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 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). 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 ac- cording 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 0 3 2 3 1 3 1 @g @g @g Γi = 0i + ij − 0j (9) 0j 2 @ @xj @t @xi A 0 2 2 2 1 2 1 @g @g @g Γi = ij + ik − jk (10) jk 2 @ @xk @xj @xi A 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 = − r2g (18) 00 2 00 3 2 2 2 2 4 1 4 1 @ g 1 2 @ g 1 2 2 R = − r2g + 00 + g 00 − r2g (19) 00 2 00 2 @t2 2 ij @xixj 2 00 3 1 3 R = − r2g (20) 0i 2 0i 2 1 2 R = − r2g : (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 r g00 = −8πGN T (22) 2 2 2 2 2 ! 2 ! 4 @ g 2 @ g @g @g r2g = 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 r g0i = 16πGN T (24) 0 2 2 00 r 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+2! T µ and ξ ! 0 as r ! 1. The field equations of Brans-Dicke theory are given as 1 1 R − g R =8πG(1 + ξ)−1T + !(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 2! + 4 0 r2g = −8πG T (27) 00 2! + 3 00 2 2 2 2 4 @ g 2 @ g 2 r2g = 00 + g 00 − (rg )2 00 @t2 ij @xixj 00 2! + 4 2 0 2 2! + 2 + 8πG ξT 00 − 8πGT ii 2! + 3 2! + 3 4 0 2 2 2! + 4 2! + 4 + 16πGg T 00 − 8πG T 00 00 2! + 3 2! + 3 0 2 12 2 2 @ξ @2ξ 2 @ξ − 2! − 2 + 2Γi (28) @ @t A @t2 00 @xi 2 1 2 3 @ ξ r2g = 16πGT 0i − 2 (29) 0i @xi@t 2 0 2 2 2! + 2 @ ξ r2g = −8πGT 00δ − 2 : (30) ij ij 2! + 3 @xi@xj It also follows immediately from the definition of ξ that we have 2 8πG 0 r2ξ = − T 00: (31) 2! + 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 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 : (35) N jx − yj 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 jx − yj2 jx − zj jy − zj (v (x − y) )2 A $ i i (42) jx − yj2 i Vi $ v (43) v (x − y )(xi − yi) W $ j j j : (44) i jx − yj2 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 yρjx − yj (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].