The Post-Newtonian Approximation in Massive Gravity Sohan Vartak CERN Summer Student Programme 2015 Abstract We briefly review the post-Newtonian expansion as a way to yield the approximate equa- tions 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). Finally, the field equations for Lorentz-violating massive gravity are derived from the Lagrangian; these equations will allow for the future determination of the PPN parameters. 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 eas- ily quantizable; in particular, we are interested in Lorentz-violating theories. We first introduce the post-Newtonian expansion, which yields the approximate equations of motion in the slow-motion, weak-field limit. Following this, we review the parameterized post-Newtonian formalism, which al- lows 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). Finally, Lorentz-violating massive gravity is briefly explored. The Lagrangian for a specific theory is used to derive the field equations, which may later be utilized in the computation of the PPN parameters. We use the conventions of Blas and Sibiryakov [3]. 2 The Post-Newtonian Expansion NOTE: This section follows the discussion 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 1 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 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. The Ricci tensor is also expanded in a series and computed in terms of the Christoffel symbols, allowing us to relate it to the metric: 2 1 2 R = − r2g (7) 00 2 00 2 2 2 2 4 1 4 1 @ g 1 2 @ g 1 2 2 R = − r2g + 00 + g 00 − r2g (8) 00 2 00 2 @t2 2 ij @xixj 2 00 3 1 3 R = − r2g (9) 0i 2 0i 2 1 2 R = − r2g : (10) ij 2 ij At this point, we make use of the Einstein equations to relate the metric components to the energy-momentum tensor (which is also expanded in a series). The resulting equations can be used to solve for the components of the metric and express it in terms of various potentials. The metric can then be used to derive the post-Newtonian equations of motion. 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 (11) 2 gij = −(1 + 2γU)δij (12) 1 1 g = (4γ + 3 + α − α + ζ − 2ξ)V + (1 + α + ζ + 2ξ)W : (13) 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 repre- sents 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 : (14) N jx − yj The correspondences F $ f are given by i Φ1 $ vivi Φ3 $ Π U $ 1 Vi $ v (15) p (v (x − y) )2 v (x − y )(xi − yi) Φ $ U Φ $ A $ i i W $ j j j (16) 2 4 ρ jx − yj2 i jx − yj2 Z (x − y) (y − z) (x − z) Φ $ d3zρ(z) j j − j (17) W jx − yj2 jx − zj jy − zj Note that for U, Φ1;2;3;4, and Vi, we have F;ii = −4πGN ρf: (18) We also define the superpotential Z 3 χ = −GN d yρjx − yj (19) which satisfies χ;ii = −2U (20) χ;0i = Vi − Wi: (21) 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[3]. Lorentz invariance is broken down to the subgroup of spatial rotations via the introduction of four scalar 0 a fields, φ ; φ where a = 1; 2; 3. The Lagrangian (including Einstein-Hilbert and matter terms LEH and Lm, respectively) is given by 1 1 0 2 42 L = R + Lm + 4 (@µφ ) − µ0 16πG 8µ0 3 ! 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 : (22) 8µ4 µ ν a We denote the massive gravity Lagrangian by LS = L − (LEH + Lm) and define 0 0 @µφ @νφ Pµν = gµν − λρ 0 0 : (23) g @λφ @ρφ p We first vary the action S = R L −gd4x with respect to the metric and find R δL 1 µνp 4 δS = δgµν − 2 gµνL δg −gd x. Thus, the condition that the action be stationary (δS = 0) implies δL 1 − g L = 0: (24) 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 : (25) 4µ4 α β µν a b We have defined δgσρ δgσα δgρα δgαβ Kab ≡ − u uρ + u uσ + uσuρu u @ φa@ φb (26) µν δgµν δgµν α δgµν α α β δgµν σ ρ @ φ0 where we have defined u ≡ pµ 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 : (27) µν δgµν 2 µν S Thus, the field equations ultimately take the form m S Gµν = 8πG Tµν + Tµν (28) 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 4 ! 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 p @ (−gµβuν − gµνuβ + 2uµuβuν)@ φa@ φb 4µ4 α 0 µ β ν 0 a X 1 X 1 − (P αβ@ φa@ φb + µ4δab) p @ (−gµλuν − gµνuλ + 2uµuλuν)@ φa@ φb 4µ4 α β µ λ ν a;b X ! ! κ X X 1 + P αβ@ φa@ φa + 3µ4 p @ (−gµλuν − gµνuλ + 2uµuλuν)@ φb@ φb = 0: (29) 4µ4 α β µ λ ν a b X 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: (30) 2µ4 λ ρ µ ν a 5 Conclusion We have found the field equations for Lorentz-violating massive gravity in the form investigated by Blas and Sibiryakov.