-Lifshitz method yield slightly different GW SETs; however, the observable quantities computed from in the absence of matter, where G is Newton’s constant, them (such as the rate at which energy is removed from g is the determinant of the four-dimensional spacetime the system by all propagating degrees of freedom) are ex- metric gαβ and R is the trace of the Ricci tensor Rαβ. actly the same. On the other hand, the Noether method By varying the action with respect to the metric, we find applied to Einstein-Æther theory does not yield a sym- the field equations for GR to be metric or gauge invariant GW SET. Symmetrization of G =0 , (2) the Noether GW SET through the Belinfante procedure αβ fails to fix these problems. This is because this proce- where Gαβ is the Einstein tensor associated with gαβ. dure relies on the action being Lorentz invariant at the level of the perturbations, while Einstein-Æther theory is constructed so as to spontaneously break this symmetry. A. Perturbed Action Method The results obtained here are relevant for a variety of reasons. First, we show for the first time how different Let us begin by using the metric decomposition methods for the calculation of the GW SET yield the same observables in three gravity theories in almost all 2 gαβ =g ˜αβ + ǫhαβ + (ǫ ), (3) cases. Second, we clarify why the Noether method and O its Belinfante improvement can fail in Lorentz-violating where ǫ 1 is an order counting parameter. One should ≪ theories. Third, we provide expressions for the GW SET think of the metric perturbation hαβ as high frequency in Einstein-Æther theory for the first time, and from ripples on the backgroundg ˜αβ, which varies slowly over this, we compute the rate of energy and linear momen- spacetime; henceforth, all quantities with an overhead tum carried away by all propagating degrees of freedom. tilde represent background quantities. Both of these Fourth, the methods presented here can be used to com- fields will be treated as independent, and thus, one could pute the rate of change of angular momentum due to vary the expanded action with respect to each of them the emission of gravitational, vector and scalar waves in separately. For our purposes, however, it will suffice to Einstein-Æther theory; in turn, this can be used to place consider the variation of the expanded action with re- constraints on this theory with eccentric binary pulsar spect to the background metric order by order in hαβ. observations. Fifth, the methods described here in detail The action expanded to second order in the metric can be straightforwardly used in other gravity theories perturbation is that activate scalar and vector propagating modes. (0) (1) 2 (2) 3 This paper is organized as follows. Section II fo- SGR = SGR + ǫSGR + ǫ SGR + ǫ , (4) cuses on the GW SET in GR, with each subsec- O tion II A, IIB, IIC, II D detailing a separate method, and where the last subsection IIE calculating physical observables (0) 1 4 derived from the GW SET. Sections III and IV apply SGR = d x g˜ R,˜ (5a) all the methods for finding the GW SET to scalar-tensor 16πG − Z p theories and to Einstein-Æther theory respectively. We (1) 1 4 αβ 1 SGR = d x g˜ h¯ R˜αβ ˜ h¯ choose the common metric signature ( , +, +, +) with −16πG − − 2 − Z p 3

αβ ¯ ˜ α ˜ βh¯ , (5b) hαβ. This is accomplished by varying Eq. (7) with respect −∇ ∇ to h¯αβ. The result is i (2) 1 4 1 ˜ ¯ ˜ α¯ SGR = d x g˜ αh h 1 16πG − 8∇ ∇ ˜ ¯ ˜ γ ˜ ¯ ˜ ¯ hαβ 2 (αhβ)γ g˜αβ h =0 . (10) Z p − ∇ ∇ − 2 1 γ αβ 1 γ αβ + ˜ βh¯αγ ˜ h¯ ˜ γ h¯αβ ˜ h¯ , (5c) 2∇ ∇ − 4∇ ∇ Using the Lorenz gauge condition, αβ ˜ αh¯ =0 , (11) and where we have introduced the trace-reversed metric ∇ perturbation and taking the trace of Eq. (10), we ﬁnd

αβ αβ 1 αβ h¯ = h g˜ h . (6) ˜ h¯ =0 . (12) − 2 ¯ Before proceeding, let us simplify the expanded action If we impose the traceless condition h = 0 on some initial further by noticing that certain terms do not contribute hypersurface, then Eq. (12) forces the metric to remain to observable quantities computed from the GW SET traceless on all subsequent hypersurfaces [28]. The com- measured at spatial infinity or under short-wavelength bination of the Lorentz gauge condition and the trace averaging. Equations (5a) and (5b) are exact, while we free condition yields the transverse-traceless (TT) gauge. have adjusted (5c) for the reasons that follow. First, Applying this condition to simplify the field equations in there is no term explicitly dependent on the background Eq. (10) gives curvature. This is because the GW SET will be later ˜ ¯TT used to calculate the rate at which energy and linear mo- hαβ =0 . (13) mentum are carried away by all propagating degrees of We are now able to compute the GW SET using freedom at spatial infinity. These rates will not depend Eq. (9). Performing the necessary calculations yields on the background curvature tensor, as the latter vanishes at spatial infinity. Second, total divergences gener- 1 µν 1 Θαβ = ˜ αh¯ ˜ βh¯µν ˜ αh¯ ˜ βh¯ ated by integration by parts also become boundary terms 32πG ∇ ∇ − 2∇ ∇ that will not contribute to observables at spatial infinity. γ δ γ Third, terms which are odd in the number of perturba- + ˜ γ h¯ ˜ h¯αβ +2 ˜ h¯ ˜ h¯ . (14) ∇ ∇ ∇[γ δ]β ∇ α tion quantities will not contribute to the GW SET af- E ter short-wavelength averaging (see App. A). From these We are free to apply the TT gauge at this point, which we (1) considerations, we note that SGR will not contribute at make use of. Since the result of the GW SET is averaged all to the GW SET, and can therefore be neglected. through the brackets, we are also able to integrate by We are now left with a simplified effective action parts and eliminate boundary terms. These two effects give the familiar result, eff (0) eff 2 (2) eff 3 SGR = SGR + ǫ SGR + ǫ . (7) O 1 ˜ TT ˜ µν Θαβ = αhµν βhTT . (15) Variation of Eq. (7) with respect to the background met- 32πG ∇ ∇ (0)eff D E ric will yield the field equations. The variation of SGR gives the Einstein tensor for the background metric, and B. Perturbed Field Equation Method (2)eff thus, the variation of SGR acts as a source, which we identify with the GW SET. The variation of the second- Let us begin by decomposing the metric gαβ as order piece of the action can be written as (1) (2) gαβ = ηαβ + hαβ + hαβ , (16) (2)eff 4 GR αβ δSGR = d x gt˜ αβ δg˜ , (8) − where η is the Minkowski metric and (h(1),h(2)) are Z p αβ αβ αβ with the GW SET first and second order metric perturbations respectively. Unlike in the previous section, the background is here not GR GR Θαβ = 2 tαβ , (9) arbitrary but chosen to be the Minkowski spacetime. We − (n) n assume that hαβ = (ǫ ), but we no longer keep track and the angle-brackets stand for short-wavelength av- explicitly of the orderO counting parameter ǫ. eraging. This averaging is necessary because the back- With this decomposition, we now expand the Einstein ground geometry contains many wavelengths of oscilla- field equations [Eq. (2)] to second order in the metric tion of the metric perturbation, and the details of the (1) perturbation. We obtain the field equations for hαβ by latter only induce higher order corrections to the metric truncating the expansion of the field equations to linear in multiple-scale analysis [7, 8]. order in the perturbation: Before we can compute the GW SET, we first need (1) to derive the first-order equations of motion for the field Gαβ[h ]=0 . (17) 4

We obtain the GW SET by truncating the expansion to 1 γ (1) γ δ (1) + ηαβ ∂γ ∂ h ∂ ∂ h =0 . (20) second order in the perturbation: 2 − γδ 2 (2) (1) (1) Gαβ h =8πΘαβ Gαβ h , (18) α¯ ≡− As in Sec. II A, imposing the Lorenz gauge ∂ hαβ = 0 h i ¯(1) (1) on the trace-reversed metric perturbation hαβ hαβ where again the angled-brackets stand for short- ≡ − (1/2) η h(1), and taking the Minkowski trace in Eq. (20) wavelength averaging1. αβ forces h(1) ηαβh(1) to satisfy a wave equation in flat We begin by expanding the field equations to first order ≡ αβ in the metric perturbation. The Ricci tensor takes the spacetime. This, in turn, allows us to refine the Lorenz form gauge into the TT gauge in flat spacetime to further simplify the field equations to (1) 1 (1)γ (1) γ (1) R = 2∂γ ∂ h ∂α∂βh ∂γ ∂ h , (19) αβ 2 (α β) − − αβ γ (1)TT h i ∂γ ∂ h =0 . (21) and plugging this and its trace into the Einstein tensor αβ of Eq. (17) gives

γ The next step is to investigate the ﬁeld equations to (1) 1 (1) 1 γ (1) 2 ∂γ ∂ h ∂α∂βh ∂γ∂ h h . The expansion of the Ricci tensor is (α β) − 2 − 2 αβ O

(2) 1 (2)γ (2) γ (2) 1 (1)γδ (1) (1) 1 (1)γδ (1) (1) R = 2∂γ∂ h ∂α∂βh ∂γ ∂ h + ∂β h 2∂ h ∂δh ∂γ h 2∂ h ∂δh αβ 2 (α β) − − αβ 2 (α γ)δ − αγ − 2 (α β)δ − αβ 1 h (1)γδ (1) (1)δ (1)γ i (1)δh γ (1) 1 (1) (1)iγ γ h(1) i ∂αh ∂βh +2∂γh ∂δh 2∂γh ∂ h + ∂γh 2∂ h ∂ h . (22) − 4 γδ α β − α βδ 4 (α β) − αβ h i h i

This equation can be simplified by imposing the Lorenz C. Landau-Lifshitz Method gauge on h(2), namely ∂αh¯(2) = 0 with h¯(2) h(2) αβ αβ αβ ≡ αβ − (2) (2) αβ (2) The Landau-Lifshitz method makes use of a formula- (1/2) ηαβ h and h η h . As before, we can ≡ αβ tion of GR in terms of the “gothic g metric,” a tensor refine this gauge into the TT gauge by setting h(2) = 0, density defined as which is compatible with R(2) = 0. With this at hand, the Ricci tensor in the TT gauge reduces to gαβ √ ggαβ . (25) ≡ − (2) 1 γ (2)TT 1 (1)γδ (1)TT R = ∂γ∂ h + ∂αhTT ∂βh αβ −2 αβ 4 γδ With this density, one can construct the 4-tensor density 1 (1)γδ (1)TT 1 (1)γδ (1)TT αµβν αβ µν αν βµ + hTT ∂ ∂ h + hTT ∂ ∂ h g g g g , (26) 2 α β γδ 2 γ δ αβ H ≡ − (1)γδ (1)TT (1)TT δ (1)TTγ hTT ∂δ∂ h + ∂ h ∂ h . (23) which has the remarkable property that − (α β)γ [δ γ]β α αµβν αβ αβ The first line of Eq. (22) (and the first term of Eq. (23)) ∂µ∂ν = 2 ( g) G + 16πG ( g) tLL , (27) (1) (2) H − − is nothing but Eq. (19) with hαβ hαβ , which will con- → αβ αβ tribute to the left-hand side of Eq. (18). Using Eq. (18), where G is exactly the Einstein tensor and tLL is known the GW SET is then (suppressing superscripts for neat- as the Landau-Lifshitz pseudotensor [10], ness) αβ 1 αβ γδ αγ βδ ( g)tLL = ∂γg ∂δg ∂γ g ∂δg 1 T T γδ − 16πG − Θαβ = ∂αhγδ ∂βhT T , (24) 32πG 1 αβ γκ δǫ γ(α β)ǫ δκ D E + g gγδ∂ǫg ∂κg 2gδǫg ∂κg ∂γ g after integrating by parts inside of the averaging scheme. 2 − This expression is identical to Eq. (15), albeit around a ǫκ αγ βδ 1 αγ βδ αβ γδ +gγδg ∂ǫg ∂κg + 2g g g g Minkowski background. 8 − ǫσ κλ (2gǫκgλσ gκλgǫσ) ∂γg ∂δg . (28) × − 1 Strictly speaking, this method is not identical to Isaacson’s orig- Substituting the GR field equations into Eq. (27) gives inal work [7, 8] because here we assume a priori that the back- αµβν αβ αβ ground is Minkowski. ∂µ∂ν = 16πG ( g) Tmat + tLL , (29) H − 5

αβ where Tmat is the matter SET. We have kept this term D. Noether Current Method αβ here for clarity, but in this paper Tmat = 0. We can now use the symmetries of αµβν to derive H Noether showed that symmetries of the action lead to some conservation laws. First, we notice that by con- conserved quantities [11], which have become known as αµβν struction has the same symmetries as the Rie- Noether currents jα. Taking a field theoretical approach mann tensor.H Since αµβν is antisymmetric in α and µ, H (for fields propagating on a Minkowski background), con- we find sider the action αµβν ∂α∂µ∂ν =0 . (30) 4 S = d x (φL, ∂αφL) , (37) H L Equation (30) implies there exists a conserved quantity, Z which we will define as where φL = (φ1, φ2,...,φL) are the fields of the spacetime and is a Lagrangian density. From Hamilton’s L αβ 1 αµβν principle, the variation of the action in Eq. (37) must T ∂µ∂ν . (31) ≡ 16πG H vanish. This leads to When this quantity is short-wavelength averaged, one re- ∂ ∂ ∂ L ∂α L δφL + ∂α L δφL =0 . covers the GW SET ∂φL − ∂(∂αφL) ∂(∂αφL) (38) Θαβ = T αβ = ( g) tαβ . (32) − LL We choose our fields to remain constant on the boundary, which leads to the second term in Eq. (38) to What we described above is fairly general, so now we vanish. This is because the variation still takes place evaluate the GW SET in terms of a metric perturbation within an integral and total derivatives become bound- from a Minkowski background. Using the expansion ary terms. The term remaining constitutes the Euler- Lagrange equations. These must be satisfied, and give g = η + h , (33) αβ αβ αβ the equations of motion for the fields. To find the conserved energy-momentum tensor, we with hαβ (ǫ), the gothic metric becomes ∼ O vary the Lagrangian density with respect to the coordi- gαβ = ηαβ h¯αβ + (h2) , (34) nates, − O ∂ ∂ ∂ ¯αβ where h is the trace reversed metric perturbation. Lα = L ∂αφL + L ∂β∂αφL . (39) Equation (28) can now be simplified and written to sec- ∂x ∂φL ∂ (∂βφL) ond order in the metric perturbation as The first term on the right-hand side can be replaced with the Euler-Lagrange equations. The result, after basic αβ 1 ¯αβ ¯γδ ¯αγ ¯βδ ( g) tLL = ∂γ h ∂δh ∂γ h ∂δh analysis, is − 16πG − 1 αβ ¯δ ¯ǫγ ¯ǫ(α β)¯γ γ ¯α ¯βδ ∂ ∂ ∂ + η ∂γ hǫ ∂δh 2∂γ h ∂ hǫ + ∂ hδ ∂γ h L L ∂αφL =0 , (40) 2 − ∂xα − ∂xβ ∂ (∂ φ ) β L 1 α β γδ 1 α β 1 αβ γ δǫ + ∂ h¯γδ∂ h¯ ∂ h∂¯ h¯ η ∂ h¯δǫ∂γh¯ α 2 − 4 − 4 which then leads to the conservation law ∂αj β = 0 with the current 1 αβ γ + η ∂ h∂¯ γh¯ . (35) 8 α α ∂ j β = δ β L ∂βφL , (41) L − ∂ (∂αφL) Inserting this expression into Eq. (31) we find Tαβ, and after short-wavelength averaging, exploiting the gauge which is known as the canonical SET. The GW SET is freedom to use the TT gauge, integrating by parts, and typically assumed to be its short-wavelength average: using the first-order field equations, we obtain α α Θ β = j β . (42) h i αβ 1 α TT β γδ Θ = ∂ h ∂ hTT . (36) Unlike the previous methods, which made use of the sym- 32πG γδ metries of the variations and expansions, there is no man-

Notice that this method does not allow us to find the date that the SET derived here is symmetric. In fact, first-order field equations, which are typically obtained even in Maxwell’s electrodynamics, the canonical SET is by the perturbed field equations method of the previous not symmetric (see Appendix B). subsection to first order in the metric perturbation. The Equation (41) can now be applied to GR to find the final result matches those found in Secs. II A and IIB. canonical SET. In this setting, the fields are the metric perturbation themselves, which propagate on a flat background, and thus

gαβ = ηαβ + hαβ , (43) 6 with hαβ (ǫ). The Lagrangian must thus be ex- where τ = t r/v is retarded time and v is the speed of panded in the∼ O fields to the appropriate order. Any quan- propagation− of the field. Moreover, the spacetime (par- tity that depends on the curvature of the background tial) derivative of the field then satisfies metric vanishes since the latter is the Minkowski metric. 2 Expanding the Lagrangian to first order leaves odd terms 1 λc ∂αφL = kα ∂τ φL + , (49) in Eq. (41), which vanish upon short-wavelength averag- −v O r2 ing. Expanding the Lagrangian in Eq. (1) to second order about a Minkowski background yields where kα is a unit normal 4-vector normal to the r = i const surface (kα ( 1,Ni)) with Ni = xi/r and x are ≡ − 1 α β αγ spatial coordinates on the 2-sphere. = ∂αh ∂ h +2∂αhβγ ∂ h L 64πG With this at hand, we can now simplify the observable α β γ αβ ˙ ˙ i 2∂ h ∂βh α ∂γ hαβ ∂ h . (44) quantities E and P in GR. The rate at which energy − − and linear momentum are removed from the system is With this Lagrangian density, we can now find the 2 equations of motion for the metric perturbation and the ˙ R ˙ TT i γδ 2 EGR = h ∂ hTT d Si , (50a) GW SET. The former can be computed from the Euler- −32πG γδ I Lagrange equations: 2 D E ˙ i R i TT j γδ 2 PGR = ∂ h ∂ hTT d Sj , (50b) −32πG γδ ∂ ∂ 1 γ TT I L ∂γ L = ∂γ ∂ hαβ =0 , (45) ∂hαβ − ∂(∂γ hαβ) 2 where h˙ αβ is the partial derivative of hαβ with respect 2 2 where we have imposed the TT gauge after variation to coordinate time t and d Si = R Ni dΩ. Incorporating of the Lagrangian. Using Eq. (44) in Eq. (41), short- the shortwave approximation into Eq. (50) gives wavelength averaging, integrating by parts and using the 2 ˙ R TT γδ TT gauge condition and Eq. (45), one finds EGR = ∂τ h ∂τ hTT dΩ , (51a) −32πG γδ Z 1 2 α α T T γδ ˙ i R i TT γδ Θ β = ∂ hγδ ∂βhT T . (46) P = N ∂τ h ∂τ hTT dΩ , (51b) 32πG −32πG γδ Z D E This expression is identical to all others found in this which are the final expressions we seek for the GW SET- section. related observables. To proceed further, one would have to specify a particular physical system, solve the field E˙ P˙ equations for the metric perturbation for that given sys- E. Derivation of Physical Quantities: and tem, and then insert the solution in the above equations to carry out the integral; all of this is system specific and Ultimately, one is interested in calculating physical, outside the scope of this paper. observable quantities from the GW SET that can be measured at spatial infinity, ι0. Two examples are the rate of energy and linear momentum transported by GWs away III. SCALAR-TENSOR THEORY from any system per unit time Following the procedure presented in the previous sec- 0i 2 E˙ = Θ d Si , (47a) tion, we now derive the GW SET in scalar-tensor the- − ∞ I ories, focusing only in the theory proposed by Jordan, i ij 2 Fierz, Brans, and Dicke as an example [15–17]. This P˙ = Θ d Sj , (47b) − ∞ theory was developed in an attempt to satisfy Mach’s I principle and its action is where Θαi is the (α, i) component of the GW SET. α These observables can be simplified through the short- 1 4 αΦ Φ SST = d x√ g ΦR ω ∇ ∇ , (52) wave approximation, which assumes the characteristic 16πG − − Φ wavelength of radiation λc is much shorter than the ob- Z server’s distance to the center of mass r, i.e. the observer where R is the Ricci scalar, Φ is the scalar field, and ω is in the so-called far-away wave zone so that r λc. is a coupling constant. In the GR limit, the scalar field When this is true, the propagating fields can be expanded≫ Φ becomes unity. The field equations for scalar-tensor as [27] theory are

2 3 1 γ λc λc λc Xαβ = Gαβ + ( α βΦ gαβ γ Φ) φL = f1,L(τ)+ f2,L(τ)+ , (48) Φ ∇ ∇ − ∇ ∇ r r O " r # ω 1 γ + αΦ βΦ gαβ γ Φ Φ =0 , (53) Φ ∇ ∇ − 2 ∇ ∇ 7 and in vacuum.

γ Yαβ = γ Φ=0 , (54) ∇ ∇ where we vary the action with respect to the metric and scalar field respectively. Lastly, we will make use of the A. Perturbed Action Method “reduced field” [29] (see Appendix C for derivation of this quantity), Let us begin by decomposing the metric as in Eq. (3) 1 1 and the scalar field via θαβ = hαβ g˜αβ h g˜αβ ϕ , (55) − 2 − Φ˜ Φ= Φ+˜ ǫ ϕ , (57) whereg ˜αβ is the background metric and Φ˜ is a background scalar field. As we will show, the field equation for the reduced field is simply where ϕ = (ǫ). The action will decompose in a similar manner toO that of Eq. (4), where the expanded action TT θαβ = 0 (56) terms are

(0) 1 4 ω α SST = d x g˜ Φ˜ R˜ ˜ αΦ˜ ˜ Φ˜ , (58a) 16πG − − ˜ ∇ ∇ Φ Z p (1) 1 4 αβ 2 αβ 1 SST = d x g˜ Φ˜ R˜ θαβ (6+2ω) ˜ ϕ (1 + ω) ˜ α ˜ βθ ˜ θ , (58b) −16πG − − ˜ − ∇ ∇ − 2 Φ Z p (2) 1 4 1 2 1 αβ 1 αβ αβ γ 1 2 SST = d x g˜ Φ˜ R˜ ϕ + R˜ θαβ ϕ + Rθ˜ θαβ R˜ θα θβγ Rθ˜ −16πG − ˜ 2 ˜ 4 − − 8 Φ Φ Z p 1 αβ 1 3 α 1 γ αβ 1 α 1 γ αβ + R˜ θθαβ + + ω ˜ αϕ ˜ ϕ + ˜ γθαβ ˜ θ ˜ αθ ˜ θ ˜ βθαγ ˜ θ . (58c) 2 φ 2 ∇ ∇ 4∇ ∇ − 8∇ ∇ − 2∇ ∇ 0

All trace terms have been taken with respect to the back- Variation of the effective action with respect to the (1) ground metric. As argued before, the SST term will not background metric contribute to the GW SET upon variation, leaving us (2)eff 4 ST αβ with the effective action δSST = d x gt˜ δg˜ (64) − αβ eff (0) eff 2 (2) eff 3 Z S = SST + ǫ SST + ǫ . (59) p ST O gives the GW SET The independent variation of the effective action with ST ST Θαβ = 2 tαβ . (65) respect to θαβ and ϕ gives the first order field equations. − In particular, variation with respect to θαβ gives Carrying out the variation, we find

γ 1 ST 1 µν 6+4ω ˜ θ 2 ˜ ˜ θ g˜ θ =0 , (60) Θ = Φ˜ ˜ αθ ˜ βθµν + ˜ αϕ ˜ βϕ αβ (α β)γ αβ αβ 32πG ∇ ∇ ˜ 2 ∇ ∇ − ∇ ∇ − 2 Φ 1 γ δ γ while variation with respect to ϕ yields ˜ αθ ˜ βθ + ˜ γ θ ˜ θαβ +4 ˜ θ ˜ θα −2∇ ∇ ∇ ∇ ∇[γ δ]β ∇ ˜ ϕ =0 . (61) σ γδ 1 γ +˜gαβ ˜ γ θδσ ˜ θ + ˜ γ θ ˜ θ ∇ ∇ 4 ∇ ∇ Notice that Eq. (60) is identical to the ﬁrst-order Einstein equations in Eq. (10) with the replacement hαβ θαβ. 1 γ δσ γ → ˜ γ θδσ ˜ θ (3+2ω) ˜ γϕ ˜ ϕ . (66) Therefore, by utilizing the TT gauge condition on θαβ, −2 ∇ ∇ − ∇ ∇ namely At this point we may impose the TT gauge condition and ˜ αθTT =0 , (62a) use integration by parts to simplify Eq. (66) with the use ∇ αβ θTT =0 , (62b) of Eqs. (61) and (63). The resulting GW SET is

TT ˜ ST Φ TT 1 the equation of motion for the reduced ﬁeld θαβ is ˜ γǫ ˜ ˜ ˜ Θαβ = αθTT βθγǫ + (6+4ω) αϕ βϕ . 32πG ∇ ∇ Φ˜ 2 ∇ ∇ ˜ TT θαβ =0 . (63) (67) 8

This result is consistent with that found in [29]forg ˜αβ This satisfies the relation → ηαβ. αµβν 2 αβ 8π αβ ∂µ∂ν = 2 ( g)Φ X + tLL ST , (74) H − Φ , B. Perturbed Field Equation Method where Xαβ = 0 are the field equations [Eq. (53)]. When the GR limit is taken, this agrees exactly with what was Let us begin by expanding the metric as in Eq. (16) found in Sec. IIC. The final term in Eq. (74) will be called and similarly expand the scalar field as the scalar-tensor pseudo-tensor and is given by [20] (1) (2) Φ= φ0 + ϕ + ϕ , (68) 1 tαβ =Φtµν + 2Φ,(αΓβ)gγδ 2gγ(αΓβ) Φ,δ LL,ST LL 8π γδ − γδ where φ0 is now the constant background field. αβ ,γ δ h σ γδ ,(α β)γ δ The expansion of the field equations to first-order +g (2Φ Γγδ Φ,σΓγδg ) 2Φ g Γγδ − − (1) (1) µα νβ γ yields the equations of motion for the fields hαβ and ϕ +g g Φ,γΓµν

1 ,α ,β αβ ,γ φ0 γ (1) (1)γ φ0 (1)γδ + 2(ω 1)Φ Φ + (2 ω)g Φ,γΦ . ∂γ ∂ θαβ = φ0 ∂γ∂(αθβ) ηαβ ∂γ ∂δθ , (69) 16πΦ − − 2 − 2 (75) γ (1) ∂γ ∂ ϕ =0 . (70) One could write this expression entirely in terms of the Working in the TT gauge as in the previous subsec- gothic metric, but this does not simplify the resulting tion [see Eq. (62)] confirms our vacuum field equation expression. [Eq. (56)] in a Minkowski background. The GW SET can now be obtained through Eq. (32) As in Eq. (18), the GW SET will be defined as the av- after expanding Eq. (74) to leading non-vanishing order. erage of the expansion of the field equations to quadratic Doing so, making use of the TT gauge, integrating by order in the linear perturbation terms: parts, and using the field equations for the linear perturbations, the final GW SET is found to be 2 2 ST (1) (1) (1) (1) 8πΘαβ Xαβ h , ϕ ,h ϕ , (71) ≡− ST φ0 γǫ TT 1 Θ = ∂ θTT ∂ θ + (6+4ω) ∂ ϕ ∂ ϕ . αβ 32πG α β γǫ φ2 α β 0 where we recall Xαβ is given by Eq. (53). The details (76) of the expansion of the field equations to second order Equation (76) is identical to the GW SETs previously in the perturbations will be omitted for brevity, but the found. procedure follows that in Sec. IIB. Once this is done, we use Eq. (71) to find the GW SET D. Noether Current Method φ 1 ΘST = 0 ∂ θγǫ ∂ θTT + (6+4ω) ∂ ϕ ∂ ϕ . αβ 32πG α TT β γǫ φ2 α β 0 The derivation of the canonical SET relies on Eq. (41), (72) which requires we expand the Lagrangian to second order We arrive at this equation using integration by parts, through the metric decomposition of Eq. (43) and the imposing the TT gauge condition, Eqs. (54) and (56) for field decomposition of Eq. (57). Doing so, we find a Minkowski background. This GW SET agrees with φ0 1 α γ αβ that in Sec. IIIA for a flat background. ST = ∂ θ ∂αθ + ∂βθαγ ∂ θ L 32πG 4 1 γ αβ (3+2ω) α ∂γ θαβ ∂ θ ∂αϕ ∂ ϕ . (77) C. Landau-Lifshitz Method −2 − φ2 0 The Landau-Lifshitz method requires the use of the With this second order Lagrangian at hand, we can αµβν tensor density or a variation of it to obtain a con- now follow the same steps as in Sec. II D to calculate Hservation law of the form of Eq. (30) in Sec. IIC. In prin- the canonical SET. The first step is to ensure the Euler- ciple, one could use the same αµβν tensor density as Lagrange equations are satisfied. For the reduced field that used in GR [see Eq. (26)],H and one would obtain θαβ, the Euler-Lagrange equations give the same rate of energy and linear momentum loss in a ∂ ∂γθTT =0 , (78) binary system [30]. But in practice, it is easier to use an γ αβ improved αµβν tensor density that simplifies the cal- H where we have used the TT gauge after varying the La- culations of the modified Landau-Lifshitz pseudo-tensor grangian. As for ϕ, the Euler-Lagrange equations lead αβ tLL,ST. Following the work of [20], we choose to

αµβν 2 αβ µν αµ βν γ =Φ g g g g . (73) ∂γ ∂ ϕ =0 . (79) H − 9

The next step is to compute the Noether’s current. Sum- combinations of these constants will typically appear in ming over the variation of all fields, we find the perturbed field equations, namely

α ∂ ∂ α c14 = c1 + c4 , (85a) j β = L ∂βθµν L ∂βϕ + η β . −∂ (∂αθµν ) − ∂ (∂αϕ) L c± = c c , (85b) 1 3 (80) ± c123 = c1 + c2 + c3 . (85c) The current in Eq. (80) gives a pseudo-tensor that is not Varying the action with respect to the metric tensor gauge invariant, but after short-wavelength averaging, and the Æther vector field yields the field equations of these terms vanish. Using the TT gauge condition and the theory in vacuum: imposing the first-order field equations, one then finds Gαβ = Sαβ , (86) ST φ0 TT γδ 1 β β γ Θ = ∂ θ ∂ θTT + (6+4ω) ∂ ϕ ∂ ϕ . βK α = λuα c4 uβ u αuγ , (87) αβ 32πG α γδ β φ2 α β ∇ − − ∇ ∇ 0 (81) where Gαβ is the Einstein tensor and Sαβ is the Æther contribution to the field equations given by The result here is consistent with those presented in the γ γ γ previous sections. Sαβ = γ K (αuβ) + K(αβ)u K(α uβ) ∇ γ −γ + c ( αuγ βu γuα uβ) 1 ∇ ∇ − ∇ ∇ γ δ E. Derivation of Physical Quantities: E˙ and P˙ + c u u γ uα δuβ + λuαuβ 4 ∇ ∇ 1 γ δ gαβ K δ γu , (88) The four different ways of deriving the GW SET in − 2 ∇ scalar-tensor theory all produced the same result, and where hence, one can use any of them to derive E˙ and P˙ . In- serting the GW SET into Eqs. (47a) and (47b), the rate α αβ δ K γ = K γδ βu . (89) of change of energy and linear momentum carried away ∇ by all propagating degrees of freedom is Variation of the action with respect to the Lagrange multiplier λ gives the constraint, 2 ˙ φ0 R TT γδ 6+4ω EST = ∂τ θγδ ∂τ θTT + ∂τ ϕ ∂τ ϕ dΩ , α − 32πG φ2 u uα = 1 . (90) Z 0 − (82a) Contracting Eq. (87) with uα, we can solve for λ to obtain 2 ˙ i φ0 R i TT γδ 6+4ω PST = N ∂τ θγδ ∂τ θTT + ∂τ ϕ ∂τ ϕ dΩ . α β α β γ − 32πG φ2 λ = u βK α + c4 u αu (u γ uβ) . (91) Z 0 ∇ ∇ ∇ (82b) Unlike in Secs. II andIII, we will use an irreducible In the GR limit, we find that the rate of energy and decomposition of all fields rather than a single reduced momentum loss is identical to that of Sec. IIE. field. This will have the effect of cleanly separating the independent modes of propagation. Expanding the metric as in Eq. (3), the Æther field can be decomposed via IV. EINSTEIN-ÆTHER THEORY uα = tα + ωα , (92)

In this section, we study Einstein-Æther theory [18] where ωα = (h) and tα = ( 1, 0, 0, 0) and it is a time- by following the work of Foster [22]. We begin with the like unit| vector| O with respect− to the background metric α β action g˜αβ, i.e.g ˜αβt t = 1. We next decompose the metric perturbation into− tensor, vector, and scalar modes as 1 4 SÆ = d x√ g (R follows 16πG − αβ Z γ δ α αβ α β (α β) 0i α β ij K γδ αu βu + λ (u uα + 1) , (83) h = t t h00 +2 i t h + i j h , (93) − ∇ ∇ P P P where α is the background spatial projector where Pi αβ αβ α β α β α β αβ =g ˜αβ + tαtβ . (94) K γǫ = c1g gγǫ + c2δ γ δ ǫ + c3δ ǫδ γ c4u u gγǫ . P − (84) The Æther, vector and tensor perturbations are further The vector uα in Eq. (83) is the Æther ﬁeld, which is unit decomposed into their transverse and longitudinal parts, timelike due to the Lagrange multiplier λ constraint. The quantities ci are coupling constants of the theory. Certain ωα = tαω + α νi + ∂iν , (95a) 0 Pi 10

h0i = γi + ∂iγ , (95b) From this, we are able to see that the wave speed for the tensor mode is ij ij 1 h = φTT + Pij [f]+2∂(iφj) + ∂i∂j φ , (95c) 2 2 1 vT . (102) ˜ ˜ k ˜ ˜ ij ≡ 1 c+ with Pij [f] g˜ij k f i j f, φTT a TT spatial − ≡i ∇i ∇ i − ∇ ∇ tensor and ν ,i = γ ,i = φ ,i = 0. We perform the same procedure to the vector modes The degrees of freedom of the theory, the Æther γi and νi. The variations become α ﬁeld u and the metric tensor gαβ, have thus i (2)eff been replaced by their decompositions (ω0,ν ,ν) and δSÆ 2 1 ˜ ˜ − − 00 i TT i i = c14∂0 (γi + νi)+ [(1 c ) γi c νi]=0 , (h ,γ ,γ,φij ,f,φ , φ), but these quantities can be fur- δγ 2△ − − ther constrained. The timelike condition on the Æther (103a) ﬁeld requires that (2)eff δSÆ 2 1 = c ∂˜ (γ + ν ) ˜ [c−γ +2c ν ]=0 . 1 i 14 0 i i i 1 i ω = h , (96) δν − 2△ 0 −2 00 (103b) to leading order in the perturbations. We also choose the We can combine these equations to give us a relation gauge conditions between the variables γi and νi,

i ∂ ω =0 , (97a) γi = c νi . (104) i − + 0i ∂ih =0 , (97b) Equipped with this relation, we are able to solve for an- i ∂i∂[j hk] =0 , (97c) other wave equation, this time for the vector mode,

2 (1 c+) c14 or equivalently ν = γ = φi = 0. ˜ ˜2 ˜ 1νi − ∂0 νi νi ≡ 2c c c− − △ 1 − + =0 . (105) A. Perturbed Action Method The wave speed for the vector mode can be read oﬀ as

The metric is decomposed as in Eq. (3) while the Æther 2c c c− α α 2 1 + field takes the form of Eq. (92) with ω replaced by ǫω . vV − . (106) ≡ 2(1 c )c We expand the action to second-order (see App. D) and − + 14 find the effective action Lastly, we investigate the scalar modes h00, φ, and F . The relevant variations are eff (0) eff 2 (2) eff 3 SÆ = SÆ + ǫ SÆ + ǫ . (98) O (2)eff δSÆ 1 ˜ (1)eff = (c14 h00 F ) , (107a) Recall that the SÆ is not important for our consider- δh00 32πG△ − ation in this paper due to the angular averaging. (2)eff δSÆ 1 We start by solving for the first-order equations of = ∂0 ∂0 ˜ (1 + c2) F + c123 ˜ φ , motions for the fields. After expanding the action, we δφ 32πG △ △ h i(107b) decompose the perturbations into the various transverse and longitudinal parts given in Eq. (95). Starting with (2)eff δSÆ 1 ˜ 1 ˜2 ij = h00 + (1 + c+ +2c2) ∂0 F the tensor mode φTT, δF 32πG △ 2 (2)eff ˜ ˜2 1 ˜ δSÆ ˜2 TT ˜ TT +(1+ c2) ∂0 φ F , (107c) ij = (1 c+)∂0 φij φij =0 , (99) △ − 2△ δφTT − − △ with F ˜ ˜ if. From the vanishing of the first two where we have focused on terms quadratic in the tensor i scalar mode≡ ∇ variations,∇ we are able to find the relations mode from the action. The ˜ in Eq. (99) is the spatial Laplacian operator associated△ with the background 1 ij h00 = F, (108a) spacetime ˜ =g ˜ ˜ i ˜ j , while the ∂˜ operator is the △ ∇ ∇ 0 c14 time derivative in this background ∂˜ = tα ˜ . We may 0 α ˜ 1+ c2 rewrite Eq. (99) in a more compact form using∇ a modified φ = F. (108b) △ − c123 wave equation, Using Eq. (108) in the final variation, Eq. (107c), we are ˜ TT 2φij =0 , (100) able to find the modified wave equation for the scalar mode where the wave operator is defined as ˜ (1 c+)(2 + 2c2 + c123)c14 ˜2 ˜ ˜ ˜2 ˜ 0F − ∂0 F F 2 (1 c+)∂0 . (101) ≡ (2 c )c − △ ≡ − − △ − 14 123 11

=0 , (109) the GW SET by using

(2)eff 4 Æ αβ δSÆ = d x gt˜ δg˜ , (111) − αβ with a propagation speed of Z p where we deﬁne the GW SET to be

Æ Æ Θαβ = 2 tαβ . (112) 2 (2 c14)c123 − vS − . (110) ≡ (1 c )(2 + 2c + c )c − + 2 123 14 To simplify the result, we decompose the resulting GW SET into its tensor (T ), vector (V ), and scalar (S) pieces. In addition to this, we make use of the equations of motion given in Eqs. (100), (105) and (109). The GW SET We know from the previous sections that we can ﬁnd is

Æ 1 ˜ ij ˜ TT (T ):Θ = αφTT βφ , (113a) αβ 32πG ∇ ∇ ij D E Æ 1 c+ ˜ i ˜ c+(2c1 c+c−) 2c14 ˜2 i (V ):Θαβ = − (2c1 c+c−) αν βνi + c+ − − tαtβ ∂0 ν νi , (113b) 16πG − ∇ ∇ 2c1 c+c− − Æ 1 2 c14 (S):Θ = − ˜ αF ˜ βF αβ 64πG c ∇ ∇ 14 2 2c14 3c2 +2c14c2 c+ +2c2c+ ˜2 3c1 2(c+ + c14) ˜ tαtβ − − ∂0 F + − F F . (113c) −c14 c123 c14 △

Note that in this situation, the GW SET has terms ex- We next expand the field equations in Eqs. (86) and (87) plicitly dependent on the tα, the Lorentz-violating back- to (h) in order to find the equations of motion for the ground Æther field. However, we will see that these terms perturbedO fields do not affect physical observables.

B. Perturbed Field Equation Method

We begin by expanding the metric as in Eq. (16) and expanding the Æther ﬁeld as

uα = tα + ω(1)α + ω(2)α . (114)

(1) (1) 1 (1)γ (1) γ (1) γ (1) γ δ (1) G S = 2∂γ∂ h ∂α∂βh ∂γ ∂ h + gαβ ∂γ∂ h ∂ ∂ h αβ − αβ 2 (α β) − − αβ − γδ h i γ (1)δ γ δ (1) 1 (1) (1)γ (1) + c t t ∂ ∂δh ∂γ ∂ t t h h¨ + t ∂ ∂γω ∂ ω˙ 1 (α β) γ − (α β)δ − 2 αβ (α β) − (α β) γ (1) (1)γ 1 (1) γ (1)δ γ δ (1) ∂γ∂ t ω c gαβ ∂γ ω˙ h¨ c t t ∂ ∂δh ∂γ ∂ t t h − (α β) − 2 − 2 − 3 (α β) γ − (α β)δ i h 1 (1) (1)γ (1) γ (1) + h¨ + t ∂ ∂γ ω + ∂ ω˙ ∂γ ∂ t ω 2 αβ (α β) (α β) − (α β) 1 γ δ ǫ (1) γ (1)δ γ (1) γ δ (1) (1) (1)γ + c tαtβ t t ∂ǫ∂ h 2t ∂δh˙ +2 t t h¨ t t t ∂ h˙ +2 t ω¨ tαtβ∂γω˙ , 4 2 γδ − γ (α β)γ − (α β) γδ (α β) − (115a)

β (1) 1 β γ (1) β (1)γ β (1) 1 (1) (1)β βK α = c t ∂γ∂ h t ∂γ∂ h + ∂β∂ ω + c ∂αh˙ + ∂α∂βω ∇ 1 2 αβ − [α β] α 2 2 12

1 β γ (1) β (1)γ (1)β 1 β γ (1) β (1) (1) c t ∂γ ∂ h t ∂γ∂ h ∂α∂βω + c t t ∂αh˙ t h¨ ω¨ . (115b) − 3 2 αβ − (α β) − 4 2 βγ − αβ − α

We are now able to decompose the first-order field equa- which agrees with Eq. (109) for a Minkowski background. tions into the modes of propagation. For the tensor We now expand the field equations to h2 to find mode, the GW SET. Retaining only the tensor modeO (T ) terms gives γ TT ¨TT ∂γ ∂ φij + c+φij =0 . (116) (T ) 1 TT ij TT TT i Θ = ∂αφ ∂βφTT 2 φ φ We notice that Eq. (116) may be written to leading order αβ 32πG ij − 2 αi β as a modified wave equation, 1 TT ij ¨TT ij ηαβ 2 φij φTT + tαtβ c+φij φTT . (123) TT −2 2φij =0 , (117) We use Eq. (117) to simplify this expression to which matches what we found in Eq. (100) with the background taken to be Minkowski. The vector modes are (T ) 1 TT ij ¨TT ij found by contracting Eqs. (115a) and (115b) with the Θαβ = ∂αφij ∂βφTT + tαtβ c+φij φTT . (124) α 32πG projector i . The results are, D E P This procedure is again repeated for the vector (V ) and 1 j scalar (S) modes to give additional terms for the GW c14 (¨γi +¨νi)+ ∂j ∂ [(1 c−) γi c−νi]=0 , (118a) 2 − − SET, 1 j c14 (¨γi +¨νi) ∂j ∂ [c−γi +2c1νi]=0 . (118b) − 2 (V ) 1 c+ i Θ = − (2c c c−) ∂αν ∂βνi αβ 16πG 1 − + Equations (118a) and (118b) are then solved simultane- 2 i +tαtβ c+ 2c4(1 c+) ν¨ νi , (125) ously to generate the relation γi = c νi, leading to the − − − + modified wave equation, (S) 1 2 (1 c+) c14 j Θαβ = (2 c14) ∂αF ∂βF 1νi − νï ∂j ∂ νi =0 , (119) 64πGc14 h − ≡ 2c1 c+c− − − c14 (1 c+)(2+2c2 + c123) − tαtβF¨ F . − c which matches Eq. (105) for a Minkowski background. 123 The scalar mode equation of motion is found by looking (126) at the scalar field equations: G(1) S(1) = 0, G(1) S(1) = ii ii 00 00 Note that all of the terms proportional to tα here differ β (1) − − 0, and βK i = 0, namely ∇ from those found in Sec IV A. This is okay, since there is no way to define a unique GW SET. However, as we will j 1 ∂j ∂ h00 (2+2c2 + c123) F¨ see in Sec. IVE, the resulting physical observables will − − 2 be the same. 1 j 1 j + ∂j ∂ F (2+2c + c ) ∂j ∂ φ¨ =0 , (120a) 2 − 2 2 123 1 (c h + F )=0 , (120b) C. Landau-Lifshitz Method 2 14 00 1 j We now construct a tensor density αµβν to obtain ∂0∂i c123∂j ∂ φ + c14 h00 + c2 F =0 . (120c) 2 a conservation law of the form of Eq. (H30) in Sec. IIC. By keeping αµβν the same as that found in Eq. (26) From Eqs. (120b) and (120c) we find the relations (as also doneH in [31] to derive the GW SET for other 1 vector-tensor theories), we obtain Eq. (27) where Gαβ is h00 = F, (121a) αβ −c14 again the Einstein tensor and tLL is the Landau-Lifshitz pseudotensor found in Eq. (28). We now substitute the i 1+ c2 ∂i∂ φ = F. (121b) field equations found in Eq. (86) which gives us (including − c123 αβ Tmat for completeness)

These combined with Eq. (120a) give the modiﬁed wave αµβν αβ αβ αβ ∂µ∂ν = 2 ( g) 8πGTmat + S + 16πG ( g) tLL equations for the scalar mode H − − αβ 1 αβ αβ (1 c+)(2 + 2c2 + c123)c14 j = 16πG ( g) Tmat + S + tLL 0F − F¨ ∂j ∂ F =0 , − 8πG ≡ (2 c14)c123 − − = 16πG ( g) T αβ + t¯αβ , (127) (122) − mat LL 13

αβ TT where t¯LL is the modified Landau-Lifshitz pseudotensor tensor mode, we only have one field φij . The resulting Euler-Lagrange equation is ¯αβ 1 αβ αβ tLL = S + tLL . (128) 1 8πG (1 c ) φ¨TT φTT =0 . (133) 32π − + ij −△ ij Following the same procedure as in Sec. IIC, we find the h i GW SET to be This is again identical to the modified wave equation of Eq. (99) around a Minkowski background. The Euler- αβ 1 αµβν ΘÆ = ∂µ∂ν . (129) Lagrange equations for the vector modes are 16πG H c14 1 As before in the Landau-Lifshitz Method sections, we (¨γi +¨νi)+ [(1 c−) γi c−νi]=0 , (134a) expand the pseudotensor in Eq. (128) to second order in 8π 16π △ − − c14 1 the metric and Æther perturbations given in Eqs. (34) (¨γi +¨νi) (c−γi +2c1νi)=0 , (134b) and (92) respectively. Once this is done, we impose the 8π − 16π △ decomposition and solve Eq. (129) for each mode. Recall for γi and νi respectively. These reduce to the same that this method provides no way to solve for the field modified field equations for the vector mode around a equations for the first order perturbations. We already Minkowski background. Using the relation γi = c+νi, calculated these equations in the previous section, and so these equations can be combined as − we can use them here to simplify the GW SET. Doing so, tensor part of the GW SET is 1νi =0 . (135)

(T ) αβ 1 α TT β ij α β TT ij The scalar Euler-Lagrange equations are ΘÆ = ∂ φ ∂ φ + t t c φ¨ φ . 32πG ij TT + ij TT D E(130) 1 (c h F )=0 , (136a) We apply this analysis to the vector and scalar modes 32π △ 14 00 − and find 1 α ∂α∂ [c123 φ +(1+ c2) F ]=0 , (136b) (V ) αβ 1 c+ i 32π △ ΘÆ = − (2c1 c+c−) ∂αν ∂βνi 1 16πG − F 2 h00 (1 + c+ +2c2) F¨ 2 i 64π △ △ − △ − +tαtβ c 2c (1 c ) ν¨ νi , (131) + − 4 − + h 2(1+ c2) φ¨ =0 , (136c) − △ i when we vary with respect to h , φ, and f respectively. (S) αβ 1 00 ΘÆ = (2 c14) ∂αF ∂βF Combining these equations gives us 64πGc14 h − c (1 c )(2+2c + c ) 14 + 2 123 0F =0 , (137) − tαtβFF¨ . − c 123 (132) with the relations from Eq. (108) found explicitly. With the equations of motion at hand, we may use Notice that the results in this section match those given Eq. (41) to solve for the GW SET. We again look at in Sec. IVB. This is because in both cases, the expansion each mode individually and use Eqs. (97), (133), (135), of the field equations was used to find the GW SET. and (137) to simplify the results 1 Θ(T ) = ∂ φTT∂ φij + c t ∂ φTTφ˙jk , D. Noether Current Method αβ 32πG α ij β TT + α β jk TT D E (138a)

The derivation of the canonical SET relies on Eq. (41), (V ) 1 c+ i Θ = − (2c c c−)∂αν ∂βνi which requires we expand the Lagrangian density to sec- αβ 32πG 1 − + ond order (see App. D) through the metric decomposition 2 i + c+ 2c 4(1 c+) tα∂βνi ν˙ , (138b) of Eq. (43) and the Æther decomposition of Eq. (92)2. − − 1 (S) We decompose the results into each tensor, vector, and Θαβ = (2c+ + c14(4+3c2 c+) scalar component before applying the Euler-Lagrange 64πGc123c14 h − equations to each individual field contribution. For the 2c2 4) ∂αF ∂βF + (2(c+ 2)+ c2(3c14c+ 2) − − − − 2 ˙ +c14(2 + c+) tα∂βF F i +4(1 + c )(1 c ) ∂βF ∂iF . (138c) 2 − 14 Pα 2 See [21] for related work on deriving the non-symmetric GW αβ αβ SET in Einstein-Æther theory using the Noether current method The above SET is indeed conserved, i.e. α(Θ(T )+Θ(V )+ without applying any perturbations. αβ ∇ Θ(S)) = 0, and thus, we can calculate the rate of change 14 of the energy and linear momentum carried away to spa- E. Derivation of Physical Quantities: E˙ , P˙ , and L˙ tial infinity by all propagating modes. If we do so, we indeed find the correct answer, i.e. the same answer as Now that we have acquired the GW SET, we can solve what one finds with all other methods to compute these for the physical observables. The first one we look at is quantities. Indeed, Foster [22] used the Noether charge E˙ , which we use Eq. (47a) to solve for. The first three method, which is related to the Noether method shown methods from Sec. IV A, IVB and IVC give E˙ as here, to find the correct E˙ . However, the above GW SET is somewhat unsatisfactory because it is clearly not sym- 2 ˙ R 1 TT ij metric in its two indices or gauge invariant, and thus, EÆ = ∂τ φij ∂τ φTT −16πG 2 vT if we tried to calculate E˙ by taking the covariant diver- I (1 c+)(2c1 c+c−) i gence with respect to the second index in the GW SET, + − − ∂τ νi ∂τ ν we would find the wrong answer. vV To fix these problems, we must apply the Belinfante 2 c14 + − ∂τ F ∂τ F . (140) procedure [12–14] (App. B). Applying this method to 4 vS c14 Einstein-Æther theory gives the GW SET The results here agree with those previously found by (T ) 1 TT TT Foster [22] using the Noether charge method [23, 24], Θ = ∂ φ ∂ φij + c t ∂ φ φ˙jk , αβ 32πG α ij β TT + (α β) jk TT which is different from the Noether current method D E (139a) adopted in this paper. Similarly, we can solve for the loss rate of linear momentum. Using the GW SET from (V ) 1 c+ i Θ = − (2c c c−) ∂αν ∂βνi αβ 16πG 1 − + Secs. IV A, IVB and IVC we find, 2 j + c+ 2c 4 (1 c+) t(α∂β)ν ν˙j , (139b) 2 − − ˙ i R i 1 TT ij 1 PÆ = N ∂τ φij ∂τ φTT (S) −16πG 2 vT Θαβ = (2c+ + c14(4+3c2 c+) Z 64πGc123c14 h − (1 c+)(2c1 c+c−) i + − − ∂τ νi ∂τ ν 2c2 4) ∂αF ∂βF + (2(c+ 2)+ c2(3c14c+ 2) − − − − vV 2 ˙ +c14(2 + c+) t(α∂β)F F 2 c14 + − ∂τ F ∂τ F . (141) i 4 vS c14 +4(1 + c2)(1 c14) ∂ F ∂iF . (139c) − P(α β) E Observe that this new GW SET is indeed symmetric, but V. CONCLUSION it is not gauge invariant because of all the terms that are proportional to t ∂ . As before, the symmetrized (α β) In this paper we studied a wide array of methods to cal- GW SET of Eq. (139) does not match that of Sec. IV A culate the GW SET in theories with propagating scalar, and IVB, but this time the differences actually do prop- vector and tensor fields. The methods include the varia- agate into observable quantities. tion of the action with respect to a generic background, Given an actual observation, however, there will be a the second-order perturbation of the field equations, the unique measured value of, for example, the energy flux calculation of a pseudotensor from the symmetries of a carried by GWs, so what went wrong? The answer is in tensor density, and the use of Noether’s theorem to de- the application of the Belinfante procedure [12, 14]. This rive a canonical GW SET. Generally, all methods yield algorithm is derived assuming the Lagrangian density is the same results, but care should be taken when deal- invariant under Lorentz transformations. Einstein-Æther ing with theories that break Lorentz symmetry. This is is indeed diffeomorphism invariant, but the solutions of because the procedure that symmetrizes the canonical this theory do spontaneously break Lorentz-symmetry. SET and makes it gauge-invariant (the so-called Belin- Therefore, when the Einstein-Æther Lagrangian density fante procedure) fails in its standard form in theories is expanded about a Lorentz-violating background solu- that are not Lorentz invariant. In addition to all of this, tion, it loses its diffeomorphism invariance, and in par- we present here and for the first time the symmetric GW ticular, it loses its Lorentz invariance, making the Belin- SET for Einstein-Æther theory, from which we calculated fante procedure inapplicable. Indeed, we find that all of the rate of energy and linear momentum carried away by the differences in observables calculated with the above all propagating degrees of freedom; the rate of energy GW SET are proportional to the Lorentz-violating back- loss had been calculated before [22] and it agrees with ground Æther field tα. One could in principle generalize the results we obtained. the Belinfante procedure to allow for the construction of The work we presented here opens the door to several SETs in Lorentz-violating theories, but this is beyond the possible future studies. One crucial ingredient in binary scope of this paper. pulsar and GW observations that has not been calculated here is the rate of angular momentum carried away by all propagating degrees of freedom. The best way to calculate this quantity is through the Landau-Lifshitz 15 pseudo-tensor, but this requires expansions to one higher tensor hαβ, the commutation of the covariant derivatives order in perturbation theory and the careful application is by definition, of short-wavelength averaging. Once this is calculated, αβ αβ α µβ β µα γ δh = δ γ h R µδγ h R µδγ h , for example in Einstein-Æther theory, one could combine ∇ ∇ ∇ ∇ − − (A3) the result with the rate of energy loss computed in this α paper to calculate the rate at which the orbital period where R βγδ is the Riemann curvature tensor that cor- and the eccentricity decay in compact binaries. These responds to the background spacetime. From the field results could then be used to constrain Einstein-Æther equations, we know that the curvature tensor is sourced theory with binary pulsar observations and GW observa- by terms of h2 . Therefore, we can see that the com- tions. The latter would require the construction of mod- mutation ofO covariant derivatives goes as els for the GWs emitted in eccentric inspirals of compact αβ αβ 3 γ δh = δ γ h + h . (A4) binaries, which in turn requires the energy and angular ∇ ∇ ∇ ∇ O momentum loss rate. For our purposes, we neglect these higher order terms, Another possible avenue for future work is to calcu- allowing us to commute covariant derivatives without the late the GW SET in more complicated theories, such as addition of curvature terms. TeVeS [32] and MOG [33, 34]. Both of these theories The second useful point of emphasis is the vanishing modify Einstein’s through the inclusion of a scalar and of total divergences, vector field with non-trivial interactions with the tensor µa1···an−1 µX =0 , (A5) sector and the matter sector. The methods studied in this h∇ i ··· paper are well-suited for the calculation of the GW SET for some n-rank tensor Xµa1 an−1 that varies on the in these more complicated theories. Once that tensor has scale of the gravitational radiation wavelength. Substi- been calculated, one could then compute the rate of en- tuting this quantity into the averaging integral generates ergy carried away by all propagating degrees of freedom four terms after integration by parts. The total diver- in this theory, and from that, one could compute the rate gence term will vanish since we turn it into a surface of orbital period decay in binary systems. Binary pulsar integral. The remaining three terms will be of (h) due observations and GW observations could then be used to to the bivector and kernel varying on scales largerO than stringently constrain these theories in a new independent the wavelengths. We show this for one term below. Let λ way from previous constraints. be the scale over which any perturbation varies while is the scale length of fluctuations of the background. WeR will assume the high-frequency limit, which tells us that ACKNOWLEDGMENTS λ . With this in mind, we see how the averaging modifies≪ R terms by looking at the scaling: N.Y and A.S. acknowledge support through the ′ ′ A µ an−1 µ ···a − ′ ′ n 1 NSF CAREER grant PHY-1250636 and NASA grants µX g µ g a −1 X ( µf) , ∇ ∼ ··· n ∇ NNX16AB98G and 80NSSC17M0041. K.Y. acknowl- ∂XA ∂f edges support from Simons Foundation and NSF grant XA , O ∂λ ∼ O ∂ PHY-1305682. R ∂XA ∂λ , (A6) ∼ O ∂λ O ∂ Appendix A: Brill-Hartle Averaging Scheme R where we have simplified notation such that A = (µ, α1, , αn−1). We have made use of the fact that ∂f This section will show some of the advantages of the is of ···(1) since it varies on scales related to the back- averaging scheme used by Isaacson [35], which he called groundO geometry. Notice the averaging process added a Brill-Hartle averaging, and we refer to as wavelength- αβ term of (λ/ ) to the original tensor being averaged. averaging. The average of some tensor X will be de- The thirdO pointR to consider is the usefulness of inte- fined in the following way, gration by parts, which says that αβ 4 α ′ β ′ A B A B X (x) d xg ′ (x, x ) g ′ (x, x ) µ X Y = X µY . (A7) ≡ α β ∇ − ∇ Z ′ ′ α β ′ ′ Notice that the total divergence becomes of (h) higher X (x ) f(x, x ) , (A1) O × than the quantities remaining. Due to this, boundary α ′ where g α′ (x, x ) is the bivector of geodesic parallel dis- terms do not need to be considered here. placement [36] that depends on the background geometry The fourth and final point to consider is that the prod- and f(x, x′) is the kernel of the integral satisfying uct of an odd number of the quantities being averaged over will vanish. This is straightforward when consid- d4xf(x, x′)=1 . (A2) ering oscillatory functions which oscillate with a single Z frequency. An integral over an odd number of these quan- There are four useful concepts to consider. The first tities vanishes, which is again what happens here due to will be the commutation of covariant derivatives. For the the fact that the averaging process involves integration. 16

Appendix B: Electromagnetic Canonical SET decomposition. Let us assume that this decomposition takes the form Consider the Lagrangian for classical electrodynamics in a source free spacetime, θαβ = hαβ + C1 g˜αβ h + C2 g˜αβ ϕ, (C2)

1 αβ whereg ˜αβ in Eq. (C2) is any background metric. Since we = FαβF , (B1) L −4 need linear order terms, we expand the action in Eq. (52) α to (h) and vary with respect to the background metric where Fαβ 2∂[αAβ], with A being the 4-vector poten- O tial. Applying≡ the canonical SET from Eq. (41) gives, to obtain the ﬁeld equations in terms of hαβ and ϕ,

α 1 α µν γα 1 ϕ j = δ βFµν F ∂βAγ F . (B2) 0= hαβ g˜αβh 2˜gαβ β −4 − −2 − − φ 0 Notice that the last term in Eq. (B2) is not gauge in- 1 γ γδ 2 + 2∂γ∂ h ∂α∂βh g˜αβ ∂γ ∂δh ∂α∂βϕ . α α α 2 (α β) − − − φ variant under the transformation A A + ∂ ǫ. The 0 solution to this is the Belinfante procedure→ [12, 13]. (C3) We here use the Guarrera and Hariton [14] implemen- tation of the Belinfante procedure. The symmetric tensor Any derivative here is with respect to the background is defined as metric. At this point, we substitute in the reduced field using Eq. (C2) αβ (α β) γ (α β)δ γ αβ ΘGH = π ∂ A + ∂δ π M A + g , (B3) − γ γ L 1 2 8C1C2 α γ αβ 0= φ0 θαβ + 2 C2 g˜αβϕ where πγ ∂ /∂(∂αA ) and M is the spin tensor for −2 − φ0 − 1+4C1 the 4-vector≡ potential.L Applying Eq. (B3) and using the 2C1 C1 1 Lorenz gauge and equations of motion in a source free + 1 g˜αβθ + φ0 ∂α∂β θ 1+4C − 1+4C − 2 medium, 1 1 1 4C1C2 γ α + C2 ϕ + ∂γ∂(αθβ) ∂αA =0 , (B4a) − φ0 − 1+4C1 α β ∂ ∂ A =0 , (B4b) φ0 α + g˜ ∂ ∂ θγδ . (C4) 2 αβ γ δ we arrive at the classical result for the electromagnetic SET We know we are looking for the only term to be θαβ. This allows for us to solve for C1 and C2 by eliminating αβ αγ β 1 αβ γδ ΘGH = F F γ g F Fγδ . (B5) the terms proportional to θ and ϕ in the first bracket, − 4 leading to Equation (B3) was found using the assumption that the theory in question is Lorentz invariant. This was neces- 1 C1 = , (C5a) sary in order to derive the correct M αβ tensors, which are −2 the generators of infinitesimal Lorentz transformations. 1 C2 = . (C5b) −φ0

Appendix C: Derivation of Scalar-Tensor Reduced With the inclusion of these constants, the linearized re- Field duced ﬁeld equations become

In accordance with the procedures outlined in [29], we 2 γ γδ θαβ ∂γ ∂(αθβ) +˜gαβ ∂γ ∂δ θ =0 . (C6) consider the notion of a “reduced field.” The reduced field − φ0 is the field which enables the field equations to be written as We are now free to impose the Lorenz gauge condition for the reduced field. This eliminates the final two terms 2 1 ∂ i in Eq. (C6), leading to the field equations for the reduced 2 + ∂i∂ = 16π , (C1) −vg ∂t A − S field. where is the linear perturbation of the reduced field, vg is the speedA of propagation of the reduced field, and the source is the combination of matter and higher order Appendix D: Expanded Action for Einstein-Æther perturbationS effects. Notice that in GR, the “reduced field” is the trace-reversed metric perturbation. Here, we state the expanded action for Einstein-Æther. For the work in Sec. III, the reduced field will need These results were obtained through the use of the xTen- to be derived. This is accomplished through the use of sor package for Mathematica [37, 38]. 17

(0) 1 4 α β α β β α α β γ SÆ = d x g˜ R˜ c ( ˜ αu˜β)( ˜ u˜ ) c ( ˜ αu˜ )( ˜ β ˜ ) c ( ˜ αu˜β)( ˜ u˜ )+ c u˜ u˜ ( ˜ αu˜ )( ˜ β u˜γ) , 16πG − − 1 ∇ ∇ − 2 ∇ ∇ ∇ − 3 ∇ ∇ 4 ∇ ∇ Z p (D1a)

(1) 1 4 1 αβ αβ α β γ α γ β SÆ = d x g˜ h R˜ h R˜αβ + ˜ α ˜ βh ˜ h + c hαγ( ˜ u˜ )( ˜ u˜β)+2˜u ( ˜ h )( ˜ u˜ ) 16πG − 2 − ∇ ∇ − 1 ∇ ∇ ∇[β α]γ ∇ Z p 1 β α γ β α α γ β α β h( ˜ β u˜α)( ˜ u˜ ) hαγ( ˜ βu˜ )( ˜ u˜ ) u˜ ( ˜ γ hαβ)( ˜ u˜ ) 2( ˜ αωβ)( ˜ u˜ ) −2 ∇ ∇ − ∇ ∇ − ∇ ∇ − ∇ ∇ α β 1 α β α β c u˜ ( ˜ αh)( ˜ βu˜ )+ h( ˜ αu˜ )( ˜ β u˜ ) 2 ( ˜ αω )( ˜ βu˜ ) − 2 ∇ ∇ 2 ∇ ∇ − ∇ ∇ α γ β α γ β 1 β α β α +c 2˜u ( ˜ h )( ˜ u˜ ) u˜ ( ˜ αhβγ)( ˜ u˜ ) h( ˜ αu˜β)( ˜ u˜ ) 2 ( ˜ αωβ)( ˜ u˜ ) 3 ∇[γ β]α ∇ − ∇ ∇ − 2 ∇ ∇ − ∇ ∇ 1 α β γ α β γ δ α β γ δ +c h u˜ u˜ ( ˜ αu˜ )( ˜ β u˜γ)+ hγδ u˜ u˜ ( ˜ αu˜ )( ˜ β u˜ )+2˜u u˜ u˜ ( ˜ αu˜ )( ˜ γ hβδ) 4 2 ∇ ∇ ∇ ∇ ∇ ∇ α β γ δ α β γ α β γ u˜ u˜ u˜ ( ˜ αu˜ )( ˜ δhβγ)+2 ω u˜ ( ˜ αu˜ )( ˜ β u˜γ)+2˜u u˜ ( ˜ αωγ)( ˜ β u˜ ) , (D1b) − ∇ ∇ ∇ ∇ ∇ ∇ i (2) 1 4 β αγ 1 αβ 1 2 1 α 1 αβ SÆ = d x g˜ hα h R˜βγ hh R˜αβ + h R˜ + ( ˜ αh)( ˜ h) ( ˜ αh)( ˜ βh ) 16πG − − 2 8 4 ∇ ∇ − 2 ∇ ∇ Z p 1 γ αβ 1 α βγ 1 β α δ γ 1 α β γδ + ( ˜ αhβγ)( ˜ h ) ( ˜ αhβγ)( ˜ h )+ c hαγ hβδ( ˜ u˜ )( ˜ u˜ ) u˜ u˜ ( ˜ αh )( ˜ β hγδ) 2 ∇ ∇ − 4 ∇ ∇ 1 4 ∇ ∇ − 4 ∇ ∇ 1 αβ γ δ 1 2 α β 1 β γ α β α δ β + hαβ h ( ˜ γ u˜δ)( ˜ u˜ ) h ( ˜ αu˜β)( ˜ u˜ ) hhαβ( ˜ γ u˜ )( ˜ u˜ ) hαβ hγ ( ˜ u˜ )( ˜ u˜δ) 4 ∇ ∇ − 8 ∇ ∇ 2 ∇ ∇ − ∇ ∇ 1 α γ β α γ β β γ α δ 1 α γ β + hhαβ( ˜ u˜ )( ˜ u˜γ) h u˜ ( ˜ h )( ˜ u˜ )+2 hα u˜ ( ˜ h )( ˜ u˜ )+ h u˜ ( ˜ β hαγ)( ˜ u˜ ) 2 ∇ ∇ − ∇(α γ)β ∇ ∇[γ β]δ ∇ 2 ∇ ∇ β γ α δ α β δ γ α β β γ α +hα u˜ ( ˜ u˜ )( ˜ βhγδ)+˜u u˜ ( ˜ h )( ˜ hα ) h( ˜ αωβ)( ˜ u˜ ) 2 hαβ( ˜ γ ω )( ˜ u˜ ) ∇ ∇ ∇[γ δ]β ∇ − ∇ ∇ − ∇ ∇ α γ β α β γ α γ β α γ β +2 ω ( ˜ h )( ˜ u˜ ) ω ( ˜ β hαγ)( ˜ u˜ )+2 hαβ( ˜ u˜ )( ˜ ωγ)+2˜u ( ˜ h )( ˜ ω ) ∇[β α]γ ∇ − ∇ ∇ ∇ ∇ ∇[β α]γ ∇ α β γ α β 1 αβ γ δ 1 α β u˜ ( ˜ βhαγ )( ˜ ω ) ( ˜ αωβ)( ˜ ω ) + c hαβ h ( ˜ γ u˜ )( ˜ δ u˜ ) u˜ u˜ ( ˜ αh)( ˜ β h) − ∇ ∇ − ∇ ∇ 2 4 ∇ ∇ − 4 ∇ ∇ αβ γ δ 1 α β 1 2 α β α β α β +h u˜ ( ˜ γ hαβ)( ˜ δu˜ ) h u˜ ( ˜ αh)( ˜ β u˜ ) h ( ˜ αu˜ )( ˜ βu˜ ) u˜ ( ˜ αh)( ˜ βω ) h( ˜ αu˜ )( ˜ βω ) ∇ ∇ − 2 ∇ ∇ − 8 ∇ ∇ − ∇ ∇ − ∇ ∇ α β α β 1 αβ δ γ 1 α β γδ ω ( ˜ αh)( ˜ βu˜ ) ( ˜ αω )( ˜ β ω ) + c hαβ h ( ˜ γ u˜δ)( ˜ u˜ ) u˜ u˜ ( ˜ αh )( ˜ β hγδ) − ∇ ∇ − ∇ ∇ 3 4 ∇ ∇ − 4 ∇ ∇ 1 2 β α α γ β 1 α γ β β γ α δ h ( ˜ αu˜β)( ˜ u˜ ) h u˜ ( ˜ h )( ˜ u˜ ) hu ( ˜ γ hαβ)( ˜ u˜ )+2 hα u˜ ( ˜ h )( ˜ u˜ ) −8 ∇ ∇ − ∇(α β)γ ∇ − 2 ∇ ∇ ∇(γ δ)β ∇ β γ α δ α β γ δ α β γ α β γ hα u˜ ( ˜ βhγδ)( ˜ u˜ )+˜u u˜ ( ˜ h )( ˜ hα )+2 ω ( ˜ h )( ˜ u˜ ) ω ( ˜ αhβγ)( ˜ u˜ ) − ∇ ∇ ∇[γ δ]β ∇ ∇[β γ]α ∇ − ∇ ∇ α β γ α γ β β α β α +2˜u ( ˜ h )( ˜ ω ) u˜ ( ˜ β hαγ)( ˜ ω ) h( ˜ αωβ)( ˜ u˜ ) ( ˜ αωβ)( ˜ ω ) ∇[β α]γ ∇ − ∇ ∇ − ∇ ∇ − ∇ ∇ 1 α β γ δ ǫ α β γ δ ǫ 1 αβ γ δ ǫ +c u˜ u˜ u˜ u˜ ( ˜ ǫhαβ)( ˜ hγδ)+2˜u u˜ u˜ u˜ ( ˜ βhα )( ˜ h ) hαβ h u˜ u˜ ( ˜ γ u˜ )( ˜ δu˜ǫ) 4 4 ∇ ∇ ∇ ∇[δ ǫ]γ − 4 ∇ ∇ 1 2 α β γ 1 γ β α β α β γ δ + h u˜ u˜ ( ˜ αu˜ )( ˜ β u˜γ)+ hhαβ u˜ u˜ ( ˜ γ u˜ )( ˜ δ u˜ )+ h u˜ u˜ u˜ ( ˜ αu˜ )( ˜ γ hβδ) 8 ∇ ∇ 2 ∇ ∇ ∇ ∇ 1 α β γ δ γ δ α β α β γ α β γ h u˜ u˜ u˜ ( ˜ αu˜ )( ˜ δhβγ)+2 hαβ u˜ ω ( ˜ γ u˜ )( ˜ δu˜ )+ h u˜ ω ( ˜ αu˜ )( ˜ βu˜γ )+4˜u ω ( ˜ u˜ )( ˜ ωγ) −2 ∇ ∇ ∇ ∇ ∇ ∇ ∇(α ∇β) γ δ α β α β γ α β γ δ α α γ δ +2 hαβ u˜ u˜ ( ˜ γ u˜ )( ˜ δω )+ h u˜ u˜ ( ˜ αu˜ )( ˜ βωγ )+4˜u u˜ ω ( ˜ u˜ )( ˜ hβδ)+˜u u˜ u˜ ( ˜ αω )( ˜ γ hβδ) ∇ ∇ ∇ ∇ ∇(α ∇γ) ∇ ∇ α β γ δ α β γ δ α β γ δ α β γ u˜ u˜ ω ( ˜ γ u˜ )( ˜ δ hαβ) 4˜u u˜ ω ( ˜ αu˜ )( ˜ h ) u˜ u˜ u˜ ( ˜ αω )( ˜ δhβγ)+˜u u˜ ( ˜ αω )( ˜ βωγ ) − ∇ ∇ − ∇ ∇[β δ]γ − ∇ ∇ ∇ ∇ α β γ +ω ω ( ˜ αu˜ )( ˜ β u˜γ) . (D1c) ∇ ∇ i 18

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