The Post-Newtonian Approximation in Gravity

Sohan Vartak CERN Summer Student Programme 2015 University of Maryland, College Park

Abstract We briefly review the post-Newtonian expansion as a way to yield the approximate equations of motion from a metric theory of gravity in the slow-motion, weak-field limit. Then, the parameterized post-Newtonian (PPN) formalism is introduced, as it provides a method of characterizing metric theories of gravity (again, in the slow-motion, weak-field limit). As we are interested in examining Lorentz-violating theories, we reproduce the calculations of the 10 PPN parameters for Einstein-aether theory and Ho˘rava gravity. Finally, the field equations for Lorentz-violating massive gravity are derived from the Lagrangian.

1 1 Introduction

Thus far, the predictions of general relativity (GR) have been in very good agreement with exper- imental observations. The problem with GR is that it does not seem compatible with a quantum theory of gravity. Therefore, many alternative theories have been proposed which may be more easily quantizable. A significant portion of these theories suggest that Lorentz invariance may not in fact be a symmetry of nature; these Lorentz-violating theories are our primary interest in this exploration of alternatives to GR. We first introduce the post-Newtonian expansion, which yields the approximate equations of motion in the slow-motion, weak-field limit. This limit is sufficiently accurate to make appropriate predictions for solar system tests of a theory. Following this, we briefly review the parameterized post-Newtonian formalism, which allows for the classification of a general metric theory of gravity based on the values of 10 parameters which appear in a post- Newtonian expansion of the metric. Since metric theories differ only in the way that matter (and possible other fields) generate the metric, these parameters allow one to characterize theories according to their physical predictions (such as Lorentz violation, momentum conservation violation, and non-linearity in the superposition principle). The PPN parameters are then computed for Einstein-aether theory and Ho˘rava gravity. Finally, Lorentz-violating massive gravity is briefly explored. The Lagrangian for a specific theory is used to derive the field equations, which are then utilized in the computation of the PPN parameters. We use the conventions of Blas and Sibiryakov [5].

2 The Post-Newtonian Expansion

NOTE: This section uses the notations of Weinberg[1]. Suppose we have a system of slowly moving particles bound by gravitational forces. Taking the slow-motion, weak-ﬁeld limit, we can expand in powers of the velocity and determine the equations of motion to one order higher than that given by Newtonian mechanics. This is the post-Newtonian expansion. The equations of motion are given by the geodesic equation:

d2xµ dxν dxλ + Γµ = 0 (1) dτ 2 νλ dτ dτ where τ denotes the proper time. This implies that we can compute the accelerations using

d2xi dxν dxλ dxν dxλ dxi = −Γi + Γ0 . (2) dt2 νλ dt dt νλ dt dt dt We must compute the Christoﬀel symbols to various orders so that we can obtain our post- Newtonian approximation. We treat the metric tensor gµν as consisting of the background Minkowski metric ηµν with the corrections being expandable in powers of the velocity. Thus we have

2 4 g00 = 1 + g00 + g00 + ... (3) 2 4 gij = −δij + gij + gij + ... (4) 1 3 g0i = g0i + g0i + ... (5)

N N where g µν denotes the term of order v in gµν. We can then apply the formula

2 1 Γµ = gµρ (g + g − g ) (6) νλ 2 ρν,λ ρλ,ν νλ,ρ to determine the Christoffel symbols. We see now that we must determine the metric in order to carry out this computation. Using the above formula and expanding the Christoffel symbols in series as well, we find the relations

2 2 1 ∂g Γi = − 00 (7) 00 2 ∂xi 4 3 2 4 1 ∂g ∂g 1 ∂g Γi = − 00 + 0i + g 00 (8) 00 2 ∂xi ∂t 2 ij ∂xj  3 2 3  3 1 ∂g ∂g ∂g Γi = 0i + ij − 0j (9) 0j 2  ∂xj ∂t ∂xi 

 2 2 2  2 1 ∂g ∂g ∂g Γi = ij + ik − jk (10) jk 2  ∂xk ∂xj ∂xi 

2 3 1 ∂g Γ0 = − 00 (11) 00 2 ∂t 2 2 1 ∂g Γ0 = − 00 (12) 0i 2 ∂xi 1 0 Γ ij = 0. (13) Next, the Ricci tensor is also expanded in a series and computed in terms of the Christoﬀel symbols:

2 2 ∂Γi R = 00 (14) 00 ∂xi 3 4 4 ∂Γi ∂Γi 2 2 2 2 R = − 0i + 00 − Γ0 Γi + Γi Γj (15) 00 ∂t ∂xi 0i 00 00 ij 2 3 3 ∂Γj ∂Γj R = − ij + 0i (16) 0i ∂t ∂xj 2 2 2 2 ∂Γ0 ∂Γk ∂Γk R = − i0 − ik + ij . (17) ij ∂xj ∂xj ∂xk Combining the above two sets of equations yields a system that we can solve for the metric µν λ components. We also impose the harmonic coordinate condition g Γ µν = 0 so that the equations are simpliﬁed:

2 1 2 R = − ∇2g (18) 00 2 00

3 2 2 2 2 4 1 4 1 ∂ g 1 2 ∂ g 1 2 2 R = − ∇2g + 00 + g 00 − ∇2g (19) 00 2 00 2 ∂t2 2 ij ∂xixj 2 00 3 1 3 R = − ∇2g (20) 0i 2 0i 2 1 2 R = − ∇2g . (21) ij 2 ij We can now make use of the Einstein equations to relate the metric components to the energy- momentum tensor (which is also expanded in a series):

0 2 2 00 ∇ g00 = −8πGN T (22) 2 2 2 2 2 ! 2 ! 4 ∂ g 2 ∂ g ∂g ∂g ∇2g = 00 + g 00 − 00 00 00 ∂t2 ij ∂xixj ∂xi ∂xi

2 0 2 ! 00 2 00 ii − 8πGN T − 2g00T + T (23)

1 2 3 0i ∇ g0i = 16πGN T (24) 0 2 2 00 ∇ gij = −8πGN δijT . (25)

These equations can be used to solve for the components of the metric and express it in terms of various potentials. The results can then be used to derive the post-Newtonian equations of motion. This computation can also be performed in Brans-Dicke theory, which introduces a scalar field in addition to the metric. We write the scalar field as G−1(1 + ξ) where G is a constant of order µ 8πG µ GN and ξ is a scalar field defined by ξ;;µ = 3+2ω T µ and ξ → 0 as r → ∞. The field equations of Brans-Dicke theory are given as

1 1 R − g R =8πG(1 + ξ)−1T + ω(1 + ξ)−2(ξ ξ − g ξ ξ ρ) µν 2 µν µν ;µ ;ν 2 µν ;ρ ; −1 ρ + (1 + ξ) (ξ;µ;ν − gµνξ;;ρ). (26)

2 4 To perform the post-Newtonian computation, we expand ξ = ξ + ξ + ... and repeat the process from earlier. We arrive at the following equations relating the metric and energy-momentum tensors:

2 2ω + 4 0 ∇2g = −8πG T (27) 00 2ω + 3 00 2 2 2 2 4 ∂ g 2 ∂ g 2 ∇2g = 00 + g 00 − (∇g )2 00 ∂t2 ij ∂xixj 00 2ω + 4 2 0 2 2ω + 2 + 8πG ξT 00 − 8πGT ii 2ω + 3 2ω + 3

4 0 2 2 2ω + 4 2ω + 4 + 16πGg T 00 − 8πG T 00 00 2ω + 3 2ω + 3  2 2 2 2 ∂ξ ∂2ξ 2 ∂ξ − 2ω − 2 + 2Γi (28)  ∂t  ∂t2 00 ∂xi

2 1 2 3 ∂ ξ ∇2g = 16πGT 0i − 2 (29) 0i ∂xi∂t 2 0 2 2 2ω + 2 ∂ ξ ∇2g = −8πGT 00δ − 2 . (30) ij ij 2ω + 3 ∂xi∂xj

It also follows immediately from the deﬁnition of ξ that we have

2 8πG 0 ∇2ξ = − T 00. (31) 2ω + 3 Once again, these equations can be solved to express both the metric and ξ in terms of various potentials up to post-Newtonian order.

3 The Parameterized Post-Newtonian Formalism

The form of the metric up to post-Newtonian order in a general theory of gravity can be summa- rized by 10 parameters (α1, α2, α3, β, γ, ζ1, ζ2, ζ3, ζ4, ξ) when the metric is given in terms of speciﬁc potential functions. This formalism is discussed in more depth by Will[2]. The parameters appear as follows:

2 g00 = 1 − 2U + 2βU + 2ξΦW − (2γ + 2 + α3 + ζ1 − 2ξ)Φ1

− 2(3γ − 2β + 1 + ζ2 + ξ)Φ2 − 2(1 + ζ3)Φ3 − 2(3γ + 3ζ4 − 2ξ)Φ4 + (ζ1 − 2ξ)A (32)

gij = −(1 + 2γU)δij (33) 1 1 g = (4γ + 3 + α − α + ζ − 2ξ)V + (1 + α + ζ + 2ξ)W . (34) 0i 2 1 2 1 i 2 2 1 i Physically, γ represents the amount of space curvature produced by a unit rest mass. β represents non-linearity in the superposition principle for gravity. ξ represents the presence of preferred- location effects. α1,2,3 represent the presence of preferred-frame effects, and ζ1,2,3,4 and α3 represent the presence of violation of total momentum conservation. Note that α3 thus contributes to two different physical effects. In general relativity, the only non-zero parameters are γ = β = 1. The potentials appearing in the metric all have the form

Z ρ(y)f F (x) = G d3y . (35) N |x − y| The correspondences F ↔ f are given by

U ↔ 1 (36)

5 Φ1 ↔ vivi (37)

Φ2 ↔ U (38)

Φ3 ↔ Π (39) p Φ ↔ (40) 4 ρ Z (x − y) (y − z) (x − z) Φ ↔ d3zρ(z) j j − j (41) W |x − y|2 |x − z| |y − z| (v (x − y) )2 A ↔ i i (42) |x − y|2 i Vi ↔ v (43) v (x − y )(xi − yi) W ↔ j j j . (44) i |x − y|2

Note that for U, Φ1,2,3,4, and Vi, we have

F,ii = −4πGN ρf. (45) We also deﬁne the superpotential Z 3 χ = −GN d yρ|x − y| (46)

which satisﬁes

χ,ii = −2U (47)

χ,0i = Vi − Wi. (48)

3.1 Einstein-aether Theory NOTE: This section uses the notations of Jacobson and Foster[3]. Einstein-aether theory is a vector-tensor theory of gravity in which spacetime is endowed with a b both a metric and a unit timelike vector ﬁeld called the aether (that is, gabu u = 1). The aether gives rise to a preferred reference frame, thereby making Einstein-aether theory a Lorentz-violating theory of gravity. There are two ﬁeld equations: one which is equivalent to the Einstein equations and another that governs the aether. The Einstein equations can be written as 1 R = (S + 8πGT )(δc δd − g gcd) (49) ab cd cd a b 2 ab where

m m m Sab = ∇m(J(a ub) − J (a ub) − J(ab)u ) m m + c1(∇mua∇ ub − ∇aum∇bu ) + c4u˙ au˙ b 1 + λu u + g (J n ∇ um) (50) a b 2 ab m n

6 a ab a b a b a b n J m = (c1g gmn + c2δ mδ n + c3δ nδ m + c4u u gmn)∇bu (51) a b a u˙ = u ∇bu . (52)

The aether ﬁeld equation is

a a ∇aJ m − c4u˙ a∇mu = λum. (53) The energy-momentum tensor is given by T ab = (ρ + ρΠ + p)vavb − pgab where va is the four- velocity, ρ is the rest-mass-energy density, Π is the internal energy density, and p is the isotropic pressure of the ﬂuid. Writing the metric tensor as gab = ηab + hab, we can compute the post- Newtonian parameters of this theory. We will also impose the gauge conditions

1 h = − (h − h ) (54) ij,j 2 00,i jj,i h0i,i = −3U,0 + θni,i (55)

where θ is an arbitrary parameter and ni = ui − h0i. First we determine ua to O(1) and its covariant derivatives up to various orders. The constraint that the aether be a unit timelike vector ﬁeld implies that up to O(1), we have 1 u0 = 1 − h . (56) 2 00 We also ﬁnd

u0 = 1 + h00 (57)

ui = ni + h0i. (58)

We now present the covariant derivatives:

∇au0 = 0 (59) 1 1 ∇ u = − h 1 − h + h + n (60) 0 i 2 00,i 2 00 0i,0 i,0 1 u˙ = − h (1 − h ) + h + n (61) i 2 00,i 00 0i,0 i,0 1 ∇ u = n + h + h . (62) j i i,j 2 ij,0 0[i,j] The ﬁrst three formulas are to O(2) while the last is to O(1.5). To O(1), we have

1 R = h (63) 00 2 00,ii T00 = ρ (64)

Tij = 0 (65) c S = 14 h (66) 00 2 00,ii

7 Sij = 0. (67)

The Einstein equation then reads c 1 − 14 h = 8πGρ (68) 2 00,ii

c14 −1 1 and so we ﬁnd to O(1) that h00 = −2U and GN = 1 − 2 G. To O(1), we have Rij = 2 hij,kk. Thus the Einstein equation reads c h = δ 14 h + 8πGρδ . (69) ij,kk ij 2 00,kk ij Taking the trace of this yields

c h = 3 14 h + 8πGρ ii,kk 2 00,kk = 3(8πGN ρ) (70)

using our earlier expression for h00,kk. Thus, we ﬁnd

hij,kk = 8πGN ρδij (71)

and so hij = −2Uδij to O(1). Working to O(1.5), we examine the aether ﬁeld equation, which has the form

J0i,0 − Jji,j = 0. (72)

c14 1 Computing, we ﬁnd that J0i,0 = − 2 χ,0ijj and Jji,j = c1ni,jj + c23nj,ji + 2 (2(c1 − c3)h0[i,j]j + (c1 + 3c2 + c3)χ,0ijj). The aether ﬁeld equation then becomes

c1 − c3 1 c1 − c3 c1ni + h0i + (2c1 + 3c2 + c3 + c4)χ,0i − h0j,j − c23nj,j = 0. (73) 2 2 ,jj 2 ,i

Taking the spatial divergence yields the relation n = Aχ where A = − 2c1+3c2+c3+c4 . i,ijj ,0iijj 2c123 Substituting this back into our previous equation and rearranging yields 1 3 ni = − ((c1 − c3)h0i − (2c1A + (c1 − c3)( + Aθ))χ,0i) (74) 2c1 2

and so we have a higher-order expression for the aether. We now seek to ﬁnd g0i to O(1.5). The Ricci tensor has components

1 1 R0i = h0i + (1 − 2Aθ)χ,0i . (75) 2 2 ,jj 2 2 1 c1−c3 c14 We also have T0i = ρvi and S0i = − (J0i,0 +Jij,j). Now, Jij,j = − h0i + − E χ,0i 2 2c1 2 ,jj where E = 1 (c2 + 3c2 + 4c c − 4(c2 − c2)Aθ). Using our above expression for J , we ﬁnd 4c1 1 3 1 4 1 3 0i,0 2 2 c1−c3 E S0i = h0i + χ,0i . The ﬁeld equation can then be written as 4c1 2 ,jj

8 c2 − c2 1 1 − 1 3 h = 16πGρv + E + Aθ − χ . (76) 2c1 0i,jj i 2 ,0ijj We thus ﬁnd

2 2 −1 c1 − c3 1 c14 h0i = 1 − E + Aθ − χ,0i + 4 1 − Vi . (77) 2c1 2 2

From the relation χ,0i = Vi − Wi, we can rewrite this as 2c1 1 1 h0i = 2 2 E + Aθ − + 2(2 − c14) Vi − E + Aθ − Wi . (78) 2c1 − c1 + c3 2 2

We now seek to ﬁnd g00 to O(2). To this order, we have

1 1 1 1 1 R00 = h00,ii + hijh00,ij − hi0,i − hii,0 − h00,ih00,i + h00,j(2hij,i − hii,j). (79) 2 2 2 ,0 4 4

Deﬁning h˜00 = g00 − 1 + 2U, imposing our gauge conditions from earlier, and using previous results, we can write 1 R = (h˜ − 2U − 2U 2 + 8Φ − 2Aθχ ) . (80) 00 2 00 2 ,00 ,ii

To this order, we also have T00 = ρ(1 + Π + vivi − 2U) and Tij = ρvivj + pδij which implies

1 c 8πG T − g T gab = − 1 − 14 (U + 2Φ − 2Φ + Φ + 3Φ ) (81) ab 2 00 ab 2 1 2 3 4 ,ii We also compute (after some algebra)

1 c 1 3 S − g S gab = 14 (−2U+h˜ −2U 2+8Φ ) − + A (2c + 3c + c + c ) + c Aθ χ . 00 2 00 ab 4 00 2 ,ii 2 2 1 2 3 4 14 ,00ii (82) Combining the above results and plugging into the ﬁeld equation, we ﬁnd

1 c (h˜ − 2U − 2U 2 + 8Φ − 2Aθχ ) = − 1 − 14 (U + 2Φ − 2Φ + Φ + 3Φ ) 2 00 2 ,00 ,ii 2 1 2 3 4 ,ii c + 14 (−2U + h˜ − 2U 2 + 8Φ ) 4 00 2 ,ii 1 3 − + A (2c + 3c + c + c ) + c Aθ χ . 2 2 1 2 3 4 14 ,00ii (83)

A simple rearrangement yields the result

2 h˜00 = 2U − 4Φ1 − 4Φ2 − 2Φ3 − 6Φ4 + Qχ,00 (84) −1 where Q = 1 − c14 ((2 − c )θ + (c + 2c − c ))A. We can now choose θ = − c1+2c3−c4 so 2 14 1 3 4 2−c14 that the last term vanishes.

9 Summarizing our results, we have found

2 g00 = 1 − 2U + 2U − 4Φ1 − 4Φ2 − 2Φ3 − 6Φ4 (85)

gij = −(1 + 2U)δij (86) 2c1 1 1 g0i = 2 2 E + Aθ − + 2(2 − c14) Vi − E + Aθ − Wi . (87) 2c1 − c1 + c3 2 2 Comparing this with the standard PPN form of the metric, we can determine the PPN parameters as

β = 1 (88) γ = 1 (89) 2 8(c3 + c1c4) α1 = − 2 2 (90) 2c1 − c1 + c3 2 2 2 (2c13 − c14) 12c13c3 + 2c1c14(1 − 2c14) + (c1 − c3)(4 − 6c13 + 7c14) α2 = − 2 2 (91) c123(2 − c14) (2 − c14)(2c1 − c1 + c3) with the remaining parameters being 0. We also have expressions for the aether:

u0 = 1 + U (92) i u = −(A + B)Vi + (A − B)Wi (93)

(2−c14)(c1−c3) where B = − 2 2 . 2c1−c1+c3

3.2 Ho˘rava Gravity NOTE: This section uses the notations of Blas and Sanctuary[4]. Ho˘rava gravity (or khronometric theory) posits the existence of an absolute time foliation of spacetime which arises from a scalar ﬁeld ϕ, called the khronon. This makes the theory Lorentz- violating. The khronon satisﬁes

µ ∂µϕ∂ ϕ > 0 (94) and it appears in the action in the form

∂ ϕ u = µ . (95) µ p ρ ∂ρϕ∂ ϕ

We see that uµ is a timelike vector. The action for Ho˘rava gravity is given by

M 2 Z √ S = − b d4x −g R + Kµν ∇ uσ∇ uρ + S (96) 2 σρ µ ν m

where Mb is an arbitrary mass parameter and

10 µν µ ν µ ν µ ν K σρ = βδ ρδ σ + λδ σ δ ρ + αu u gσρ (97) with α, β, and λ being free dimensionless constants. The khronon energy-momentum tensor χ Tµν is given by

1 T χ = − ∇ K uρ + Kρ u − K ρu + g Kρ ∇ uσ µν ρ (µν) (µ ν) (µ ν) 2 µν σ ρ ρ σ ρ σ ρ + αaµaν + 2∇ρK (µ uν) − uµuνu ∇ρK σ − 2αaσu(µ∇ν)u + αa aρuµuν (98)

µ µν ρ ρ where K σ = K σρ ∇νu and aµ = u ∇ρuµ. The ﬁeld equation that results from varying the action with respect to the metric is

−2 χ m Gµν − Mb Tµν + Tµν = 0 (99) 1 where Gµν = Rµν − 2 gµνR is the Einstein tensor. The equation of motion for the khronon ﬁeld is 1 µν σ σ ∇µ √ P [∇σK − αaσ∇νu ] = 0 (100) X ν µν µν µ ν ρ where P = g − u u and X = ∂ρϕ∂ ϕ. In performing computations, we expand the metric and khronon as gµν = ηµν + hµν and ϕ = t + χ. The source is assumed to have the same energy-momentum tensor as that used in the Einstein-aether calculation:

T µν = (ρ + ρΠ + p)vµvν − pgµν. (101) In computing the PPN parameters for this theory, we impose the gauge conditions

2 1 2 2 ∂ h = − ∂ h − ∂ h (102) j ij 2 i 00 i kk 3 2 ∂ih0i = Γ∂0h00 (103)

α2+2αβ+6αλ−2β2−6βλ−6β−6λ where Γ is an arbitrary constant. We will later set Γ = 2(α−2)(β+λ) to move to the PPN gauge. Note that since χ now denotes part of the khronon, we denote the superpotential by H. Finding g00 to O(1) using the ﬁeld equation, we ﬁnd

2 ∆h00 = 8πGN ρ (104) −2 α −1 where 8πGN = Mb 1 − 2 . Solving gij to O(1), we ﬁnd

2 ∆hij = 8πGN ρδij. (105) Note that these two equations are identical to those found in the Einstein-aether calculation. The khronon is to be found to O(1.5), and its equation of motion to this order is

3 2 3λ + α + β 2 (∆χ − Γ∂ h ) = − ∂ h . (106) 0 00 2(λ + β) 0 00

11 The solution is then given by

3 (α − 2β)(2 + β + 3λ) χ = H˙ (107) 2(α − 2)(β + λ) R 3 where H = −GN d yρ|x − y|. Solving the ﬁeld equation once more (this time determining g0i) to O(1.5), we ﬁnd

3 2 −1 ∆h0i = (β − 1) 8πGN ρvi(α − 2) − [−2 + α + Γ(1 − β)]∂0∂ih00 . (108)

Finally, we solve g00 to O(2). From the ﬁeld equation, we ﬁnd

4 2 2 2 2 ∆h00 =∂ih00∂ih00 − h00∆h00 − 4∆Φ1 + 4∆Φ2 − 2∆Φ3 − 6∆Φ4 −α2 + 2β(3 + β − 2Γ) + 2(3 + 3β − 2Γ)λ + 2α(β(Γ − 1) + (Γ − 3)λ) + ∆∂2H. (109) (α − 2)(β + λ) 0

With the aforementioned choice of Γ, the last term vanishes in the PPN gauge. Summarizing our results, we have found

2 g00 = 1 − 2U + 2U − 4Φ1 + 4Φ2 − 2Φ3 − 6Φ4 (110)

gij = −(1 + 2U)δij (111) 1 1 g = (7 + α − α )V + (1 + α )W (112) 0i 2 1 2 i 2 2 i (α − 2β)(2 + β + 3λ) χ = H.˙ (113) 2(α − 2)(β + λ)

Comparing these to the standard PPN form of the metric, we have found that the only nonzero PPN parameters in Ho˘rava gravity are

β = 1 (114) γ = 1 (115) 4(α − 2β) α = (116) 1 β − 1 (α − 2β)[−β(3 + β + 3λ) − λ + α(1 + β + 2λ)] α = . (117) 2 (α − 2)(β − 1)(β + λ)

4 Lorentz-Violating Massive Gravity

As their name suggests, massive gravity theories endow the graviton with mass. We examine a speciﬁc form of Lorentz-violating massive gravity as introduced by Blas and Sibiryakov[5]. Lorentz invariance is broken down to the subgroup of spatial rotations via the introduction of four scalar ﬁelds, φ0, φa where a = 1, 2, 3. The Lagrangian (including Einstein-Hilbert and matter terms) is given by

12 L = LEH + LS 1 1 0 2 42 = R + Lm + 4 (∂µφ ) − µ0 16πG 8µ0 ! κ0 X − (∂ φ0)2 − µ4 P µν∂ φa∂ φa + 3µ4 4µ4 µ 0 µ ν 0 a 1 X 2 − P µν∂ φa∂ φb + µ4δab 8µ4 µ ν a,b !2 κ X + P µν∂ φa∂ φa + 3µ4 (118) 8µ4 µ ν a where Lm denotes the matter Lagrangian and

0 0 ∂µφ ∂νφ Pµν = gµν − λρ 0 0 . (119) g ∂λφ ∂ρφ √ We ﬁrst vary the action S = R L −gd4x with respect to the metric:

√ Z δ(L −g) δS = δgµνd4x δgµν √ Z √ δL δ −g = −g + L δgµνd4x δgµν δgµν Z √ δL 1√ = −g − −gg L δgµνd4x δgµν 2 µν Z δL 1 √ = − g L δgµν −gd4x. (120) δgµν 2 µν

Thus, the condition that the action be stationary δS = 0 implies

δL 1 − g L = 0. (121) δgµν 2 µν Computing, we ﬁnd

δLS κ0 X 1 X = − (∂ φ0)2 − µ4 Kaa − P αβ∂ φa∂ φb + µ4δab Kab δgµν 4µ4 α 0 µν 4µ4 α ν µν 0 a a,b ! ! κ X X + P αβ∂ φa∂ φa + 3µ4 Kbb . (122) 4µ4 α β µν a b We have deﬁned

δgσρ δgσα δgρα δgαβ Kab ≡ − u uρ + u uσ + uσuρu u ∂ φa∂ φb (123) µν δgµν δgµν α δgµν α α β δgµν σ ρ

13 ∂ φ0 where we have deﬁned u ≡ √µ with X ≡ gαβ∂ φ0∂ φ0. Note that a and b are not tensor µ X α β indices, but merely labeling indices; thus, the summation convention does not apply to them. We can now deﬁne our energy-momentum tensor as

δL 1 T S ≡ −2 S − g L . (124) µν δgµν 2 µν S Thus, the ﬁeld equations ultimately take the form

m S Gµν = 8πG Tµν + Tµν (125) 1 m where Gµν ≡ Rµν − 2 gµνR is the familiar Einstein tensor and Tµν denotes the matter energy- momentum tensor. Using the Euler-Lagrange equations ∂ ∂L − ∂L = 0 and noting that the second term vanishes µ ∂(∂µφ) ∂φ for all φ ﬁelds, we ﬁnd for φ0

! 1 κ0 X (∂ φ0)2 − µ4 ∂ δµβ∂ φ0 − ∂ δµα∂ φ0 P βλ∂ φa∂ φa + 3µ4 2µ4 α 0 µ β 2µ4 µ α β λ 0 0 a ! κ0 X 1 − (∂ φ0)2 − µ4 √ ∂ (−gµβuν − gµνuβ + 2uµuβuν)∂ φa∂ φb 4µ4 α 0 µ β ν 0 a X 1 X 1 − (P αβ∂ φa∂ φb + µ4δab) √ ∂ (−gµλuν − gµνuλ + 2uµuλuν)∂ φa∂ φb 4µ4 α β µ λ ν a,b X ! ! κ X X 1 + P αβ∂ φa∂ φa + 3µ4 √ ∂ (−gµλuν − gµνuλ + 2uµuλuν)∂ φb∂ φb = 0. 4µ4 α β µ λ ν a b X (126)

For φi, we ﬁnd

κ0 0 2 4 µν i − 4 (∂λφ ) − µ0 ∂µP ∂νφ 2µ0 1 X − (P λρ∂ φa∂ φb + µ4δab)∂ δia[P µν∂ φb + δab(P µν∂ φi)] 2µ4 λ ρ µ ν ν a,b ib µν a ab µν i +δ [P ∂νφ + δ (P ∂νφ )] ! κ X + P λρ∂ φa∂ φa + 3µ4 ∂ P µν∂ φi = 0. (127) 2µ4 λ ρ µ ν a

5 Conclusion

The parameterized post-Newtonian formalism provides a useful way of classifying metric theories of gravity according to their observable effects in the slow-motion, weak-field limit. We have found the field equations for Lorentz-violating massive gravity in the form investigated by Blas and Sibiryakov.

14 In the future, these equations can be used to determine the PPN parameters for Lorentz-violating massive gravity. This would allow for a proper characterization of the theory within the PPN formalism and provide a step towards empirical solar system tests of the theory.

References

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[3] B. Z. Foster and T. Jacobson, Phys. Rev. D 73 (2006) 064015 [arXiv:gr-qc/0509083].

[4] D. Blas and H. Sanctuary, Phys. Rev. D 84 (2011) 064004 [arXiv:gr-gc/1105.5149]

[5] D. Blas and S. Sibiryakov, JETP 147 (2015) 3 [arXiv:1410.2408 [hep-th]]